���� JFIF  XX �� �� �     $.' ",#(7),01444'9=82<.342  2!!22222222222222222222222222222222222222222222222222�� ��" �� 4     ��   �� �,�PG"Z_�4�˷����kjز�Z�,F+��_z�,�© �����zh6�٨�ic�fu��� #ډb���_�N� ?� �wQ���5-�~�I���8��� �TK<5o�Iv-� ����k�_U_����� ~b�M��d��� �Ӝ�U�Hh��?]��E�w��Q���k�{��_}qFW7HTՑ��Y��F� ?_�'ϔ��_�Ջt� �=||I �� 6�έ"�����D���/[�k�9�� �Y�8 ds|\���Ҿp6�Ҵ���]��.����6� z<�v��@]�i% �� $j��~ �g��J>��no����pM[me�i$[�� �� s�o�ᘨ�˸ nɜG-�ĨU�ycP� 3.DB�li�;� �hj���x 7Z^�N�h��� ���N3u{�:j �x�힞��#M &��jL P@ _���� P�� &��o8 ������9 �����@Sz 6�t7#O�ߋ � s}Yf�T� ��lmr����Z)'N��k�۞p ����w\�T ȯ?�8` �O��i{wﭹW�[�r�� ��Q4F�׊�� �3m&L�=��h3� ���z~��#� \�l :�F,j@�� ʱ�wQT����8�"kJO��� 6�֚l���� }��� R�>ډK���]��y����&����p�}b�� ;N�1�m�r$� |��7�>e�@ B�TM*-i H��g�D�)� E�m�|�ؘbҗ�a ��Ҿ���� t4��� o���G��*oCN�rP���Q��@z,|?W[0 �����:�n,j WiE��W� �$~/�hp\��?��{(�0���+�Y8rΟ�+����>S-S�� ��VN;� }�s?.����� w �9��˟<���Mq4�Wv' ��{)0�1mB ��V����W[� ����8�/<� �%���wT^�5���b��)iM� p g�N�&ݝ� �VO~� q���u���9� ����!��J27��� �$ O-���! �: �%H��� ـ ����y�ΠM=t{!S�� oK8������ t<����è :a�� ����[���� �ա�H���~��w��Qz`�p o�^ �� ��Q��n�  �,uu�C� $ ^���,� �����8�#��:�6��e�|~� ��!�3� 3.�\0�� q��o�4`.|� ����y�Q�`~;�d�ׯ,��O�Zw�������`73�v�܋�< ���Ȏ�� ـ4k��5�K�a�u�=9Yd��$>x�A�&�� j0� ���vF��� Y� |�y��� ~�6�@c��1vOp �Ig�� ��4��l�OD� ��L����� R���c���j�_�uX 6��3?nk��Wy�f;^*B� ��@ �~a�`��Eu������ +� �� 6�L��.ü>��}y���}_�O�6�͐�:�Yr G�X��kG�� ���l^w�� �~㒶sy� �Iu�!� W ��X��N�7BV��O��!X�2����wvG�R�f�T#�����t�/?���%8�^�W�aT ��G�cL�M���I��(J����1~�8�?aT ���]����AS�E��(��*E}� 2�� #I/�׍qz��^t�̔��� b�Yz4x ���t�){ OH� �+(E��A&�N�������XT��o��"�XC�� '���)}�J�z�p� ��~5�}�^����+�6����w��c��Q�| Lp�d�H��}�(�.|����k��c4^� "�����Z?ȕ ��a< �L�!0 39C� �Eu� C�F�Ew�ç ;�n?�*o���B�8�bʝ���'#Rqf�� �M}7����]��� �s2tcS{�\icTx;�\��7K���P ���ʇ Z O-��~�� c>"��?�� �����P ��E��O�8��@�8��G��Q�g�a�Վ���󁶠 �䧘��_%#r�>� 1�z�a�� eb��qcP ѵ��n���#L��� =��׀t� L�7�` ��V��� A{�C:�g���e@ �w1 Xp 3�c3�ġ���� p��M"'-�@n4���fG� �B3�DJ�8[Jo�ߐ���gK)ƛ��$���� � ��8�3�����+���� �����6�ʻ���� ���S�kI�*KZlT _`�� �?��K� ���QK�d ����B`�s}�>���` ��*�>��,*@J�d�oF*� ���弝��O}�k��s��]��y�ߘ ��c1G�V���<=�7��7����6 �q�PT��tXԀ�!9*4�4Tހ 3XΛex�46�� �Y��D ����� �BdemDa����\�_l,� �G�/���֌7���Y�](�xTt^%�GE�����4�}bT ���ڹ�����; Y)���B�Q��u��>J/J � ⮶.�XԄ��j�ݳ� +E��d ��r�5�_D �1 �� o�� �B�x�΢�#� ��<��W�����8���R6�@ g�M�.��� dr�D��>(otU��@ x=��~v���2� ӣ�d�oBd ��3�eO�6�㣷�� ���ݜ 6��6Y��Qz`�� S��{���\P �~z m5{J/L��1������<�e�ͅPu� b�]�ϔ ���'�� ����f�b� Zpw��c`"��i���BD@:)ִ�:�]��h v�E� w���T�l ��P� ��"Ju�}��وV J��G6��. J/�Qgl߭�e�����@�z�Zev2u� )]կ��� ��7x�� �s�M�-<ɯ�c��r� v�����@��$�ޮ}lk���a�� �'����>x��O\�Z Fu>��� ��ck#��&:��`�$ �ai�>2Δ����l���oF[h� �lE�ܺ�Π k:)���` �� $[6�����9�����kOw�\|��� 8}������ބ:��񶐕� �I�A1/� =�2[�,�!��.}gN#�u����b ��� ~� �݊��}34q��� �d�E��L c��$ ��"�[q�U�硬g^��%B � z���r�p J�ru%v\h 1Y�ne` ǥ:g�� �pQM~�^� Xi� ��`S�:V2 9.�P���V� ?B�k�� AEvw%�_�9C�Q����wKekP ؠ�\� ;Io d�{ ߞo�c1eP��� �\� `����E=���@K<�Y�� �eڼ�J ���w����{av�F�'�M�@ /J��+9p ���|]���� �Iw &` ��8���& M�hg ��[�{ ��Xj�� %��Ӓ� $��(��� �ʹN��� <>�I���RY� ��K2�NPlL�ɀ )��&e� ���B+ь����( � �JTx ���_?EZ� }@ 6�U���뙢ط�z��dWI� n` D����噥�[��uV��"�G& Ú����2 g�}&m� �?ċ �"����Om#� ������� � ��{� ON��"S�X ��Ne��ysQ���@ Fn��Vg��� dX�~nj� ]J�<�K]: ��FW�� b�������62 �=��5f����JKw� �bf�X� 55��~J �%^� ���:�-�QIE��P��v�nZum� z � ~ə ���� ���ة����;�f��\v��� g�8�1��f2 4;�V���ǔ�)��� �9���1\�� c��v�/'Ƞ�w����� ��$�4�R-��t�� �� e�6�/�ġ �̕Ecy�J���u�B���<�W�ַ~�w[B1L۲�-JS΂�{���΃���� ��A��20�c# �� @    0!1@AP"#2Q`$3V�%45a6�FRUq���   � ���^7ׅ,$n� ������+��F�`��2X'��0vM��p�L=������ 5��8������u�p~���.�`r�����\��� O��,ư�0oS ��_�M�����l���4�kv\JSd���x���SW�<��Ae�IX����������$I���w�:S���y���›R��9�Q[���,�5�;�@]�%���u�@ *ro�lbI �� ��+���%m:�͇ZV�����u�̉����θau<�fc�.����{�4Ա� �Q����*�Sm��8\ujqs]{kN���)qO�y�_*dJ�b�7���yQqI&9�ԌK!�M}�R�;�� ����S�T���1���i[U�ɵz�]��U)V�S6���3$K{� ߊ<�(� E]Զ[ǼENg�����'�\?#)Dkf��J���o��v���'�%ƞ�&K�u� !��b�35LX�Ϸ��63$K�a�;�9>,R��W��3�3� d�JeTYE.Mϧ��-�o�j3+y��y^�c�������VO�9NV\nd�1 ��!͕_)a�v;����թ�M�lWR1��)El��P;��yوÏ�u 3�k�5Pr6<�⒲l�!˞*��u־�n�!�l:����UNW ��%��Chx8vL'��X�@��*��)���̮��ˍ��� � ��D-M�+J�U�kvK����+�x8��cY������?�Ԡ��~3mo��|�u@[XeY�C�\Kp�x8�oC�C�&����N�~3-H���� ��MX�s�u<`���~"WL��$8ξ��3���a�)|:@�m�\���^�`�@ҷ)�5p+��6���p�%i)P M���ngc�����#0Aruz���RL+xSS?���ʮ}()#�t��mˇ!��0}}y����<�e� �-ή�Ԩ��X������ MF���ԙ~l L.3���}�V뽺�v��� ��멬��Nl�)�2����^�Iq��a��M��qG��T�����c3#������3U�Ǎ���}��לS�|qa��ڃ�+���-��2�f����/��bz��ڐ�� �ݼ[2�ç����k�X�2�* �Z�d���J�G����M*9W���s{��w���T��x��y,�in�O�v��]���n����P�$� JB@=4�OTI�n��e�22a\����q�d���%�$��(���:���: /*�K[PR�fr\nڙdN���F�n�$�4� [�� U�zƶ����� �mʋ���,�ao�u 3�z� �x��Kn����\[��VFmbE;�_U��&V�Gg�]L�۪&#n%�$ɯ� dG���D�TI=�%+AB�Ru#��b4�1�»x�cs�YzڙJG��f��Il� �d�eF'T� iA��T���uC�$����Y��H?����[!G`}���ͪ� �纤Hv\������j�Ex�K���!���OiƸ�Yj�+u-<���'q����uN�*�r\��+�]���<�wOZ.fp�ێ��,-*)V?j-kÊ#�`�r��dV����(�ݽBk�����G�ƛk�QmUڗe��Z���f}|����8�8��a���i��3'J�����~G_�^���d�8w������ R�`(�~�.��u���l�s+g�bv���W���lGc}��u���afE~1�Ue������Z�0�8�=e�� f@/�jqEKQQ�J� �oN��J���W5~M>$6�Lt�;$ʳ{���^��6�{����v6���ķܰg�V�cnn �~z�x�«�,2�u�?cE+Ș�H؎�%�Za�)���X>uW�Tz�Nyo����s���FQƤ��$��*�&�LLXL)�1�" L��eO��ɟ�9=���:t��Z���c��Ž���Y?�ӭV�wv�~,Y��r�ۗ�|�y��GaF�����C�����.�+� ���v1���fήJ�����]�S��T��B��n5sW}y�$��~z�'�c ��8 ��� ,! �p��VN�S��N�N�q��y8z˱�A��4��*��'������2n<�s���^ǧ˭P�Jޮɏ�U�G�L�J�*#��<�V��t7�8����TĜ>��i}K%,���)[��z�21z ?�N�i�n1?T�I�R#��m-�����������������1����lA�`��fT5+��ܐ�c�q՝��ʐ��,���3�f2U�եmab��#ŠdQ�y>\��)�SLY����w#��.���ʑ�f��� ,"+�w�~�N�'�c�O�3F�������N<���)j��&��,-� �љ���֊�_�zS���TǦ����w�>��?�������n��U仆�V���e�����0���$�C�d���rP �m�׈e�Xm�Vu� �L��.�bֹ��� �[Դaզ���*��\y�8�Է:�Ez\�0�Kq�C b��̘��cө���Q��=0Y��s�N��S.��� 3.���O�o:���#���v7�[#߫ ��5�܎�L���Er4���9n��COWlG�^��0k�%<���ZB���aB_���������'=��{i�v�l�$�uC���mƎҝ{�c㱼�y]���W�i ��ߧc��m�H� m�"�"�����;Y�ߝ�Z�Ǔ�����:S#��|}�y�,/k�Ld� TA�(�AI$+I3��;Y*���Z��}|��ӧO��d�v��..#:n��f>�>���ȶI�TX��� 8��y����"d�R�|�)0���=���n4��6ⲑ�+��r<�O�܂~zh�z����7ܓ�HH�Ga롏���nCo�>������a ���~]���R���̲c?�6(�q�;5%� |�uj�~z8R =X��I�V=�|{v�Gj\gc��q����z�؋%M�ߍ����1y��#��@f^���^�>N��� ��#x#۹��6�Y~�?�dfPO��{��P�4��V��u1E1J �*|���%�� �JN��`eWu�zk M6���q t[�� ��g�G���v��WIG��u_ft����5�j�"�Y�:T��ɐ���*�;� e5���4����q$C��2d�}���� _S�L#m�Yp��O�.�C�;��c����Hi#֩%+) �Ӎ��ƲV���SYź��g |���tj��3�8���r|���V��1#;.SQ�A[���S������#���`n�+���$��$ I �P\[�@�s��(�ED�z���P��])8�G#��0B��[ى��X�II�q<��9�~[Z멜�Z�⊔IWU&A>�P~�#��dp<�?����7���c��'~���5 ��+$���lx@�M�dm��n<=e�dyX��?{�|Aef ,|n3�<~z�ƃ�uۧ�����P��Y,�ӥQ�*g�#먙R�\���;T��i,��[9Qi歉����c>]9�� ��"�c��P�� �Md?٥��If�ت�u��k��/����F��9�c*9��Ǎ:�ØF���z�n*�@|I�ށ9����N3{'��[�'ͬ�Ҳ4��#}��!�V� Fu��,�,mTIk���v C�7v���B�6k�T9��1�*l� '~��ƞF��lU��'�M ����][ΩũJ_�{�i�I�n��$�� �L�� j��O�dx�����kza۪��#�E��Cl����x˘�o�����V���ɞ�ljr��)�/,�߬h�L��#��^��L�ф�,íMƁe�̩�NB�L�����iL����q�}��(��q��6IçJ$�W�E$��:������=#����(�K�B����zђ <��K(�N�۫K�w��^O{!����) �H���>x�������lx�?>Պ�+�>�W���,Ly!_�D���Ō�l���Q�!�[ �S����J��1��Ɛ�Y}��b,+�Lo�x�ɓ)����=�y�oh�@�꥟/��I��ѭ=��P�y9��� �ۍYӘ�e+�p�Jnϱ?V\SO%�(�t� ���=?MR�[Ș�����d�/ ��n�l��B�7j� ��!