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-/* @(#)s_expm1.c 5.1 93/09/24 */
-/*
- * ====================================================
- * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
- *
- * Developed at SunPro, a Sun Microsystems, Inc. business.
- * Permission to use, copy, modify, and distribute this
- * software is freely granted, provided that this notice
- * is preserved.
- * ====================================================
- */
-/* Modified by Naohiko Shimizu/Tokai University, Japan 1997/08/25,
- for performance improvement on pipelined processors.
- */
-
-/* expm1(x)
- * Returns exp(x)-1, the exponential of x minus 1.
- *
- * Method
- * 1. Argument reduction:
- * Given x, find r and integer k such that
- *
- * x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658
- *
- * Here a correction term c will be computed to compensate
- * the error in r when rounded to a floating-point number.
- *
- * 2. Approximating expm1(r) by a special rational function on
- * the interval [0,0.34658]:
- * Since
- * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ...
- * we define R1(r*r) by
- * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r)
- * That is,
- * R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
- * = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
- * = 1 - r^2/60 + r^4/2520 - r^6/100800 + ...
- * We use a special Reme algorithm on [0,0.347] to generate
- * a polynomial of degree 5 in r*r to approximate R1. The
- * maximum error of this polynomial approximation is bounded
- * by 2**-61. In other words,
- * R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
- * where Q1 = -1.6666666666666567384E-2,
- * Q2 = 3.9682539681370365873E-4,
- * Q3 = -9.9206344733435987357E-6,
- * Q4 = 2.5051361420808517002E-7,
- * Q5 = -6.2843505682382617102E-9;
- * (where z=r*r, and the values of Q1 to Q5 are listed below)
- * with error bounded by
- * | 5 | -61
- * | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2
- * | |
- *
- * expm1(r) = exp(r)-1 is then computed by the following
- * specific way which minimize the accumulation rounding error:
- * 2 3
- * r r [ 3 - (R1 + R1*r/2) ]
- * expm1(r) = r + --- + --- * [--------------------]
- * 2 2 [ 6 - r*(3 - R1*r/2) ]
- *
- * To compensate the error in the argument reduction, we use
- * expm1(r+c) = expm1(r) + c + expm1(r)*c
- * ~ expm1(r) + c + r*c
- * Thus c+r*c will be added in as the correction terms for
- * expm1(r+c). Now rearrange the term to avoid optimization
- * screw up:
- * ( 2 2 )
- * ({ ( r [ R1 - (3 - R1*r/2) ] ) } r )
- * expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
- * ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 )
- * ( )
- *
- * = r - E
- * 3. Scale back to obtain expm1(x):
- * From step 1, we have
- * expm1(x) = either 2^k*[expm1(r)+1] - 1
- * = or 2^k*[expm1(r) + (1-2^-k)]
- * 4. Implementation notes:
- * (A). To save one multiplication, we scale the coefficient Qi
- * to Qi*2^i, and replace z by (x^2)/2.
- * (B). To achieve maximum accuracy, we compute expm1(x) by
- * (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf)
- * (ii) if k=0, return r-E
- * (iii) if k=-1, return 0.5*(r-E)-0.5
- * (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E)
- * else return 1.0+2.0*(r-E);
- * (v) if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1)
- * (vi) if k <= 20, return 2^k((1-2^-k)-(E-r)), else
- * (vii) return 2^k(1-((E+2^-k)-r))
- *
- * Special cases:
- * expm1(INF) is INF, expm1(NaN) is NaN;
- * expm1(-INF) is -1, and
- * for finite argument, only expm1(0)=0 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 expm1(x) overflow
- *
- * 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.
