| /* |
| * ==================================================== |
| * 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. |
| * ==================================================== |
| */ |
| |
| /* Expansions and modifications for 128-bit long double are |
| Copyright (C) 2001 Stephen L. Moshier <moshier@na-net.ornl.gov> |
| and are incorporated herein by permission of the author. The author |
| reserves the right to distribute this material elsewhere under different |
| copying permissions. These modifications are distributed here under |
| the following terms: |
| |
| This library is free software; you can redistribute it and/or |
| modify it under the terms of the GNU Lesser General Public |
| License as published by the Free Software Foundation; either |
| version 2.1 of the License, or (at your option) any later version. |
| |
| This library is distributed in the hope that it will be useful, |
| but WITHOUT ANY WARRANTY; without even the implied warranty of |
| MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU |
| Lesser General Public License for more details. |
| |
| You should have received a copy of the GNU Lesser General Public |
| License along with this library; if not, see |
| <http://www.gnu.org/licenses/>. */ |
| |
| /* __ieee754_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 <math.h> |
| #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 = 1.0e300L, |
| tiny = 1.0e-300L; |
| |
| /* 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 |
| __ieee754_powl (long double x, long double y) |
| { |
| long double z, ax, z_h, z_l, p_h, p_l; |
| long double y1, t1, t2, r, s, t, u, v, w; |
| long double s2, s_h, s_l, t_h, t_l, ay; |
| int32_t i, j, k, yisint, n; |
| uint32_t ix, iy; |
| int32_t hx, hy, hax; |
| double ohi, xhi, xlo, yhi, ylo; |
| uint32_t lx, ly, lj; |
| |
| ldbl_unpack (x, &xhi, &xlo); |
| EXTRACT_WORDS (hx, lx, xhi); |
| ix = hx & 0x7fffffff; |
| |
| ldbl_unpack (y, &yhi, &ylo); |
| EXTRACT_WORDS (hy, ly, yhi); |
| iy = hy & 0x7fffffff; |
| |
| /* y==zero: x**0 = 1 */ |
| if ((iy | ly) == 0) |
| return one; |
| |
| /* 1.0**y = 1; -1.0**+-Inf = 1 */ |
| if (x == one) |
| return one; |
| if (x == -1.0L && ((iy - 0x7ff00000) | ly) == 0) |
| return one; |
| |
| /* +-NaN return x+y */ |
| if ((ix >= 0x7ff00000 && ((ix - 0x7ff00000) | lx) != 0) |
| || (iy >= 0x7ff00000 && ((iy - 0x7ff00000) | ly) != 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) |
| { |
| uint32_t low_ye; |
| |
| GET_HIGH_WORD (low_ye, ylo); |
| if ((low_ye & 0x7fffffff) >= 0x43400000) /* Low part >= 2^53 */ |
| yisint = 2; /* even integer y */ |
| else if (iy >= 0x3ff00000) /* 1.0 */ |
| { |
| if (__floorl (y) == y) |
| { |
| z = 0.5 * y; |
| if (__floorl (z) == z) |
| yisint = 2; |
| else |
| yisint = 1; |
| } |
| } |
| } |
| |
| ax = fabsl (x); |
| |
| /* special value of y */ |
| if (ly == 0) |
| { |
| if (iy == 0x7ff00000) /* y is +-inf */ |
| { |
| if (ax > one) |
| /* (|x|>1)**+-inf = inf,0 */ |
| return (hy >= 0) ? y : zero; |
| else |
| /* (|x|<1)**-,+inf = inf,0 */ |
| return (hy < 0) ? -y : zero; |
| } |
| if (ylo == 0.0) |
| { |
| if (iy == 0x3ff00000) |
| { /* y is +-1 */ |
| if (hy < 0) |
| return one / x; |
| else |
| return x; |
| } |
| if (hy == 0x40000000) |
| return x * x; /* y is 2 */ |
| if (hy == 0x3fe00000) |
| { /* y is 0.5 */ |
| if (hx >= 0) /* x >= +0 */ |
| return __ieee754_sqrtl (x); |
| } |
| } |
| } |
| |
| /* special value of x */ |
| if (lx == 0) |
| { |
| if (ix == 0x7ff00000 || ix == 0 || (ix == 0x3ff00000 && xlo == 0.0)) |
| { |
| z = ax; /*x is +-0,+-inf,+-1 */ |
| if (hy < 0) |
