# Coverage for pygeodesy/fmath.py : 95%

## 267 statements 254 run 13 missing 72 excluded

Hot-keys on this page

r m x p toggle line displays

j k next/prev highlighted chunk

0 (zero) top of page

1 (one) first highlighted chunk

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 | # -*- coding: utf-8 -*- u'''Utilities using precision floating point summation. ''' # make sure int/int division yields float quotient, see .basics from __future__ import division as _; del _ # PYCHOK semicolon from pygeodesy.basics import _copysign, copysign0, isint, isscalar, len2 from pygeodesy.constants import EPS0, EPS02, EPS1, NAN, PI, PI_2, PI_4, \ _0_0, _0_5, _1_0, _N_1_0, _1_3rd, _1_5, \ _2_0, _N_2_0, _2_3rd, _3_0, isnear0, isnear1, \ _isfinite, remainder as _remainder from pygeodesy.errors import _IsnotError, LenError, _TypeError, _ValueError, \ _xError, _xkwds_get, _xkwds_pop from pygeodesy.fsums import _2float, _Powers, Fsum, _fsum, fsum, fsum1_, \ _pow_op_, Fmt, unstr from pygeodesy.interns import MISSING, _few_, _h_, _negative_, _not_scalar_, \ _singular_, _too_ from pygeodesy.lazily import _ALL_LAZY, _sys_version_info2 # from pygeodesy.streprs import Fmt, unstr # from .fsums from pygeodesy.units import Int_ from math import fabs, sqrt # pow from operator import mul as _mul # in .triaxials __all__ = _ALL_LAZY.fmath __version__ = '22.11.04' # sqrt(2) <https://WikiPedia.org/wiki/Square_root_of_2> _0_4142 = 0.414213562373095 # sqrt(_2_0) - _1_0 class Fdot(Fsum): '''Precision dot product. ''' def __init__(self, a, *b, **name): '''New L{Fdot} precision dot product M{sum(a[i] * b[i] for i=0..len(a))}. @arg a: Iterable, list, tuple, etc. (C{scalar}s). @arg b: Other values (C{scalar}s), all positional. @kwarg name: Optional name (C{str}). @raise OverflowError: Partial C{2sum} overflow. @raise LenError: Unequal C{len(B{a})} and C{len(B{b})}. @see: Function L{fdot} and method L{Fsum.fadd}. ''' Fsum.__init__(self, **name) self.fadd(_map_a_x_b(a, b, Fdot)) class Fhorner(Fsum): '''Precision polynomial evaluation using the Horner form. ''' def __init__(self, x, *cs, **name): '''New L{Fhorner} evaluation of the polynomial M{sum(cs[i] * x**i for i=0..len(cs))}. @arg x: Polynomial argument (C{scalar}). @arg cs: Polynomial coeffients (C{scalar} or C{Fsum} instances), all positional. @kwarg name: Optional name (C{str}). @raise OverflowError: Partial C{2sum} overflow. @raise TypeError: Non-scalar B{C{x}}. @raise ValueError: Non-finite B{C{x}}. @see: Function L{fhorner} and methods L{Fsum.fadd} and L{Fsum.fmul}. ''' Fsum.__init__(self, *cs[-1:], **name) if len(cs) > 1: x = _2float(x=x) _a = self._fadd # (other, op) _f = self._finite # (other, op) op = Fhorner.__name__ ps = self._ps for c in reversed(cs[:-1]): # multiply-accumulate ps[:] = [_f(p * x, op) for p in ps] _a(c, op) # assert self._ps is ps class Fhypot(Fsum): '''Precision hypotenuse of summation. ''' def __init__(self, *xs, **power_name_RESIDUAL): '''New L{Fhypot} hypotenuse of (the I{power} of) several C{scalar} or C{Fsum} values. @arg xs: One or more values to include (each C{scalar} or an C{Fsum} instance). @kwarg power_name_RESIDUAL: Optional exponent and root order C{B{power}=2}, C{B{name}=NN} and C{B{RESIDUAL}=None}, see L{Fsum.__init__}. ''' try: p = _xkwds_pop(power_name_RESIDUAL, power=2) Fsum.