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8301396: Port fdlibm expm1 to Java
Reviewed-by: bpb
This commit is contained in:
Joe Darcy
parent
3be5317b59
commit
34493248c0
4 changed files with 504 additions and 19 deletions
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@ -985,4 +985,213 @@ class FdLibm {
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}
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}
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}
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/* expm1(x)
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* Returns exp(x)-1, the exponential of x minus 1.
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*
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* Method
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* 1. Argument reduction:
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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 ~ 0.34658
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*
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* Here a correction term c will be computed to compensate
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* the error in r when rounded to a floating-point number.
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*
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* 2. Approximating expm1(r) by a special rational function on
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* the interval [0,0.34658]:
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* Since
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* r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ...
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* we define R1(r*r) by
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* r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r)
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* That is,
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* R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
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* = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
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* = 1 - r^2/60 + r^4/2520 - r^6/100800 + ...
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* We use a special Reme algorithm on [0,0.347] to generate
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* a polynomial of degree 5 in r*r to approximate R1. The
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* maximum error of this polynomial approximation is bounded
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* by 2**-61. In other words,
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* R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
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* where Q1 = -1.6666666666666567384E-2,
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* Q2 = 3.9682539681370365873E-4,
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* Q3 = -9.9206344733435987357E-6,
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* Q4 = 2.5051361420808517002E-7,
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* Q5 = -6.2843505682382617102E-9;
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* (where z=r*r, and the values of Q1 to Q5 are listed below)
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* with error bounded by
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* | 5 | -61
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* | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2
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* | |
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*
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* expm1(r) = exp(r)-1 is then computed by the following
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* specific way which minimize the accumulation rounding error:
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* 2 3
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* r r [ 3 - (R1 + R1*r/2) ]
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* expm1(r) = r + --- + --- * [--------------------]
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* 2 2 [ 6 - r*(3 - R1*r/2) ]
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*
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* To compensate the error in the argument reduction, we use
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* expm1(r+c) = expm1(r) + c + expm1(r)*c
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* ~ expm1(r) + c + r*c
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* Thus c+r*c will be added in as the correction terms for
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* expm1(r+c). Now rearrange the term to avoid optimization
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* screw up:
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* ( 2 2 )
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* ({ ( r [ R1 - (3 - R1*r/2) ] ) } r )
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* expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
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* ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 )
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* ( )
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*
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* = r - E
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* 3. Scale back to obtain expm1(x):
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* From step 1, we have
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* expm1(x) = either 2^k*[expm1(r)+1] - 1
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* = or 2^k*[expm1(r) + (1-2^-k)]
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* 4. Implementation notes:
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* (A). To save one multiplication, we scale the coefficient Qi
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* to Qi*2^i, and replace z by (x^2)/2.
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* (B). To achieve maximum accuracy, we compute expm1(x) by
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* (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf)
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* (ii) if k=0, return r-E
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* (iii) if k=-1, return 0.5*(r-E)-0.5
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* (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E)
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* else return 1.0+2.0*(r-E);
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* (v) if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1)
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* (vi) if k <= 20, return 2^k((1-2^-k)-(E-r)), else
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* (vii) return 2^k(1-((E+2^-k)-r))
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*
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* Special cases:
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* expm1(INF) is INF, expm1(NaN) is NaN;
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* expm1(-INF) is -1, and
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* for finite argument, only expm1(0)=0 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 expm1(x) overflow
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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 Expm1 {
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private static final double one = 1.0;
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private static final double huge = 1.0e+300;
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private static final double tiny = 1.0e-300;
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private static final double o_threshold = 0x1.62e42fefa39efp9; // 7.09782712893383973096e+02
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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 invln2 = 0x1.71547652b82fep0; // 1.44269504088896338700e+00
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// scaled coefficients related to expm1
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private static final double Q1 = -0x1.11111111110f4p-5; // -3.33333333333331316428e-02
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private static final double Q2 = 0x1.a01a019fe5585p-10; // 1.58730158725481460165e-03
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private static final double Q3 = -0x1.4ce199eaadbb7p-14; // -7.93650757867487942473e-05
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private static final double Q4 = 0x1.0cfca86e65239p-18; // 4.00821782732936239552e-06
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private static final double Q5 = -0x1.afdb76e09c32dp-23; // -2.01099218183624371326e-07
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static double compute(double x) {
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double y, hi, lo, c=0, t, e, hxs, hfx, r1;
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int k, 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 & 0x8000_0000; // sign bit of x
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y = Math.abs(x);
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hx &= 0x7fff_ffff; // high word of |x|
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// filter out huge and non-finite argument
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if (hx >= 0x4043_687A) { // if |x| >= 56*ln2
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if (hx >= 0x4086_2E42) { // if |x| >= 709.78...
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if (hx >= 0x7ff_00000) {
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if (((hx & 0xf_ffff) | __LO(x)) != 0) {
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return x + x; // NaN
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} else {
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return (xsb == 0)? x : -1.0; // exp(+-inf)={inf,-1}
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}
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}
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if (x > o_threshold) {
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return huge*huge; // overflow
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}
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}
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if (xsb != 0) { // x < -56*ln2, return -1.0 with inexact
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if (x + tiny < 0.0) { // raise inexact
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return tiny - one; // return -1
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}
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}
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}
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// argument reduction
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if (hx > 0x3fd6_2e42) { // if |x| > 0.5 ln2
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if (hx < 0x3FF0_A2B2) { // and |x| < 1.5 ln2
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if (xsb == 0) {
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hi = x - ln2_hi;
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lo = ln2_lo;
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k = 1;
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} else {
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hi = x + ln2_hi;
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lo = -ln2_lo;
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k = -1;
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}
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} else {
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k = (int)(invln2*x + ((xsb == 0) ? 0.5 : -0.5));
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t = k;
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hi = x - t*ln2_hi; // t*ln2_hi is exact here
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lo = t*ln2_lo;
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}
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x = hi - lo;
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c = (hi - x) - lo;
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} else if (hx < 0x3c90_0000) { // when |x| < 2**-54, return x
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t = huge + x; // return x with inexact flags when x != 0
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return x - (t - (huge + x));
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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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hfx = 0.5*x;
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hxs = x*hfx;
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r1 = one + hxs*(Q1 + hxs*(Q2 + hxs*(Q3 + hxs*(Q4 + hxs*Q5))));
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t = 3.0 - r1*hfx;
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e = hxs *((r1 - t)/(6.0 - x*t));
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if (k == 0) {
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return x - (x*e - hxs); // c is 0
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} else {
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e = (x*(e - c) - c);
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e -= hxs;
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if (k == -1) {
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return 0.5*(x - e) - 0.5;
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}
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if (k == 1) {
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if (x < -0.25) {
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return -2.0*(e - (x + 0.5));
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} else {
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return one + 2.0*(x - e);
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}
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}
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if (k <= -2 || k > 56) { // suffice to return exp(x) - 1
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y = one - (e - x);
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y = __HI(y, __HI(y) + (k << 20)); // add k to y's exponent
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return y - one;
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}
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t = one;
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if (k < 20) {
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t = __HI(t, 0x3ff0_0000 - (0x2_00000 >> k)); // t = 1-2^-k
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y = t - ( e - x);
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y = __HI(y, __HI(y) + (k << 20)); // add k to y's exponent
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} else {
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t = __HI(t, ((0x3ff - k) << 20)); // 2^-k
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y = x - (e + t);
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y += one;
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y = __HI(y, __HI(y) + (k << 20)); // add k to y's exponent
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}
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}
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return y;
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}
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}
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}
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