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src/java.base/share/classes/java/lang/FdLibm.java

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*** 1,8 **** - /* ! * Copyright (c) 1998, 2001, Oracle and/or its affiliates. All rights reserved. * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER. * * This code is free software; you can redistribute it and/or modify it * under the terms of the GNU General Public License version 2 only, as * published by the Free Software Foundation. Oracle designates this --- 1,7 ---- /* ! * Copyright (c) 1998, 2015, Oracle and/or its affiliates. All rights reserved. * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER. * * This code is free software; you can redistribute it and/or modify it * under the terms of the GNU General Public License version 2 only, as * published by the Free Software Foundation. Oracle designates this
*** 22,128 **** * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA * or visit www.oracle.com if you need additional information or have any * questions. */ ! /* __ieee754_hypot(x,y) * * Method : ! * If (assume round-to-nearest) z=x*x+y*y * has error less than sqrt(2)/2 ulp, than * sqrt(z) has error less than 1 ulp (exercise). * ! * So, compute sqrt(x*x+y*y) with some care as * follows to get the error below 1 ulp: * ! * Assume x>y>0; * (if possible, set rounding to round-to-nearest) * 1. if x > 2y use ! * x1*x1+(y*y+(x2*(x+x1))) for x*x+y*y ! * where x1 = x with lower 32 bits cleared, x2 = x-x1; else * 2. if x <= 2y use ! * t1*y1+((x-y)*(x-y)+(t1*y2+t2*y)) ! * where t1 = 2x with lower 32 bits cleared, t2 = 2x-t1, ! * y1= y with lower 32 bits chopped, y2 = y-y1. * * NOTE: scaling may be necessary if some argument is too * large or too tiny * * Special cases: * hypot(x,y) is INF if x or y is +INF or -INF; else * hypot(x,y) is NAN if x or y is NAN. * * Accuracy: ! * hypot(x,y) returns sqrt(x^2+y^2) with error less * than 1 ulps (units in the last place) */ ! #include "fdlibm.h" ! #ifdef __STDC__ ! double __ieee754_hypot(double x, double y) ! #else ! double __ieee754_hypot(x,y) ! double x, y; ! #endif ! { ! double a=x,b=y,t1,t2,y1,y2,w; ! int j,k,ha,hb; ! ! ha = __HI(x)&0x7fffffff; /* high word of x */ ! hb = __HI(y)&0x7fffffff; /* high word of y */ ! if(hb > ha) {a=y;b=x;j=ha; ha=hb;hb=j;} else {a=x;b=y;} ! __HI(a) = ha; /* a <- |a| */ ! __HI(b) = hb; /* b <- |b| */ ! if((ha-hb)>0x3c00000) {return a+b;} /* x/y > 2**60 */ ! k=0; ! if(ha > 0x5f300000) { /* a>2**500 */ ! if(ha >= 0x7ff00000) { /* Inf or NaN */ ! w = a+b; /* for sNaN */ ! if(((ha&0xfffff)|__LO(a))==0) w = a; ! if(((hb^0x7ff00000)|__LO(b))==0) w = b; ! return w; } ! /* scale a and b by 2**-600 */ ! ha -= 0x25800000; hb -= 0x25800000; k += 600; ! __HI(a) = ha; ! __HI(b) = hb; ! } ! if(hb < 0x20b00000) { /* b < 2**-500 */ ! if(hb <= 0x000fffff) { /* subnormal b or 0 */ ! if((hb|(__LO(b)))==0) return a; ! t1=0; ! __HI(t1) = 0x7fd00000; /* t1=2^1022 */ b *= t1; a *= t1; k -= 1022; ! } else { /* scale a and b by 2^600 */ ! ha += 0x25800000; /* a *= 2^600 */ ! hb += 0x25800000; /* b *= 2^600 */ k -= 600; - __HI(a) = ha; - __HI(b) = hb; } } ! /* medium size a and b */ ! w = a-b; ! if (w>b) { t1 = 0; ! __HI(t1) = ha; ! t2 = a-t1; ! w = sqrt(t1*t1-(b*(-b)-t2*(a+t1))); } else { ! a = a+a; y1 = 0; ! __HI(y1) = hb; y2 = b - y1; t1 = 0; ! __HI(t1) = ha+0x00100000; t2 = a - t1; ! w = sqrt(t1*y1-(w*(-w)-(t1*y2+t2*b))); } - if(k!