package java.lang; /** * Port of the "Freely Distributable Math Library", version 4.3, from C to Java. */ class FdLibm { /** * Return the low-order 32 bits of the double argument as an int. */ private static int __LO(double x) { long transducer = Double.doubleToLongBits(x); return (int)transducer; } /** * Return the a double with its low-order bits reset. */ private static double __LO(double x, int low) { long transX = Double.doubleToLongBits(x); return Double.longBitsToDouble((transX & 0xFFFF_FFFF_0000_0000L)|low ); } /** * Return the high-order 32 bits of the double argument as an int. */ private static int __HI(double x) { long transducer = Double.doubleToLongBits(x); return (int)(transducer >> 32); } /** * Return the a double with its high-order bits reset. */ private static double __HI(double x, int high) { long transX = Double.doubleToLongBits(x); return Double.longBitsToDouble((transX & 0x0000_0000_FFFF_FFFFL)|( ((long)high)) << 32 ); } /** * 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) ** 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 * * 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. */ public static class Pow { static final double bp[] = {1.0, 1.5,}; static final double dp_h[] = { 0.0, 0x1.2b8034p-1,}; // 5.84962487220764160156e-01 static final double dp_l[] = { 0.0, 0x1.cfdeb43cfd006p-27,}; // 1.35003920212974897128e-08 static final double zero = 0.0; static final double one = 1.0; static final double two = 2.0; static final double two53 = 0x1.0p53; // 9007199254740992.0 static final double huge = 1.0e300; static final double tiny = 1.0e-300; /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */ static final double L1 = 0x1.3333333333303p-1; // 5.99999999999994648725e-01 static final double L2 = 0x1.b6db6db6fabffp-2; // 4.28571428578550184252e-01 static final double L3 = 0x1.55555518f264dp-2; // 3.33333329818377432918e-01 static final double L4 = 0x1.17460a91d4101p-2; // 2.72728123808534006489e-01 static final double L5 = 0x1.d864a93c9db65p-3; // 2.30660745775561754067e-01 static final double L6 = 0x1.a7e284a454eefp-3; // 2.06975017800338417784e-01 static final double P1 = 0x1.555555555553ep-3; // 1.66666666666666019037e-01 static final double P2 = -0x1.6c16c16bebd93p-9; // -2.77777777770155933842e-03 static final double P3 = 0x1.1566aaf25de2cp-14; // 6.61375632143793436117e-05 static final double P4 = -0x1.bbd41c5d26bf1p-20; // -1.65339022054652515390e-06 static final double P5 = 0x1.6376972bea4d0p-25; // 4.13813679705723846039e-08 static final double lg2 = 0x1.62e42fefa39efp-1; // 6.93147180559945286227e-01 static final double lg2_h = 0x1.62e43p-1; // 6.93147182464599609375e-01 static final double lg2_l = -0x1.05c610ca86c39p-29; // -1.90465429995776804525e-09 static final double ovt = 8.0085662595372944372e-0017; // -(1024-log2(ovfl+.5ulp)) static final double cp = 0x1.ec709dc3a03fdp-1; // 9.61796693925975554329e-01 = 2/(3ln2) static final double cp_h = 0x1.ec709ep-1; // 9.61796700954437255859e-01 = (float)cp static final double cp_l = -0x1.e2fe0145b01f5p-28; // -7.02846165095275826516e-09 = tail of cp_h static final double ivln2 = 0x1.71547652b82fep0; // 1.44269504088896338700e+00 = 1/ln2 static final double ivln2_h = 0x1.715476p0; // 1.44269502162933349609e+00 = 24 bits of 1/ln2 static final double ivln2_l = 0x1.4ae0bf85ddf44p-26; // 1.92596299112661746887e-08 = 1/ln2 tail public static 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; int i, j, k, yisint, n; int hx, hy, ix, iy; /*unsigned*/ int lx, ly; hx = __HI(x); lx = __LO(x); hy = __HI(y); ly = __LO(y); ix = hx & 0x7fffffff; iy = hy & 0x7fffffff; /* y==zero: x**0 = 1 */ if (y == 0.0) return 1.0; /* +-NaN return x+y */ if (Double.isNaN(x) || Double.isNaN(y)) 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) return y - y; /* inf**+-1 is NaN */ else if (ix >= 0x3ff00000) /* (|x| > 1)**+-inf = inf,0 */ return (hy >= 0) ? y: zero; else /* (|x| < 1)**-,+inf = inf,0 */ return (hy < 0) ? -y: zero; } 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 Math.sqrt(x); } } ax = Math.abs(x); /* special value of x */ if (lx == 0) { if (ix == 0x7ff00000 || ix==0 || ix == 0x3ff00000){ 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 = -1.0*z; /* (x<0)**odd = -(|x|**odd) */ } return z; } } n = (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; t1 =__LO(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; ix = __HI(ax); } n += ((ix) >> 20) - 0x3ff; j = ix & 0x000fffff; /* determine interval */ ix = j | 0x3ff00000; /* normalize ix */ if(j <= 0x3988E) k=0; /* |x| > 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; t_h = __LO(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; p_h = __LO(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); t1 = __LO(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; y1 = __LO(y1, 0); p_l = (y-y1) * t1 + y *t2; p_h = y1 * t1; z = p_l + p_h; j = __HI(z); i = __LO(z); if (j >= 0x40900000) { /* z >= 1024 */ if (((j - 0x40900000) | i)!=0) /* if z > 1024 */ return s*huge*huge; /* overflow */ else { if (p_l+ovt>z-p_h) return s*huge*huge; /* overflow */ } } else if ((j & 0x7fffffff) >= 0x4090cc00 ) { /* z <= -1075 */ if (((j-0xc090cc00)|i)!=0) /* z < -1075 */ 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 = j + (0x00100000 >> (k+1)); k = ((n & 0x7fffffff) >> 20)-0x3ff; /* new k for n */ t = zero; t = __HI(t, (n & ~(0x000fffff >> k)) ); n = ((n & 0x000fffff)|0x00100000) >> (20-k); if (j < 0) n = -n; p_h -= t; } t = p_l+p_h; t = __LO(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); j = __HI(z); j += (n << 20); if ((j >> 20) <= 0) z = Math.scalb(z, n); /* subnormal output */ else { int z_hi = __HI(z); z_hi += (n << 20); z = __HI(z, z_hi); } return s * z; } } }