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8301392: Port fdlibm log1p to Java
Reviewed-by: bpb
This commit is contained in:
Joe Darcy
parent
f696785fd3
commit
ee0f5b5ed0
4 changed files with 409 additions and 15 deletions
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@ -60,6 +60,7 @@ package java.lang;
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class FdLibm {
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// Constants used by multiple algorithms
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private static final double INFINITY = Double.POSITIVE_INFINITY;
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private static final double TWO54 = 0x1.0p54; // 1.80143985094819840000e+16
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private FdLibm() {
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throw new UnsupportedOperationException("No FdLibm instances for you.");
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@ -779,11 +780,10 @@ class FdLibm {
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* shown.
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*/
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static class Log10 {
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private static double two54 = 0x1.0p54; // 1.80143985094819840000e+16;
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private static double ivln10 = 0x1.bcb7b1526e50ep-2; // 4.34294481903251816668e-01
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private static final double ivln10 = 0x1.bcb7b1526e50ep-2; // 4.34294481903251816668e-01
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private static double log10_2hi = 0x1.34413509f6p-2; // 3.01029995663611771306e-01;
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private static double log10_2lo = 0x1.9fef311f12b36p-42; // 3.69423907715893078616e-13;
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private static final double log10_2hi = 0x1.34413509f6p-2; // 3.01029995663611771306e-01;
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private static final double log10_2lo = 0x1.9fef311f12b36p-42; // 3.69423907715893078616e-13;
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private Log10() {
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throw new UnsupportedOperationException();
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@ -799,13 +799,13 @@ class FdLibm {
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k=0;
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if (hx < 0x0010_0000) { /* x < 2**-1022 */
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if (((hx & 0x7fff_ffff) | lx) == 0) {
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return -two54/0.0; /* log(+-0)=-inf */
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return -TWO54/0.0; /* log(+-0)=-inf */
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}
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if (hx < 0) {
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return (x - x)/0.0; /* log(-#) = NaN */
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}
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k -= 54;
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x *= two54; /* subnormal number, scale up x */
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x *= TWO54; /* subnormal number, scale up x */
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hx = __HI(x);
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}
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@ -822,4 +822,167 @@ class FdLibm {
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return z + y * log10_2hi;
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}
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}
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/**
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* Returns the natural logarithm of the sum of the argument and 1.
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*
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* Method :
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* 1. Argument Reduction: find k and f such that
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* 1+x = 2^k * (1+f),
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* where sqrt(2)/2 < 1+f < sqrt(2) .
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*
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* Note. If k=0, then f=x is exact. However, if k!=0, then f
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* may not be representable exactly. In that case, a correction
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* term is need. Let u=1+x rounded. Let c = (1+x)-u, then
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* log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
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* and add back the correction term c/u.
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* (Note: when x > 2**53, one can simply return log(x))
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*
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* 2. Approximation of log1p(f).
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* Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
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* = 2s + 2/3 s**3 + 2/5 s**5 + .....,
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* = 2s + s*R
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* We use a special Reme algorithm on [0,0.1716] to generate
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* a polynomial of degree 14 to approximate R The maximum error
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* of this polynomial approximation is bounded by 2**-58.45. In
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* other words,
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* 2 4 6 8 10 12 14
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* R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s +Lp6*s +Lp7*s
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* (the values of Lp1 to Lp7 are listed in the program)
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* and
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* | 2 14 | -58.45
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* | Lp1*s +...+Lp7*s - R(z) | <= 2
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* | |
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* Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
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* In order to guarantee error in log below 1ulp, we compute log
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* by
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* log1p(f) = f - (hfsq - s*(hfsq+R)).
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*
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* 3. Finally, log1p(x) = k*ln2 + log1p(f).
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* = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
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* Here ln2 is split into two floating point number:
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* ln2_hi + ln2_lo,
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* where n*ln2_hi is always exact for |n| < 2000.
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*
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* Special cases:
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* log1p(x) is NaN with signal if x < -1 (including -INF) ;
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* log1p(+INF) is +INF; log1p(-1) is -INF with signal;
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* log1p(NaN) is that NaN with no signal.
