/* * ==================================================== * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. * * Developed at SunPro, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this * software is freely granted, provided that this notice * is preserved. * ==================================================== */ /* * Copyright (c) 2008 Stephen L. Moshier * * Permission to use, copy, modify, and distribute this software for any * purpose with or without fee is hereby granted, provided that the above * copyright notice and this permission notice appear in all copies. * * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE. */ /* powl(x,y) return x**y * * n * Method: Let x = 2 * (1+f) * 1. Compute and return log2(x) in two pieces: * log2(x) = w1 + w2, * where w1 has 113-53 = 60 bit trailing zeros. * 2. Perform y*log2(x) = n+y' by simulating muti-precision * arithmetic, where |y'|<=0.5. * 3. Return x**y = 2**n*exp(y'*log2) * * Special cases: * 1. (anything) ** 0 is 1 * 2. (anything) ** 1 is itself * 3. (anything) ** NAN is NAN * 4. NAN ** (anything except 0) is NAN * 5. +-(|x| > 1) ** +INF is +INF * 6. +-(|x| > 1) ** -INF is +0 * 7. +-(|x| < 1) ** +INF is +0 * 8. +-(|x| < 1) ** -INF is +INF * 9. +-1 ** +-INF is NAN * 10. +0 ** (+anything except 0, NAN) is +0 * 11. -0 ** (+anything except 0, NAN, odd integer) is +0 * 12. +0 ** (-anything except 0, NAN) is +INF * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF * 14. -0 ** (odd integer) = -( +0 ** (odd integer) ) * 15. +INF ** (+anything except 0,NAN) is +INF * 16. +INF ** (-anything except 0,NAN) is +0 * 17. -INF ** (anything) = -0 ** (-anything) * 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer) * 19. (-anything except 0 and inf) ** (non-integer) is NAN * */ #include #include "math_private.h" static const long double bp[] = { 1.0L, 1.5L, }; /* log_2(1.5) */ static const long double dp_h[] = { 0.0, 5.8496250072115607565592654282227158546448E-1L }; /* Low part of log_2(1.5) */ static const long double dp_l[] = { 0.0, 1.0579781240112554492329533686862998106046E-16L }; static const long double zero = 0.0L, one = 1.0L, two = 2.0L, two113 = 1.0384593717069655257060992658440192E34L, huge = 1.0e3000L, tiny = 1.0e-3000L; /* 3/2 log x = 3 z + z^3 + z^3 (z^2 R(z^2)) z = (x-1)/(x+1) 1 <= x <= 1.25 Peak relative error 2.3e-37 */ static const long double LN[] = { -3.0779177200290054398792536829702930623200E1L, 6.5135778082209159921251824580292116201640E1L, -4.6312921812152436921591152809994014413540E1L, 1.2510208195629420304615674658258363295208E1L, -9.9266909031921425609179910128531667336670E-1L }; static const long double LD[] = { -5.129862866715009066465422805058933131960E1L, 1.452015077564081884387441590064272782044E2L, -1.524043275549860505277434040464085593165E2L, 7.236063513651544224319663428634139768808E1L, -1.494198912340228235853027849917095580053E1L /* 1.0E0 */ }; /* exp(x) = 1 + x - x / (1 - 2 / (x - x^2 R(x^2))) 0 <= x <= 0.5 Peak relative error 5.7e-38 */ static const long double PN[] = { 5.081801691915377692446852383385968225675E8L, 9.360895299872484512023336636427675327355E6L, 4.213701282274196030811629773097579432957E4L, 5.201006511142748908655720086041570288182E1L, 9.088368420359444263703202925095675982530E-3L, }; static const long double PD[] = { 3.049081015149226615468111430031590411682E9L, 1.069833887183886839966085436512368982758E8L, 8.259257717868875207333991924545445705394E5L, 1.872583833284143212651746812884298360922E3L, /* 1.0E0 */ }; static const long double /* ln 2 */ lg2 = 6.9314718055994530941723212145817656807550E-1L, lg2_h = 6.9314718055994528622676398299518041312695E-1L, lg2_l = 2.3190468138462996154948554638754786504121E-17L, ovt = 8.0085662595372944372e-0017L, /* 2/(3*log(2)) */ cp = 9.6179669392597560490661645400126142495110E-1L, cp_h = 9.6179669392597555432899980587535537779331E-1L, cp_l = 5.0577616648125906047157785230014751039424E-17L; long double powl(long double x, long double y) { long double z, ax, z_h, z_l, p_h, p_l; long double yy1, t1, t2, r, s, t, u, v, w; long double s2, s_h, s_l, t_h, t_l; int32_t i, j, k, yisint, n; u_int32_t ix, iy; int32_t hx, hy; ieee_quad_shape_type o, p, q; p.value = x; hx = p.parts32.mswhi; ix = hx & 0x7fffffff; q.value = y; hy = q.parts32.mswhi; iy = hy & 0x7fffffff; /* y==zero: x**0 = 1 */ if ((iy | q.parts32.mswlo | q.parts32.lswhi | q.parts32.lswlo) == 