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GeezerVison is showing github.com — cpython/Modules/cmathmodule.c at 3.14 · python/cpython Expand file tree/ Copy pathcmathmodule.cMore file actions 1315 lines (1113 loc) · 39.1 KB / Copy pathcmathmodule.cFile metadata and controls1315 lines (1113 loc) · 39.1 KB 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999 1000 /* Complex math module */ /* much code borrowed from mathmodule.c */ #ifndef Py_BUILD_CORE_BUILTIN # define Py_BUILD_CORE_MODULE 1 #endif #include "Python.h" #include "pycore_complexobject.h" // _Py_c_neg() #include "pycore_pymath.h" // _PY_SHORT_FLOAT_REPR /* we need DBL_MAX, DBL_MIN, DBL_EPSILON, DBL_MANT_DIG and FLT_RADIX from float.h. We assume that FLT_RADIX is either 2 or 16. */ #include <float.h> /* For _Py_log1p with workarounds for buggy handling of zeros. */ #include "_math.h" #include "clinic/cmathmodule.c.h" /*[clinic input] module cmath [clinic start generated code]*/ /*[clinic end generated code: output=da39a3ee5e6b4b0d input=308d6839f4a46333]*/ /*[python input] class Py_complex_protected_converter(Py_complex_converter): def modify(self): return 'errno = 0;' class Py_complex_protected_return_converter(CReturnConverter): type = "Py_complex" def render(self, function, data): self.declare(data) data.return_conversion.append(""" if (errno == EDOM) { PyErr_SetString(PyExc_ValueError, "math domain error"); goto exit; } else if (errno == ERANGE) { PyErr_SetString(PyExc_OverflowError, "math range error"); goto exit; } else { return_value = PyComplex_FromCComplex(_return_value); } """.strip()) [python start generated code]*/ /*[python end generated code: output=da39a3ee5e6b4b0d input=8b27adb674c08321]*/ #if (FLT_RADIX != 2 && FLT_RADIX != 16) #error "Modules/cmathmodule.c expects FLT_RADIX to be 2 or 16" #endif #ifndef M_LN2 #define M_LN2 (0.6931471805599453094) /* natural log of 2 */ #endif #ifndef M_LN10 #define M_LN10 (2.302585092994045684) /* natural log of 10 */ #endif /* CM_LARGE_DOUBLE is used to avoid spurious overflow in the sqrt, log, inverse trig and inverse hyperbolic trig functions. Its log is used in the evaluation of exp, cos, cosh, sin, sinh, tan, and tanh to avoid unnecessary overflow. */ #define CM_LARGE_DOUBLE (DBL_MAX/4.) #define CM_SQRT_LARGE_DOUBLE (sqrt(CM_LARGE_DOUBLE)) #define CM_LOG_LARGE_DOUBLE (log(CM_LARGE_DOUBLE)) #define CM_SQRT_DBL_MIN (sqrt(DBL_MIN)) /* CM_SCALE_UP is an odd integer chosen such that multiplication by 2**CM_SCALE_UP is sufficient to turn a subnormal into a normal. CM_SCALE_DOWN is (-(CM_SCALE_UP+1)/2). These scalings are used to compute square roots accurately when the real and imaginary parts of the argument are subnormal. */ #if FLT_RADIX==2 #define CM_SCALE_UP (2*(DBL_MANT_DIG/2) + 1) #elif FLT_RADIX==16 #define CM_SCALE_UP (4*DBL_MANT_DIG+1) #endif #define CM_SCALE_DOWN (-(CM_SCALE_UP+1)/2) /* forward declarations */ static Py_complex cmath_asinh_impl(PyObject *, Py_complex); static Py_complex cmath_atanh_impl(PyObject *, Py_complex); static Py_complex cmath_cosh_impl(PyObject *, Py_complex); static Py_complex cmath_sinh_impl(PyObject *, Py_complex); static Py_complex cmath_sqrt_impl(PyObject *, Py_complex); static Py_complex cmath_tanh_impl(PyObject *, Py_complex); static PyObject * math_error(void); /* Code to deal with