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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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* 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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* Note: Assuming log() return accurate answer, the following
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* algorithm can be used to compute log1p(x) to within a few ULP:
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*
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* u = 1+x;
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* if(u==1.0) return x ; else
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* return log(u)*(x/(u-1.0));
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*
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* See HP-15C Advanced Functions Handbook, p.193.
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*/
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static class Log1p {
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private static final double ln2_hi = 0x1.62e42feep-1; // 6.93147180369123816490e-01
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private static final double ln2_lo = 0x1.a39ef35793c76p-33; // 1.90821492927058770002e-10
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private static final double Lp1 = 0x1.5555555555593p-1; // 6.666666666666735130e-01
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private static final double Lp2 = 0x1.999999997fa04p-2; // 3.999999999940941908e-01
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private static final double Lp3 = 0x1.2492494229359p-2; // 2.857142874366239149e-01
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private static final double Lp4 = 0x1.c71c51d8e78afp-3; // 2.222219843214978396e-01
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private static final double Lp5 = 0x1.7466496cb03dep-3; // 1.818357216161805012e-01
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private static final double Lp6 = 0x1.39a09d078c69fp-3; // 1.531383769920937332e-01
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private static final double Lp7 = 0x1.2f112df3e5244p-3; // 1.479819860511658591e-01
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public static double compute(double x) {
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double hfsq, f=0, c=0, s, z, R, u;
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int k, hx, hu=0, ax;
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hx = __HI(x); /* high word of x */
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ax = hx & 0x7fff_ffff;
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k = 1;
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if (hx < 0x3FDA_827A) { /* x < 0.41422 */
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if (ax >= 0x3ff0_0000) { /* x <= -1.0 */
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if (x == -1.0) /* log1p(-1)=-inf */
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return -INFINITY;
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else
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return Double.NaN; /* log1p(x < -1) = NaN */
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}
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if (ax < 0x3e20_0000) { /* |x| < 2**-29 */
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if (TWO54 + x > 0.0 /* raise inexact */
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&& ax < 0x3c90_0000) /* |x| < 2**-54 */
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return x;
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else
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return x - x*x*0.5;
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}
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if (hx > 0 || hx <= 0xbfd2_bec3) { /* -0.2929 < x < 0.41422 */
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k=0;
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f=x;
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hu=1;
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}
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}
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if (hx >= 0x7ff0_0000) {
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return x + x;
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}
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if (k != 0) {
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if (hx < 0x4340_0000) {
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u = 1.0 + x;
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hu = __HI(u); /* high word of u */
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k = (hu >> 20) - 1023;
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c = (k > 0)? 1.0 - (u-x) : x-(u-1.0); /* correction term */
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c /= u;
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} else {
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u = x;
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hu = __HI(u); /* high word of u */
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k = (hu >> 20) - 1023;
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c = 0;
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}
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hu &= 0x000f_ffff;
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if (hu < 0x6_a09e) {
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u = __HI(u, hu | 0x3ff0_0000); /* normalize u */
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} else {
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k += 1;
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u = __HI(u, hu | 0x3fe0_0000); /* normalize u/2 */
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hu = (0x0010_0000 - hu) >> 2;
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}
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f = u - 1.0;
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}
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hfsq = 0.5*f*f;
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if (hu == 0) { /* |f| < 2**-20 */
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if (f == 0.0) {
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if (k == 0) {
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return 0.0;
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} else {
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c += k * ln2_lo;
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return k * ln2_hi + c;
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}
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}
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R = hfsq * (1.0 - 0.66666666666666666*f);
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if (k == 0) {
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return f - R;
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} else {
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return k * ln2_hi - ((R-(k * ln2_lo+c)) - f);
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}
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}
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s = f/(2.0 + f);
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z = s * s;
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R = z * (Lp1 + z * (Lp2 + z * (Lp3 + z * (Lp4 + z * (Lp5 + z * (Lp6 + z*Lp7))))));
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if (k == 0) {
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return f - (hfsq - s*(hfsq + R));
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} else {
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return k * ln2_hi - ((hfsq - (s*(hfsq + R) + (k * ln2_lo+c))) - f);
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}
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}
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}
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}
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