0) return one; /* 1.0**y = 1; -1.0**+-Inf = 1 */ if (x == one) return one; if (x == -1.0L && iy == 0x7fff0000 && (q.parts32.mswlo | q.parts32.lswhi | q.parts32.lswlo) == 0) return one; /* +-NaN return x+y */ if ((ix > 0x7fff0000) || ((ix == 0x7fff0000) && ((p.parts32.mswlo | p.parts32.lswhi | p.parts32.lswlo) != 0)) || (iy > 0x7fff0000) || ((iy == 0x7fff0000) && ((q.parts32.mswlo | q.parts32.lswhi | q.parts32.lswlo) != 0))) return x + y; /* determine if y is an odd int when x < 0 * yisint = 0 ... y is not an integer * yisint = 1 ... y is an odd int * yisint = 2 ... y is an even int */ yisint = 0; if (hx < 0) { if (iy >= 0x40700000) /* 2^113 */ yisint = 2; /* even integer y */ else if (iy >= 0x3fff0000) /* 1.0 */ { if (floorl (y) == y) { z = 0.5 * y; if (floorl (z) == z) yisint = 2; else yisint = 1; } } } /* special value of y */ if ((q.parts32.mswlo | q.parts32.lswhi | q.parts32.lswlo) == 0) { if (iy == 0x7fff0000) /* y is +-inf */ { if (((ix - 0x3fff0000) | p.parts32.mswlo | p.parts32.lswhi | p.parts32.lswlo) == 0) return y - y; /* +-1**inf is NaN */ else if (ix >= 0x3fff0000) /* (|x|>1)**+-inf = inf,0 */ return (hy >= 0) ? y : zero; else /* (|x|<1)**-,+inf = inf,0 */ return (hy < 0) ? -y : zero; } if (iy == 0x3fff0000) { /* y is +-1 */ if (hy < 0) return one / x; else return x; } if (hy == 0x40000000) return x * x; /* y is 2 */ if (hy == 0x3ffe0000) { /* y is 0.5 */ if (hx >= 0) /* x >= +0 */ return sqrtl (x); } } ax = fabsl (x); /* special value of x */ if ((p.parts32.mswlo | p.parts32.lswhi | p.parts32.lswlo) == 0) { if (ix == 0x7fff0000 || ix == 0 || ix == 0x3fff0000) { z = ax; /*x is +-0,+-inf,+-1 */ if (hy < 0) z = one / z; /* z = (1/|x|) */ if (hx < 0) { if (((ix - 0x3fff0000) | yisint) == 0) { z = (z - z) / (z - z); /* (-1)**non-int is NaN */ } else if (yisint == 1) z = -z; /* (x<0)**odd = -(|x|**odd) */ } return z; } } /* (x<0)**(non-int) is NaN */ if (((((u_int32_t) hx >> 31) - 1) | yisint) == 0) return (x - x) / (x - x); /* |y| is huge. 2^-16495 = 1/2 of smallest representable value. If (1 - 1/131072)^y underflows, y > 1.4986e9 */ if (iy > 0x401d654b) { /* if (1 - 2^-113)^y underflows, y > 1.1873e38 */ if (iy > 0x407d654b) { if (ix <= 0x3ffeffff) return (hy < 0) ? huge * huge : tiny * tiny; if (ix >= 0x3fff0000) return (hy > 0) ? huge * huge : tiny * tiny; } /* over/underflow if x is not close to one */ if (ix < 0x3ffeffff) return (hy < 0) ? huge * huge : tiny * tiny; if (ix > 0x3fff0000) return (hy > 0) ? huge * huge : tiny * tiny; } n = 0; /* take care subnormal number */ if (ix < 0x00010000) { ax *= two113; n -= 113; o.value = ax; ix = o.parts32.mswhi; } n += ((ix) >> 16) - 0x3fff; j = ix & 0x0000ffff; /* determine interval */ ix = j | 0x3fff0000; /* normalize ix */ if (j <= 0x3988) k = 0; /* |x|> 31) - 1) | (yisint - 1)) == 0) s = -one; /* (-ve)**(odd int) */ /* split up y into yy1+y2 and compute (yy1+y2)*(t1+t2) */ yy1 = y; o.value = yy1; o.parts32.lswlo = 0; o.parts32.lswhi &= 0xf8000000; yy1 = o.value; p_l = (y - yy1) * t1 + y * t2; p_h = yy1 * t1; z = p_l + p_h; o.value = z; j = o.parts32.mswhi; if (j >= 0x400d0000) /* z >= 16384 */ { /* if z > 16384 */ if (((j - 0x400d0000) | o.parts32.mswlo | o.parts32.lswhi | o.parts32.lswlo) != 0) return s * huge * huge; /* overflow */ else { if (p_l + ovt > z - p_h) return s * huge * huge; /* overflow */ } } else if ((j & 0x7fffffff) >= 0x400d01b9) /* z <= -16495 */ { /* z < -16495 */ if (((j - 0xc00d01bc) | o.parts32.mswlo | o.parts32.lswhi | o.parts32.lswlo) != 0) return s * tiny * tiny; /* underflow */ else { if (p_l <= z - p_h) return s * tiny * tiny; /* underflow */ } } /* compute 2**(p_h+p_l) */ i = j & 0x7fffffff; k = (i >> 16) - 0x3fff; n = 0; if (i > 0x3ffe0000) { /* if |z| > 0.5, set n = [z+0.5] */ n = floorl (z + 0.5L); t = n; p_h -= t; } t = p_l + p_h; o.value = t; o.parts32.lswlo = 0; o.parts32.lswhi &= 0xf8000000; t = o.value; u = t * lg2_h; v = (p_l - (t - p_h)) * lg2 + t * lg2_l; z = u + v; w = v - (z - u); /* exp(z) */ t = z * z; u = PN[0] + t * (PN[1] + t * (PN[2] + t * (PN[3] + t * PN[4]))); v = PD[0] + t * (PD[1] + t * (PD[2] + t * (PD[3] + t))); t1 = z - t * u / v; r = (z * t1) / (t1 - two) - (w + z * w); z = one - (r - z); o.value = z; j = o.parts32.mswhi; j += (n << 16); if ((j >> 16) <= 0) z = scalbnl (z, n); /* subnormal output */ else { o.parts32.mswhi = j; z = o.value; } return s * z; } DEF_STD(powl);