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Diffstat (limited to 'src/math/pow.c')
| -rw-r--r-- | src/math/pow.c | 326 |
1 files changed, 326 insertions, 0 deletions
diff --git a/src/math/pow.c b/src/math/pow.c new file mode 100644 index 00000000..f843645d --- /dev/null +++ b/src/math/pow.c @@ -0,0 +1,326 @@ +/* origin: FreeBSD /usr/src/lib/msun/src/e_pow.c */ +/* + * ==================================================== + * Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved. + * + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ +/* pow(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 53-24 = 29 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 except 1) ** NAN is NAN, 1 ** NAN is 1 + * 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 1 + * 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 + * + * Accuracy: + * pow(x,y) returns x**y nearly rounded. In particular + * pow(integer,integer) + * always returns the correct integer provided it is + * representable. + * + * Constants : + * The hexadecimal values are the intended ones for the following + * constants. The decimal values may be used, provided that the + * compiler will convert from decimal to binary accurately enough + * to produce the hexadecimal values shown. + */ + +#include "libm.h" + +static const double +bp[] = {1.0, 1.5,}, +dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */ +dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */ +zero = 0.0, +one = 1.0, +two = 2.0, +two53 = 9007199254740992.0, /* 0x43400000, 0x00000000 */ +huge = 1.0e300, +tiny = 1.0e-300, +/* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */ +L1 = 5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */ +L2 = 4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */ +L3 = 3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */ +L4 = 2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */ +L5 = 2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */ +L6 = 2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */ +P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */ +P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */ +P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */ +P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */ +P5 = 4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */ +lg2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */ +lg2_h = 6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */ +lg2_l = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */ +ovt = 8.0085662595372944372e-017, /* -(1024-log2(ovfl+.5ulp)) */ +cp = 9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */ +cp_h = 9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */ +cp_l = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/ +ivln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */ +ivln2_h = 1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/ +ivln2_l = 1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/ + +double pow(double x, double y) +{ + double z,ax,z_h,z_l,p_h,p_l; + double y1,t1,t2,r,s,t,u,v,w; + int32_t i,j,k,yisint,n; + int32_t hx,hy,ix,iy; + uint32_t lx,ly; + + EXTRACT_WORDS(hx, lx, x); + EXTRACT_WORDS(hy, ly, y); + ix = hx & 0x7fffffff; + iy = hy & 0x7fffffff; + + /* y == zero: x**0 = 1 */ + if ((iy|ly) == 0) + return one; + + /* x == 1: 1**y = 1, even if y is NaN */ + if (hx == 0x3ff00000 && lx == 0) + return one; + + /* y != zero: result is NaN if either arg is NaN */ + if (ix > 0x7ff00000 || (ix == 0x7ff00000 && lx != 0) || + iy > 0x7ff00000 || (iy == 0x7ff00000 && ly != 0)) + return (x+0.0)+(y+0.0); // FIXME: 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 >= 0x43400000) + yisint = 2; /* even integer y */ + else if (iy >= 0x3ff00000) { + k = (iy>>20) - 0x3ff; /* exponent */ + if (k > 20) { + j = ly>>(52-k); + if ((j<<(52-k)) == ly) + yisint = 2 - (j&1); + } else if (ly == 0) { + j = iy>>(20-k); + if ((j<<(20-k)) == iy) + yisint = 2 - (j&1); + } + } + } + + /* special value of y */ + if (ly == 0) { + if (iy == 0x7ff00000) { /* y is +-inf */ + if (((ix-0x3ff00000)|lx) == 0) /* (-1)**+-inf is 1 */ + return one; + else if (ix >= 0x3ff00000) /* (|x|>1)**+-inf = inf,0 */ + return hy >= 0 ? y : zero; + else /* (|x|<1)**+-inf = 0,inf */ + return hy < 0 ? -y : zero; + } + if (iy == 0x3ff00000) { /* y is +-1 */ + if (hy < 0) + return one/x; + return x; + } + if (hy == 0x40000000) /* y is 2 */ + return x*x; + if (hy == 0x3fe00000) { /* y is 0.5 */ + if (hx >= 0) /* x >= +0 */ + return sqrt(x); + } + } + + ax = fabs(x); + /* special value of x */ + if (lx == 0) { + if (ix == 0x7ff00000 || ix == 0 || ix == 0x3ff00000) { /* x is +-0,+-inf,+-1 */ + z = ax; + if (hy < 0) /* z = (1/|x|) */ + z = one/z; + 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; + } + } + + /* CYGNUS LOCAL + fdlibm-5.3 fix: This used to be + n = (hx>>31)+1; + but ANSI C says a right shift of a signed negative quantity is + implementation defined. */ + n = ((uint32_t)hx>>31) - 1; + + /* (x<0)**(non-int) is NaN */ + if ((n|yisint) == 0) + return (x-x)/(x-x); + + s = one; /* s (sign of result -ve**odd) = -1 else = 1 */ + if ((n|(yisint-1)) == 0) + s = -one;/* (-ve)**(odd int) */ + + /* |y| is huge */ + if (iy > 0x41e00000) { /* if |y| > 2**31 */ + if (iy > 0x43f00000) { /* if |y| > 2**64, must o/uflow */ + 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 ? s*huge*huge : s*tiny*tiny; + if (ix > 0x3ff00000) + return hy > 0 ? s*huge*huge : s*tiny*tiny; + /* now |1-x| is tiny <= 2**-20, suffice to compute + log(x) by x-x^2/2+x^3/3-x^4/4 */ + t = ax - one; /* t has 20 trailing zeros */ + w = (t*t)*(0.5 - t*(0.3333333333333333333333-t*0.25)); + u = ivln2_h*t; /* ivln2_h has 21 sig. bits */ + v = t*ivln2_l - w*ivln2; + t1 = u + v; + SET_LOW_WORD(t1, 0); + t2 = v - (t1-u); + } else { + double ss,s2,s_h,s_l,t_h,t_l; + n = 0; + /* take care subnormal number */ + if (ix < 0x00100000) { + ax *= two53; + n -= 53; + GET_HIGH_WORD(ix,ax); + } + n += ((ix)>>20) - 0x3ff; + j = ix & 0x000fffff; + /* determine interval */ + ix = j | 0x3ff00000; /* normalize ix */ + if (j <= 0x3988E) /* |x|<sqrt(3/2) */ + k = 0; + else if (j < 0xBB67A) /* |x|<sqrt(3) */ + k = 1; + else { + k = 0; + n += 1; + ix -= 0x00100000; + } + SET_HIGH_WORD(ax, ix); + + /* compute ss = 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]); + ss = u*v; + s_h = ss; + SET_LOW_WORD(s_h, 0); + /* t_h=ax+bp[k] High */ + t_h = zero; + SET_HIGH_WORD(t_h, ((ix>>1)|0x20000000) + 0x00080000 + (k<<18)); + t_l = ax - (t_h-bp[k]); + s_l = v*((u-s_h*t_h)-s_h*t_l); + /* compute log(ax) */ + s2 = ss*ss; + r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6))))); + r += s_l*(s_h+ss); + s2 = s_h*s_h; + t_h = 3.0 + s2 + r; + SET_LOW_WORD(t_h, 0); + t_l = r - ((t_h-3.0)-s2); + /* u+v = ss*(1+...) */ + u = s_h*t_h; + v = s_l*t_h + t_l*ss; + /* 2/(3log2)*(ss+...) */ + p_h = u + v; + SET_LOW_WORD(p_h, 0); + 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) = (ss+..)*2/(3*log2) = n + dp_h + z_h + z_l */ + t = (double)n; + t1 = ((z_h + z_l) + dp_h[k]) + t; + SET_LOW_WORD(t1, 0); + t2 = z_l - (((t1 - t) - dp_h[k]) - z_h); + } + + /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */ + y1 = y; + SET_LOW_WORD(y1, 0); + p_l = (y-y1)*t1 + y*t2; + p_h = y1*t1; + z = p_l + p_h; + EXTRACT_WORDS(j, i, z); + if (j >= 0x40900000) { /* z >= 1024 */ + if (((j-0x40900000)|i) != 0) /* if z > 1024 */ + return s*huge*huge; /* overflow */ + if (p_l + ovt > z - p_h) + return s*huge*huge; /* overflow */ + } else if ((j&0x7fffffff) >= 0x4090cc00) { /* z <= -1075 */ // FIXME: instead of abs(j) use unsigned j + if (((j-0xc090cc00)|i) != 0) /* z < -1075 */ + return s*tiny*tiny; /* underflow */ + 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 = j + (0x00100000>>(k+1)); + k = ((n&0x7fffffff)>>20) - 0x3ff; /* new k for n */ + t = zero; + SET_HIGH_WORD(t, n & ~(0x000fffff>>k)); + n = ((n&0x000fffff)|0x00100000)>>(20-k); + if (j < 0) + n = -n; + p_h -= t; + } + t = p_l + p_h; + SET_LOW_WORD(t, 0); + u = t*lg2_h; + v = (p_l-(t-p_h))*lg2 + t*lg2_l; + z = u + v; + w = v - (z-u); + t = z*z; + t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5)))); + r = (z*t1)/(t1-two) - (w + z*w); + z = one - (r-z); + GET_HIGH_WORD(j, z); + j += n<<20; + if ((j>>20) <= 0) /* subnormal output */ + z = scalbn(z,n); + else + SET_HIGH_WORD(z, j); + return s*z; +} |