=0) { - t1 = 1.0; - __HI(t1) += (k<<20); - return t1*w; - } else return w; } --- 21,495 ---- * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA * or visit www.oracle.com if you need additional information or have any * questions. */ ! package java.lang; ! ! /** ! * Port of the "Freely Distributable Math Library", version 5.3, from C to Java. ! * ! * <p>The C version of fdlibm relied on the idiom of pointer aliasing ! * a 64-bit double floating-point value as a two-element array of ! * 32-bit integers and reading and writing the two halves of the ! * double independently. This coding pattern was problematic to C ! * optimizers and not directly expressible in Java. Therefore, rather ! * than a memory level overlay, if portions of a double need to be ! * operated on as integer values, the standard library methods for ! * bitwise floating-point to integer conversion, ! * Double.longBitsToDouble and Double.doubleToRawLongBits, are directly ! * or indirectly used . ! * ! * <p>The C version of fdlibm also took some pains to signal the ! * correct IEEE 754 exceptional conditions divide by zero, invalid, ! * overflow and underflow. For example, overflow would be signaled by ! * {@code huge * huge} where {@code huge} was a large constant that ! * would overflow when squared. Since IEEE floating-point exceptional ! * handling is not supported natively in the JVM, such coding patterns ! * have been omitted from this port. For example, rather than {@code ! * return huge * huge}, this port will use {@code return INFINITY}. ! */ ! class FdLibm { ! // Constants used by multiple algorithms ! private static final double INFINITY = Double.POSITIVE_INFINITY; ! ! private FdLibm() { ! throw new UnsupportedOperationException("No instances for you."); ! } ! ! /** ! * Return the low-order 32 bits of the double argument as an int. ! */ ! private static int __LO(double x) { ! long transducer = Double.doubleToRawLongBits(x); ! return (int)transducer; ! } ! ! /** ! * Return a double with its low-order bits of the second argument ! * and the high-order bits of the first argument.. ! */ ! private static double __LO(double x, int low) { ! long transX = Double.doubleToRawLongBits(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.doubleToRawLongBits(x); ! return (int)(transducer >> 32); ! } ! ! /** ! * Return a double with its high-order bits of the second argument ! * and the low-order bits of the first argument.. ! */ ! private static double __HI(double x, int high) { ! long transX = Double.doubleToRawLongBits(x); ! return Double.longBitsToDouble((transX & 0x0000_0000_FFFF_FFFFL)|( ((long)high)) << 32 ); ! } ! ! /** ! * hypot(x,y) * * Method : ! * If (assume round-to-nearest) z = x*x + y*y * has error less than sqrt(2)/2 ulp, than * sqrt(z) has error less than 1 ulp (exercise). * ! * So, compute sqrt(x*x + y*y) with some care as * follows to get the error below 1 ulp: * ! * Assume x > y > 0; * (if possible, set rounding to round-to-nearest) * 1. if x > 2y use ! * x1*x1 + (y*y + (x2*(x + x1))) for x*x + y*y ! * where x1 = x with lower 32 bits cleared, x2 = x - x1; else * 2. if x <= 2y use ! * t1*y1 + ((x-y) * (x-y) + (t1*y2 + t2*y)) ! * where t1 = 2x with lower 32 bits cleared, t2 = 2x - t1, ! * y1= y with lower 32 bits chopped, y2 = y - y1. * * NOTE: scaling may be necessary if some argument is too * large or too tiny * * Special cases: * hypot(x,y) is INF if x or y is +INF or -INF; else * hypot(x,y) is NAN if x or y is NAN. * * Accuracy: ! * hypot(x,y) returns sqrt(x^2 + y^2) with error less * than 1 ulps (units in the last place) */ + public static class Hypot { + public static final double TWO_MINUS_600 = 0x1.0p-600; + public static final double TWO_PLUS_600 = 0x1.0p+600; ! public static double compute(double x, double y) { ! double a = Math.abs(x); ! double b = Math.abs(y); ! if (!Double.isFinite(a) || !Double.isFinite(b)) { ! if (a == INFINITY || b == INFINITY) ! return INFINITY; ! else ! return a + b; // Propagate NaN significand bits ! } ! ! if (b > a) { ! double tmp = a; ! a = b; ! b = tmp; ! } ! assert a >= b; ! ! int ha = __HI(a); // high word of a ! int hb = __HI(b); // high word of b ! ! if ((ha - hb) > 0x3c00000) { ! return a + b; // x / y > 2**60 ! } ! ! int k = 0; ! if (a > 0x1.0p500) { // a > 2**500 ! // scale a and b by 2**-600 ! ha -= 0x25800000; ! hb -= 0x25800000; ! a = a * TWO_MINUS_600; ! b = b * TWO_MINUS_600; ! k += 600; } ! double t1, t2; ! if (b < 0x1.0p-500) { // b < 2**-500 ! if (hb <= Double.MIN_NORMAL) { // subnormal b or 0 */ ! if (b == 0.0) ! return a; ! t1 = 0x1.0p1022; // t1 = 2^1022 b *= t1; a *= t1; k -= 1022; ! } else { // scale a and b by 2^600 ! ha += 0x25800000; // a *= 2^600 ! hb += 0x25800000; // b *= 2^600 ! a = a * TWO_PLUS_600; ! b = b * TWO_PLUS_600; k -= 600; } } ! // medium size a and b ! double w = a - b; ! if (w > b) { t1 = 0; ! t1 = __HI(t1, ha); ! t2 = a - t1; ! w = Math.sqrt(t1*t1 - (b*(-b) - t2 * (a + t1))); } else { ! double y1, y2; ! a = a + a; y1 = 0; ! y1 = __HI(y1, hb); y2 = b - y1; t1 = 0; ! t1 = __HI(t1, ha + 0x00100000); t2 = a - t1; ! w = Math.sqrt(t1*y1 - (w*(-w) - (t1*y2 + t2*b))); ! } ! if (k != 0) { ! return Math.powerOfTwoD(k) * w; ! } else ! return w; ! } ! } ! ! /** ! * Compute 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. ! */ ! public static class Pow { ! public static strictfp double compute(final double x, final double y) { ! double z; ! double r, s, t, u, v, w; ! int i, j, k, n; ! ! // y == zero: x**0 = 1 ! if (y == 0.0) ! return 1.0; ! ! // +/-NaN return x + y to propagate NaN significands ! if (Double.isNaN(x) || Double.isNaN(y)) ! return x + y; ! ! final double y_abs = Math.abs(y); ! double x_abs = Math.abs(x); ! // Special values of y ! if (y == 2.0) { ! return x * x; ! } else if (y == 0.5) { ! if (x >= -Double.MAX_VALUE) // Handle x == -infinity later ! return Math.sqrt(x + 0.0); // Add 0.0 to properly handle x == -0.0 ! } else if (y_abs == 1.0) { // y is +/-1 ! return (y == 1.0) ? x : 1.0 / x; ! } else if (y_abs == INFINITY) { // y is +/-infinity ! if (x_abs == 1.0) ! return y - y; // inf**+/-1 is NaN ! else if (x_abs > 1.0) // (|x| > 1)**+/-inf = inf, 0 ! return (y >= 0) ? y : 0.0; ! else // (|x| < 1)**-/+inf = inf, 0 ! return (y < 0) ? -y : 0.0; ! } ! ! final int hx = __HI(x); ! int ix = hx & 0x7fffffff; ! ! /* ! * When x < 0, determine if y is an odd integer: ! * y_is_int = 0 ... y is not an integer ! * y_is_int = 1 ... y is an odd int ! * y_is_int = 2 ... y is an even int ! */ ! int y_is_int = 0; ! if (hx < 0) { ! if (y_abs >= 0x1.0p53) // |y| >= 2^53 = 9.007199254740992E15 ! y_is_int = 2; // y is an even integer since ulp(2^53) = 2.0 ! else if (y_abs >= 1.0) { // |y| >= 1.0 ! long y_abs_as_long = (long) y_abs; ! if ( ((double) y_abs_as_long) == y_abs) { ! y_is_int = 2 - (int)(y_abs_as_long & 0x1L); ! } ! } ! } ! ! // Special value of x ! if (x_abs == 0.0 || ! x_abs == INFINITY || ! x_abs == 1.0) { ! z = x_abs; // x is +/-0, +/-inf, +/-1 ! if (y < 0.0) ! z = 1.0/z; // z = (1/|x|) ! if (hx < 0) { ! if (((ix - 0x3ff00000) | y_is_int) == 0) { ! z = (z-z)/(z-z); // (-1)**non-int is NaN ! } else if (y_is_int == 1) ! z = -1.0 * z; // (x < 0)**odd = -(|x|**odd) ! } ! return z; ! } ! ! n = (hx >> 31) + 1; ! ! // (x < 0)**(non-int) is NaN ! if ((n | y_is_int) == 0) ! return (x-x)/(x-x); ! ! s = 1.0; // s (sign of result -ve**odd) = -1 else = 1 ! if ( (n | (y_is_int - 1)) == 0) ! s = -1.0; // (-ve)**(odd int) ! ! double p_h, p_l, t1, t2; ! // |y| is huge ! if (y_abs > 0x1.0p31) { // if |y| > 2**31 ! final double INV_LN2 = 0x1.7154_7652_b82fep0; // 1.44269504088896338700e+00 = 1/ln2 ! final double INV_LN2_H = 0x1.715476p0; // 1.44269502162933349609e+00 = 24 bits of 1/ln2 ! final double INV_LN2_L = 0x1.4ae0_bf85_ddf44p-26; // 1.92596299112661746887e-08 = 1/ln2 tail ! ! // Over/underflow if x is not close to one ! if (x_abs < 0x1.fffffp-1) // |x| < 0.9999995231628418 ! return (y < 0.0) ? s * INFINITY : s * 0.0; ! if (x_abs > 1.0) // |x| > 1.0 ! return (y > 0.0) ? s * INFINITY : s * 0.0; ! /* ! * now |1-x| is tiny <= 2**-20, sufficient to compute ! * log(x) by x - x^2/2 + x^3/3 - x^4/4 ! */ ! t = x_abs - 1.0; // t has 20 trailing zeros ! w = (t * t) * (0.5 - t * (0.3333333333333333333333 - t * 0.25)); ! u = INV_LN2_H * t; // INV_LN2_H has 21 sig. bits ! v = t * INV_LN2_L - w * INV_LN2; ! t1 = u + v; ! t1 =__LO(t1, 0); ! t2 = v - (t1 - u); ! } else { ! final double CP = 0x1.ec70_9dc3_a03fdp-1; // 9.61796693925975554329e-01 = 2/(3ln2) ! final double CP_H = 0x1.ec709ep-1; // 9.61796700954437255859e-01 = (float)cp ! final double CP_L = -0x1.e2fe_0145_b01f5p-28; // -7.02846165095275826516e-09 = tail of CP_H ! ! double z_h, z_l, ss, s2, s_h, s_l, t_h, t_l; ! n = 0; ! // Take care of subnormal