�;ӥ�/�[-���A�>� dN�sLj ��,ɪv��=1c�.SQ�O3�U���ƀ�ܽ�E����������̻��9G�ϷD�7(�}��Ävӌ\� y�_0[w ���<΍>����a_��[0+�L��F.�޺��f�>oN�T����q;���y\��bՃ��y�jH�<|q-eɏ�_?_9+P���Hp$�����[ux�K w�Mw��N�ی'$Y2�=��q���KB��P��~�� ����Yul:�[<����F1�2�O���5=d����]Y�sw:���Ϯ���E��j,_Q��X��z`H1,#II ��d�wr��P˂@�ZJV����y$�\y�{}��^~���[:N����ߌ�U�������O��d�����ؾe��${p>G��3c���Ė�lʌ�� ת��[��`ϱ�-W����dg�I��ig2��� ��}s ��ؤ(%#sS@���~���3�X�nRG�~\jc3�v��ӍL��M[JB�T��s3}��j�Nʖ��W����;7� �ç?=X�F=-�=����q�ߚ���#���='�c��7���ڑW�I(O+=:uxq�������������e2�zi+�kuG�R��������0�&e�n���iT^J����~\jy���p'dtG��s����O��3����9* �b#Ɋ�� p������[Bws�T�>d4�ۧs���nv�n���U���_�~,�v����ƜJ1��s�� �QIz�� )�(lv8M���U=�;����56��G���s#�K���MP�=��LvyGd��}�VwWBF�'�à �?MH�U�g2�� ����!�p�7Q��j��ڴ����=��j�u��� Jn�A s���uM������e��Ɔ�Ҕ�!) '��8Ϣ�ٔ� �ޝ(��Vp���צ֖d=�IC�J�Ǡ{q������kԭ�߸���i��@K����u�|�p=..�*+����x�����z[Aqġ#s2a�Ɗ���RR�)*HRsi�~�a &f��M��P����-K�L@��Z��Xy�'x�{}��Zm+���:�)�) IJ�-i�u���� ���ܒH��'� L(7�y�GӜq���� j��� 6ߌg1�g�o���,kر���tY�?W,���p���e���f�OQS��!K�۟cҒA�|ս�j�>��=⬒��˧L[�� �߿2JaB~R��u�:��Q�] �0H~���]�7��Ƽ�I���( }��cq '�ήET���q�?f�ab���ӥvr� �)o��-Q��_'����ᴎo��K������;��V���o��%���~OK ����*��b�f:���-ťIR��`B�5!RB@���ï�� �u �̯e\�_U�_������� g�ES��3������� QT��a�� ��x����U<~�c?�*�#]�MW,[8O�a�x��]�1bC|踤�P��lw5V%�)�{t�<��d��5���0i�XSU��m:��Z�┵�i�"��1�^B�-��P�hJ��&)O��*�D��c�W��vM��)����}���P��ܗ-q����\mmζZ-l@�}��a��E�6��F�@��&Sg@���ݚ�M����� ȹ 4����#p�\H����dYDo�H���"��\��..R�B�H�z_�/5˘����6��KhJR��P�mƶi�m���3� ,#c�co��q�a)*P t����R�m�k�7x�D�E�\Y�閣_X�<���~�)���c[[�BP����6�Yq���S��0����%_����;��Àv�~�| VS؇ ��'O0��F0��\���U�-�d@�����7�SJ*z��3n��y��P����O��������� m�~�P�3|Y��ʉr#�C�<�G~�.,! ���bqx���h~0=��!ǫ�jy����l� O,�[B��~��|9��ٱ����Xly�#�i�B��g%�S��������tˋ���e���ې��\[d�t)��.+u�|1 ������#�~Oj����hS�%��i.�~X���I�H�m��0n���c�1uE�q��cF�RF�o���7� �O�ꮧ� ���ۛ{��ʛi5�rw?׌#Qn�TW��~?y$��m\�\o����%W� ?=>S�N@�� �Ʈ���R����N�)�r"C�:��:����� �����#��qb��Y�. �6[��2K����2u�Ǧ�HYR��Q�MV��� �G�$��Q+.>�����nNH��q�^��� ����q��mM��V��D�+�-�#*�U�̒ ���p욳��u:�������IB���m� ��PV@O���r[b= �� ��1U�E��_Nm�yKbN�O���U�}�the�`�|6֮P>�\2�P�V���I�D�i�P�O;�9�r�mAHG�W�S]��J*�_�G��+kP�2����Ka�Z���H�'K�x�W�MZ%�O�YD�Rc+o��?�q��Ghm��d�S�oh�\�D�|:W������UA�Qc yT�q� �����~^�H��/��#p�CZ���T�I�1�ӏT����4��"�ČZ�����}��`w�#�*,ʹ�� ��0�i��課�Om�*�da��^gJ݅{���l�e9uF#T�ֲ��̲�ٞC"�q���ߍ ոޑ�o#�XZTp����@ o�8��(jd��xw�]�,f���`~� |,s��^����f�1���t��|��m�򸄭/ctr��5s��7�9Q�4�H1꠲BB@ l9@���C�����+�wp�xu�£Yc�9��?`@#�o�mH�s2��)�=��2�.�l����jg�9$�Y�S�%*L������R�Y������7Z���,*=�䷘$�������arm�o�ϰ���UW.|�r�uf����IGw�t����Zwo��~5 ��YյhO+=8fF�)�W�7�L9lM�̘·Y���֘YLf�큹�pRF���99.A �"wz��=E\Z���'a� 2��Ǚ�#;�'}�G���*��l��^"q��+2FQ� hj��kŦ��${���ޮ-�T�٭cf�|�3#~�RJ����t��$b�(R��(����r���dx� >U b�&9,>���%E\� Ά�e�$��'�q't��*�א���ެ�b��-|d���SB�O�O��$�R+�H�)�܎�K��1m`;�J�2�Y~9��O�g8=vqD`K[�F)k�[���1m޼c��n���]s�k�z$@��)!I �x՝"v��9=�ZA=`Ɠi �:�E��)` 7��vI��}d�YI�_ �o�:ob���o ���3Q��&D&�2=�� �Ά��;>�h����y.*ⅥS������Ӭ�+q&����j|UƧ��� �}���J0��WW< ۋS�)jQR�j���Ư��rN)�Gű�4Ѷ(�S)Ǣ�8��i��W52���No˓� ۍ%�5brOn�L�;�n��\G����=�^U�dI���8$�&���h��'���+�(������cȁ߫k�l��S^���cƗjԌE�ꭔ��gF���Ȓ��@���}O���*;e�v�WV���YJ\�]X'5��ղ�k�F��b 6R�o՜m��i N�i���� >J����?��lPm�U��}>_Z&�KK��q�r��I�D�Չ~�q�3fL�:S�e>���E���-G���{L�6p�e,8��������QI��h��a�Xa��U�A'���ʂ���s�+טIjP�-��y�8ۈZ?J$��W�P� ��R�s�]��|�l(�ԓ��sƊi��o(��S0 ��Y� 8�T97.�����WiL��c�~�dxc�E|�2!�X�K�Ƙਫ਼�$((�6�~|d9u+�qd�^3�89��Y�6L�.I�����?���iI�q���9�)O/뚅����O���X��X�V��ZF[�یgQ�L��K1���RҖr@v�#��X�l��F���Нy�S�8�7�kF!A��sM���^rkp�jP�DyS$N���q�� nxҍ!U�f�!eh�i�2�m ���`�Y�I�9r�6� �TF���C}/�y�^���Η���5d�'��9A-��J��>{�_l+�`��A���[�'��յ�ϛ#w:݅�%��X�}�&�PSt�Q�"�-��\縵�/����$Ɨh�Xb�*�y��BS����;W�ջ_mc�����vt?2}1�;qS�d�d~u:2k5�2�R�~�z+|HE!)�Ǟl��7`��0�<�,�2*���Hl-��x�^����'_TV�gZA�'j� ^�2Ϊ��N7t�����?w�� �x1��f��Iz�C-Ȗ��K�^q�;���-W�DvT�7��8�Z�������� hK�(P:��Q- �8�n�Z���܃e貾�<�1�YT<�,�����"�6{ / �?�͟��|1�:�#g��W�>$����d��J��d�B�� =��jf[��%rE^��il:��B���x���Sּ�1հ��,�=��*�7 fcG��#q� �eh?��2�7�����,�!7x��6�n�LC�4x��},Geǝ�tC.��vS �F�43��zz\��;QYC,6����~;RYS/6���|2���5���v��T��i����������mlv��������&� �nRh^ejR�LG�f���? �ۉҬܦƩ��|��Ȱ����>3����!v��i�ʯ�>�v��オ�X3e���_1z�Kȗ\<������!�8���V��]��?b�k41�Re��T�q��mz��TiOʦ�Z��Xq���L������q"+���2ۨ��8}�&N7XU7Ap�d�X��~�׿��&4e�o�F��� �H�� ��O���č�c�� 懴�6���͉��+)��v;j��ݷ�� �UV�� i��� j���Y9GdÒJ1��詞�����V?h��l�� ��l�cGs�ځ�������y�Ac���� �\V3�? �� ܙg�>qH�S,�E�W�[�㺨�uch�⍸�O�}���a��>�q�6�n6� ���N6�q�� ���� N    ! 1AQaq�0@����"2BRb�#Pr���3C`��Scst���$4D���%Td��  ? � ��N����a��3��m���C���w��������xA�m�q�m��� m������$����4n淿t'��C"w��zU=D�\R+w�p+Y�T�&�պ@��ƃ��3ޯ?�Aﶂ��aŘ���@-�����Q�=���9D��ռ�ѻ@��M�V��P��܅�G5�f�Y<�u=,EC)�<�Fy'�"�&�չ�X~f��l�KԆV��?�� �W�N����=(� �;���{�r����ٌ�Y���h{�١������jW����P���Tc�����X�K�r��}���w�R��%��?���E��m�� �Y�q|����\lEE4� ��r���}�lsI�Y������f�$�=�d�yO����p�����yBj8jU�o�/�S��?�U��*������ˍ�0����� �u�q�m [�?f����a�� )Q�>����6#������� ?����0UQ����,IX���(6ڵ[�DI�MNލ�c&���υ�j\��X�R|,4��� j������T�hA�e��^���d���b<����n�� �즇�=!���3�^�`j�h�ȓr��jẕ�c�,ٞX����-����a�ﶔ���#�$��]w�O��Ӫ�1y%��L�Y<�wg#�ǝ�̗`�x�xa�t�w��»1���o7o5��>�m뭛C���Uƃߜ}�C���y1Xνm�F8�jI���]����H���ۺиE@I�i;r�8ӭ���� V�F�Շ| ��&?�3|x�B�MuS�Ge�=Ӕ�#BE5G�� ���Y!z��_e��q�р/W>|-�Ci߇�t�1ޯќd�R3�u��g�=0 5��[?�#͏��q�cf���H��{ ?u�=?�?ǯ���}Z��z���hmΔ�BFTW�����<�q� (v� ��!��z���iW]*�J�V�z��gX֧A�q�&��/w���u�gYӘa���; �i=����g:��?2�dž6�ى�k�4�>�Pxs����}������G�9� �3 ���)gG�R<>r h�$��'nc�h�P��Bj��J�ҧH� -��N1���N��?��~��}-q!=��_2hc�M��l�vY%UE�@|�v����M2�.Y[|y�"Eï��K�ZF,�ɯ?,q�?v�M 80jx�"�;�9vk�����+ ֧�� �ȺU��?�%�vcV��mA�6��Qg^M��� �A}�3�nl� QRN�l8�kkn�'�����(��M�7m9و�q���%ޟ���*h$Zk"��$�9��: �?U8�Sl��,,|ɒ��xH(ѷ����Gn�/Q�4�P��G�%��Ա8�N��!� �&�7�;���eKM7�4��9R/%����l�c>�x;������>��C�:�����t��h?aKX�bhe�ᜋ^�$�Iհ �hr7%F$�E��Fd���t��5���+�(M6�t����Ü�UU|zW�=a�Ts�Tg������dqP�Q����b'�m���1{|Y����X�N��b �P~��F^F:����k6�"�j!�� �I�r�`��1&�-$�Bevk:y���#y w��I0��x��=D�4��tU���P�ZH��ڠ底taP��6����b>�xa� ���Q�#� WeF��ŮNj�p�J* mQ�N��� �*I�-*�ȩ�F�g�3 �5��V�ʊ�ɮ�a��5F���O@{���NX��?����H�]3��1�Ri_u��������ѕ�� ����0��� F��~��:60�p�͈�S��qX#a�5>���`�o&+�<2�D����: �������ڝ�$�nP���*)�N�|y�Ej�F�5ټ�e���ihy�Z �>���k�bH�a�v��h�-#���!�Po=@k̆IEN��@��}Ll?j�O������߭�ʞ���Q|A07x���wt!xf���I2?Z��<ץ�T���cU�j��]�� 陎Ltl �}5�ϓ��$�,��O�mˊ�;�@O��jE��j(�ا,��LX���LO���Ц�90�O �.����a��nA���7������j4 ��W��_ٓ���zW�jcB������y՗+EM�)d���N�g6�y1_x��p�$Lv :��9�"z��p���ʙ$��^��JԼ*�ϭ����o���=x�Lj�6�J��u82�A�H�3$�ٕ@�=Vv�]�'�qEz�;I˼��)��=��ɯ���x �/�W(V���p�����$ �m�������u�����񶤑Oqˎ�T����r��㠚x�sr�GC��byp�G��1ߠ�w e�8�$⿄����/�M{*}��W�]˷.�CK\�ުx���/$�WP w���r� |i���&�}�{�X� �>��$-��l���?-z���g����lΆ���(F���h�vS*���b���߲ڡn,|)mrH[���a�3�ר�[1��3o_�U�3�TC�$��(�=�)0�kgP���� ��u�^=��4 �WYCҸ:��vQ�ר�X�à��tk�m,�t*��^�,�}D*� �"(�I��9R����>`�`��[~Q]�#af��i6l��8���6�:,s�s�N6�j"�A4���IuQ��6E,�GnH��zS�HO�uk�5$�I�4��ؤ�Q9�@��C����wp �BGv[]�u�Ov��� 0I4���\��y�����Q�Ѹ��~>Z��8�T��a��q�ޣ;z��a���/��S��I:�ܫ_�|������>=Z����8:�S��U�I�J��"IY���8%b8���H��:�QO�6�;7�I�S��J��ҌAά3��>c���E+&jf$eC+�z�;��V����� �r���ʺ������my�e���aQ�f&��6�ND ��.:��NT�vm�<- u���ǝ\MvZY�N�NT��-A�>jr!S��n�O 1�3�Ns�%�3D@���`������ܟ 1�^c<���� �a�ɽ�̲�Xë#�w�|y�cW�=�9I*H8�p�^(4���՗�k��arOcW�tO�\�ƍR��8����'�K���I�Q�����?5�>[�}��yU�ײ -h��=��% q�ThG�2�)���"ו3]�!kB��*p�FDl�A���,�eEi�H�f�Ps�����5�H:�Փ~�H�0Dت�D�I����h�F3�������c��2���E��9�H��5�zԑ�ʚ�i�X�=:m�xg�hd(�v����׊�9iS��O��d@0ڽ���:�p�5�h-��t�&���X�q�ӕ,��ie�|���7A�2���O%P��E��htj��Y1��w�Ѓ!