__init__(self, **power_name_RESIDUAL) if xs: self._facc(_Powers(p, xs), up=False) # PYCHOK None self._fset(self._fpow(_1_0 / p, _pow_op_), asis=True) except Exception as X: raise self._ErrorX(X, xs, power=p) class Fpolynomial(Fsum): '''Precision polynomial evaluation. ''' def __init__(self, x, *cs, **name): '''New L{Fpolynomial} evaluation of the polynomial M{sum(cs[i] * x**i for i=0..len(cs))}. @arg x: Polynomial argument (C{scalar}). @arg cs: Polynomial coeffients (C{scalar}s), all positional. @kwarg name: Optional name (C{str}). @raise OverflowError: Partial C{2sum} overflow. @raise TypeError: Non-scalar B{C{x}}. @raise ValueError: Non-finite B{C{x}}. @see: Function L{fpolynomial} and method L{Fsum.fadd}. ''' Fsum.__init__(self, *cs[:1], **name) n = len(cs) - 1 if n > 0: self.fadd(_map_a_x_b(cs[1:], fpowers(x, n), Fpolynomial)) class Fpowers(Fsum): '''Precision summation or powers, optimized for C{power=2}. ''' def __init__(self, power, *xs, **name_RESIDUAL): '''New L{Fpowers} sum of (the I{power} of) several C{scalar} or C{Fsum} values. @arg power: The exponent (C{scalar} or C{Fsum}). @arg xs: One or more values to include (each C{scalar} or an C{Fsum} instance). @kwarg power_name_RESIDUAL: Optional exponent and root order C{B{power}=2}, C{B{name}=NN} and C{B{RESIDUAL}=None}, see L{Fsum.__init__}. ''' try: Fsum.__init__(self, **name_RESIDUAL) if xs: self._facc(_Powers(power, xs), up=False) # PYCHOK None except Exception as X: raise self._ErrorX(X, xs, power=power) class Fn_rt(Fsum): '''Precision n-th root of summation. ''' def __init__(self, root, *xs, **name_RESIDUAL): '''New L{Fn_rt} root of the precision sum of several C{scalar} or C{Fsum} values. @arg root: The order (C{scalar} or C{Fsum}). @arg xs: Values to include (each C{scalar} or an C{Fsum} instance). @kwarg name_RESIDUAL: See L{Fsum.__init__}. ''' try: Fsum.__init__(self, *xs, **name_RESIDUAL) self._fset(self._fpow(_1_0 / root, _pow_op_), asis=True) except Exception as X: raise self._ErrorX(X, xs, root=root) class Fcbrt(Fn_rt): '''Precision cubic root of summation. ''' def __init__(self, *xs, **name_RESIDUAL): '''New L{Fcbrt} cubic root of the precision sum of several C{scalar} or C{Fsum} values. @arg xs: Values to include (each C{scalar} or an C{Fsum} instance). @kwarg name_RESIDUAL: See L{Fsum.__init__}. ''' Fn_rt.__init__(self, _3_0, *xs, **name_RESIDUAL) class Fsqrt(Fn_rt): '''Precision square root of summation. ''' def __init__(self, *xs, **name_RESIDUAL): '''New L{Fsqrt} square root of the precision sum of several C{scalar} or C{Fsum} values. @arg xs: Values to include (each C{scalar} or an C{Fsum} instance). @kwarg name_RESIDUAL: See L{Fsum.__init__}. ''' Fn_rt.__init__(self, _2_0, *xs, **name_RESIDUAL) try: from math import cbrt # Python 3.11+ def cbrt2(x): '''Compute the cube root I{squared} M{x**(2/3)}. ''' return cbrt(x)**2 # cbrt(-0.0*2) == -0.0 except ImportError: # Python 3.10- def cbrt(x): '''Compute the cube root M{x**(1/3)}. @arg x: Value (C{scalar}). @return: Cubic root (C{float}). @see: Functions L{cbrt2} and L{sqrt3}. ''' # <https://archive.lib.MSU.edu/crcmath/math/math/r/r021.htm> # simpler and more accurate than Ken Turkowski's CubeRoot, see # <https://People.FreeBSD.org/~lstewart/references/apple_tr_kt32_cuberoot.pdf> return _copysign(pow(fabs(x), _1_3rd), x) # cbrt(-0.0) == -0.0 def cbrt2(x): # PYCHOK attr '''Compute the cube root I{squared} M{x**(2/3)}. @arg x: Value (C{scalar}). @return: Cube root I{squared} (C{float}). @see: Functions L{cbrt} and L{sqrt3}. ''' return pow(fabs(x), _2_3rd) # XXX pow(fabs(x), _1_3rd)**2 def euclid(x, y): '''I{Appoximate} the norm M{sqrt(x**2 + y**2)} by M{max(abs(x), abs(y)) + min(abs(x), abs(y)) * 0.4142...