- */
-
-#include <errno.h>
-#include <float.h>
-#include <math.h>
-#include <math_private.h>
-#define one Q[0]
-static const double
- huge = 1.0e+300,
- tiny = 1.0e-300,
- o_threshold = 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */
- ln2_hi = 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
- ln2_lo = 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
- invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
-/* scaled coefficients related to expm1 */
- Q[] = { 1.0, -3.33333333333331316428e-02, /* BFA11111 111110F4 */
- 1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */
- -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */
- 4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */
- -2.01099218183624371326e-07 }; /* BE8AFDB7 6E09C32D */
-
-double
-__expm1 (double x)
-{
- double y, hi, lo, c, t, e, hxs, hfx, r1, h2, h4, R1, R2, R3;
- int32_t k, xsb;
- u_int32_t hx;
-
- GET_HIGH_WORD (hx, x);
- xsb = hx & 0x80000000; /* sign bit of x */
- if (xsb == 0)
- y = x;
- else
- y = -x; /* y = |x| */
- hx &= 0x7fffffff; /* high word of |x| */
-
- /* filter out huge and non-finite argument */
- if (hx >= 0x4043687A) /* if |x|>=56*ln2 */
- {
- if (hx >= 0x40862E42) /* if |x|>=709.78... */
- {
- if (hx >= 0x7ff00000)
- {
- u_int32_t low;
- GET_LOW_WORD (low, x);
- if (((hx & 0xfffff) | low) != 0)
- return x + x; /* NaN */
- else
- return (xsb == 0) ? x : -1.0; /* exp(+-inf)={inf,-1} */
- }
- if (x > o_threshold)
- {
- __set_errno (ERANGE);
- return huge * huge; /* overflow */
- }
- }
- if (xsb != 0) /* x < -56*ln2, return -1.0 with inexact */
- {
- math_force_eval (x + tiny); /* raise inexact */
- return tiny - one; /* return -1 */
- }
- }
-
- /* argument reduction */
- if (hx > 0x3fd62e42) /* if |x| > 0.5 ln2 */
- {
- if (hx < 0x3FF0A2B2) /* and |x| < 1.5 ln2 */
- {
- if (xsb == 0)
- {
- hi = x - ln2_hi; lo = ln2_lo; k = 1;
- }
- else
- {
- hi = x + ln2_hi; lo = -ln2_lo; k = -1;
- }
- }
- else
- {
- k = invln2 * x + ((xsb == 0) ? 0.5 : -0.5);
- t = k;
- hi = x - t * ln2_hi; /* t*ln2_hi is exact here */
- lo = t * ln2_lo;
- }
- x = hi - lo;
- c = (hi - x) - lo;
- }
- else if (hx < 0x3c900000) /* when |x|<2**-54, return x */
- {
- math_check_force_underflow (x);
- t = huge + x; /* return x with inexact flags when x!=0 */
- return x - (t - (huge + x));
- }
- else
- k = 0;
-
- /* x is now in primary range */
- hfx = 0.5 * x;
- hxs = x * hfx;
- R1 = one + hxs * Q[1]; h2 = hxs * hxs;
- R2 = Q[2] + hxs * Q[3]; h4 = h2 * h2;
- R3 = Q[4] + hxs * Q[5];
- r1 = R1 + h2 * R2 + h4 * R3;
- t = 3.0 - r1 * hfx;
- e = hxs * ((r1 - t) / (6.0 - x * t));
- if (k == 0)
- return x - (x * e - hxs); /* c is 0 */
- else
- {
- e = (x * (e - c) - c);
- e -= hxs;
- if (k == -1)
- return 0.5 * (x - e) - 0.5;
- if (k == 1)
- {
- if (x < -0.25)
- return -2.0 * (e - (x + 0.5));
- else
- return one + 2.0 * (x - e);
- }
- if (k <= -2 || k > 56) /* suffice to return exp(x)-1 */
- {
- u_int32_t high;
- y = one - (e - x);
- GET_HIGH_WORD (high, y);
- SET_HIGH_WORD (y, high + (k << 20)); /* add k to y's exponent */
- return y - one;
- }
- t = one;
- if (k < 20)
- {
- u_int32_t high;
- SET_HIGH_WORD (t, 0x3ff00000 - (0x200000 >> k)); /* t=1-2^-k */
- y = t - (e - x);
- GET_HIGH_WORD (high, y);
- SET_HIGH_WORD (y, high + (k << 20)); /* add k to y's exponent */
- }
- else
- {
- u_int32_t high;
- SET_HIGH_WORD (t, ((0x3ff - k) << 20)); /* 2^-k */
- y = x - (e + t);
- y += one;
- GET_HIGH_WORD (high, y);
- SET_HIGH_WORD (y, high + (k << 20)); /* add k to y's exponent */
- }
- }
- return y;
-}
-weak_alias (__expm1, expm1)
-#ifdef NO_LONG_DOUBLE
-strong_alias (__expm1, __expm1l)
-weak_alias (__expm1, expm1l)
-#endif
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