/* pow.c * * Power function * * * * SYNOPSIS: * * double x, y, z, pow(); * * z = pow( x, y ); * * * * DESCRIPTION: * * Computes x raised to the yth power. Analytically, * * x**y = exp( y log(x) ). * * Following Cody and Waite, this program uses a lookup table * of 2**-i/16 and pseudo extended precision arithmetic to * obtain an extra three bits of accuracy in both the logarithm * and the exponential. * * * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * IEEE -26,26 30000 4.2e-16 7.7e-17 * IEEE 0,8700 30000 1.5e-14 2.1e-15 * 0.99 < x < 1.01, 0 < y < 8700, uniformly distributed. * * * ERROR MESSAGES: * * message condition value returned * pow overflow x**y > MAXNUM INFINITY * pow underflow x**y < 1/MAXNUM 0.0 * pow domain x<0 and y noninteger 0.0 * */ /* Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1995, 2000 by Stephen L. Moshier */ /* Modified for mingw 2002-09-27 Danny Smith */ #ifdef __MINGW32__ #include "cephes_mconf.h" #else #include "mconf.h" static char fname[] = {"pow"}; #endif #ifndef _SET_ERRNO #define _SET_ERRNO(x) #endif #define SQRTH 0.70710678118654752440 #ifdef UNK static uD P[4] = { { { 4.97778295871696322025E-1 } }, { { 3.73336776063286838734E0 } }, { { 7.69994162726912503298E0 } }, { { 4.66651806774358464979E0 } } }; static uD Q[4] = { { { 9.33340916416696166113E0 } }, { { 2.79999886606328401649E1 } }, { { 3.35994905342304405431E1 } }, { { 1.39995542032307539578E1 } } }; /* 2^(-i/16), IEEE precision */ static uD A[17] = { { { 1.00000000000000000000E0 } }, { { 9.57603280698573700036E-1 } }, { { 9.17004043204671215328E-1 } }, { { 8.78126080186649726755E-1 } }, { { 8.40896415253714502036E-1 } }, { { 8.05245165974627141736E-1 } }, { { 7.71105412703970372057E-1 } }, { { 7.38413072969749673113E-1 } }, { { 7.07106781186547572737E-1 } }, { { 6.77127773468446325644E-1 } }, { { 6.48419777325504820276E-1 } }, { { 6.20928906036742001007E-1 } }, { { 5.94603557501360513449E-1 } }, { { 5.69394317378345782288E-1 } }, { { 5.45253866332628844837E-1 } }, { { 5.22136891213706877402E-1 } }, { { 5.00000000000000000000E-1 } } }; static uD B[9] = { { { 0.00000000000000000000E0 } }, { { 1.64155361212281360176E-17 } }, { { 4.09950501029074826006E-17 } }, { { 3.97491740484881042808E-17 } }, { { -4.83364665672645672553E-17 } }, { { 1.26912513974441574796E-17 } }, { { 1.99100761573282305549E-17 } }, { { -1.52339103990623557348E-17 } }, { { 0.00000000000000000000E0 } } }; static uD R[7] = { { { 1.49664108433729301083E-5 } }, { { 1.54010762792771901396E-4 } }, { { 1.33335476964097721140E-3 } }, { { 9.61812908476554225149E-3 } }, { { 5.55041086645832347466E-2 } }, { { 2.40226506959099779976E-1 } }, { { 6.93147180559945308821E-1 } } }; #define douba(k) A[k].d #define doubb(k) B[k].d #define MEXP 16383.0 #ifdef DENORMAL #define MNEXP -17183.0 #else #define MNEXP -16383.0 #endif #endif #ifdef IBMPC static const uD P[4] = { { { 0x5cf0,0x7f5b,0xdb99,0x3fdf } }, { { 0xdf15,0xea9e,0xddef,0x400d } }, { { 0xeb6f,0x7f78,0xccbd,0x401e } }, { { 0x9b74,0xb65c,0xaa83,0x4012 } } }; static const uD Q[4] = { { { 0x914e,0x9b20,0xaab4,0x4022 } }, { { 0xc9f5,0x41c1,0xffff,0x403b } }, { { 0x6402,0x1b17,0xccbc,0x4040 } }, { { 0xe92e,0x918a,0xffc5,0x402b } } }; static const uD A[17] = { { { 0x0000,0x0000,0x0000,0x3ff0 } }, { { 0x90da,0xa2a4,0xa4af,0x3fee } }, { { 0xa487,0xdcfb,0x5818,0x3fed } }, { { 0x529c,0xdd85,0x199b,0x3fec } }, { { 0xd3ad,0x995a,0xe89f,0x3fea } }, { { 0xf090,0x82a3,0xc491,0x3fe9 } }, { { 0xa0db,0x422a,0xace5,0x3fe8 } }, { { 0x0187,0x73eb,0xa114,0x3fe7 } }, { { 0x3bcd,0x667f,0xa09e,0x3fe6 } }, { { 0x5429,0xdd48,0xab07,0x3fe5 } }, { { 0x2a27,0xd536,0xbfda,0x3fe4 } }, { { 0x3422,0x4c12,0xdea6,0x3fe3 } }, { { 0xb715,0x0a31,0x06fe,0x3fe3 } }, { { 0x6238,0x6e75,0x387a,0x3fe2 } }, { { 0x517b,0x3c7d,0x72b8,0x3fe1 } }, { { 0x890f,0x6cf9,0xb558,0x3fe0 } }, { { 0x0000,0x0000,0x0000,0x3fe0 } } }; static const uD B[9] = { { { 0x0000,0x0000,0x0000,0x0000 } }, { { 0x3707,0xd75b,0xed02,0x3c72 } }, { { 0xcc81,0x345d,0xa1cd,0x3c87 } }, { { 0x4b27,0x5686,0xe9f1,0x3c86 } }, { { 0x6456,0x13b2,0xdd34,0xbc8b } }, { { 0x42e2,0xafec,0x4397,0x3c6d } }, { { 0x82e4,0xd231,0xf46a,0x3c76 } }, { { 0x8a76,0xb9d7,0x9041,0xbc71 } }, { { 0x0000,0x0000,0x0000,0x0000 } } }; static const uD R[7] = { { { 0x937f,0xd7f2,0x6307,0x3eef } }, { { 0x9259,0x60fc,0x2fbe,0x3f24 } }, { { 0xef1d,0xc84a,0xd87e,0x3f55 } }, { { 0x33b7,0x6ef1,0xb2ab,0x3f83 } }, { { 0x1a92,0xd704,0x6b08,0x3fac } }, { { 0xc56d,0xff82,0xbfbd,0x3fce } }, { { 0x39ef,0xfefa,0x2e42,0x3fe6 } } }; #define douba(k) (A[(k)].d) #define doubb(k) (B[(k)].d) #define MEXP 16383.0 #ifdef DENORMAL #define MNEXP -17183.0 #else #define MNEXP -16383.0 #endif #endif #ifdef MIEEE static uD P[4] = { { { 0x3fdf,0xdb99,0x7f5b,0x5cf0 } }, { { 0x400d,0xddef,0xea9e,0xdf15 } }, { { 0x401e,0xccbd,0x7f78,0xeb6f } }, { { 0x4012,0xaa83,0xb65c,0x9b74 } } }; static uD Q[4] = { { { 0x4022,0xaab4,0x9b20,0x914e } }, { { 0x403b,0xffff,0x41c1,0xc9f5 } }, { { 0x4040,0xccbc,0x1b17,0x6402 } }, { { 0x402b,0xffc5,0x918a,0xe92e } } }; static unsigned short A[17] = { { { 0x3ff0,0x0000,0x0000,0x0000 } }, { { 0x3fee,0xa4af,0xa2a4,0x90da } }, { { 0x3fed,0x5818,0xdcfb,0xa487 } }, { { 0x3fec,0x199b,0xdd85,0x529c } }, { { 0x3fea,0xe89f,0x995a,0xd3ad } }, { { 0x3fe9,0xc491,0x82a3,0xf090 } }, { { 0x3fe8,0xace5,0x422a,0xa0db } }, { { 0x3fe7,0xa114,0x73eb,0x0187 } }, { { 0x3fe6,0xa09e,0x667f,0x3bcd } }, { { 0x3fe5,0xab07,0xdd48,0x5429 } }, { { 0x3fe4,0xbfda,0xd536,0x2a27 } }, { { 0x3fe3,0xdea6,0x4c12,0x3422 } }, { { 0x3fe3,0x06fe,0x0a31,0xb715 } }, { { 0x3fe2,0x387a,0x6e75,0x6238 } }, { { 0x3fe1,0x72b8,0x3c7d,0x517b } }, { { 0x3fe0,0xb558,0x6cf9,0x890f } }, { { 0x3fe0,0x0000,0x0000,0x0000 } } }; static uD B[9] = { { { 0x0000,0x0000,0x0000,0x0000 } }, { { 0x3c72,0xed02,0xd75b,0x3707 } }, { { 0x3c87,0xa1cd,0x345d,0xcc81 } }, { { 0x3c86,0xe9f1,0x5686,0x4b27 } }, { { 0xbc8b,0xdd34,0x13b2,0x6456 } }, { { 0x3c6d,0x4397,0xafec,0x42e2 } }, { { 0x3c76,0xf46a,0xd231,0x82e4 } }, { { 0xbc71,0x9041,0xb9d7,0x8a76 } }, { { 0x0000,0x0000,0x0000,0x0000 } } }; static uD R[7] = { { { 0x3eef,0x6307,0xd7f2,0x937f } }, { { 0x3f24,0x2fbe,0x60fc,0x9259 } }, { { 0x3f55,0xd87e,0xc84a,0xef1d } }, { { 0x3f83,0xb2ab,0x6ef1,0x33b7 } }, { { 0x3fac,0x6b08,0xd704,0x1a92 } }, { { 0x3fce,0xbfbd,0xff82,0xc56d } }, { { 0x3fe6,0x2e42,0xfefa,0x39ef } } }; #define douba(k) (A[(k)].d) #define doubb(k) (B[(k)].d) #define MEXP 16383.0 #ifdef DENORMAL #define MNEXP -17183.0 #else #define MNEXP -16383.0 #endif #endif /* log2(e) - 1 */ #define LOG2EA 0.44269504088896340736 #define F W #define Fa Wa #define Fb Wb #define G W #define Ga Wa #define Gb u #define H W #define Ha Wb #define Hb Wb #ifdef __MINGW32__ static __inline__ double reduc( double ); extern double __powi ( double, int ); extern double pow ( double x, double y); #else /* __MINGW32__ */ extern double floor ( double ); extern double fabs ( double ); extern double frexp ( double, int * ); extern double ldexp ( double, int ); extern double polevl ( double, void *, int ); extern double p1evl ( double, uD *, int ); extern double __powi ( double, int ); extern int signbit ( double ); extern int isnan ( double ); extern int isfinite ( double ); static double reduc ( double ); extern double MAXNUM; #ifdef INFINITIES extern double INFINITY; #endif #ifdef NANS extern double NAN; #endif #ifdef MINUSZERO extern double NEGZERO; #endif #endif /* __MINGW32__ */ double pow(double x, double y) { double w, z, W, Wa, Wb, ya, yb, u; /* double F, Fa, Fb, G, Ga, Gb, H, Ha, Hb */ double aw, ay, wy; int e, i, nflg, iyflg, yoddint; if (y == 0.0) return (1.0); #ifdef NANS if (isnan(x) || isnan(y)) { _SET_ERRNO (EDOM); return (x + y); } #endif if (y == 1.0) return (x); #ifdef INFINITIES if (!isfinite(y) && (x == 1.0 || x == -1.0)) { mtherr( "pow", DOMAIN ); #ifdef NANS return( NAN ); #else return( INFINITY ); #endif } #endif if (x == 1.0) return (1.0); if (y >= MAXNUM) { _SET_ERRNO (ERANGE); #ifdef INFINITIES if (x > 1.0) return (INFINITY); #else if (x > 1.0) return (MAXNUM); #endif