����  ���� ࢽ��My�7�\�a�@�ţ�J �4�Ȼ�F�@o�̒?4�wx��)��]�P��~�����u�����5�����7X ��9��^ܩ�U;Iꭆ 5 �������eK2�7(�{|��Y׎ �V��\"���Z�1� Z�����}��(�Ǝ"�1S���_�vE30>���p;� ΝD��%x�W�?W?v����o�^V�i�d��r[��/&>�~`�9Wh��y�;���R�� � ;;ɮT��?����r$�g1�K����A��C��c��K��l:�'��3 c�ﳯ*"t8�~l��)���m��+U,z��`( �>yJ�?����h>��]��v��ЍG*�{`��;y]��I�T� ;c��NU�fo¾h���/$���|NS���1�S�"�H��V���T���4��uhǜ�]�v;���5�͠x��'C\�SBpl���h}�N����� A�Bx���%��ޭ�l��/����T��w�ʽ]D�=����K���ž�r㻠l4�S�O?=�k �M:� ��c�C�a�#ha���)�ѐxc�s���gP�iG�� {+���x���Q���I= �� z��ԫ+ �8"�k�ñ�j=|����c ��y��CF��/ ��*9ж�h{ �?4�o� ��k�m�Q�N�x��;�Y��4膚�a�w?�6�> e]�����Q�r�:����g�,i"�����ԩA� *M�<�G��b�if��l^M��5� �Ҩ�{����6J��ZJ�����P�*�����Y���ݛu�_4�9�I8�7���������,^ToR���m4�H��?�N�S�ѕw��/S��甍�@�9H�S�T��t�ƻ���ʒU��*{Xs�@����f��� ��֒Li�K{H�w^���������Ϥm�tq���s� ���ք��f:��o~s��g�r��ט� �S�ѱC�e]�x���a��) ���(b-$(�j>�7q�B?ӕ�F��hV25r[7 Y� }L�R��}����*sg+��x�r�2�U=�*'WS��ZDW]�WǞ�<��叓���{�$�9Ou4��y�90-�1�'*D`�c�^o?(�9��u���ݐ��'PI&� f�Jݮ�������:wS����jfP1F:X �H�9dԯ�� �˝[�_54 �}*;@�ܨ�� ð�yn�T���?�ןd�#���4rG�ͨ��H�1�|-#���Mr�S3��G�3�����)�.᧏3v�z֑��r����$G"�`j �1t��x0<Ɔ�Wh6�y�6��,œ�Ga��gA����y��b��)� �h�D��ß�_�m��ü �gG;��e�v��ݝ�nQ� ��C����-�*��o���y�a��M��I�>�<���]obD��"�:���G�A��-\%LT�8���c�)��+y76���o�Q�#*{�(F�⽕�y����=���rW�\p���۩�c���A���^e6��K������ʐ�cVf5$�'->���ՉN"���F�"�UQ@�f��Gb~��#�&�M=��8�ט�JNu9��D��[̤�s�o�~��� ��� G��9T�tW^g5y$b��Y'��س�Ǵ�=��U-2 #�MC�t(�i� �lj�@Q 5�̣i�*�O����s�x�K�f��}\��M{E�V�{�υ��Ƈ�����);�H����I��fe�Lȣr�2��>��W� I�Ȃ6������i��k�� �5�YOxȺ����>��Y�f5'��|��H+��98pj�n�.O�y�������jY��~��i�w'������l�;�s�2��Y��:'lg�ꥴ)o#'Sa�a�K��Z� �m��}�`169�n���"���x��I ��*+� }F<��cГ���F�P�������ֹ*�PqX�x۩��,� ��N�� �4<-����%����:��7����W���u�`����� $�?�I��&����o��o��`v�>��P��"��l���4��5'�Z�gE���8���?��[�X�7(��.Q�-��*���ތL@̲����v��.5���[��=�t\+�CNܛ��,g�SQnH����}*F�G16���&:�t��4ُ"A��̣��$�b �|����#rs��a�����T�� ]�<�j��B S�('$�ɻ� �wP;�/�n��?�ݜ��x�F��yUn�~mL*-�������Xf�wd^�a�}��f�,=t�׵i�.2/wpN�Ep8�OР���•��R�FJ� 55TZ��T �ɭ�<��]��/�0�r�@�f��V��V����Nz�G��^���7hZi����k��3�,kN�e|�vg�1{9]_i��X5y7� 8e]�U����'�-2,���e"����]ot�I��Y_��n�(JҼ��1�O ]bXc���Nu�No��pS���Q_���_�?i�~�x h5d'�(qw52] ��'ޤ�q��o1�R!���`ywy�A4u���h<קy���\[~�4�\ X�Wt/� 6�����n�F�a8��f���z �3$�t(���q��q�x��^�XWeN'p<-v�!�{�(>ӽDP7��ո0�y)�e$ٕv�Ih'Q�EA�m*�H��RI��=:��� ���4牢) �%_iN�ݧ�l]� �Nt���G��H�L��� ɱ�g<���1V�,�J~�ٹ�"K��Q�� 9�HS�9�?@��k����r�;we݁�]I�!{ �@�G�[�"��`���J:�n]�{�cA�E����V��ʆ���#��U9�6����j�#Y�m\��q�e4h�B�7��C�������d<�?J����1g:ٳ���=Y���D�p�ц� ׈ǔ��1�]26؜oS�'��9�V�FVu�P�h�9�xc�oq�X��p�o�5��Ա5$�9W�V(�[Ak�aY錎qf;�'�[�|���b�6�Ck��)��#a#a˙��8���=äh�4��2��C��4tm^ �n'c� ��]GQ$[Wҿ��i���vN�{Fu ��1�gx��1┷���N�m��{j-,��x�� Ūm�ЧS�[�s���Gna���䑴�� x�p 8<������97�Q���ϴ�v�aϚG��Rt�Һ׈�f^\r��WH�JU�7Z���y)�vg=����n��4�_)y��D'y�6�]�c�5̪ �\� �PF�k����&�c;��cq�$~T�7j ���nç]�<�g ":�to�t}�159�<�/�8������m�b�K#g'I'.W����� 6��I/��>v��\�MN��g���m�A�yQL�4u�Lj�j9��#44�t��l^�}L����n��R��!��t��±]��r��h6ٍ>�yҏ�N��fU�� ���� Fm@�8}�/u��jb9������he:A�y�ծw��GpΧh�5����l}�3p468��)U��d��c����;Us/�֔�YX�1�O2��uq�s��`hwg�r~�{ R��mhN��؎*q 42�*th��>�#���E����#��Hv�O����q�}����� 6�e��\�,Wk�#���X��b>��p}�դ��3���T5��†��6��[��@ �P�y*n��|'f�֧>�lư΂�̺����SU�'*�q�p�_S�����M�� '��c�6��� ��m�� ySʨ;M��r���Ƌ�m�Kxo,���Gm�P��A�G�:��i��w�9�}M(�^�V��$ǒ�ѽ�9���|���� �a����J�SQ�a���r�B;����}���ٻ֢�2�%U���c�#�g���N�a�ݕ�'�v�[�OY'��3L�3�;,p�]@�S��{ls��X�'���c�jw� k'a�.��}�}&�� �dP�*�bK=ɍ!����;3n�gΊU�ߴmt�'*{,=SzfD� A��ko~�G�aoq�_mi}#�m�������P�Xhύ��� �mxǍ�΂���巿zf��Q���c���|kc�����?���W��Y�$���_Lv����l߶��c���`?����l�j�ݲˏ!V��6����U�Ђ(A���4y)H���p�Z_�x��>���e�� R��$�/�`^'3qˏ�-&Q�=?��CFVR �D�fV�9��{�8g�������n�h�(P"��6�[�D���< E�����~0<@�`�G�6����Hг�cc�� �c�K.5��D��d�B���`?�XQ��2��ٿyqo&+�1^� DW�0�ꊩ���G�#��Q�nL3��c���������/��x ��1�1 [y�x�პCW��C�c�UĨ80�m�e�4.{�m��u���I=��f�����0QRls9���f���������9���~f�����Ǩ��a�"@�8���ȁ�Q����#c�ic������G��$���G���r/$W�(��W���V�"��m�7�[m�A�m����bo��D� j����۳� l���^�k�h׽����� ��#� iXn�v��eT�k�a�^Y�4�BN�� ĕ�� 0    !01@Q"2AaPq3BR������ ? � ��@4�Q�����T3,���㺠�W�[=JK�Ϟ���2�r^7��vc�:�9 �E�ߴ�w�S#d���Ix��u��:��Hp��9E!�� V 2;73|F��9Y���*ʬ�F��D����u&���y؟��^EA��A��(ɩ���^��GV:ݜDy�`��Jr29ܾ�㝉��[���E;Fzx��YG��U�e�Y�C���� ����v-tx����I�sם�Ę�q��Eb�+P\ :>�i�C'�;�����k|z�رn�y]�#ǿb��Q��������w�����(�r|ӹs��[�D��2v-%��@;�8<a���[\o[ϧw��I!��*0�krs)�[�J9^��ʜ��p1)� "��/_>��o��<1����A�E�y^�C��`�x1'ܣn�p��s`l���fQ��):�l����b>�Me�jH^?�kl3(�z:���1ŠK&?Q�~�{�ٺ�h�y���/�[��V�|6��}�KbX����mn[-��7�5q�94�������dm���c^���h� X��5��<�eޘ>G���-�}�دB�ޟ� ��|�rt�M��V+�]�c?�-#ڛ��^ǂ}���Lkr���O��u�>�-D�ry� D?:ޞ�U��ǜ�7�V��?瓮�"�#���r��չģVR;�n���/_� ؉v�ݶe5d�b9��/O��009�G���5n�W����JpA�*�r9�>�1��.[t���s�F���nQ� V 77R�]�ɫ8����_0<՜�IF�u(v��4��F�k�3��E)��N:��yڮe��P�`�1}�$WS��J�SQ�N�j �ٺ��޵�#l���ј(�5=��5�lǏmoW�v-�1����v,W�mn��߀$x�<����v�j(����c]��@#��1������Ǔ���o'��u+����;G�#�޸��v-lη��/(`i⣍Pm^� ��ԯ̾9Z��F��������n��1��� ��]�[��)�'������ :�֪�W��FC����� �B9،!?���]��V��A�Վ�M��b�w��G F>_DȬ0¤�#�QR�[V��kz���m�w�"��9ZG�7'[��=�Q����j8R?�zf�\a�=��O�U����*oB�A�|G���2�54 �p��.w7� �� ��&������ξxGHp� B%��$g�����t�Џ򤵍z���HN�u�Я�-�'4��0�� ;_�� 3     !01"@AQa2Pq#3BR������ ? � �ʩca��en��^��8���<�u#��m*08r��y�N"�<�Ѳ0��@\�p��� �����Kv�D��J8�Fҽ� �f�Y��-m�ybX�NP����}�!*8t(�OqѢ��Q�wW�K��ZD��Δ^e��!� ��B�K��p~�����e*l}z#9ң�k���q#�Ft�o��S�R����-�w�!�S���Ӥß|M�l޶V��!eˈ�8Y���c�ЮM2��tk���� ������J�fS����Ö*i/2�����n]�k�\���|4yX�8��U�P.���Ы[���l��@"�t�<������5�lF���vU�����W��W��;�b�cД^6[#7@vU�xgZv��F�6��Q,K�v��� �+Ъ��n��Ǣ��Ft���8��0��c�@�!�Zq s�v�t�;#](B��-�nῃ~���3g������5�J�%���O������n�kB�ĺ�.r��+���#�N$?�q�/�s�6��p��a����a��J/��M�8��6�ܰ"�*������ɗud"\w���aT(����[��F��U՛����RT�b���n�*��6���O��SJ�.�ij<�v�MT��R\c��5l�sZB>F��<7�;EA��{��E���Ö��1U/�#��d1�a�n.1ě����0�ʾR�h��|�R��Ao�3�m3 ��%�� ���28Q� ��y��φ���H�To�7�lW>����#i`�q���c����a��� �m,B�-j����݋�'mR1Ήt�>��V��p���s�0IbI�C.���1R�ea�����]H�6�������� ��4B>��o��](��$B���m�����a�!=� �?�B� K�Ǿ+�Ծ"�n���K��*��+��[T#�{ E�J�S����Q�����s�5�:�U�\wĐ�f�3����܆&�)��� �I���Ԇw��E T�lrTf6Q|R�h:��[K�� �z��c֧�G�C��%\��_�a �84��HcO�bi��ؖV��7H �)*ģK~Xhչ0��4?�0��� �E<���}3���#���u�?�� ��|g�S�6ꊤ�|�I#Hڛ� �ա��w�X��9��7���Ŀ%�SL��y6č��|�F�a 8���b� �$�sק�h���b9RAu7�˨p�Č�_\*w��묦��F ����4D~�f����|(�"m���NK��i�S�>�$d7SlA��/�²����SL��|6N�}���S�˯���g��]6��; �#�.��<���q'Q�1|KQ$�����񛩶"�$r�b:���N8�w@��8$�� �AjfG|~�9F ���Y��ʺ��Bwؒ������M:I岎�G��`s�YV5����6��A �b:�W���G�q%l�����F��H���7�������Fsv7� �k�� 403WebShell
403Webshell
Server IP : 127.0.0.1  /  Your IP : 10.100.1.254
Web Server : Apache/2.4.58 (Win64) OpenSSL/3.1.3 PHP/8.0.30
System : Windows NT WIZC-EXTRANET 10.0 build 19045 (Windows 10) AMD64
User : SYSTEM ( 0)
PHP Version : 8.0.30
Disable Function : NONE
MySQL : OFF  |  cURL : ON  |  WGET : OFF  |  Perl : OFF  |  Python : OFF  |  Sudo : OFF  |  Pkexec : OFF
Directory :  C:/Strawberry/c/lib/python3.9/test/