special values (infinities, NaNs, etc.). */ /* special_type takes a double and returns an integer code indicating the type of the double as follows: */ enum special_types { ST_NINF, /* 0, negative infinity */ ST_NEG, /* 1, negative finite number (nonzero) */ ST_NZERO, /* 2, -0. */ ST_PZERO, /* 3, +0. */ ST_POS, /* 4, positive finite number (nonzero) */ ST_PINF, /* 5, positive infinity */ ST_NAN /* 6, Not a Number */ }; static enum special_types special_type(double d) { if (isfinite(d)) { if (d != 0) { if (copysign(1., d) == 1.) return ST_POS; else return ST_NEG; } else { if (copysign(1., d) == 1.) return ST_PZERO; else return ST_NZERO; } } if (isnan(d)) return ST_NAN; if (copysign(1., d) == 1.) return ST_PINF; else return ST_NINF; } #define SPECIAL_VALUE(z, table) \ if (!isfinite((z).real) || !isfinite((z).imag)) { \ errno = 0; \ return table[special_type((z).real)] \ [special_type((z).imag)]; \ } #define P Py_MATH_PI #define P14 0.25*Py_MATH_PI #define P12 0.5*Py_MATH_PI #define P34 0.75*Py_MATH_PI #define INF Py_INFINITY #define N Py_NAN #define U -9.5426319407711027e33 /* unlikely value, used as placeholder */ /* First, the C functions that do the real work. Each of the c_* functions computes and returns the C99 Annex G recommended result and also sets errno as follows: errno = 0 if no floating-point exception is associated with the result; errno = EDOM if C99 Annex G recommends raising divide-by-zero or invalid for this result; and errno = ERANGE where the overflow floating-point signal should be raised. */ static Py_complex acos_special_values[7][7] = { { {P34,INF}, {P,INF}, {P,INF}, {P,-INF}, {P,-INF}, {P34,-INF}, {N,INF} }, { {P12,INF}, {U,U}, {U,U}, {U,U}, {U,U}, {P12,-INF}, {N,N} }, { {P12,INF}, {U,U}, {P12,0.}, {P12,-0.}, {U,U}, {P12,-INF}, {P12,N} }, { {P12,INF}, {U,U}, {P12,0.}, {P12,-0.}, {U,U}, {P12,-INF}, {P12,N} }, { {P12,INF}, {U,U}, {U,U}, {U,U}, {U,U}, {P12,-INF}, {N,N} }, { {P14,INF}, {0.,INF}, {0.,INF}, {0.,-INF}, {0.,-INF}, {P14,-INF}, {N,INF} }, { {N,INF}, {N,N}, {N,N}, {N,N}, {N,N}, {N,-INF}, {N,N} } }; /*[clinic input] cmath.acos -> Py_complex_protected z: Py_complex_protected / Return the arc cosine of z. [clinic start generated code]*/ static Py_complex cmath_acos_impl(PyObject *module, Py_complex z) /*[clinic end generated code: output=40bd42853fd460ae input=bd6cbd78ae851927]*/ { Py_complex s1, s2, r; SPECIAL_VALUE(z, acos_special_values); if (fabs(z.real) > CM_LARGE_DOUBLE || fabs(z.imag) > CM_LARGE_DOUBLE) { /* avoid unnecessary overflow for large arguments */ r.real = atan2(fabs(z.imag), z.real); r.imag = -copysign(log(hypot(z.real/2., z.imag/2.)) + M_LN2*2., z.imag); } else { s1.real = 1.-z.real; s1.imag = -z.imag; s1 = cmath_sqrt_impl(module, s1); s2.real = 1.+z.real; s2.imag = z.imag; s2 = cmath_sqrt_impl(module, s2); r.real = 2.*atan2(s1.real, s2.real); r.imag = asinh(s2.real*s1.imag - s2.imag*s1.real); } errno = 0; return r; } static Py_complex acosh_special_values[7][7] = { { {INF,-P34}, {INF,-P}, {INF,-P}, {INF,P}, {INF,P}, {INF,P34}, {INF,N} }, { {INF,-P12}, {U,U}, {U,U}, {U,U}, {U,U}, {INF,P12}, {N,N} }, { {INF,-P12}, {U,U}, {0.,-P12}, {0.,P12}, {U,U}, {INF,P12}, {N,P12} }, { {INF,-P12}, {U,U}, {0.,-P12}, {0.,P12}, {U,U}, {INF,P12}, {N,P12} }, { {INF,-P12}, {U,U}, {U,U}, {U,U}, {U,U}, {INF,P12}, {N,N} }, { {INF,-P14}, {INF,-0.