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Diffstat (limited to 'REORG.TODO/sysdeps/ieee754/dbl-64/s_expm1.c')
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diff --git a/REORG.TODO/sysdeps/ieee754/dbl-64/s_expm1.c b/REORG.TODO/sysdeps/ieee754/dbl-64/s_expm1.c new file mode 100644 index 0000000000..54d771007a --- /dev/null +++ b/REORG.TODO/sysdeps/ieee754/dbl-64/s_expm1.c @@ -0,0 +1,262 @@ +/* @(#)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 |