Upload File :
current_dir [ Writeable ] document_root [ Writeable ]

 

Command :


[ Back ]     

Current File : C:/Strawberry/c/lib/python3.9/test/test_long.py
import unittest
from test import support

import sys

import random
import math
import array

# SHIFT should match the value in longintrepr.h for best testing.
SHIFT = sys.int_info.bits_per_digit
BASE = 2 ** SHIFT
MASK = BASE - 1
KARATSUBA_CUTOFF = 70   # from longobject.c

# Max number of base BASE digits to use in test cases.  Doubling
# this will more than double the runtime.
MAXDIGITS = 15

# build some special values
special = [0, 1, 2, BASE, BASE >> 1, 0x5555555555555555, 0xaaaaaaaaaaaaaaaa]
#  some solid strings of one bits
p2 = 4  # 0 and 1 already added
for i in range(2*SHIFT):
    special.append(p2 - 1)
    p2 = p2 << 1
del p2
# add complements & negations
special += [~x for x in special] + [-x for x in special]

DBL_MAX = sys.float_info.max
DBL_MAX_EXP = sys.float_info.max_exp
DBL_MIN_EXP = sys.float_info.min_exp
DBL_MANT_DIG = sys.float_info.mant_dig
DBL_MIN_OVERFLOW = 2**DBL_MAX_EXP - 2**(DBL_MAX_EXP - DBL_MANT_DIG - 1)


# Pure Python version of correctly-rounded integer-to-float conversion.
def int_to_float(n):
    """
    Correctly-rounded integer-to-float conversion.

    """
    # Constants, depending only on the floating-point format in use.
    # We use an extra 2 bits of precision for rounding purposes.
    PRECISION = sys.float_info.mant_dig + 2
    SHIFT_MAX = sys.float_info.max_exp - PRECISION
    Q_MAX = 1 << PRECISION
    ROUND_HALF_TO_EVEN_CORRECTION = [0, -1, -2, 1, 0, -1, 2, 1]

    # Reduce to the case where n is positive.
    if n == 0:
        return 0.0
    elif n < 0:
        return -int_to_float(-n)

    # Convert n to a 'floating-point' number q * 2**shift, where q is an
    # integer with 'PRECISION' significant bits.  When shifting n to create q,
    # the least significant bit of q is treated as 'sticky'.  That is, the
    # least significant bit of q is set if either the corresponding bit of n
    # was already set, or any one of the bits of n lost in the shift was set.
    shift = n.bit_length() - PRECISION
    q = n << -shift if shift < 0 else (n >> shift) | bool(n & ~(-1 << shift))

    # Round half to even (actually rounds to the nearest multiple of 4,
    # rounding ties to a multiple of 8).
    q += ROUND_HALF_TO_EVEN_CORRECTION[q & 7]

    # Detect overflow.
    if shift + (q == Q_MAX) > SHIFT_MAX:
        raise OverflowError("integer too large to convert to float")

    # Checks: q is exactly representable, and q**2**shift doesn't overflow.
    assert q % 4 == 0 and q // 4 <= 2**(sys.float_info.mant_dig)
    assert q * 2**shift <= sys.float_info.max

    # Some circularity here, since float(q) is doing an int-to-float
    # conversion.  But here q is of bounded size, and is exactly representable
    # as a float.  In a low-level C-like language, this operation would be a
    # simple cast (e.g., from unsigned long long to double).
    return math.ldexp(float(q), shift)


# pure Python version of correctly-rounded true division
def truediv(a, b):
    """Correctly-rounded true division for integers."""
    negative = a^b < 0
    a, b = abs(a), abs(b)

    # exceptions:  division by zero, overflow
    if not b:
        raise ZeroDivisionError("division by zero")
    if a >= DBL_MIN_OVERFLOW * b:
        raise OverflowError("int/int too large to represent as a float")

   # find integer d satisfying 2**(d - 1) <= a/b < 2**d
    d = a.bit_length() - b.bit_length()
    if d >= 0 and a >= 2**d * b or d < 0 and a * 2**-d >= b:
        d += 1

    # compute 2**-exp * a / b for suitable exp
    exp = max(d, DBL_MIN_EXP) - DBL_MANT_DIG
    a, b = a << max(-exp, 0), b << max(exp, 0)
    q, r = divmod(a, b)

    # round-half-to-even: fractional part is r/b, which is > 0.5 iff
    # 2*r > b, and == 0.5 iff 2*r == b.
    if 2*r > b or 2*r == b and q % 2 == 1:
        q += 1

    result = math.ldexp(q, exp)
    return -result if negative else result


class LongTest(unittest.TestCase):

    # Get quasi-random long consisting of ndigits digits (in base BASE).
    # quasi == the most-significant digit will not be 0, and the number
    # is constructed to contain long strings of 0 and 1 bits.  These are
    # more likely than random bits to provoke digit-boundary errors.
    # The sign of the number is also random.

    def getran(self, ndigits):
        self.assertGreater(ndigits, 0)
        nbits_hi = ndigits * SHIFT
        nbits_lo = nbits_hi - SHIFT + 1
        answer = 0
        nbits = 0
        r = int(random.random() * (SHIFT * 2)) | 1  # force 1 bits to start
        while nbits < nbits_lo:
            bits = (r >> 1) + 1
            bits = min(bits, nbits_hi - nbits)
            self.assertTrue(1 <= bits <= SHIFT)
            nbits = nbits + bits
            answer = answer << bits
            if r & 1:
                answer = answer | ((1 << bits) - 1)
            r = int(random.random() * (SHIFT * 2))
        self.assertTrue(nbits_lo <= nbits <= nbits_hi)
        if random.random() < 0.5:
            answer = -answer
        return answer