}, {INF,-0.}, {INF,0.}, {INF,0.}, {INF,P14}, {INF,N} }, { {INF,N}, {N,N}, {N,N}, {N,N}, {N,N}, {INF,N}, {N,N} } }; /*[clinic input] cmath.acosh = cmath.acos Return the inverse hyperbolic cosine of z. [clinic start generated code]*/ static Py_complex cmath_acosh_impl(PyObject *module, Py_complex z) /*[clinic end generated code: output=3e2454d4fcf404ca input=3f61bee7d703e53c]*/ { Py_complex s1, s2, r; SPECIAL_VALUE(z, acosh_special_values); if (fabs(z.real) > CM_LARGE_DOUBLE || fabs(z.imag) > CM_LARGE_DOUBLE) { /* avoid unnecessary overflow for large arguments */ r.real = log(hypot(z.real/2., z.imag/2.)) + M_LN2*2.; r.imag = atan2(z.imag, z.real); } else { s1.real = z.real - 1.; s1.imag = z.imag; s1 = cmath_sqrt_impl(module, s1); s2.real = z.real + 1.; s2.imag = z.imag; s2 = cmath_sqrt_impl(module, s2); r.real = asinh(s1.real*s2.real + s1.imag*s2.imag); r.imag = 2.*atan2(s1.imag, s2.real); } errno = 0; return r; } /*[clinic input] cmath.asin = cmath.acos Return the arc sine of z. [clinic start generated code]*/ static Py_complex cmath_asin_impl(PyObject *module, Py_complex z) /*[clinic end generated code: output=3b264cd1b16bf4e1 input=be0bf0cfdd5239c5]*/ { /* asin(z) = -i asinh(iz) */ Py_complex s, r; s.real = -z.imag; s.imag = z.real; s = cmath_asinh_impl(module, s); r.real = s.imag; r.imag = -s.real; return r; } static Py_complex asinh_special_values[7][7] = { { {-INF,-P14}, {-INF,-0.}, {-INF,-0.}, {-INF,0.}, {-INF,0.}, {-INF,P14}, {-INF,N} }, { {-INF,-P12}, {U,U}, {U,U}, {U,U}, {U,U}, {-INF,P12}, {N,N} }, { {-INF,-P12}, {U,U}, {-0.,-0.}, {-0.,0.}, {U,U}, {-INF,P12}, {N,N} }, { {INF,-P12}, {U,U}, {0.,-0.}, {0.,0.}, {U,U}, {INF,P12}, {N,N} }, { {INF,-P12}, {U,U}, {U,U}, {U,U}, {U,U}, {INF,P12}, {N,N} }, { {INF,-P14}, {INF,-0.}, {INF,-0.}, {INF,0.}, {INF,0.}, {INF,P14}, {INF,N} }, { {INF,N}, {N,N}, {N,-0.}, {N,0.}, {N,N}, {INF,N}, {N,N} } }; /*[clinic input] cmath.asinh = cmath.acos Return the inverse hyperbolic sine of z. [clinic start generated code]*/ static Py_complex cmath_asinh_impl(PyObject *module, Py_complex z) /*[clinic end generated code: output=733d8107841a7599 input=5c09448fcfc89a79]*/ { Py_complex s1, s2, r; SPECIAL_VALUE(z, asinh_special_values); if (fabs(z.real) > CM_LARGE_DOUBLE || fabs(z.imag) > CM_LARGE_DOUBLE) { if (z.imag >= 0.) { r.real = copysign(log(hypot(z.real/2., z.imag/2.)) + M_LN2*2., z.real); } else { r.real = -copysign(log(hypot(z.real/2., z.imag/2.)) + M_LN2*2., -z.real); } r.imag = atan2(z.imag, fabs(z.real)); } else { s1.real = 1.+z.imag; s1.imag = -z.real; s1 = cmath_sqrt_impl(module, s1); s2.real = 1.-z.imag; s2.imag = z.real; s2 = cmath_sqrt_impl(module, s2); r.real = asinh(s1.real*s2.imag-s2.real*s1.imag); r.imag = atan2(z.imag, s1.real*s2.real-s1.imag*s2.imag); } errno = 0; return r; } /*[clinic input] cmath.atan = cmath.acos Return the arc tangent of z. [clinic start generated code]*/ static Py_complex cmath_atan_impl(PyObject *module, Py_complex z) /*[clinic end generated code: output=b6bfc497058acba4 input=3b21ff7d5eac632a]*/ { /* atan(z) = -i atanh(iz) */ Py_complex s, r; s.real = -z.imag; s.imag = z.real; s = cmath_atanh_impl(module, s); r.real = s.imag; r.imag = -s.real; return r; } static Py_complex atanh_special_values[7][7] = { { {-0.,-P12}, {-0.,-P12}, {-0.,-P12}, {-0.,P12}, {-0.,P12}, {-0.,P12}, {-0.,N} }, { {-0.,-P12}, {U,U}, {U,U}, {U,U}, {U,U}, {-0.,P12}, {N,N} }, { {-0.,-P12}, {U,U}, {-0.,-0.