| z = one / z; /* z = (1/|x|) */ |
| if (hx < 0) |
| { |
| if (((ix - 0x3ff00000) | 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 > 0x41d654b0) |
| { |
| /* if (1 - 2^-113)^y underflows, y > 1.1873e38 */ |
| if (iy > 0x47d654b0) |
| { |
| if (ix <= 0x3fefffff) |
| return (hy < 0) ? huge * huge : tiny * tiny; |
| if (ix >= 0x3ff00000) |
| return (hy > 0) ? huge * huge : tiny * tiny; |
| } |
| /* over/underflow if x is not close to one */ |
| if (ix < 0x3fefffff) |
| return (hy < 0) ? huge * huge : tiny * tiny; |
| if (ix > 0x3ff00000) |
| return (hy > 0) ? huge * huge : tiny * tiny; |
| } |
| |
| ay = y > 0 ? y : -y; |
| if (ay < 0x1p-117) |
| y = y < 0 ? -0x1p-117 : 0x1p-117; |
| |
| n = 0; |
| /* take care subnormal number */ |
| if (ix < 0x00100000) |
| { |
| ax *= two113; |
| n -= 113; |
| ohi = ldbl_high (ax); |
| GET_HIGH_WORD (ix, ohi); |
| } |
| n += ((ix) >> 20) - 0x3ff; |
| j = ix & 0x000fffff; |
| /* determine interval */ |
| ix = j | 0x3ff00000; /* normalize ix */ |
| if (j <= 0x39880) |
| k = 0; /* |x|<sqrt(3/2) */ |
| else if (j < 0xbb670) |
| k = 1; /* |x|<sqrt(3) */ |
| else |
| { |
| k = 0; |
| n += 1; |
| ix -= 0x00100000; |
| } |
| |
| ohi = ldbl_high (ax); |
| GET_HIGH_WORD (hax, ohi); |
| ax = __scalbnl (ax, ((int) ((ix - hax) * 2)) >> 21); |
| |
| /* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */ |
| u = ax - bp[k]; /* bp[0]=1.0, bp[1]=1.5 */ |
| v = one / (ax + bp[k]); |
| s = u * v; |
| s_h = ldbl_high (s); |
| |
| /* t_h=ax+bp[k] High */ |
| t_h = ax + bp[k]; |
| t_h = ldbl_high (t_h); |
| t_l = ax - (t_h - bp[k]); |
| s_l = v * ((u - s_h * t_h) - s_h * t_l); |
| /* compute log(ax) */ |
| s2 = s * s; |
| u = LN[0] + s2 * (LN[1] + s2 * (LN[2] + s2 * (LN[3] + s2 * LN[4]))); |
| v = LD[0] + s2 * (LD[1] + s2 * (LD[2] + s2 * (LD[3] + s2 * (LD[4] + s2)))); |
| r = s2 * s2 * u / v; |
| r += s_l * (s_h + s); |
| s2 = s_h * s_h; |
| t_h = 3.0 + s2 + r; |
| t_h = ldbl_high (t_h); |
| t_l = r - ((t_h - 3.0) - s2); |
| /* u+v = s*(1+...) */ |
| u = s_h * t_h; |
| v = s_l * t_h + t_l * s; |
| /* 2/(3log2)*(s+...) */ |
| p_h = u + v; |
| p_h = ldbl_high (p_h); |
| p_l = v - (p_h - u); |
| z_h = cp_h * p_h; /* cp_h+cp_l = 2/(3*log2) */ |
| z_l = cp_l * p_h + p_l * cp + dp_l[k]; |
| /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */ |
| t = (long double) n; |
| t1 = (((z_h + z_l) + dp_h[k]) + t); |
| t1 = ldbl_high (t1); |
| t2 = z_l - (((t1 - t) - dp_h[k]) - z_h); |
| |
| /* s (sign of result -ve**odd) = -1 else = 1 */ |
| s = one; |
| if (((((u_int32_t) hx >> 31) - 1) | (yisint - 1)) == 0) |
| s = -one; /* (-ve)**(odd int) */ |
| |
| /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */ |
| y1 = ldbl_high (y); |
| p_l = (y - y1) * t1 + y * t2; |
| p_h = y1 * t1; |
| z = p_l + p_h; |
| ohi = ldbl_high (z); |
| EXTRACT_WORDS (j, lj, ohi); |
| if (j >= 0x40d00000) /* z >= 16384 */ |
| { |
| /* if z > 16384 */ |
| if (((j - 0x40d00000) | lj) != 0) |
| return s * huge * huge; /* overflow */ |
| else |
| { |
| if (p_l + ovt > z - p_h) |
| return s * huge * huge; /* overflow */ |
| } |
| } |
| else if ((j & 0x7fffffff) >= 0x40d01b90) /* z <= -16495 */ |
| { |
| /* z < -16495 */ |
| if (((j - 0xc0d01bc0) | lj) != 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 >> 20) - 0x3ff; |
| n = 0; |
| if (i > 0x3fe00000) |
| { /* if |z| > 0.5, set n = [z+0.5] */ |
| n = __floorl (z + 0.5L); |
| t = n; |
| p_h -= t; |
| } |
| t = p_l + p_h; |
| t = ldbl_high (t); |
| 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); |
| z = __scalbnl (z, n); |
| return s * z; |
| } |
| strong_alias (__ieee754_powl, __powl_finite) |