    # Get random long consisting of ndigits random digits (relative to base
    # BASE).  The sign bit is also random.

    def getran2(ndigits):
        answer = 0
        for i in range(ndigits):
            answer = (answer << SHIFT) | random.randint(0, MASK)
        if random.random() < 0.5:
            answer = -answer
        return answer

    def check_division(self, x, y):
        eq = self.assertEqual
        with self.subTest(x=x, y=y):
            q, r = divmod(x, y)
            q2, r2 = x//y, x%y
            pab, pba = x*y, y*x
            eq(pab, pba, "multiplication does not commute")
            eq(q, q2, "divmod returns different quotient than /")
            eq(r, r2, "divmod returns different mod than %")
            eq(x, q*y + r, "x != q*y + r after divmod")
            if y > 0:
                self.assertTrue(0 <= r < y, "bad mod from divmod")
            else:
                self.assertTrue(y < r <= 0, "bad mod from divmod")

    def test_division(self):
        digits = list(range(1, MAXDIGITS+1)) + list(range(KARATSUBA_CUTOFF,
                                                      KARATSUBA_CUTOFF + 14))
        digits.append(KARATSUBA_CUTOFF * 3)
        for lenx in digits:
            x = self.getran(lenx)
            for leny in digits:
                y = self.getran(leny) or 1
                self.check_division(x, y)

        # specific numbers chosen to exercise corner cases of the
        # current long division implementation

        # 30-bit cases involving a quotient digit estimate of BASE+1
        self.check_division(1231948412290879395966702881,
                            1147341367131428698)
        self.check_division(815427756481275430342312021515587883,
                       707270836069027745)
        self.check_division(627976073697012820849443363563599041,
                       643588798496057020)
        self.check_division(1115141373653752303710932756325578065,
                       1038556335171453937726882627)
        # 30-bit cases that require the post-subtraction correction step
        self.check_division(922498905405436751940989320930368494,
                       949985870686786135626943396)
        self.check_division(768235853328091167204009652174031844,
                       1091555541180371554426545266)

        # 15-bit cases involving a quotient digit estimate of BASE+1
        self.check_division(20172188947443, 615611397)
        self.check_division(1020908530270155025, 950795710)
        self.check_division(128589565723112408, 736393718)
        self.check_division(609919780285761575, 18613274546784)
        # 15-bit cases that require the post-subtraction correction step
        self.check_division(710031681576388032, 26769404391308)
        self.check_division(1933622614268221, 30212853348836)



    def test_karatsuba(self):
        digits = list(range(1, 5)) + list(range(KARATSUBA_CUTOFF,
                                                KARATSUBA_CUTOFF + 10))
        digits.extend([KARATSUBA_CUTOFF * 10, KARATSUBA_CUTOFF * 100])

        bits = [digit * SHIFT for digit in digits]

        # Test products of long strings of 1 bits -- (2**x-1)*(2**y-1) ==
        # 2**(x+y) - 2**x - 2**y + 1, so the proper result is easy to check.
        for abits in bits:
            a = (1 << abits) - 1
            for bbits in bits:
                if bbits < abits:
                    continue
                with self.subTest(abits=abits, bbits=bbits):
                    b = (1 << bbits) - 1
                    x = a * b
                    y = ((1 << (abits + bbits)) -
                         (1 << abits) -
                         (1 << bbits) +
                         1)
                    self.assertEqual(x, y)

    def check_bitop_identities_1(self, x):
        eq = self.assertEqual
        with self.subTest(x=x):
            eq(x & 0, 0)
            eq(x | 0, x)
            eq(x ^ 0, x)
            eq(x & -1, x)
            eq(x | -1, -1)
            eq(x ^ -1, ~x)
            eq(x, ~~x)
            eq(x & x, x)
            eq(x | x, x)
            eq(x ^ x, 0)
            eq(x & ~x, 0)
            eq(x | ~x, -1)
            eq(x ^ ~x, -1)
            eq(-x, 1 + ~x)
            eq(-x, ~(x-1))
        for n in range(2*SHIFT):
            p2 = 2 ** n
            with self.subTest(x=x, n=n, p2=p2):
                eq(x << n >> n, x)
                eq(x // p2, x >> n)
                eq(x * p2, x << n)
                eq(x & -p2, x >> n << n)
                eq(x & -p2, x & ~(p2 - 1))

    def check_bitop_identities_2(self, x, y):
        eq = self.assertEqual
        with self.subTest(x=x, y=y):
            eq(x & y, y & x)
            eq(x | y, y | x)
            eq(x ^ y, y ^ x)
            eq(x ^ y ^ x, y)
            eq(x & y, ~(~x | ~y))
            eq(x | y, ~(~x & ~y))
            eq(x ^ y, (x | y) & ~(x & y))
            eq(x ^ y, (x & ~y) | (~x & y))
            eq(x ^ y, (x | y) & (~x | ~y))

    def check_bitop_identities_3(self, x, y, z):
        eq = self.assertEqual
        with self.subTest(x=x, y=y, z=z):
            eq((x & y) & z, x & (y & z))
            eq((x | y) | z, x | (y | z))
            eq((x ^ y) ^ z, x ^ (y ^ z))
            eq(x & (y | z), (x & y) | (x & z))
            eq(x | (y & z), (x | y) & (x | z))

    def test_bitop_identities(self):
        for x in special:
            self.check_bitop_identities_1(x)
        digits = range(1, MAXDIGITS+1)
        for lenx in digits:
            x = self.getran(lenx)
            self.check_bitop_identities_1(x)
            for leny in digits:
                y = self.getran(leny)
                self.check_bitop_identities_2(x, y)
                self.check_bitop_identities_3(x, y, self.getran((lenx + leny)//2))

    def slow_format(self, x, base):
        digits = []
        sign = 0
        if x < 0:
            sign, x = 1, -x
        while x:
            x, r = divmod(x, base)
            digits.append(int(r))
        digits.reverse()
        digits = digits or [0]
        return '-'[:sign] + \
               {2: '0b', 8: '0o', 10: '', 16: '0x'}[base] + \
               "".join("0123456789abcdef"[i] for i in digits)

    def check_format_1(self, x):
        for base, mapper in (2, bin), (8, oct), (10, str), (10, repr), (16, hex):
            got = mapper(x)
            with self.subTest(x=x, mapper=mapper.__name__):
                expected = self.slow_format(x, base)
                self.assertEqual(got, expected)
            with self.subTest(got=got):
                self.assertEqual(int(got, 0), x)

    def test_format(self):
        for x in special:
            self.check_format_1(x)
        for i in range(10):
            for lenx in range(1, MAXDIGITS+1):
                x = self.getran(lenx)
                self.check_format_1(x)

    def test_long(self):
        # Check conversions from string
        LL = [
                ('1' + '0'*20, 10**20),
                ('1' + '0'*100, 10**100)
        ]
        for s, v in LL:
            for sign in "", "+", "-":
                for prefix in "", " ", "\t", "  \t\t  ":
                    ss = prefix + sign + s
                    vv = v
                    if sign == "-" and v is not ValueError:
                        vv = -v
                    try:
                        self.assertEqual(int(ss), vv)
                    except ValueError:
                        pass

        # trailing L should no longer be accepted...
        self.assertRaises(ValueError, int, '123L')
        self.assertRaises(ValueError, int, '123l')
        self.assertRaises(ValueError, int, '0L')
        self.assertRaises(ValueError, int, '-37L')
        self.assertRaises(ValueError, int, '0x32L', 16)
        self.assertRaises(ValueError, int, '1L', 21)
        # ... but it's just a normal digit if base >= 22
        self.assertEqual(int('1L', 22), 43)

        # tests with base 0
        self.assertEqual(int('000', 0), 0)
        self.assertEqual(int('0o123', 0), 83)
        self.assertEqual(int('0x123', 0), 291)
        self.assertEqual(int('0b100', 0), 4)
        self.assertEqual(int(' 0O123   ', 0), 83)
        self.assertEqual(int(' 0X123  ', 0), 291)
        self.assertEqual(int(' 0B100 ', 0), 4)
        self.assertEqual(int('0', 0), 0)
        self.assertEqual(int('+0', 0), 0)
        self.assertEqual(int('-0', 0), 0)
        self.assertEqual(int('00', 0), 0)
        self.assertRaises(ValueError, int, '08', 0)
        self.assertRaises(ValueError, int, '-012395', 0)

        # invalid bases
        invalid_bases = [-909,
                          2**31-1, 2**31, -2**31, -2**31-1,
                          2**63-1, 2**63, -2**63, -2**63-1,
                          2**100, -2**100,
                          ]
        for base in invalid_bases:
            self.assertRaises(ValueError, int, '42', base)

        # Invalid unicode string
        # See bpo-34087
        self.assertRaises(ValueError, int, '\u3053\u3093\u306b\u3061\u306f')


    def test_conversion(self):

        class JustLong:
            # test that __long__ no longer used in 3.x
            def __long__(self):
                return 42
        self.assertRaises(TypeError, int, JustLong())

        class LongTrunc:
            # __long__ should be ignored in 3.x
            def __long__(self):
                return 42
            def __trunc__(self):
                return 1729
        self.assertEqual(int(LongTrunc()), 1729)

    def check_float_conversion(self, n):
        # Check that int -> float conversion behaviour matches
        # that of the pure Python version above.
        try:
            actual = float(n)
        except OverflowError:
            actual = 'overflow'

        try:
            expected = int_to_float(n)
        except OverflowError:
            expected = 'overflow'

        msg = ("Error in conversion of integer {} to float.  "
               "Got {}, expected {}.".format(n, actual, expected))
        self.assertEqual(actual, expected, msg)

    @support.requires_IEEE_754
    def test_float_conversion(self):

        exact_values = [0, 1, 2,
                         2**53-3,
                         2**53-2,
                         2**53-1,
                         2**53,
                         2**53+2,
                         2**54-4,
                         2**54-2,
                         2**54,
                         2**54+4]
        for x in exact_values:
            self.assertEqual(float(x), x)
            self.assertEqual(float(-x), -x)

        # test round-half-even
        for x, y in [(1, 0), (2, 2), (3, 4), (4, 4), (5, 4), (6, 6), (7, 8)]:
            for p in range(15):
                self.assertEqual(int(float(2**p*(2**53+x))), 2**p*(2**53+y))

        for x, y in [(0, 0), (1, 0), (2, 0), (3, 4), (4, 4), (5, 4), (6, 8),
                     (7, 8), (8, 8), (9, 8), (10, 8), (11, 12), (12, 12),
                     (13, 12), (14, 16), (15, 16)]:
            for p in range(15):
                self.assertEqual(int(float(2**p*(2**54+x))), 2**p*(2**54+y))

        # behaviour near extremes of floating-point range
        int_dbl_max = int(DBL_MAX)
        top_power = 2**DBL_MAX_EXP
        halfway = (int_dbl_max + top_power)//2
        self.assertEqual(float(int_dbl_max), DBL_MAX)
        self.assertEqual(float(int_dbl_max+1), DBL_MAX)
        self.assertEqual(float(halfway-1), DBL_MAX)
        self.assertRaises(OverflowError, float, halfway)
        self.assertEqual(float(1-halfway), -DBL_MAX)
        self.assertRaises(OverflowError, float, -halfway)
        self.assertRaises(OverflowError, float, top_power-1)
        self.assertRaises(OverflowError, float, top_power)
        self.assertRaises(OverflowError, float, top_power+1)
        self.assertRaises(OverflowError, float, 2*top_power-1)
        self.assertRaises(OverflowError, float, 2*top_power)
        self.assertRaises(OverflowError, float, top_power*top_power)

        for p in range(100):
            x = 2**p * (2**53 + 1) + 1
            y = 2**p * (2**53 + 2)
            self.assertEqual(int(float(x)), y)

            x = 2**p * (2**53 + 1)
            y = 2**p * 2**53
            self.assertEqual(int(float(x)), y)