}, {-0.,0.}, {U,U}, {-0.,P12}, {-0.,N} }, { {0.,-P12}, {U,U}, {0.,-0.}, {0.,0.}, {U,U}, {0.,P12}, {0.,N} }, { {0.,-P12}, {U,U}, {U,U}, {U,U}, {U,U}, {0.,P12}, {N,N} }, { {0.,-P12}, {0.,-P12}, {0.,-P12}, {0.,P12}, {0.,P12}, {0.,P12}, {0.,N} }, { {0.,-P12}, {N,N}, {N,N}, {N,N}, {N,N}, {0.,P12}, {N,N} } }; /*[clinic input] cmath.atanh = cmath.acos Return the inverse hyperbolic tangent of z. [clinic start generated code]*/ static Py_complex cmath_atanh_impl(PyObject *module, Py_complex z) /*[clinic end generated code: output=e83355f93a989c9e input=2b3fdb82fb34487b]*/ { Py_complex r; double ay, h; SPECIAL_VALUE(z, atanh_special_values); /* Reduce to case where z.real >= 0., using atanh(z) = -atanh(-z). */ if (z.real < 0.) { return _Py_c_neg(cmath_atanh_impl(module, _Py_c_neg(z))); } ay = fabs(z.imag); if (z.real > CM_SQRT_LARGE_DOUBLE || ay > CM_SQRT_LARGE_DOUBLE) { /* if abs(z) is large then we use the approximation atanh(z) ~ 1/z +/- i*pi/2 (+/- depending on the sign of z.imag) */ h = hypot(z.real/2., z.imag/2.); /* safe from overflow */ r.real = z.real/4./h/h; r.imag = copysign(Py_MATH_PI/2., z.imag); errno = 0; } else if (z.real == 1. && ay < CM_SQRT_DBL_MIN) { /* C99 standard says: atanh(1+/-0.) should be inf +/- 0i */ if (ay == 0.) { r.real = INF; r.imag = z.imag; errno = EDOM; } else { r.real = -log(sqrt(ay)/sqrt(hypot(ay, 2.))); r.imag = copysign(atan2(2., -ay)/2, z.imag); errno = 0; } } else { r.real = m_log1p(4.*z.real/((1-z.real)*(1-z.real) + ay*ay))/4.; r.imag = -atan2(-2.*z.imag, (1-z.real)*(1+z.real) - ay*ay)/2.; errno = 0; } return r; } /*[clinic input] cmath.cos = cmath.acos Return the cosine of z. [clinic start generated code]*/ static Py_complex cmath_cos_impl(PyObject *module, Py_complex z) /*[clinic end generated code: output=fd64918d5b3186db input=6022e39b77127ac7]*/ { /* cos(z) = cosh(iz) */ Py_complex r; r.real = -z.imag; r.imag = z.real; r = cmath_cosh_impl(module, r); return r; } /* cosh(infinity + i*y) needs to be dealt with specially */ static Py_complex cosh_special_values[7][7] = { { {INF,N}, {U,U}, {INF,0.}, {INF,-0.}, {U,U}, {INF,N}, {INF,N} }, { {N,N}, {U,U}, {U,U}, {U,U}, {U,U}, {N,N}, {N,N} }, { {N,0.}, {U,U}, {1.,0.}, {1.,-0.}, {U,U}, {N,0.}, {N,0.} }, { {N,0.}, {U,U}, {1.,-0.}, {1.,0.}, {U,U}, {N,0.}, {N,0.} }, { {N,N}, {U,U}, {U,U}, {U,U}, {U,U}, {N,N}, {N,N} }, { {INF,N}, {U,U}, {INF,-0.}, {INF,0.}, {U,U}, {INF,N}, {INF,N} }, { {N,N}, {N,N}, {N,0.}, {N,0.}, {N,N}, {N,N}, {N,N} } }; /*[clinic input] cmath.cosh = cmath.acos Return the hyperbolic cosine of z. [clinic start generated code]*/ static Py_complex cmath_cosh_impl(PyObject *module, Py_complex z) /*[clinic end generated code: output=2e969047da601bdb input=d6b66339e9cc332b]*/ { Py_complex r; double x_minus_one; /* special treatment for cosh(+/-inf + iy) if y is not a NaN */ if (!isfinite(z.real) || !isfinite(z.imag)) { if (isinf(z.real) && isfinite(z.imag) && (z.imag != 0.)) { if (z.real > 0) { r.real = copysign(INF, cos(z.imag)); r.imag = copysign(INF, sin(z.imag)); } else { r.real = copysign(INF, cos(z.imag)); r.imag = -copysign(INF, sin(z.imag)); } } else { r = cosh_special_values[special_type(z.real)] [special_type(z.imag)]; } /* need to set