}. @arg x: X component (C{scalar}). @arg y: Y component (C{scalar}). @return: Appoximate norm (C{float}). @see: Function L{euclid_}. ''' x, y = fabs(x), fabs(y) if y > x: x, y = y, x return x + y * _0_4142 # XXX * _0_5 before 20.10.02 def euclid_(*xs): '''I{Appoximate} the norm M{sqrt(sum(x**2 for x in xs))} by cascaded L{euclid}. @arg xs: X arguments, positional (C{scalar}s). @return: Appoximate norm (C{float}). @see: Function L{euclid}. ''' e = _0_0 for x in sorted(map(fabs, xs)): # XXX not reverse=True # e = euclid(x, e) if x > e: e, x = x, e if x: e += x * _0_4142 return e def facos1(x): '''Fast approximation of L{pygeodesy.acos1}C{(B{x})}. @see: U{ShaderFastLibs.h<https://GitHub.com/michaldrobot/ ShaderFastLibs/blob/master/ShaderFastMathLib.h>}. ''' a = fabs(x) if a < EPS0: r = PI_2 elif a < EPS1: H = Fhorner(-a, 1.5707288, 0.2121144, 0.0742610, 0.0187293) r = float(H * sqrt(_1_0 - a)) if x < 0: r = PI - r else: r = PI if x < 0 else _0_0 return r def fasin1(x): # PYCHOK no cover '''Fast approximation of L{pygeodesy.asin1}C{(B{x})}. @see: L{facos1}. ''' return PI_2 - facos1(x) def fatan(x): '''Fast approximation of C{atan(B{x})}. ''' a = fabs(x) if a < _1_0: r = fatan1(a) if a else _0_0 elif a > _1_0: r = PI_2 - fatan1(_1_0 / a) # == fatan2(a, _1_0) else: r = PI_4 if x < 0: # copysign0(r, x) r = -r return r def fatan1(x): '''Fast approximation of C{atan(B{x})} for C{0 <= B{x} <= 1}, I{unchecked}. @see: U{ShaderFastLibs.h<https://GitHub.com/michaldrobot/ShaderFastLibs/ blob/master/ShaderFastMathLib.h>} and U{Efficient approximations for the arctangent function<http://www-Labs.IRO.UMontreal.CA/ ~mignotte/IFT2425/Documents/EfficientApproximationArctgFunction.pdf>}, IEEE Signal Processing Magazine, 111, May 2006. ''' # Eq (9): PI_4 * x - x * (abs(x) - 1) * (0.2447 + 0.0663 * abs(x)), for -1 < x < 1 # PI_4 * x - (x**2 - x) * (0.2447 + 0.0663 * x), for 0 < x - 1 # x * (1.0300981633974482 + x * (-0.1784 - x * 0.0663)) H = Fhorner(x, _0_0, 1.0300982, -0.1784, -0.0663) return float(H) def fatan2(y, x): '''Fast approximation of C{atan2(B{y}, B{x})}. @see: U{fastApproximateAtan(x, y)<https://GitHub.com/CesiumGS/cesium/blob/ master/Source/Shaders/Builtin/Functions/fastApproximateAtan.glsl>} and L{fatan1}. ''' b, a = fabs(y), fabs(x) if a < b: r = (PI_2 - fatan1(a / b)) if a else PI_2 elif b < a: r = fatan1(b / a) if b else _0_0 elif a: # == b != 0 r = PI_4 else: # a == b == 0 return _0_0 if x < 0: r = PI - r if y < 0: # copysign0(r, y) r = -r return r def favg(v1, v2, f=_0_5): '''Return the average of two values. @arg v1: One value (C{scalar}). @arg v2: Other value (C{scalar}). @kwarg f: Optional fraction (C{float}). @return: M{v1 + f * (v2 - v1)} (C{float}). ''' # @raise ValueError: Fraction out of range. # ''' # if not 0 <= f <= 1: # XXX restrict fraction? # raise _ValueError(fraction=f) # v1 + f * (v2 - v1) == v1 * (1 - f) + v2 * f return fsum1_(v1, -f * v1, f * v2) def fdot(a, *b): '''Return the precision dot product M{sum(a[i] * b[i] for i=0..len(a))}. @arg a: Iterable, list, tuple, etc. (C{scalar}s). @arg b: All positional arguments (C{scalar}s). @return: Dot product (C{float}). @raise LenError: Unequal C{len(B{a})} and C{len(B{b})}. @see: Class L{Fdot} and U{Algorithm 5.10 