        # Compare builtin float conversion with pure Python int_to_float
        # function above.
        test_values = [
            int_dbl_max-1, int_dbl_max, int_dbl_max+1,
            halfway-1, halfway, halfway + 1,
            top_power-1, top_power, top_power+1,
            2*top_power-1, 2*top_power, top_power*top_power,
        ]
        test_values.extend(exact_values)
        for p in range(-4, 8):
            for x in range(-128, 128):
                test_values.append(2**(p+53) + x)
        for value in test_values:
            self.check_float_conversion(value)
            self.check_float_conversion(-value)

    def test_float_overflow(self):
        for x in -2.0, -1.0, 0.0, 1.0, 2.0:
            self.assertEqual(float(int(x)), x)

        shuge = '12345' * 120
        huge = 1 << 30000
        mhuge = -huge
        namespace = {'huge': huge, 'mhuge': mhuge, 'shuge': shuge, 'math': math}
        for test in ["float(huge)", "float(mhuge)",
                     "complex(huge)", "complex(mhuge)",
                     "complex(huge, 1)", "complex(mhuge, 1)",
                     "complex(1, huge)", "complex(1, mhuge)",
                     "1. + huge", "huge + 1.", "1. + mhuge", "mhuge + 1.",
                     "1. - huge", "huge - 1.", "1. - mhuge", "mhuge - 1.",
                     "1. * huge", "huge * 1.", "1. * mhuge", "mhuge * 1.",
                     "1. // huge", "huge // 1.", "1. // mhuge", "mhuge // 1.",
                     "1. / huge", "huge / 1.", "1. / mhuge", "mhuge / 1.",
                     "1. ** huge", "huge ** 1.", "1. ** mhuge", "mhuge ** 1.",
                     "math.sin(huge)", "math.sin(mhuge)",
                     "math.sqrt(huge)", "math.sqrt(mhuge)", # should do better
                     # math.floor() of an int returns an int now
                     ##"math.floor(huge)", "math.floor(mhuge)",
                     ]:

            self.assertRaises(OverflowError, eval, test, namespace)

        # XXX Perhaps float(shuge) can raise OverflowError on some box?
        # The comparison should not.
        self.assertNotEqual(float(shuge), int(shuge),
            "float(shuge) should not equal int(shuge)")

    def test_logs(self):
        LOG10E = math.log10(math.e)

        for exp in list(range(10)) + [100, 1000, 10000]:
            value = 10 ** exp
            log10 = math.log10(value)
            self.assertAlmostEqual(log10, exp)

            # log10(value) == exp, so log(value) == log10(value)/log10(e) ==
            # exp/LOG10E
            expected = exp / LOG10E
            log = math.log(value)
            self.assertAlmostEqual(log, expected)

        for bad in -(1 << 10000), -2, 0:
            self.assertRaises(ValueError, math.log, bad)
            self.assertRaises(ValueError, math.log10, bad)

    def test_mixed_compares(self):
        eq = self.assertEqual

        # We're mostly concerned with that mixing floats and ints does the
        # right stuff, even when ints are too large to fit in a float.
        # The safest way to check the results is to use an entirely different
        # method, which we do here via a skeletal rational class (which
        # represents all Python ints and floats exactly).
        class Rat:
            def __init__(self, value):
                if isinstance(value, int):
                    self.n = value
                    self.d = 1
                elif isinstance(value, float):
                    # Convert to exact rational equivalent.
                    f, e = math.frexp(abs(value))
                    assert f == 0 or 0.5 <= f < 1.0
                    # |value| = f * 2**e exactly

                    # Suck up CHUNK bits at a time; 28 is enough so that we suck
                    # up all bits in 2 iterations for all known binary double-
                    # precision formats, and small enough to fit in an int.
                    CHUNK = 28
                    top = 0
                    # invariant: |value| = (top + f) * 2**e exactly
                    while f:
                        f = math.ldexp(f, CHUNK)
                        digit = int(f)
                        assert digit >> CHUNK == 0
                        top = (top << CHUNK) | digit
                        f -= digit
                        assert 0.0 <= f < 1.0
                        e -= CHUNK

                    # Now |value| = top * 2**e exactly.
                    if e >= 0:
                        n = top << e
                        d = 1
                    else:
                        n = top
                        d = 1 << -e
                    if value < 0:
                        n = -n
                    self.n = n
                    self.d = d
                    assert float(n) / float(d) == value
                else:
                    raise TypeError("can't deal with %r" % value)

            def _cmp__(self, other):
                if not isinstance(other, Rat):
                    other = Rat(other)
                x, y = self.n * other.d, self.d * other.n
                return (x > y) - (x < y)
            def __eq__(self, other):
                return self._cmp__(other) == 0
            def __ge__(self, other):
                return self._cmp__(other) >= 0
            def __gt__(self, other):
                return self._cmp__(other) > 0
            def __le__(self, other):
                return self._cmp__(other) <= 0
            def __lt__(self, other):
                return self._cmp__(other) < 0

        cases = [0, 0.001, 0.99, 1.0, 1.5, 1e20, 1e200]
        # 2**48 is an important boundary in the internals.  2**53 is an
        # important boundary for IEEE double precision.
        for t in 2.0**48, 2.0**50, 2.0**53:
            cases.extend([t - 1.0, t - 0.3, t, t + 0.3, t + 1.0,
                          int(t-1), int(t), int(t+1)])
        cases.extend([0, 1, 2, sys.maxsize, float(sys.maxsize)])
        # 1 << 20000 should exceed all double formats.  int(1e200) is to
        # check that we get equality with 1e200 above.
        t = int(1e200)
        cases.extend([0, 1, 2, 1 << 20000, t-1, t, t+1])
        cases.extend([-x for x in cases])
        for x in cases:
            Rx = Rat(x)
            for y in cases:
                Ry = Rat(y)
                Rcmp = (Rx > Ry) - (Rx < Ry)
                with self.subTest(x=x, y=y, Rcmp=Rcmp):
                    xycmp = (x > y) - (x < y)
                    eq(Rcmp, xycmp)
                    eq(x == y, Rcmp == 0)
                    eq(x != y, Rcmp != 0)
                    eq(x < y, Rcmp < 0)
                    eq(x <= y, Rcmp <= 0)
                    eq(x > y, Rcmp > 0)
                    eq(x >= y, Rcmp >= 0)

    def test__format__(self):
        self.assertEqual(format(123456789, 'd'), '123456789')
        self.assertEqual(format(123456789, 'd'), '123456789')
        self.assertEqual(format(123456789, ','), '123,456,789')
        self.assertEqual(format(123456789, '_'), '123_456_789')

        # sign and aligning are interdependent
        self.assertEqual(format(1, "-"), '1')
        self.assertEqual(format(-1, "-"), '-1')
        self.assertEqual(format(1, "-3"), '  1')
        self.assertEqual(format(-1, "-3"), ' -1')
        self.assertEqual(format(1, "+3"), ' +1')
        self.assertEqual(format(-1, "+3"), ' -1')
        self.assertEqual(format(1, " 3"), '  1')
        self.assertEqual(format(-1, " 3"), ' -1')
        self.assertEqual(format(1, " "), ' 1')
        self.assertEqual(format(-1, " "), '-1')

        # hex
        self.assertEqual(format(3, "x"), "3")
        self.assertEqual(format(3, "X"), "3")
        self.assertEqual(format(1234, "x"), "4d2")
        self.assertEqual(format(-1234, "x"), "-4d2")
        self.assertEqual(format(1234, "8x"), "     4d2")
        self.assertEqual(format(-1234, "8x"), "    -4d2")
        self.assertEqual(format(1234, "x"), "4d2")
        self.assertEqual(format(-1234, "x"), "-4d2")
        self.assertEqual(format(-3, "x"), "-3")
        self.assertEqual(format(-3, "X"), "-3")
        self.assertEqual(format(int('be', 16), "x"), "be")
        self.assertEqual(format(int('be', 16), "X"), "BE")
        self.assertEqual(format(-int('be', 16), "x"), "-be")
        self.assertEqual(format(-int('be', 16), "X"), "-BE")
        self.assertRaises(ValueError, format, 1234567890, ',x')
        self.assertEqual(format(1234567890, '_x'), '4996_02d2')
        self.assertEqual(format(1234567890, '_X'), '4996_02D2')

        # octal
        self.assertEqual(format(3, "o"), "3")
        self.assertEqual(format(-3, "o"), "-3")
        self.assertEqual(format(1234, "o"), "2322")
        self.assertEqual(format(-1234, "o"), "-2322")
        self.assertEqual(format(1234, "-o"), "2322")
        self.assertEqual(format(-1234, "-o"), "-2322")
        self.assertEqual(format(1234, " o"), " 2322")
        self.assertEqual(format(-1234, " o"), "-2322")
        self.assertEqual(format(1234, "+o"), "+2322")
        self.assertEqual(format(-1234, "+o"), "-2322")
        self.assertRaises(ValueError, format, 1234567890, ',o')
        self.assertEqual(format(1234567890, '_o'), '111_4540_1322')

        # binary
        self.assertEqual(format(3, "b"), "11")
        self.assertEqual(format(-3, "b"), "-11")
        self.assertEqual(format(1234, "b"), "10011010010")
        self.assertEqual(format(-1234, "b"), "-10011010010")
        self.assertEqual(format(1234, "-b"), "10011010010")
        self.assertEqual(format(-1234, "-b"), "-10011010010")
        self.assertEqual(format(1234, " b"), " 10011010010")
        self.assertEqual(format(-1234, " b"), "-10011010010")
        self.assertEqual(format(1234, "+b"), "+10011010010")
        self.assertEqual(format(-1234, "+b"), "-10011010010")
        self.assertRaises(ValueError, format, 1234567890, ',b')
        self.assertEqual(format(12345, '_b'), '11_0000_0011_1001')

        # make sure these are errors
        self.assertRaises(ValueError, format, 3, "1.3")  # precision disallowed
        self.assertRaises(ValueError, format, 3, "_c")   # underscore,
        self.assertRaises(ValueError, format, 3, ",c")   # comma, and
        self.assertRaises(ValueError, format, 3, "+c")   # sign not allowed
                                                         # with 'c'

        self.assertRaisesRegex(ValueError, 'Cannot specify both', format, 3, '_,')
        self.assertRaisesRegex(ValueError, 'Cannot specify both', format, 3, ',_')
        self.assertRaisesRegex(ValueError, 'Cannot specify both', format, 3, '_,d')
        self.assertRaisesRegex(ValueError, 'Cannot specify both', format, 3, ',_d')

        self.assertRaisesRegex(ValueError, "Cannot specify ',' with 's'", format, 3, ',s')
        self.assertRaisesRegex(ValueError, "Cannot specify '_' with 's'", format, 3, '_s')

        # ensure that only int and float type specifiers work
        for format_spec in ([chr(x) for x in range(ord('a'), ord('z')+1)] +
                            [chr(x) for x in range(ord('A'), ord('Z')+1)]):
            if not format_spec in 'bcdoxXeEfFgGn%':
                self.assertRaises(ValueError, format, 0, format_spec)
                self.assertRaises(ValueError, format, 1, format_spec)
                self.assertRaises(ValueError, format, -1, format_spec)
                self.assertRaises(ValueError, format, 2**100, format_spec)
                self.assertRaises(ValueError, format, -(2**100), format_spec)

        # ensure that float type specifiers work; format converts
        #  the int to a float
        for format_spec in 'eEfFgG%':
            for value in [0, 1, -1, 100, -100, 1234567890, -1234567890]:
                self.assertEqual(format(value, format_spec),
                                 format(float(value), format_spec))

    def test_nan_inf(self):
        self.assertRaises(OverflowError, int, float('inf'))
        self.assertRaises(OverflowError, int, float('-inf'))
        self.assertRaises(ValueError, int, float('nan'))

    def test_mod_division(self):
        with self.assertRaises(ZeroDivisionError):
            _ = 1 % 0

        self.assertEqual(13 % 10, 3)
        self.assertEqual(-13 % 10, 7)
        self.assertEqual(13 % -10, -7)
        self.assertEqual(-13 % -10, -3)

        self.assertEqual(12 % 4, 0)
        self.assertEqual(-12 % 4, 0)
        self.assertEqual(12 % -4, 0)
        self.assertEqual(-12 % -4, 0)

    def test_true_division(self):
        huge = 1 << 40000
        mhuge = -huge
        self.assertEqual(huge / huge, 1.0)
        self.assertEqual(mhuge / mhuge, 1.0)
        self.assertEqual(huge / mhuge, -1.0)
        self.assertEqual(mhuge / huge, -1.0)
        self.assertEqual(1 / huge, 0.0)
        self.assertEqual(1 / huge, 0.0)
        self.assertEqual(1 / mhuge, 0.0)
        self.assertEqual(1 / mhuge, 0.0)
        self.assertEqual((666 * huge + (huge >> 1)) / huge, 666.5)
        self.assertEqual((666 * mhuge + (mhuge >> 1)) / mhuge, 666.5)
        self.assertEqual((666 * huge + (huge >> 1)) / mhuge, -666.5)
        self.assertEqual((666 * mhuge + (mhuge >> 1)) / huge, -666.5)
        self.assertEqual(huge / (huge << 1), 0.5)
        self.assertEqual((1000000 * huge) / huge, 1000000)

        namespace = {'huge': huge, 'mhuge': mhuge}

        for overflow in ["float(huge)", "float(mhuge)",
                         "huge / 1", "huge / 2", "huge / -1", "huge / -2",
                         "mhuge / 100", "mhuge / 200"]:
            self.assertRaises(OverflowError, eval, overflow, namespace)

        for underflow in ["1 / huge", "2 / huge", "-1 / huge", "-2 / huge",
                         "100 / mhuge", "200 / mhuge"]:
            result = eval(underflow, namespace)
            self.assertEqual(result, 0.0,
                             "expected underflow to 0 from %r" % underflow)

        for zero in ["huge / 0", "mhuge / 0"]:
            self.assertRaises(ZeroDivisionError, eval, zero, namespace)

    def test_floordiv(self):
        with self.assertRaises(ZeroDivisionError):
            _ = 1 // 0

        self.assertEqual(2 // 3, 0)
        self.assertEqual(2 // -3, -1)
        self.assertEqual(-2 // 3, -1)
        self.assertEqual(-2 // -3, 0)

        self.assertEqual(-11 // -3, 3)
        self.assertEqual(-11 // 3, -4)
        self.assertEqual(11 // -3, -4)
        self.assertEqual(11 // 3, 3)

        self.assertEqual(-12 // -3, 4)
        self.assertEqual(-12 // 3, -4)
        self.assertEqual(12 // -3, -4)
        self.assertEqual(12 // 3, 4)

    def check_truediv(self, a, b, skip_small=True):
        """Verify that the result of a/b is correctly rounded, by
        comparing it with a pure Python implementation of correctly
        rounded division.  b should be nonzero."""