if (x > 0.0 && x < 1.0) return (0.0); if (x < -1.0) { #ifdef INFINITIES return (INFINITY); #else return (MAXNUM); #endif } if (x > -1.0 && x < 0.0) return (0.0); } if (y <= -MAXNUM) { _SET_ERRNO (ERANGE); if (x > 1.0) return (0.0); #ifdef INFINITIES if (x > 0.0 && x < 1.0) return (INFINITY); #else if (x > 0.0 && x < 1.0) return (MAXNUM); #endif if (x < -1.0) return (0.0); #ifdef INFINITIES if (x > -1.0 && x < 0.0) return (INFINITY); #else if (x > -1.0 && x < 0.0) return (MAXNUM); #endif } if (x >= MAXNUM) { #if INFINITIES if (y > 0.0) return (INFINITY); #else if (y > 0.0) return (MAXNUM); #endif return (0.0); } /* Set iyflg to 1 if y is an integer. */ iyflg = 0; w = floor(y); if (w == y) iyflg = 1; /* Test for odd integer y. */ yoddint = 0; if (iyflg) { ya = fabs(y); ya = floor(0.5 * ya); yb = 0.5 * fabs(w); if (ya != yb) yoddint = 1; } if (x <= -MAXNUM) { if (y > 0.0) { #ifdef INFINITIES if (yoddint) return (-INFINITY); return (INFINITY); #else if (yoddint) return (-MAXNUM); return (MAXNUM); #endif } if (y < 0.0) { #ifdef MINUSZERO if (yoddint) return (NEGZERO); #endif return (0.0); } } nflg = 0; /* flag = 1 if x<0 raised to integer power */ if (x <= 0.0) { if (x == 0.0) { if (y < 0.0) { #ifdef MINUSZERO if (signbit(x) && yoddint) return (-INFINITY); #endif #ifdef INFINITIES return (INFINITY); #else return (MAXNUM); #endif } if (y > 0.0) { #ifdef MINUSZERO if (signbit(x) && yoddint) return (NEGZERO); #endif return (0.0); } return (1.0); } else { if (iyflg == 0) { /* noninteger power of negative number */ mtherr(fname, DOMAIN); _SET_ERRNO (EDOM); #ifdef NANS return (NAN); #else return (0.0L); #endif } nflg = 1; } } /* Integer power of an integer. */ if (iyflg) { i = w; w = floor(x); if( (w == x) && (fabs(y) < 32768.0) ) { w = __powi(x, (int) y); return (w); } } if (nflg) x = fabs(x); /* For results close to 1, use a series expansion. */ w = x - 1.0; aw = fabs(w); ay = fabs(y); wy = w * y; ya = fabs(wy); if ((aw <= 1.0e-3 && ay <= 1.0) || (ya <= 1.0e-3 && ay >= 1.0)) { z = (((((w*(y-5.)/720. + 1./120.)*w*(y-4.) + 1./24.)*w*(y-3.) + 1./6.)*w*(y-2.) + 0.5)*w*(y-1.) )*wy + wy + 1.; goto done; } /* These are probably too much trouble. */ #if 0 w = y * log(x); if (aw > 1.0e-3 && fabs(w) < 1.0e-3) { z = ((((((w/7. + 1.)*w/6. + 1.)*w/5. + 1.)*w/4. + 1.)*w/3. + 1.)*w/2. + 1.)*w + 1.; goto done; } if (ya <= 1.0e-3 && aw <= 1.0e-4) { z = (((((wy*1./720. + (-w*1./48. + 1./120.) )*wy + ((w*17./144. - 1./12.)*w + 1./24.) )*wy + (((-w*5./16. + 7./24.)*w - 1./4.)*w + 1./6.) )*wy + ((((w*137./360. - 5./12.)*w + 11./24.)*w - 1./2.)*w + 1./2.) )*wy + (((((-w*1./6. + 1./5.)*w - 1./4)*w + 1./3.)*w -1./2.)