B{DotK} <https://www.TUHH.De/ti3/paper/rump/OgRuOi05.pdf>}. ''' return fsum(_map_a_x_b(a, b, fdot)) def fdot3(a, b, c, start=0): '''Return the precision dot product M{start + sum(a[i] * b[i] * c[i] for i=0..len(a))}. @arg a: Iterable, list, tuple, etc. (C{scalar}s). @arg b: Iterable, list, tuple, etc. (C{scalar}s). @arg c: Iterable, list, tuple, etc. (C{scalar}s). @kwarg start: Optional bias (C{scalar}). @return: Dot product (C{float}). @raise LenError: Unequal C{len(B{a})}, C{len(B{b})} and/or C{len(B{c})}. @raise OverflowError: Partial C{2sum} overflow. ''' def _mul3(a, b, c): # map function return a * b * c def _muly(a, b, c, start): yield start for abc in map(_mul3, a, b, c): yield abc if not len(a) == len(b) == len(c): raise LenError(fdot3, a=len(a), b=len(b), c=len(c)) return fsum(_muly(a, b, c, start) if start else map(_mul3, a, b, c)) def fhorner(x, *cs): '''Evaluate the polynomial M{sum(cs[i] * x**i for i=0..len(cs))} using the Horner form. @arg x: Polynomial argument (C{scalar}). @arg cs: Polynomial coeffients (C{scalar}s). @return: Horner value (C{float}). @raise OverflowError: Partial C{2sum} overflow. @raise TypeError: Non-scalar B{C{x}}. @raise ValueError: No B{C{cs}} coefficients or B{C{x}} is not finite. @see: Function L{fpolynomial} and class L{Fhorner}. ''' H = Fhorner(x, *cs) return float(H) def fidw(xs, ds, beta=2): '''Interpolate using using U{Inverse Distance Weighting <https://WikiPedia.org/wiki/Inverse_distance_weighting>} (IDW). @arg xs: Known values (C{scalar}s). @arg ds: Non-negative distances (C{scalar}s). @kwarg beta: Inverse distance power (C{int}, 0, 1, 2, or 3). @return: Interpolated value C{x} (C{float}). @raise LenError: Unequal or zero C{len(B{ds})} and C{len(B{xs})}. @raise ValueError: Invalid B{C{beta}}, negative B{C{ds}} value, weighted B{C{ds}} below L{EPS}. @note: Using C{B{beta}=0} returns the mean of B{C{xs}}. ''' n, xs = len2(xs) d, ds = len2(ds) if n != d or n < 1: raise LenError(fidw, xs=n, ds=d) d, x = min(zip(ds, xs)) if d > EPS0 and n > 1: b = -Int_(beta=beta, low=0, high=3) if b < 0: ds = tuple(d**b for d in ds) d = fsum(ds, floats=True) if isnear0(d): # PYCHOK no cover n = Fmt.PAREN(fsum='ds') raise _ValueError(n, d, txt=_singular_) x = Fdot(xs, *ds).fover(d) else: # b == 0 x = fsum(xs, floats=True) / n # fmean(xs) elif d < 0: # PYCHOK no cover n = Fmt.SQUARE(ds=ds.index(d)) raise _ValueError(n, d, txt=_negative_) return x def fmean(xs): '''Compute the accurate mean M{sum(xs[i] for i=0..len(xs)) / len(xs)}. @arg xs: Values (C{scalar} or L{Fsum} instances). @return: Mean value (C{float}). @raise OverflowError: Partial C{2sum} overflow. @raise ValueError: No B{C{xs}} values. ''' n, xs = len2(xs) if n > 0: return fsum(xs) / n # if n > 1 else _2float(index=0, xs=xs[0]) raise _ValueError(xs=xs) def fmean_(*xs): '''Compute the accurate mean M{sum(xs[i] for i=0..len(xs)) / len(xs)}. @arg xs: Values (C{scalar} or L{Fsum} instances). @return: Mean value (C{float}). @raise OverflowError: Partial C{2sum} overflow. @raise ValueError: No B{C{xs}} values. ''' return fmean(xs) def fpolynomial(x, *cs, **over): '''Evaluate the polynomial M{sum(cs[i] * x**i for i=0..len(cs)) [/ over]}. @arg x: Polynomial argument (C{scalar}). @arg cs: Polynomial coeffients (C{scalar}s), all positional. @kwarg over: Optional, final divisor (C{scalar} @return: Polynomial value (C{float}). @raise OverflowError: Partial C{2sum} overflow. @raise TypeError: Non-scalar