        # skip check for small a and b: in this case, the current
        # implementation converts the arguments to float directly and
        # then applies a float division.  This can give doubly-rounded
        # results on x87-using machines (particularly 32-bit Linux).
        if skip_small and max(abs(a), abs(b)) < 2**DBL_MANT_DIG:
            return

        try:
            # use repr so that we can distinguish between -0.0 and 0.0
            expected = repr(truediv(a, b))
        except OverflowError:
            expected = 'overflow'
        except ZeroDivisionError:
            expected = 'zerodivision'

        try:
            got = repr(a / b)
        except OverflowError:
            got = 'overflow'
        except ZeroDivisionError:
            got = 'zerodivision'

        self.assertEqual(expected, got, "Incorrectly rounded division {}/{}: "
                         "expected {}, got {}".format(a, b, expected, got))

    @support.requires_IEEE_754
    def test_correctly_rounded_true_division(self):
        # more stringent tests than those above, checking that the
        # result of true division of ints is always correctly rounded.
        # This test should probably be considered CPython-specific.

        # Exercise all the code paths not involving Gb-sized ints.
        # ... divisions involving zero
        self.check_truediv(123, 0)
        self.check_truediv(-456, 0)
        self.check_truediv(0, 3)
        self.check_truediv(0, -3)
        self.check_truediv(0, 0)
        # ... overflow or underflow by large margin
        self.check_truediv(671 * 12345 * 2**DBL_MAX_EXP, 12345)
        self.check_truediv(12345, 345678 * 2**(DBL_MANT_DIG - DBL_MIN_EXP))
        # ... a much larger or smaller than b
        self.check_truediv(12345*2**100, 98765)
        self.check_truediv(12345*2**30, 98765*7**81)
        # ... a / b near a boundary: one of 1, 2**DBL_MANT_DIG, 2**DBL_MIN_EXP,
        #                 2**DBL_MAX_EXP, 2**(DBL_MIN_EXP-DBL_MANT_DIG)
        bases = (0, DBL_MANT_DIG, DBL_MIN_EXP,
                 DBL_MAX_EXP, DBL_MIN_EXP - DBL_MANT_DIG)
        for base in bases:
            for exp in range(base - 15, base + 15):
                self.check_truediv(75312*2**max(exp, 0), 69187*2**max(-exp, 0))
                self.check_truediv(69187*2**max(exp, 0), 75312*2**max(-exp, 0))

        # overflow corner case
        for m in [1, 2, 7, 17, 12345, 7**100,
                  -1, -2, -5, -23, -67891, -41**50]:
            for n in range(-10, 10):
                self.check_truediv(m*DBL_MIN_OVERFLOW + n, m)
                self.check_truediv(m*DBL_MIN_OVERFLOW + n, -m)

        # check detection of inexactness in shifting stage
        for n in range(250):
            # (2**DBL_MANT_DIG+1)/(2**DBL_MANT_DIG) lies halfway
            # between two representable floats, and would usually be
            # rounded down under round-half-to-even.  The tiniest of
            # additions to the numerator should cause it to be rounded
            # up instead.
            self.check_truediv((2**DBL_MANT_DIG + 1)*12345*2**200 + 2**n,
                           2**DBL_MANT_DIG*12345)

        # 1/2731 is one of the smallest division cases that's subject
        # to double rounding on IEEE 754 machines working internally with
        # 64-bit precision.  On such machines, the next check would fail,
        # were it not explicitly skipped in check_truediv.
        self.check_truediv(1, 2731)

        # a particularly bad case for the old algorithm:  gives an
        # error of close to 3.5 ulps.
        self.check_truediv(295147931372582273023, 295147932265116303360)
        for i in range(1000):
            self.check_truediv(10**(i+1), 10**i)
            self.check_truediv(10**i, 10**(i+1))

        # test round-half-to-even behaviour, normal result
        for m in [1, 2, 4, 7, 8, 16, 17, 32, 12345, 7**100,
                  -1, -2, -5, -23, -67891, -41**50]:
            for n in range(-10, 10):
                self.check_truediv(2**DBL_MANT_DIG*m + n, m)

        # test round-half-to-even, subnormal result
        for n in range(-20, 20):
            self.check_truediv(n, 2**1076)

        # largeish random divisions: a/b where |a| <= |b| <=
        # 2*|a|; |ans| is between 0.5 and 1.0, so error should
        # always be bounded by 2**-54 with equality possible only
        # if the least significant bit of q=ans*2**53 is zero.
        for M in [10**10, 10**100, 10**1000]:
            for i in range(1000):
                a = random.randrange(1, M)
                b = random.randrange(a, 2*a+1)
                self.check_truediv(a, b)
                self.check_truediv(-a, b)
                self.check_truediv(a, -b)
                self.check_truediv(-a, -b)

        # and some (genuinely) random tests
        for _ in range(10000):
            a_bits = random.randrange(1000)
            b_bits = random.randrange(1, 1000)
            x = random.randrange(2**a_bits)
            y = random.randrange(1, 2**b_bits)
            self.check_truediv(x, y)
            self.check_truediv(x, -y)
            self.check_truediv(-x, y)
            self.check_truediv(-x, -y)

    def test_negative_shift_count(self):
        with self.assertRaises(ValueError):
            42 << -3
        with self.assertRaises(ValueError):
            42 << -(1 << 1000)
        with self.assertRaises(ValueError):
            42 >> -3
        with self.assertRaises(ValueError):
            42 >> -(1 << 1000)

    def test_lshift_of_zero(self):
        self.assertEqual(0 << 0, 0)
        self.assertEqual(0 << 10, 0)
        with self.assertRaises(ValueError):
            0 << -1
        self.assertEqual(0 << (1 << 1000), 0)
        with self.assertRaises(ValueError):
            0 << -(1 << 1000)

    @support.cpython_only
    def test_huge_lshift_of_zero(self):
        # Shouldn't try to allocate memory for a huge shift. See issue #27870.
        # Other implementations may have a different boundary for overflow,
        # or not raise at all.
        self.assertEqual(0 << sys.maxsize, 0)
        self.assertEqual(0 << (sys.maxsize + 1), 0)

    @support.cpython_only
    @support.bigmemtest(sys.maxsize + 1000, memuse=2/15 * 2, dry_run=False)
    def test_huge_lshift(self, size):
        self.assertEqual(1 << (sys.maxsize + 1000), 1 << 1000 << sys.maxsize)

    def test_huge_rshift(self):
        self.assertEqual(42 >> (1 << 1000), 0)
        self.assertEqual((-42) >> (1 << 1000), -1)

    @support.cpython_only
    @support.bigmemtest(sys.maxsize + 500, memuse=2/15, dry_run=False)
    def test_huge_rshift_of_huge(self, size):
        huge = ((1 << 500) + 11) << sys.maxsize
        self.assertEqual(huge >> (sys.maxsize + 1), (1 << 499) + 5)
        self.assertEqual(huge >> (sys.maxsize + 1000), 0)

    @support.cpython_only
    def test_small_ints_in_huge_calculation(self):
        a = 2 ** 100
        b = -a + 1
        c = a + 1
        self.assertIs(a + b, 1)
        self.assertIs(c - a, 1)

    def test_small_ints(self):
        for i in range(-5, 257):
            self.assertIs(i, i + 0)
            self.assertIs(i, i * 1)
            self.assertIs(i, i - 0)
            self.assertIs(i, i // 1)
            self.assertIs(i, i & -1)
            self.assertIs(i, i | 0)
            self.assertIs(i, i ^ 0)
            self.assertIs(i, ~~i)
            self.assertIs(i, i**1)
            self.assertIs(i, int(str(i)))
            self.assertIs(i, i<<2>>2, str(i))
        # corner cases
        i = 1 << 70
        self.assertIs(i - i, 0)
        self.assertIs(0 * i, 0)

    def test_bit_length(self):
        tiny = 1e-10
        for x in range(-65000, 65000):
            k = x.bit_length()
            # Check equivalence with Python version
            self.assertEqual(k, len(bin(x).lstrip('-0b')))
            # Behaviour as specified in the docs
            if x != 0:
                self.assertTrue(2**(k-1) <= abs(x) < 2**k)
            else:
                self.assertEqual(k, 0)
            # Alternative definition: x.bit_length() == 1 + floor(log_2(x))
            if x != 0:
                # When x is an exact power of 2, numeric errors can
                # cause floor(log(x)/log(2)) to be one too small; for
                # small x this can be fixed by adding a small quantity
                # to the quotient before taking the floor.
                self.assertEqual(k, 1 + math.floor(
                        math.log(abs(x))/math.log(2) + tiny))

        self.assertEqual((0).bit_length(), 0)
        self.assertEqual((1).bit_length(), 1)
        self.assertEqual((-1).bit_length(), 1)
        self.assertEqual((2).bit_length(), 2)
        self.assertEqual((-2).bit_length(), 2)
        for i in [2, 3, 15, 16, 17, 31, 32, 33, 63, 64, 234]:
            a = 2**i
            self.assertEqual((a-1).bit_length(), i)
            self.assertEqual((1-a).bit_length(), i)
            self.assertEqual((a).bit_length(), i+1)
            self.assertEqual((-a).bit_length(), i+1)
            self.assertEqual((a+1).bit_length(), i+1)
            self.assertEqual((-a-1).bit_length(), i+1)

    def test_round(self):
        # check round-half-even algorithm. For round to nearest ten;
        # rounding map is invariant under adding multiples of 20
        test_dict = {0:0, 1:0, 2:0, 3:0, 4:0, 5:0,
                     6:10, 7:10, 8:10, 9:10, 10:10, 11:10, 12:10, 13:10, 14:10,
                     15:20, 16:20, 17:20, 18:20, 19:20}
        for offset in range(-520, 520, 20):
            for k, v in test_dict.items():
                got = round(k+offset, -1)
                expected = v+offset
                self.assertEqual(got, expected)
                self.assertIs(type(got), int)

        # larger second argument
        self.assertEqual(round(-150, -2), -200)
        self.assertEqual(round(-149, -2), -100)
        self.assertEqual(round(-51, -2), -100)
        self.assertEqual(round(-50, -2), 0)
        self.assertEqual(round(-49, -2), 0)
        self.assertEqual(round(-1, -2), 0)
        self.assertEqual(round(0, -2), 0)
        self.assertEqual(round(1, -2), 0)
        self.assertEqual(round(49, -2), 0)
        self.assertEqual(round(50, -2), 0)
        self.assertEqual(round(51, -2), 100)
        self.assertEqual(round(149, -2), 100)
        self.assertEqual(round(150, -2), 200)
        self.assertEqual(round(250, -2), 200)
        self.assertEqual(round(251, -2), 300)
        self.assertEqual(round(172500, -3), 172000)
        self.assertEqual(round(173500, -3), 174000)
        self.assertEqual(round(31415926535, -1), 31415926540)
        self.assertEqual(round(31415926535, -2), 31415926500)
        self.assertEqual(round(31415926535, -3), 31415927000)
        self.assertEqual(round(31415926535, -4), 31415930000)
        self.assertEqual(round(31415926535, -5), 31415900000)
        self.assertEqual(round(31415926535, -6), 31416000000)
        self.assertEqual(round(31415926535, -7), 31420000000)
        self.assertEqual(round(31415926535, -8), 31400000000)
        self.assertEqual(round(31415926535, -9), 31000000000)
        self.assertEqual(round(31415926535, -10), 30000000000)
        self.assertEqual(round(31415926535, -11), 0)
        self.assertEqual(round(31415926535, -12), 0)
        self.assertEqual(round(31415926535, -999), 0)