*w ) )*wy + wy + 1.0; goto done; } #endif /* separate significand from exponent */ x = frexp(x, &e); #if 0 /* For debugging, check for gross overflow. */ if ((e * y) > (MEXP + 1024)) goto overflow; #endif /* Find significand of x in antilog table A[]. */ i = 1; if (x <= douba(9)) i = 9; if (x <= douba(i + 4)) i += 4; if (x <= douba(i + 2)) i += 2; if (x >= douba(1)) i = -1; i += 1; /* Find (x - A[i])/A[i] * in order to compute log(x/A[i]): * * log(x) = log( a x/a ) = log(a) + log(x/a) * * log(x/a) = log(1+v), v = x/a - 1 = (x-a)/a */ x -= douba(i); x -= doubb(i/2); x /= douba(i); /* rational approximation for log(1+v): * * log(1+v) = v - v**2/2 + v**3 P(v) / Q(v) */ z = x*x; w = x * ( z * polevl( x, P, 3 ) / p1evl( x, Q, 4 ) ); w = w - ldexp( z, -1 ); /* w - 0.5 * z */ /* Convert to base 2 logarithm: * multiply by log2(e) */ w = w + LOG2EA * w; /* Note x was not yet added in * to above rational approximation, * so do it now, while multiplying * by log2(e). */ z = w + LOG2EA * x; z = z + x; /* Compute exponent term of the base 2 logarithm. */ w = -i; w = ldexp( w, -4 ); /* divide by 16 */ w += e; /* Now base 2 log of x is w + z. */ /* Multiply base 2 log by y, in extended precision. */ /* separate y into large part ya * and small part yb less than 1/16 */ ya = reduc(y); yb = y - ya; F = z * y + w * yb; Fa = reduc(F); Fb = F - Fa; G = Fa + w * ya; Ga = reduc(G); Gb = G - Ga; H = Fb + Gb; Ha = reduc(H); w = ldexp(Ga + Ha, 4); /* Test the power of 2 for overflow */ if (w > MEXP) { #ifndef INFINITIES mtherr(fname, OVERFLOW); #endif #ifdef INFINITIES if (nflg && yoddint) return (-INFINITY); return (INFINITY); #else if (nflg && yoddint) return (-MAXNUM); return (MAXNUM); #endif } if (w < (MNEXP - 1)) { #ifndef DENORMAL mtherr(fname, UNDERFLOW); #endif #ifdef MINUSZERO if (nflg && yoddint) return (NEGZERO); #endif return (0.0); } e = w; Hb = H - Ha; if (Hb > 0.0) { e += 1; Hb -= 0.0625; } /* Now the product y * log2(x) = Hb + e/16.0. * * Compute base 2 exponential of Hb, * where -0.0625 <= Hb <= 0. */ z = Hb * polevl(Hb, R, 6); /* z = 2**Hb - 1 */ /* Express e/16 as an integer plus a negative number of 16ths. * Find lookup table entry for the fractional power of 2. */ if (e < 0) i = 0; else i = 1; i = e/16 + i; e = 16*i - e; w = douba(e); z = w + w * z; /* 2**-e * ( 1 + (2**Hb-1) ) */ z = ldexp(z, i); /* multiply by integer power of 2 */ done: /* Negate if odd integer power of negative number */ if (nflg && yoddint) { #ifdef MINUSZERO if (z == 0.0) z = NEGZERO; else #endif z = -z; } return (z); } /* Find a multiple of 1/16 that is within 1/16 of x. */ static double reduc(double x) { double t; t = ldexp(x, 4); t = floor(t); t = ldexp(t, -4); return (t); }