errno = EDOM if y is +/- infinity and x is not a NaN */ if (isinf(z.imag) && !isnan(z.real)) errno = EDOM; else errno = 0; return r; } if (fabs(z.real) > CM_LOG_LARGE_DOUBLE) { /* deal correctly with cases where cosh(z.real) overflows but cosh(z) does not. */ x_minus_one = z.real - copysign(1., z.real); r.real = cos(z.imag) * cosh(x_minus_one) * Py_MATH_E; r.imag = sin(z.imag) * sinh(x_minus_one) * Py_MATH_E; } else { r.real = cos(z.imag) * cosh(z.real); r.imag = sin(z.imag) * sinh(z.real); } /* detect overflow, and set errno accordingly */ if (isinf(r.real) || isinf(r.imag)) errno = ERANGE; else errno = 0; return r; } /* exp(infinity + i*y) and exp(-infinity + i*y) need special treatment for finite y */ static Py_complex exp_special_values[7][7] = { { {0.,0.}, {U,U}, {0.,-0.}, {0.,0.}, {U,U}, {0.,0.}, {0.,0.} }, { {N,N}, {U,U}, {U,U}, {U,U}, {U,U}, {N,N}, {N,N} }, { {N,N}, {U,U}, {1.,-0.}, {1.,0.}, {U,U}, {N,N}, {N,N} }, { {N,N}, {U,U}, {1.,-0.}, {1.,0.}, {U,U}, {N,N}, {N,N} }, { {N,N}, {U,U}, {U,U}, {U,U}, {U,U}, {N,N}, {N,N} }, { {INF,N}, {U,U}, {INF,-0.}, {INF,0.}, {U,U}, {INF,N}, {INF,N} }, { {N,N}, {N,N}, {N,-0.}, {N,0.}, {N,N}, {N,N}, {N,N} } }; /*[clinic input] cmath.exp = cmath.acos Return the exponential value e**z. [clinic start generated code]*/ static Py_complex cmath_exp_impl(PyObject *module, Py_complex z) /*[clinic end generated code: output=edcec61fb9dfda6c input=8b9e6cf8a92174c3]*/ { Py_complex r; double l; if (!isfinite(z.real) || !isfinite(z.imag)) { if (isinf(z.real) && isfinite(z.imag) && (z.imag != 0.)) { if (z.real > 0) { r.real = copysign(INF, cos(z.imag)); r.imag = copysign(INF, sin(z.imag)); } else { r.real = copysign(0., cos(z.imag)); r.imag = copysign(0., sin(z.imag)); } } else { r = exp_special_values[special_type(z.real)] [special_type(z.imag)]; } /* need to set errno = EDOM if y is +/- infinity and x is not a NaN and not -infinity */ if (isinf(z.imag) && (isfinite(z.real) || (isinf(z.real) && z.real > 0))) errno = EDOM; else errno = 0; return r; } if (z.real > CM_LOG_LARGE_DOUBLE) { l = exp(z.real-1.); r.real = l*cos(z.imag)*Py_MATH_E; r.imag = l*sin(z.imag)*Py_MATH_E; } else { l = exp(z.real); r.real = l*cos(z.imag); r.imag = l*sin(z.imag); } /* detect overflow, and set errno accordingly */ if (isinf(r.real) || isinf(r.imag)) errno = ERANGE; else errno = 0; return r; } static Py_complex log_special_values[7][7] = { { {INF,-P34}, {INF,-P}, {INF,-P}, {INF,P}, {INF,P}, {INF,P34}, {INF,N} }, { {INF,-P12}, {U,U}, {U,U}, {U,U}, {U,U}, {INF,P12}, {N,N} }, { {INF,-P12}, {U,U}, {-INF,-P}, {-INF,P}, {U,U}, {INF,P12}, {N,N} }, { {INF,-P12}, {U,U}, {-INF,-0.}, {-INF,0.}, {U,U}, {INF,P12}, {N,N} }, { {INF,-P12}, {U,U}, {U,U}, {U,U}, {U,U}, {INF,P12}, {N,N} }, { {INF,-P14}, {INF,-0.}, {INF,-0.}, {INF,0.}, {INF,0.}, {INF,P14}, {INF,N} }, { {INF,N}, {N,N}, {N,N}, {N,N}, {N,N}, {INF,N}, {N,N} } }; static Py_complex c_log(Py_complex z) { /* The usual formula for the real part is log(hypot(z.real, z.imag)). There are four situations where this formula is potentially problematic: (1) the absolute value of z is subnormal. Then hypot is subnormal, so has fewer than the usual number of bits of accuracy, hence may have large relative error. This then gives a large absolute error in the log. This can be solved by rescaling z by a suitable power