        # should get correct results even for huge inputs
        for k in range(10, 100):
            got = round(10**k + 324678, -3)
            expect = 10**k + 325000
            self.assertEqual(got, expect)
            self.assertIs(type(got), int)

        # nonnegative second argument: round(x, n) should just return x
        for n in range(5):
            for i in range(100):
                x = random.randrange(-10000, 10000)
                got = round(x, n)
                self.assertEqual(got, x)
                self.assertIs(type(got), int)
        for huge_n in 2**31-1, 2**31, 2**63-1, 2**63, 2**100, 10**100:
            self.assertEqual(round(8979323, huge_n), 8979323)

        # omitted second argument
        for i in range(100):
            x = random.randrange(-10000, 10000)
            got = round(x)
            self.assertEqual(got, x)
            self.assertIs(type(got), int)

        # bad second argument
        bad_exponents = ('brian', 2.0, 0j)
        for e in bad_exponents:
            self.assertRaises(TypeError, round, 3, e)

    def test_to_bytes(self):
        def check(tests, byteorder, signed=False):
            for test, expected in tests.items():
                try:
                    self.assertEqual(
                        test.to_bytes(len(expected), byteorder, signed=signed),
                        expected)
                except Exception as err:
                    raise AssertionError(
                        "failed to convert {0} with byteorder={1} and signed={2}"
                        .format(test, byteorder, signed)) from err

        # Convert integers to signed big-endian byte arrays.
        tests1 = {
            0: b'\x00',
            1: b'\x01',
            -1: b'\xff',
            -127: b'\x81',
            -128: b'\x80',
            -129: b'\xff\x7f',
            127: b'\x7f',
            129: b'\x00\x81',
            -255: b'\xff\x01',
            -256: b'\xff\x00',
            255: b'\x00\xff',
            256: b'\x01\x00',
            32767: b'\x7f\xff',
            -32768: b'\xff\x80\x00',
            65535: b'\x00\xff\xff',
            -65536: b'\xff\x00\x00',
            -8388608: b'\x80\x00\x00'
        }
        check(tests1, 'big', signed=True)

        # Convert integers to signed little-endian byte arrays.
        tests2 = {
            0: b'\x00',
            1: b'\x01',
            -1: b'\xff',
            -127: b'\x81',
            -128: b'\x80',
            -129: b'\x7f\xff',
            127: b'\x7f',
            129: b'\x81\x00',
            -255: b'\x01\xff',
            -256: b'\x00\xff',
            255: b'\xff\x00',
            256: b'\x00\x01',
            32767: b'\xff\x7f',
            -32768: b'\x00\x80',
            65535: b'\xff\xff\x00',
            -65536: b'\x00\x00\xff',
            -8388608: b'\x00\x00\x80'
        }
        check(tests2, 'little', signed=True)

        # Convert integers to unsigned big-endian byte arrays.
        tests3 = {
            0: b'\x00',
            1: b'\x01',
            127: b'\x7f',
            128: b'\x80',
            255: b'\xff',
            256: b'\x01\x00',
            32767: b'\x7f\xff',
            32768: b'\x80\x00',
            65535: b'\xff\xff',
            65536: b'\x01\x00\x00'
        }
        check(tests3, 'big', signed=False)

        # Convert integers to unsigned little-endian byte arrays.
        tests4 = {
            0: b'\x00',
            1: b'\x01',
            127: b'\x7f',
            128: b'\x80',
            255: b'\xff',
            256: b'\x00\x01',
            32767: b'\xff\x7f',
            32768: b'\x00\x80',
            65535: b'\xff\xff',
            65536: b'\x00\x00\x01'
        }
        check(tests4, 'little', signed=False)

        self.assertRaises(OverflowError, (256).to_bytes, 1, 'big', signed=False)
        self.assertRaises(OverflowError, (256).to_bytes, 1, 'big', signed=True)
        self.assertRaises(OverflowError, (256).to_bytes, 1, 'little', signed=False)
        self.assertRaises(OverflowError, (256).to_bytes, 1, 'little', signed=True)
        self.assertRaises(OverflowError, (-1).to_bytes, 2, 'big', signed=False)
        self.assertRaises(OverflowError, (-1).to_bytes, 2, 'little', signed=False)
        self.assertEqual((0).to_bytes(0, 'big'), b'')
        self.assertEqual((1).to_bytes(5, 'big'), b'\x00\x00\x00\x00\x01')
        self.assertEqual((0).to_bytes(5, 'big'), b'\x00\x00\x00\x00\x00')
        self.assertEqual((-1).to_bytes(5, 'big', signed=True),
                         b'\xff\xff\xff\xff\xff')
        self.assertRaises(OverflowError, (1).to_bytes, 0, 'big')

    def test_from_bytes(self):
        def check(tests, byteorder, signed=False):
            for test, expected in tests.items():
                try:
                    self.assertEqual(
                        int.from_bytes(test, byteorder, signed=signed),
                        expected)
                except Exception as err:
                    raise AssertionError(
                        "failed to convert {0} with byteorder={1!r} and signed={2}"
                        .format(test, byteorder, signed)) from err

        # Convert signed big-endian byte arrays to integers.
        tests1 = {
            b'': 0,
            b'\x00': 0,
            b'\x00\x00': 0,
            b'\x01': 1,
            b'\x00\x01': 1,
            b'\xff': -1,
            b'\xff\xff': -1,
            b'\x81': -127,
            b'\x80': -128,
            b'\xff\x7f': -129,
            b'\x7f': 127,
            b'\x00\x81': 129,
            b'\xff\x01': -255,
            b'\xff\x00': -256,
            b'\x00\xff': 255,
            b'\x01\x00': 256,
            b'\x7f\xff': 32767,
            b'\x80\x00': -32768,
            b'\x00\xff\xff': 65535,
            b'\xff\x00\x00': -65536,
            b'\x80\x00\x00': -8388608
        }
        check(tests1, 'big', signed=True)

        # Convert signed little-endian byte arrays to integers.
        tests2 = {
            b'': 0,
            b'\x00': 0,
            b'\x00\x00': 0,
            b'\x01': 1,
            b'\x00\x01': 256,
            b'\xff': -1,
            b'\xff\xff': -1,
            b'\x81': -127,
            b'\x80': -128,
            b'\x7f\xff': -129,
            b'\x7f': 127,
            b'\x81\x00': 129,
            b'\x01\xff': -255,
            b'\x00\xff': -256,
            b'\xff\x00': 255,
            b'\x00\x01': 256,
            b'\xff\x7f': 32767,
            b'\x00\x80': -32768,
            b'\xff\xff\x00': 65535,
            b'\x00\x00\xff': -65536,
            b'\x00\x00\x80': -8388608
        }
        check(tests2, 'little', signed=True)

        # Convert unsigned big-endian byte arrays to integers.
        tests3 = {
            b'': 0,
            b'\x00': 0,
            b'\x01': 1,
            b'\x7f': 127,
            b'\x80': 128,
            b'\xff': 255,
            b'\x01\x00': 256,
            b'\x7f\xff': 32767,
            b'\x80\x00': 32768,
            b'\xff\xff': 65535,
            b'\x01\x00\x00': 65536,
        }
        check(tests3, 'big', signed=False)

        # Convert integers to unsigned little-endian byte arrays.
        tests4 = {
            b'': 0,
            b'\x00': 0,
            b'\x01': 1,
            b'\x7f': 127,
            b'\x80': 128,
            b'\xff': 255,
            b'\x00\x01': 256,
            b'\xff\x7f': 32767,
            b'\x00\x80': 32768,
            b'\xff\xff': 65535,
            b'\x00\x00\x01': 65536,
        }
        check(tests4, 'little', signed=False)

        class myint(int):
            pass

        self.assertIs(type(myint.from_bytes(b'\x00', 'big')), myint)
        self.assertEqual(myint.from_bytes(b'\x01', 'big'), 1)
        self.assertIs(
            type(myint.from_bytes(b'\x00', 'big', signed=False)), myint)
        self.assertEqual(myint.from_bytes(b'\x01', 'big', signed=False), 1)
        self.assertIs(type(myint.from_bytes(b'\x00', 'little')), myint)
        self.assertEqual(myint.from_bytes(b'\x01', 'little'), 1)
        self.assertIs(type(myint.from_bytes(
            b'\x00', 'little', signed=False)), myint)
        self.assertEqual(myint.from_bytes(b'\x01', 'little', signed=False), 1)
        self.assertEqual(
            int.from_bytes([255, 0, 0], 'big', signed=True), -65536)
        self.assertEqual(
            int.from_bytes((255, 0, 0), 'big', signed=True), -65536)
        self.assertEqual(int.from_bytes(
            bytearray(b'\xff\x00\x00'), 'big', signed=True), -65536)
        self.assertEqual(int.from_bytes(
            bytearray(b'\xff\x00\x00'), 'big', signed=True), -65536)
        self.assertEqual(int.from_bytes(
            array.array('B', b'\xff\x00\x00'), 'big', signed=True), -65536)
        self.assertEqual(int.from_bytes(
            memoryview(b'\xff\x00\x00'), 'big', signed=True), -65536)
        self.assertRaises(ValueError, int.from_bytes, [256], 'big')
        self.assertRaises(ValueError, int.from_bytes, [0], 'big\x00')
        self.assertRaises(ValueError, int.from_bytes, [0], 'little\x00')
        self.assertRaises(TypeError, int.from_bytes, "", 'big')
        self.assertRaises(TypeError, int.from_bytes, "\x00", 'big')
        self.assertRaises(TypeError, int.from_bytes, 0, 'big')
        self.assertRaises(TypeError, int.from_bytes, 0, 'big', True)
        self.assertRaises(TypeError, myint.from_bytes, "", 'big')
        self.assertRaises(TypeError, myint.from_bytes, "\x00", 'big')
        self.assertRaises(TypeError, myint.from_bytes, 0, 'big')
        self.assertRaises(TypeError, int.from_bytes, 0, 'big', True)

        class myint2(int):
            def __new__(cls, value):
                return int.__new__(cls, value + 1)

        i = myint2.from_bytes(b'\x01', 'big')
        self.assertIs(type(i), myint2)
        self.assertEqual(i, 2)

        class myint3(int):
            def __init__(self, value):
                self.foo = 'bar'

        i = myint3.from_bytes(b'\x01', 'big')
        self.assertIs(type(i), myint3)
        self.assertEqual(i, 1)
        self.assertEqual(getattr(i, 'foo', 'none'), 'bar')

    def test_access_to_nonexistent_digit_0(self):
        # http://bugs.python.org/issue14630: A bug in _PyLong_Copy meant that
        # ob_digit[0] was being incorrectly accessed for instances of a
        # subclass of int, with value 0.
        class Integer(int):
            def __new__(cls, value=0):
                self = int.__new__(cls, value)
                self.foo = 'foo'
                return self

        integers = [Integer(0) for i in range(1000)]
        for n in map(int, integers):
            self.assertEqual(n, 0)

    def test_shift_bool(self):
        # Issue #21422: ensure that bool << int and bool >> int return int
        for value in (True, False):
            for shift in (0, 2):
                self.assertEqual(type(value << shift), int)
                self.assertEqual(type(value >> shift), int)

    def test_as_integer_ratio(self):
        class myint(int):
            pass
        tests = [10, 0, -10, 1, sys.maxsize + 1, True, False, myint(42)]
        for value in tests:
            numerator, denominator = value.as_integer_ratio()
            self.assertEqual((numerator, denominator), (int(value), 1))
            self.assertEqual(type(numerator), int)
            self.assertEqual(type(denominator), int)


if __name__ == "__main__":
    unittest.main()

Youez - 2016 - github.com/yon3zu
LinuXploit