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8139688: Port fdlibm exp to Java
Reviewed-by: bpb, nadezhin
This commit is contained in:
parent
2e27b2e68a
commit
be91309965
6 changed files with 441 additions and 15 deletions
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@ -1,5 +1,5 @@
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/*
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* Copyright (c) 1998, 2015, Oracle and/or its affiliates. All rights reserved.
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* Copyright (c) 1998, 2016, Oracle and/or its affiliates. All rights reserved.
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* DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER.
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*
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* This code is free software; you can redistribute it and/or modify it
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@ -79,7 +79,8 @@ class FdLibm {
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*/
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private static double __LO(double x, int low) {
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long transX = Double.doubleToRawLongBits(x);
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return Double.longBitsToDouble((transX & 0xFFFF_FFFF_0000_0000L)|low );
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return Double.longBitsToDouble((transX & 0xFFFF_FFFF_0000_0000L) |
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(low & 0x0000_0000_FFFF_FFFFL));
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}
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/**
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@ -96,7 +97,8 @@ class FdLibm {
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*/
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private static double __HI(double x, int high) {
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long transX = Double.doubleToRawLongBits(x);
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return Double.longBitsToDouble((transX & 0x0000_0000_FFFF_FFFFL)|( ((long)high)) << 32 );
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return Double.longBitsToDouble((transX & 0x0000_0000_FFFF_FFFFL) |
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( ((long)high)) << 32 );
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}
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/**
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@ -580,4 +582,152 @@ class FdLibm {
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return s * z;
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}
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}
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/**
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* Returns the exponential of x.
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*
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* Method
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* 1. Argument reduction:
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* Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
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* Given x, find r and integer k such that
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*
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* x = k*ln2 + r, |r| <= 0.5*ln2.
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*
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* Here r will be represented as r = hi-lo for better
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* accuracy.
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*
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* 2. Approximation of exp(r) by a special rational function on
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* the interval [0,0.34658]:
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* Write
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* R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
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* We use a special Reme algorithm on [0,0.34658] to generate
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* a polynomial of degree 5 to approximate R. The maximum error
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* of this polynomial approximation is bounded by 2**-59. In
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* other words,
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* R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
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* (where z=r*r, and the values of P1 to P5 are listed below)
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* and
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* | 5 | -59
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* | 2.0+P1*z+...+P5*z - R(z) | <= 2
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* | |
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* The computation of exp(r) thus becomes
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* 2*r
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* exp(r) = 1 + -------
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* R - r
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* r*R1(r)
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* = 1 + r + ----------- (for better accuracy)
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* 2 - R1(r)
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* where
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* 2 4 10
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* R1(r) = r - (P1*r + P2*r + ... + P5*r ).
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*
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* 3. Scale back to obtain exp(x):
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* From step 1, we have
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* exp(x) = 2^k * exp(r)
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*
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* Special cases:
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* exp(INF) is INF, exp(NaN) is NaN;
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* exp(-INF) is 0, and
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* for finite argument, only exp(0)=1 is exact.
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*
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* Accuracy:
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* according to an error analysis, the error is always less than
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* 1 ulp (unit in the last place).
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*
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* Misc. info.
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* For IEEE double
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* if x > 7.09782712893383973096e+02 then exp(x) overflow
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* if x < -7.45133219101941108420e+02 then exp(x) underflow
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*
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* Constants:
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* The hexadecimal values are the intended ones for the following
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* constants. The decimal values may be used, provided that the
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* compiler will convert from decimal to binary accurately enough
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* to produce the hexadecimal values shown.
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*/
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static class Exp {
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private static final double one = 1.0;
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private static final double[] half = {0.5, -0.5,};
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private static final double huge = 1.0e+300;
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private static final double twom1000= 0x1.0p-1000; // 9.33263618503218878990e-302 = 2^-1000
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private static final double o_threshold= 0x1.62e42fefa39efp9; // 7.09782712893383973096e+02
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private static final double u_threshold= -0x1.74910d52d3051p9; // -7.45133219101941108420e+02;
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private static final double[] ln2HI ={ 0x1.62e42feep-1, // 6.93147180369123816490e-01
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-0x1.62e42feep-1}; // -6.93147180369123816490e-01
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private static final double[] ln2LO ={ 0x1.a39ef35793c76p-33, // 1.90821492927058770002e-10
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-0x1.a39ef35793c76p-33}; // -1.90821492927058770002e-10
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private static final double invln2 = 0x1.71547652b82fep0; // 1.44269504088896338700e+00
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private static final double P1 = 0x1.555555555553ep-3; // 1.66666666666666019037e-01
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private static final double P2 = -0x1.6c16c16bebd93p-9; // -2.77777777770155933842e-03
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private static final double P3 = 0x1.1566aaf25de2cp-14; // 6.61375632143793436117e-05
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private static final double P4 = -0x1.bbd41c5d26bf1p-20; // -1.65339022054652515390e-06
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private static final double P5 = 0x1.6376972bea4d0p-25; // 4.13813679705723846039e-08
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// should be able to forgo strictfp due to controlled over/underflow
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public static strictfp double compute(double x) {
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double y;
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double hi = 0.0;
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double lo = 0.0;
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double c;
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double t;
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int k = 0;
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int xsb;
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/*unsigned*/ int hx;
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hx = __HI(x); /* high word of x */
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xsb = (hx >> 31) & 1; /* sign bit of x */
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hx &= 0x7fffffff; /* high word of |x| */
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/* filter out non-finite argument */
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if (hx >= 0x40862E42) { /* if |x| >= 709.78... */
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if (hx >= 0x7ff00000) {
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if (((hx & 0xfffff) | __LO(x)) != 0)
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return x + x; /* NaN */
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else
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return (xsb == 0) ? x : 0.0; /* exp(+-inf) = {inf, 0} */
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}
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if (x > o_threshold)
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return huge * huge; /* overflow */
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if (x < u_threshold) // unsigned compare needed here?
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return twom1000 * twom1000; /* underflow */
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}
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/* argument reduction */
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if (hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
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if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
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hi = x - ln2HI[xsb];
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lo=ln2LO[xsb];
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k = 1 - xsb - xsb;
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} else {
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k = (int)(invln2 * x + half[xsb]);
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t = k;
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hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */
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lo = t*ln2LO[0];
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}
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x = hi - lo;
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} else if (hx < 0x3e300000) { /* when |x|<2**-28 */
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if (huge + x > one)
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return one + x; /* trigger inexact */
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} else {
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k = 0;
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}
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/* x is now in primary range */
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t = x * x;
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c = x - t*(P1 + t*(P2 + t*(P3 + t*(P4 + t*P5))));
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if (k == 0)
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return one - ((x*c)/(c - 2.0) - x);
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else
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y = one - ((lo - (x*c)/(2.0 - c)) - hi);
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if(k >= -1021) {
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y = __HI(y, __HI(y) + (k << 20)); /* add k to y's exponent */
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return y;
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} else {
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y = __HI(y, __HI(y) + ((k + 1000) << 20)); /* add k to y's exponent */
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return y * twom1000;
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}
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}
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}
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}
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