B{C{x}}. @raise ValueError: No B{C{cs}} coefficients or B{C{x}} is not finite. @see: Function L{fhorner} and class L{Fpolynomial}. ''' P = Fpolynomial(x, *cs) d = _xkwds_get(over, over=0) if over else 0 return P.fover(d) if d else float(P) def fpowers(x, n, alts=0): '''Return a series of powers M{[x**i for i=1..n]}. @arg x: Value (C{scalar}). @arg n: Highest exponent (C{int}). @kwarg alts: Only alternating powers, starting with this exponent (C{int}). @return: Powers of B{C{x}} (C{float}s or C{int}s). @raise TypeError: Non-scalar B{C{x}} or B{C{n}} not C{int}. @raise ValueError: Non-finite B{C{x}} or non-positive B{C{n}}. ''' if not isint(n): raise _IsnotError(int.__name__, n=n) elif n < 1: raise _ValueError(n=n) p = t = x if isint(x) else _2float(x=x) ps = [p] a_ = ps.append for _ in range(1, n): p *= t a_(p) if alts > 0: # x**2, x**4, ... # ps[alts-1::2] chokes PyChecker ps = ps[slice(alts-1, None, 2)] return ps try: from math import prod as fprod # Python 3.8 except ImportError: def fprod(xs, start=1): '''Iterable product, like C{math.prod} or C{numpy.prod}. @arg xs: Terms to be multiplied, an iterable, list, tuple, etc. (C{scalar}s). @kwarg start: Initial term, also the value returned for an empty B{C{xs}} (C{scalar}). @return: The product (C{float}). @see: U{NumPy.prod<https://docs.SciPy.org/doc/ numpy/reference/generated/numpy.prod.html>}. ''' return freduce(_mul, xs, start) def frange(start, number, step=1): '''Generate a range of C{float}s. @arg start: First value (C{float}). @arg number: The number of C{float}s to generate (C{int}). @kwarg step: Increment value (C{float}). @return: A generator (C{float}s). @see: U{NumPy.prod<https://docs.SciPy.org/doc/ numpy/reference/generated/numpy.arange.html>}. ''' if not isint(number): raise _IsnotError(int.__name__, number=number) for i in range(number): yield start + i * step try: from functools import reduce as freduce except ImportError: try: freduce = reduce # PYCHOK expected except NameError: # Python 3+ def freduce(f, xs, *start): '''For missing C{functools.reduce}. ''' if start: r = v = start[0] else: r, v = 0, MISSING for v in xs: r = f(r, v) if v is MISSING: raise _TypeError(xs=(), start=MISSING) return r def fremainder(x, y): '''Remainder in range C{[-B{y / 2}, B{y / 2}]}. @arg x: Numerator (C{scalar}). @arg y: Modulus, denominator (C{scalar}). @return: Remainder (C{scalar}, preserving signed 0.0) or C{NAN} for any non-finite B{C{x}}. @raise ValueError: Infinite or near-zero B{C{y}}. @see: I{Karney}'s U{Math.remainder<https://PyPI.org/ project/geographiclib/>} and Python 3.7+ U{math.remainder<https://docs.Python.org/3/ library/math.html#math.remainder>}. ''' # with Python 2.7.16 and 3.7.3 on macOS 10.13.6 and # with Python 3.10.2 on macOS 12.2.1 M1 arm64 native # fmod( 0, 360) == 0.0 # fmod( 360, 360) == 0.0 # fmod(-0, 360) == 0.0 # fmod(-0.0, 360) == -0.0 # fmod(-360, 360) == -0.0 # however, using the % operator ... # 0 % 360 == 0 # 360 % 360 == 0 # 360.0 % 360 == 0.0 # -0 % 360 == 0 # -360 % 360 == 0 == (-360) % 360 # -0.0 % 360 == 0.0 == (-0.0) % 360 # -360.0 % 360 == 0.0 == (-360.0) % 360 # On Windows 32-bit with python 2.7, math.fmod(-0.0, 360) # == +0.0. This fixes this bug. See also Math::AngNormalize # in the C++ library, Math.sincosd has a similar fix. if _isfinite(x): try: r = _remainder(x, y) if x else x except Exception as e: raise _xError(e, unstr(fremainder, x, y)) else: # handle