of 2. (2) the absolute value of z is greater than DBL_MAX (e.g. when both z.real and z.imag are within a factor of 1/sqrt(2) of DBL_MAX) Again, rescaling solves this. (3) the absolute value of z is close to 1. In this case it's difficult to achieve good accuracy, at least in part because a change of 1ulp in the real or imaginary part of z can result in a change of billions of ulps in the correctly rounded answer. (4) z = 0. The simplest thing to do here is to call the floating-point log with an argument of 0, and let its behaviour (returning -infinity, signaling a floating-point exception, setting errno, or whatever) determine that of c_log. So the usual formula is fine here. */ Py_complex r; double ax, ay, am, an, h; SPECIAL_VALUE(z, log_special_values); ax = fabs(z.real); ay = fabs(z.imag); if (ax > CM_LARGE_DOUBLE || ay > CM_LARGE_DOUBLE) { r.real = log(hypot(ax/2., ay/2.)) + M_LN2; } else if (ax < DBL_MIN && ay < DBL_MIN) { if (ax > 0. || ay > 0.) { /* catch cases where hypot(ax, ay) is subnormal */ r.real = log(hypot(ldexp(ax, DBL_MANT_DIG), ldexp(ay, DBL_MANT_DIG))) - DBL_MANT_DIG*M_LN2; } else { /* log(+/-0. +/- 0i) */ r.real = -INF; r.imag = atan2(z.imag, z.real); errno = EDOM; return r; } } else { h = hypot(ax, ay); if (0.71 <= h && h <= 1.73) { am = ax > ay ? ax : ay; /* max(ax, ay) */ an = ax > ay ? ay : ax; /* min(ax, ay) */ r.real = m_log1p((am-1)*(am+1)+an*an)/2.; } else { r.real = log(h); } } r.imag = atan2(z.imag, z.real); errno = 0; return r; } /*[clinic input] cmath.log10 = cmath.acos Return the base-10 logarithm of z. [clinic start generated code]*/ static Py_complex cmath_log10_impl(PyObject *module, Py_complex z) /*[clinic end generated code: output=2922779a7c38cbe1 input=cff5644f73c1519c]*/ { Py_complex r; int errno_save; r = c_log(z); errno_save = errno; /* just in case the divisions affect errno */ r.real = r.real / M_LN10; r.imag = r.imag / M_LN10; errno = errno_save; return r; } /*[clinic input] cmath.sin = cmath.acos Return the sine of z. [clinic start generated code]*/ static Py_complex cmath_sin_impl(PyObject *module, Py_complex z) /*[clinic end generated code: output=980370d2ff0bb5aa input=2d3519842a8b4b85]*/ { /* sin(z) = -i sin(iz) */ Py_complex s, r; s.real = -z.imag; s.imag = z.real; s = cmath_sinh_impl(module, s); r.real = s.imag; r.imag = -s.real; return r; } /* sinh(infinity + i*y) needs to be dealt with specially */ static Py_complex sinh_special_values[7][7] = { { {INF,N}, {U,U}, {-INF,-0.}, {-INF,0.}, {U,U}, {INF,N}, {INF,N} }, { {N,N}, {U,U}, {U,U}, {U,U}, {U,U}, {N,N}, {N,N} }, { {0.,N}, {U,U}, {-0.,-0.}, {-0.,0.}, {U,U}, {0.,N}, {0.,N} }, { {0.,N}, {U,U}, {0.,-0.}, {0.,0.}, {U,U}, {0.,N}, {0.,N} }, { {N,N}, {U,U}, {U,U}, {U,U}, {U,U}, {N,N}, {N,N} }, { {INF,N}, {U,U}, {INF,-0.}, {INF,0.}, {U,U}, {INF,N}, {INF,N} }, { {N,N}, {N,N}, {N,-0.}, {N,0.