numbers ! if (ix < 0x00100000) { ! x_abs *= 0x1.0p53; // 2^53 = 9007199254740992.0 ! n -= 53; ! ix = __HI(x_abs); ! } ! n += ((ix) >> 20) - 0x3ff; ! j = ix & 0x000fffff; ! // Determine interval ! ix = j | 0x3ff00000; // Normalize ix ! if (j <= 0x3988E) ! k = 0; // |x| <sqrt(3/2) ! else if (j < 0xBB67A) ! k = 1; // |x| <sqrt(3) ! else { ! k = 0; ! n += 1; ! ix -= 0x00100000; ! } ! x_abs = __HI(x_abs, ix); ! ! // Compute ss = s_h + s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) ! ! final double BP[] = {1.0, ! 1.5}; ! final double DP_H[] = {0.0, ! 0x1.2b80_34p-1}; // 5.84962487220764160156e-01 ! final double DP_L[] = {0.0, ! 0x1.cfde_b43c_fd006p-27};// 1.35003920212974897128e-08 ! ! // Poly coefs for (3/2)*(log(x)-2s-2/3*s**3 ! final double L1 = 0x1.3333_3333_33303p-1; // 5.99999999999994648725e-01 ! final double L2 = 0x1.b6db_6db6_fabffp-2; // 4.28571428578550184252e-01 ! final double L3 = 0x1.5555_5518_f264dp-2; // 3.33333329818377432918e-01 ! final double L4 = 0x1.1746_0a91_d4101p-2; // 2.72728123808534006489e-01 ! final double L5 = 0x1.d864_a93c_9db65p-3; // 2.30660745775561754067e-01 ! final double L6 = 0x1.a7e2_84a4_54eefp-3; // 2.06975017800338417784e-01 ! u = x_abs - BP[k]; // BP[0]=1.0, BP[1]=1.5 ! v = 1.0 / (x_abs + BP[k]); ! ss = u * v; ! s_h = ss; ! s_h = __LO(s_h, 0); ! // t_h=x_abs + BP[k] High ! t_h = 0.0; ! t_h = __HI(t_h, ((ix >> 1) | 0x20000000) + 0x00080000 + (k << 18) ); ! t_l = x_abs - (t_h - BP[k]); ! s_l = v * ((u - s_h * t_h) - s_h * t_l); ! // Compute log(x_abs) ! 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(x_abs) = (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) ! double 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 * INFINITY; // Overflow ! else { ! final double OVT = 8.0085662595372944372e-0017; // -(1024-log2(ovfl+.5ulp)) ! if (p_l + OVT > z - p_h) ! return s * INFINITY; // Overflow ! } ! } else if ((j & 0x7fffffff) >= 0x4090cc00 ) { // z <= -1075 ! if (((j - 0xc090cc00) | i)!=0) // z < -1075 ! return s * 0.0; // Underflow ! else { ! if (p_l <= z - p_h) ! return s * 0.0; // Underflow ! } ! } ! /* ! * Compute 2**(p_h+p_l) ! */ ! // Poly coefs for (3/2)*(log(x)-2s-2/3*s**3 ! final double P1 = 0x1.5555_5555_5553ep-3; // 1.66666666666666019037e-01 ! final double P2 = -0x1.6c16_c16b_ebd93p-9; // -2.77777777770155933842e-03 ! final double P3 = 0x1.1566_aaf2_5de2cp-14; // 6.61375632143793436117e-05 ! final double P4 = -0x1.bbd4_1c5d_26bf1p-20; // -1.65339022054652515390e-06 ! final double P5 = 0x1.6376_972b_ea4d0p-25; // 4.13813679705723846039e-08 ! final double LG2 = 0x1.62e4_2fef_a39efp-1; // 6.93147180559945286227e-01 ! final double LG2_H = 0x1.62e43p-1; // 6.93147182464599609375e-01 ! final double LG2_L = -0x1.05c6_10ca_86c39p-29; // -1.90465429995776804525e-09 ! 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; ! 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 - 2.0) - (w + z * w); ! z = 1.0 - (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; ! } } }
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