| Index: src/base/ieee754.cc
|
| diff --git a/src/base/ieee754.cc b/src/base/ieee754.cc
|
| index e642b6327a9b29360db3c09ebc0235b15b3c9f0e..4d11bdafe8b89e26a737a3c332f39fc7d6b45b0f 100644
|
| --- a/src/base/ieee754.cc
|
| +++ b/src/base/ieee754.cc
|
| @@ -389,152 +389,6 @@
|
| return pi - (z - pi_lo); /* atan(+,-) */
|
| default: /* case 3 */
|
| return (z - pi_lo) - pi; /* atan(-,-) */
|
| - }
|
| -}
|
| -
|
| -/* exp(x)
|
| - * Returns the exponential of x.
|
| - *
|
| - * Method
|
| - * 1. Argument reduction:
|
| - * Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
|
| - * Given x, find r and integer k such that
|
| - *
|
| - * x = k*ln2 + r, |r| <= 0.5*ln2.
|
| - *
|
| - * Here r will be represented as r = hi-lo for better
|
| - * accuracy.
|
| - *
|
| - * 2. Approximation of exp(r) by a special rational function on
|
| - * the interval [0,0.34658]:
|
| - * Write
|
| - * R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
|
| - * We use a special Remes algorithm on [0,0.34658] to generate
|
| - * a polynomial of degree 5 to approximate R. The maximum error
|
| - * of this polynomial approximation is bounded by 2**-59. In
|
| - * other words,
|
| - * R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
|
| - * (where z=r*r, and the values of P1 to P5 are listed below)
|
| - * and
|
| - * | 5 | -59
|
| - * | 2.0+P1*z+...+P5*z - R(z) | <= 2
|
| - * | |
|
| - * The computation of exp(r) thus becomes
|
| - * 2*r
|
| - * exp(r) = 1 + -------
|
| - * R - r
|
| - * r*R1(r)
|
| - * = 1 + r + ----------- (for better accuracy)
|
| - * 2 - R1(r)
|
| - * where
|
| - * 2 4 10
|
| - * R1(r) = r - (P1*r + P2*r + ... + P5*r ).
|
| - *
|
| - * 3. Scale back to obtain exp(x):
|
| - * From step 1, we have
|
| - * exp(x) = 2^k * exp(r)
|
| - *
|
| - * Special cases:
|
| - * exp(INF) is INF, exp(NaN) is NaN;
|
| - * exp(-INF) is 0, and
|
| - * for finite argument, only exp(0)=1 is exact.
|
| - *
|
| - * Accuracy:
|
| - * according to an error analysis, the error is always less than
|
| - * 1 ulp (unit in the last place).
|
| - *
|
| - * Misc. info.
|
| - * For IEEE double
|
| - * if x > 7.09782712893383973096e+02 then exp(x) overflow
|
| - * if x < -7.45133219101941108420e+02 then exp(x) underflow
|
| - *
|
| - * 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.
|
| - */
|
| -double exp(double x) {
|
| - static const double
|
| - one = 1.0,
|
| - halF[2] = {0.5, -0.5},
|
| - o_threshold = 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */
|
| - u_threshold = -7.45133219101941108420e+02, /* 0xc0874910, 0xD52D3051 */
|
| - ln2HI[2] = {6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
|
| - -6.93147180369123816490e-01}, /* 0xbfe62e42, 0xfee00000 */
|
| - ln2LO[2] = {1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
|
| - -1.90821492927058770002e-10}, /* 0xbdea39ef, 0x35793c76 */
|
| - invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
|
| - P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
|
| - P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
|
| - P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
|
| - P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
|
| - P5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */
|
| -
|
| - static volatile double
|
| - huge = 1.0e+300,
|
| - twom1000 = 9.33263618503218878990e-302, /* 2**-1000=0x01700000,0*/
|
| - two1023 = 8.988465674311579539e307; /* 0x1p1023 */
|
| -
|
| - double y, hi = 0.0, lo = 0.0, c, t, twopk;
|
| - int32_t k = 0, xsb;
|
| - u_int32_t hx;
|
| -
|
| - GET_HIGH_WORD(hx, x);
|
| - xsb = (hx >> 31) & 1; /* sign bit of x */
|
| - hx &= 0x7fffffff; /* high word of |x| */
|
| -
|
| - /* filter out non-finite argument */
|
| - if (hx >= 0x40862E42) { /* if |x|>=709.78... */
|
| - if (hx >= 0x7ff00000) {
|
| - u_int32_t lx;
|
| - GET_LOW_WORD(lx, x);
|
| - if (((hx & 0xfffff) | lx) != 0)
|
| - return x + x; /* NaN */
|
| - else
|
| - return (xsb == 0) ? x : 0.0; /* exp(+-inf)={inf,0} */
|
| - }
|
| - if (x > o_threshold) return huge * huge; /* overflow */
|
| - if (x < u_threshold) return twom1000 * twom1000; /* underflow */
|
| - }
|
| -
|
| - /* argument reduction */
|
| - if (hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
|
| - if (hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
|
| - hi = x - ln2HI[xsb];
|
| - lo = ln2LO[xsb];
|
| - k = 1 - xsb - xsb;
|
| - } else {
|
| - k = static_cast<int>(invln2 * x + halF[xsb]);
|
| - t = k;
|
| - hi = x - t * ln2HI[0]; /* t*ln2HI is exact here */
|
| - lo = t * ln2LO[0];
|
| - }
|
| - STRICT_ASSIGN(double, x, hi - lo);
|
| - } else if (hx < 0x3e300000) { /* when |x|<2**-28 */
|
| - if (huge + x > one) return one + x; /* trigger inexact */
|
| - } else {
|
| - k = 0;
|
| - }
|
| -
|
| - /* x is now in primary range */
|
| - t = x * x;
|
| - if (k >= -1021) {
|
| - INSERT_WORDS(twopk, 0x3ff00000 + (k << 20), 0);
|
| - } else {
|
| - INSERT_WORDS(twopk, 0x3ff00000 + ((k + 1000) << 20), 0);
|
| - }
|
| - c = x - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5))));
|
| - if (k == 0) {
|
| - return one - ((x * c) / (c - 2.0) - x);
|
| - } else {
|
| - y = one - ((lo - (x * c) / (2.0 - c)) - hi);
|
| - }
|
| - if (k >= -1021) {
|
| - if (k == 1024) return y * 2.0 * two1023;
|
| - return y * twopk;
|
| - } else {
|
| - return y * twopk * twom1000;
|
| }
|
| }
|
|
|
|
|