}, {N,N}, {N,N}, {N,N} } }; /*[clinic input] cmath.sinh = cmath.acos Return the hyperbolic sine of z. [clinic start generated code]*/ static Py_complex cmath_sinh_impl(PyObject *module, Py_complex z) /*[clinic end generated code: output=38b0a6cce26f3536 input=d2d3fc8c1ddfd2dd]*/ { Py_complex r; double x_minus_one; /* special treatment for sinh(+/-inf + iy) if y is finite and nonzero */ if (!isfinite(z.real) || !isfinite(z.imag)) { if (isinf(z.real) && isfinite(z.imag) && (z.imag != 0.)) { if (z.real > 0) { r.real = copysign(INF, cos(z.imag)); r.imag = copysign(INF, sin(z.imag)); } else { r.real = -copysign(INF, cos(z.imag)); r.imag = copysign(INF, sin(z.imag)); } } else { r = sinh_special_values[special_type(z.real)] [special_type(z.imag)]; } /* need to set errno = EDOM if y is +/- infinity and x is not a NaN */ if (isinf(z.imag) && !isnan(z.real)) errno = EDOM; else errno = 0; return r; } if (fabs(z.real) > CM_LOG_LARGE_DOUBLE) { x_minus_one = z.real - copysign(1., z.real); r.real = cos(z.imag) * sinh(x_minus_one) * Py_MATH_E; r.imag = sin(z.imag) * cosh(x_minus_one) * Py_MATH_E; } else { r.real = cos(z.imag) * sinh(z.real); r.imag = sin(z.imag) * cosh(z.real); } /* detect overflow, and set errno accordingly */ if (isinf(r.real) || isinf(r.imag)) errno = ERANGE; else errno = 0; return r; } static Py_complex sqrt_special_values[7][7] = { { {INF,-INF}, {0.,-INF}, {0.,-INF}, {0.,INF}, {0.,INF}, {INF,INF}, {N,INF} }, { {INF,-INF}, {U,U}, {U,U}, {U,U}, {U,U}, {INF,INF}, {N,N} }, { {INF,-INF}, {U,U}, {0.,-0.}, {0.,0.}, {U,U}, {INF,INF}, {N,N} }, { {INF,-INF}, {U,U}, {0.,-0.}, {0.,0.}, {U,U}, {INF,INF}, {N,N} }, { {INF,-INF}, {U,U}, {U,U}, {U,U}, {U,U}, {INF,INF}, {N,N} }, { {INF,-INF}, {INF,-0.}, {INF,-0.}, {INF,0.}, {INF,0.}, {INF,INF}, {INF,N} }, { {INF,-INF}, {N,N}, {N,N}, {N,N}, {N,N}, {INF,INF}, {N,N} } }; /*[clinic input] cmath.sqrt = cmath.acos Return the square root of z. [clinic start generated code]*/ static Py_complex cmath_sqrt_impl(PyObject *module, Py_complex z) /*[clinic end generated code: output=b6507b3029c339fc input=7088b166fc9a58c7]*/ { /* Method: use symmetries to reduce to the case when x = z.real and y = z.imag are nonnegative. Then the real part of the result is given by s = sqrt((x + hypot(x, y))/2) and the imaginary part is d = (y/2)/s If either x or y is very large then there's a risk of overflow in computation of the expression x + hypot(x, y). We can avoid this by rewriting the formula for s as: s = 2*sqrt(x/8 + hypot(x/8, y/8)) This costs us two extra multiplications/divisions, but avoids the overhead of checking for x and y large. If both x and y are subnormal then hypot(x, y) may also be subnormal, so will lack full precision. We solve this by rescaling x and y by a sufficiently large power of 2 to ensure that x and y are normal. */ Py_complex r; double s,d; double ax, ay; SPECIAL_VALUE(z, sqrt_special_values); if (z.real == 0. && z.imag == 0.) { r.real = 0.; r.imag = z.imag; return r; } ax = fabs(z.real); ay = fabs(z.imag); if (ax < DBL_MIN && ay < DBL_MIN) { /* here we catch cases where hypot(ax, ay) is subnormal */ ax = ldexp(ax, CM_SCALE_UP); s = ldexp(sqrt(ax + hypot(ax, ldexp(ay, CM_SCALE_UP))), CM_SCALE_DOWN); } else { ax /= 8.; s = 2.*sqrt(ax + hypot(ax, ay/8.)); } d = ay/(2.*s); if (z.real >= 0.) { r.real = s; r.imag = copysign(d, z.imag); } else { r.real = d; r.imag = copysign(s, z.imag); } errno = 0; return r; } /*[clinic input] cmath.tan = cmath.acos Return the tangent of z. [clinic start generated code]*/ static Py_complex cmath_tan_impl(PyObject *module, Py_complex z) /*[clinic end generated code: output=7c5f13158a72eb13 input=fc167e528767888e]*/ { /* tan(z) = -i tanh(iz) */ Py_complex s, r; s.real = -z.imag; s.imag = z.real; s = cmath_tanh_impl(module, s); r.real = s.imag; r.imag = -s.real; return r; } /* tanh(infinity + i*y) needs to be dealt with specially */ static Py_complex tanh_special_values[7][7] = { { {-1.,0.