x INF and NINF as NAN r = NAN return r if _sys_version_info2 < (3, 8): # PYCHOK no cover from math import hypot # OK in Python 3.7- def hypot_(*xs): '''Compute the norm M{sqrt(sum(x**2 for x in xs))}. Similar to Python 3.8+ n-dimension U{math.hypot <https://docs.Python.org/3.8/library/math.html#math.hypot>}, but exceptions, C{nan} and C{infinite} values are handled differently. @arg xs: X arguments (C{scalar}s), all positional. @return: Norm (C{float}). @raise OverflowError: Partial C{2sum} overflow. @raise ValueError: Invalid or no B{C{xs}} values. @note: The Python 3.8+ Euclidian distance U{math.dist <https://docs.Python.org/3.8/library/math.html#math.dist>} between 2 I{n}-dimensional points I{p1} and I{p2} can be computed as M{hypot_(*((c1 - c2) for c1, c2 in zip(p1, p2)))}, provided I{p1} and I{p2} have the same, non-zero length I{n}. ''' h, x2 = _h_x2(xs) return (h * sqrt(x2)) if x2 else _0_0 elif _sys_version_info2 < (3, 10): # In Python 3.8 and 3.9 C{math.hypot} is inaccurate, see # U{agdhruv<https://GitHub.com/geopy/geopy/issues/466>}, # U{cffk<https://Bugs.Python.org/issue43088>} and module # U{geomath.py<https://PyPI.org/project/geographiclib/1.52>} def hypot(x, y): '''Compute the norm M{sqrt(x**2 + y**2)}. @arg x: X argument (C{scalar}). @arg y: Y argument (C{scalar}). @return: C{sqrt(B{x}**2 + B{y}**2)} (C{float}). ''' if x: h = sqrt(x**2 + y**2) if y else fabs(x) elif y: h = fabs(y) else: h = _0_0 return h from math import hypot as hypot_ # PYCHOK in Python 3.8 and 3.9 else: from math import hypot # PYCHOK in Python 3.10+ hypot_ = hypot def _h_x2(xs): '''(INTERNAL) Helper for L{hypot_} and L{hypot2_}. ''' if xs: n, xs = len2(xs) if n > 0: h = float(max(map(fabs, xs))) if h < EPS0: x2 = _0_0 else: # math.fsum, see C{_hypot21_} below x2 = _fsum(_x2_h2(_1_0, xs, h, _N_1_0)) return h, x2 raise _ValueError(xs=xs, txt=_too_(_few_)) def hypot1(x): '''Compute the norm M{sqrt(1 + x**2)}. @arg x: Argument (C{scalar}). @return: Norm (C{float}). ''' return hypot(_1_0, x) if x else _1_0 def hypot2(x, y): '''Compute the I{squared} norm M{x**2 + y**2}. @arg x: X argument (C{scalar}). @arg y: Y argument (C{scalar}). @return: C{B{x}**2 + B{y}**2} (C{float}). ''' if x: if y: if fabs(x) < fabs(y): x, y = y, x h2 = x**2 * ((y / x)**2 + _1_0) else: h2 = x**2 elif y: h2 = y**2 else: h2 = _0_0 return h2 def hypot2_(*xs): '''Compute the I{squared} norm C{sum(x**2 for x in B{xs})}. @arg xs: X arguments (C{scalar}s), all positional. @return: Squared norm (C{float}). @raise OverflowError: Partial C{2sum} overflow. @raise ValueError: Invalid or no B{C{xs}} value. @see: Function L{hypot_}. ''' h, x2 = _h_x2(xs) return (h**2 * x2) if x2 else _0_0 def _hypot21_(*xs): # PYCHOK in .triaxials '''(INTERNAL) Compute M{sum(x**2 for x is xs) - 1} with C{abs(x)} assumed to be near 1. ''' # math.fsum, see C{_h_x2} above return _fsum(_x2_h2(_1_0, xs, _0_0, _N_2_0)) def _map_a_x_b(a, b, where): '''(INTERNAL) Yield B{C{a * b}}. ''' n = len(b) if len(a) != n: # PYCHOK no cover raise LenError(where, a=len(a), b=n) return map(_mul, a, b) if n > 3 else _map_a_x_b1(a, b) def _map_a_x_b1(a, b): '''(INTERNAL) Yield B{C{a * b}}, 1-primed. ''' yield _1_0 for ab in map(_mul, a, b): if ab: yield ab yield _N_1_0 def norm2(x, y): '''Normalize a 2-dimensional vector. @arg x: X component (C{scalar}). @arg y: Y component (C{scalar}). @return: 2-Tuple C{(x, y)}, normalized. @raise ValueError: Invalid B{C{x}} or B{C{y}} or zero norm. ''' h = hypot(x, y) if not h: x = y = _0_0 # pass? elif not isnear1(h): try: x, y = x / h, y / h except Exception as e: raise _xError(e, x=x, y=y, h=h) return x, y def norm_(*xs): '''Normalize all n-dimensional vector components. @arg xs: Components (C{scalar}s), all positional. @return: Yield each component, normalized. @raise ValueError: Invalid or insufficent B{C{xs}} or zero norm. ''' h = hypot_(*xs) if h: try: for i, x in enumerate(xs): yield x / h except Exception as e: raise _xError(e, Fmt.SQUARE(xs=i), x, _h_, h) else: for _ in xs: yield _0_0 def sqrt0(x): '''Compute the square root iff C{B{x} >} L{EPS02}. @arg x: Value (C{scalar}). @return: Square root (C{float}) or C{0.0}. @note: Any C{B{x} <} L{EPS02} I{including} C{B{x} < 0} returns C{0.0}. ''' return sqrt(x) if x > EPS02 else (_0_0 if x < EPS02 else EPS0) def sqrt3(x): '''Compute the square root, I{cubed} M{sqrt(x)**3} or M{sqrt(x**3)}. @arg x: Value (C{scalar}). @return: Square root I{cubed} (C{float}). @raise ValueError: Negative B{C{x}}. @see: Functions L{cbrt} and L{cbrt2}. ''' if x < 0: raise _ValueError(x=x, txt=_negative_) return pow(x, _1_5) if x else _0_0 def sqrt_a(h, b): '''Compute C{I{a}} side of a right-angled triangle from C{sqrt(B{h}**2 - B{b}**2)}. @arg h: Hypotenuse or outer annulus radius (C{scalar}). @arg b: Triangle side or inner annulus radius (C{scalar}). @return: C{copysign(I{a}, B{h})} or C{unsigned 0.0} (C{float}). @raise TypeError: Non-scalar B{C{h}} or B{C{b}}. @raise ValueError: If C{abs(B{h}) < abs(B{b})}. @see: Inner tangent chord B{I{d}} of an U{annulus <https://WikiPedia.org/wiki/Annulus_(mathematics)>} and function U{annulus_area<https://People.SC.FSU.edu/ ~jburkardt/py_src/geometry/geometry.py>}. ''' try: if not (isscalar(h) and isscalar(b)): raise TypeError(_not_scalar_) elif fabs(h) < fabs(b): raise ValueError('abs(h) < abs(b)') if isnear0(h): # PYCHOK no cover c, b = fabs(h), fabs(b) d = c - b a = copysign0(sqrt((c + b) * d), h) if d > 0 else _0_0 else: c = float(h) s = _1_0 - (b / c)**2 a = (sqrt(s) * c) if 0 < s < 1 else (c if s else _0_0) except Exception as x: raise _xError(x, h=h, b=b) return a def _x2_h2(s, xs, h, e): '''(INTERNAL) Yield M{(x / h)**2 for x in xs}. ''' yield s if h in (_0_0, _1_0): for x in xs: if x: yield x**2 else: for x in xs: if x: yield (x / h)**2 yield e # **) MIT License # # Copyright (C) 2016-2023 -- mrJean1 at Gmail -- All Rights Reserved. # # Permission is hereby granted, free of charge, to any person obtaining a # copy of this software and associated documentation files (the "Software"), # to deal in the Software without restriction, including without limitation # the rights to use, copy, modify, merge, publish, distribute, sublicense, # and/or sell copies of the Software, and to permit persons to whom the # Software is furnished to do so, subject to the following conditions: # # The above copyright notice and this permission notice shall be included # in all copies or substantial portions of the Software. # # THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS # OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, # FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL # THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR # OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, # ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR # OTHER DEALINGS IN THE SOFTWARE.