Add a "fork" of musl as //fusl.

fusl/src/math/pow.c

+/* 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.  1 ** (anything)  is 1
+ *      3.  (anything except 1) ** 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 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, raise divbyzero
+ *      13. -0 ** (-anything except 0, NAN, odd integer)  is +INF, raise divbyzero
+ *      14. -0 ** (+odd integer) is -0
+ *      15. -0 ** (-odd integer) is -INF, raise divbyzero
+ *      16. +INF ** (+anything except 0,NAN) is +INF
+ *      17. +INF ** (-anything except 0,NAN) is +0
+ *      18. -INF ** (+odd integer) is -INF
+ *      19. -INF ** (anything) = -0 ** (-anything), (anything except odd integer)
+ *      20. (anything) ** 1 is (anything)
+ *      21. (anything) ** -1 is 1/(anything)
+ *      22. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
+ *      23. (-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 */
+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;
+
+        /* x**0 = 1, even if x is NaN */
+        if ((iy|ly) == 0)
+                return 1.0;
+        /* 1**y = 1, even if y is NaN */
+        if (hx == 0x3ff00000 && lx == 0)
+                return 1.0;
+        /* NaN if either arg is NaN */
+        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) {
+                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 1.0;
+                        else if (ix >= 0x3ff00000) /* (|x|>1)**+-inf = inf,0 */
+                                return hy >= 0 ? y : 0.0;
+                        else                       /* (|x|<1)**+-inf = 0,inf */
+                                return hy >= 0 ? 0.0 : -y;
+                }
+                if (iy == 0x3ff00000) {    /* y is +-1 */
+                        if (hy >= 0)
+                                return x;
+                        y = 1/x;
+#if FLT_EVAL_METHOD!=0
+                        {
+                                union {double f; uint64_t i;} u = {y};
+                                uint64_t i = u.i & -1ULL/2;
+                                if (i>>52 == 0 && (i&(i-1)))
+                                        FORCE_EVAL((float)y);
+                        }
+#endif
+                        return y;
+                }
+                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 = 1.0/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;
+                }
+        }
+
+        s = 1.0; /* sign of result */
+        if (hx < 0) {
+                if (yisint == 0) /* (x<0)**(non-int) is NaN */
+                        return (x-x)/(x-x);
+                if (yisint == 1) /* (x<0)**(odd int) */
+                        s = -1.0;
+        }
+
+        /* |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 - 1.0;       /* 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 = 1.0/(ax+bp[k]);
+                ss = u*v;
+                s_h = ss;
+                SET_LOW_WORD(s_h, 0);
+                /* t_h=ax+bp[k] High */
+                t_h = 0.0;
+                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 = 0.0;
+                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)/(t1-2.0) - (w + z*w);
+        z = 1.0 - (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;
+}