}, {U,U}, {-1.,-0.}, {-1.,0.}, {U,U}, {-1.,0.}, {-1.,0.} }, { {N,N}, {U,U}, {U,U}, {U,U}, {U,U}, {N,N}, {N,N} }, { {-0.0,N}, {U,U}, {-0.,-0.}, {-0.,0.}, {U,U}, {-0.0,N}, {-0.,N} }, { {0.0,N}, {U,U}, {0.,-0.}, {0.,0.}, {U,U}, {0.0,N}, {0.,N} }, { {N,N}, {U,U}, {U,U}, {U,U}, {U,U}, {N,N}, {N,N} }, { {1.,0.}, {U,U}, {1.,-0.}, {1.,0.}, {U,U}, {1.,0.}, {1.,0.} }, { {N,N}, {N,N}, {N,-0.}, {N,0.}, {N,N}, {N,N}, {N,N} } }; /*[clinic input] cmath.tanh = cmath.acos Return the hyperbolic tangent of z. [clinic start generated code]*/ static Py_complex cmath_tanh_impl(PyObject *module, Py_complex z) /*[clinic end generated code: output=36d547ef7aca116c input=22f67f9dc6d29685]*/ { /* Formula: tanh(x+iy) = (tanh(x)(1+tan(y)^2) + i tan(y)(1-tanh(x))^2) / (1+tan(y)^2 tanh(x)^2) To avoid excessive roundoff error, 1-tanh(x)^2 is better computed as 1/cosh(x)^2. When abs(x) is large, we approximate 1-tanh(x)^2 by 4 exp(-2*x) instead, to avoid possible overflow in the computation of cosh(x). */ Py_complex r; double tx, ty, cx, txty, denom; /* special treatment for tanh(+/-inf + iy) if y is finite and nonzero */ if (!isfinite(z.real) || !isfinite(z.imag)) { if (isinf(z.real) && isfinite(z.imag) && (z.imag != 0.)) { if (z.real > 0) { r.real = 1.0; r.imag = copysign(0., 2.*sin(z.imag)*cos(z.imag)); } else { r.real = -1.0; r.imag = copysign(0., 2.*sin(z.imag)*cos(z.imag)); } } else { r = tanh_special_values[special_type(z.real)] [special_type(z.imag)]; } /* need to set errno = EDOM if z.imag is +/-infinity and z.real is finite */ if (isinf(z.imag) && isfinite(z.real)) errno = EDOM; else errno = 0; return r; } /* danger of overflow in 2.*z.imag !*/ if (fabs(z.real) > CM_LOG_LARGE_DOUBLE) { r.real = copysign(1., z.real); r.imag = 4.*sin(z.imag)*cos(z.imag)*exp(-2.*fabs(z.real)); } else { tx = tanh(z.real); ty = tan(z.imag); cx = 1./cosh(z.real); txty = tx*ty; denom = 1. + txty*txty; r.real = tx*(1.+ty*ty)/denom; r.imag = ((ty/denom)*cx)*cx; } errno = 0; return r; } /*[clinic input] cmath.log z as x: Py_complex base as y_obj: object = NULL / log(z[, base]) -> the logarithm of z to the given base. If the base is not specified, returns the natural logarithm (base e) of z. [clinic start generated code]*/ static PyObject * cmath_log_impl(PyObject *module, Py_complex x, PyObject *y_obj) /*[clinic end generated code: output=4effdb7d258e0d94 input=eb25de0757baf4a0]*/ { Py_complex y; errno = 0; x = c_log(x); if (y_obj != NULL) { y = PyComplex_AsCComplex(y_obj); if (PyErr_Occurred()) { return NULL; } y = c_log(y); x = _Py_c_quot(x, y); } if (errno != 0) return math_error(); return PyComplex_FromCComplex(x); } /* And now the glue to make them available from Python: */ static PyObject * math_error(void) { if (errno == EDOM) PyErr_SetString(PyExc_ValueError, "math domain error"); else if (errno == ERANGE) PyErr_SetString(PyExc_OverflowError, "math range error"); else /* Unexpected math error */ PyErr_SetFromErrno(PyExc_ValueError); return NULL; } /*[clinic input] cmath.phase z: Py_complex / Return argument, also known as the phase angle, of a complex. [clinic start generated code]*/ |
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