diff options
author | Samuel Thibault <samuel.thibault@ens-lyon.org> | 2016-08-20 19:50:45 +0200 |
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committer | Samuel Thibault <samuel.thibault@ens-lyon.org> | 2016-08-20 19:50:45 +0200 |
commit | 4dd9e35bfd35d3138bc44169baba098005bad51e (patch) | |
tree | a4939c43a9c3fe00eb27f023e14acc5e1fe8808c /sysdeps/ieee754/ldbl-96/e_jnl.c | |
parent | bd42a4599d1b6f77bcfe1e4f67b7cbd9e1cb2dfd (diff) | |
parent | f76453c31593957fec1a99b986bfa5506618b79c (diff) |
Merge commit 'refs/top-bases/t/bigmem' into t/bigmem
Diffstat (limited to 'sysdeps/ieee754/ldbl-96/e_jnl.c')
-rw-r--r-- | sysdeps/ieee754/ldbl-96/e_jnl.c | 522 |
1 files changed, 272 insertions, 250 deletions
diff --git a/sysdeps/ieee754/ldbl-96/e_jnl.c b/sysdeps/ieee754/ldbl-96/e_jnl.c index fa8e27efec..2f3a452f55 100644 --- a/sysdeps/ieee754/ldbl-96/e_jnl.c +++ b/sysdeps/ieee754/ldbl-96/e_jnl.c @@ -57,6 +57,7 @@ */ #include <errno.h> +#include <float.h> #include <math.h> #include <math_private.h> @@ -70,7 +71,7 @@ __ieee754_jnl (int n, long double x) { u_int32_t se, i0, i1; int32_t i, ix, sgn; - long double a, b, temp, di; + long double a, b, temp, di, ret; long double z, w; /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x) @@ -81,7 +82,7 @@ __ieee754_jnl (int n, long double x) ix = se & 0x7fff; /* if J(n,NaN) is NaN */ - if (__builtin_expect ((ix == 0x7fff) && ((i0 & 0x7fffffff) != 0), 0)) + if (__glibc_unlikely ((ix == 0x7fff) && ((i0 & 0x7fffffff) != 0))) return x + x; if (n < 0) { @@ -95,195 +96,206 @@ __ieee754_jnl (int n, long double x) return (__ieee754_j1l (x)); sgn = (n & 1) & (se >> 15); /* even n -- 0, odd n -- sign(x) */ x = fabsl (x); - if (__builtin_expect ((ix | i0 | i1) == 0 || ix >= 0x7fff, 0)) - /* if x is 0 or inf */ - b = zero; - else if ((long double) n <= x) - { - /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */ - if (ix >= 0x412D) - { /* x > 2**302 */ + { + SET_RESTORE_ROUNDL (FE_TONEAREST); + if (__glibc_unlikely ((ix | i0 | i1) == 0 || ix >= 0x7fff)) + /* if x is 0 or inf */ + return sgn == 1 ? -zero : zero; + else if ((long double) n <= x) + { + /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */ + if (ix >= 0x412D) + { /* x > 2**302 */ - /* ??? This might be a futile gesture. - If x exceeds X_TLOSS anyway, the wrapper function - will set the result to zero. */ + /* ??? This might be a futile gesture. + If x exceeds X_TLOSS anyway, the wrapper function + will set the result to zero. */ - /* (x >> n**2) - * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) - * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) - * Let s=sin(x), c=cos(x), - * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then - * - * n sin(xn)*sqt2 cos(xn)*sqt2 - * ---------------------------------- - * 0 s-c c+s - * 1 -s-c -c+s - * 2 -s+c -c-s - * 3 s+c c-s - */ - long double s; - long double c; - __sincosl (x, &s, &c); - switch (n & 3) - { - case 0: - temp = c + s; - break; - case 1: - temp = -c + s; - break; - case 2: - temp = -c - s; - break; - case 3: - temp = c - s; - break; - } - b = invsqrtpi * temp / __ieee754_sqrtl (x); - } - else - { - a = __ieee754_j0l (x); - b = __ieee754_j1l (x); - for (i = 1; i < n; i++) - { - temp = b; - b = b * ((long double) (i + i) / x) - a; /* avoid underflow */ - a = temp; - } - } - } - else - { - if (ix < 0x3fde) - { /* x < 2**-33 */ - /* x is tiny, return the first Taylor expansion of J(n,x) - * J(n,x) = 1/n!*(x/2)^n - ... - */ - if (n >= 400) /* underflow, result < 10^-4952 */ - b = zero; - else - { - temp = x * 0.5; - b = temp; - for (a = one, i = 2; i <= n; i++) - { - a *= (long double) i; /* a = n! */ - b *= temp; /* b = (x/2)^n */ - } - b = b / a; - } - } - else - { - /* use backward recurrence */ - /* x x^2 x^2 - * J(n,x)/J(n-1,x) = ---- ------ ------ ..... - * 2n - 2(n+1) - 2(n+2) - * - * 1 1 1 - * (for large x) = ---- ------ ------ ..... - * 2n 2(n+1) 2(n+2) - * -- - ------ - ------ - - * x x x - * - * Let w = 2n/x and h=2/x, then the above quotient - * is equal to the continued fraction: - * 1 - * = ----------------------- - * 1 - * w - ----------------- - * 1 - * w+h - --------- - * w+2h - ... - * - * To determine how many terms needed, let - * Q(0) = w, Q(1) = w(w+h) - 1, - * Q(k) = (w+k*h)*Q(k-1) - Q(k-2), - * When Q(k) > 1e4 good for single - * When Q(k) > 1e9 good for double - * When Q(k) > 1e17 good for quadruple - */ - /* determine k */ - long double t, v; - long double q0, q1, h, tmp; - int32_t k, m; - w = (n + n) / (long double) x; - h = 2.0L / (long double) x; - q0 = w; - z = w + h; - q1 = w * z - 1.0L; - k = 1; - while (q1 < 1.0e11L) - { - k += 1; - z += h; - tmp = z * q1 - q0; - q0 = q1; - q1 = tmp; - } - m = n + n; - for (t = zero, i = 2 * (n + k); i >= m; i -= 2) - t = one / (i / x - t); - a = t; - b = one; - /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n) - * Hence, if n*(log(2n/x)) > ... - * single 8.8722839355e+01 - * double 7.09782712893383973096e+02 - * long double 1.1356523406294143949491931077970765006170e+04 - * then recurrent value may overflow and the result is - * likely underflow to zero - */ - tmp = n; - v = two / x; - tmp = tmp * __ieee754_logl (fabsl (v * tmp)); + /* (x >> n**2) + * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) + * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) + * Let s=sin(x), c=cos(x), + * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then + * + * n sin(xn)*sqt2 cos(xn)*sqt2 + * ---------------------------------- + * 0 s-c c+s + * 1 -s-c -c+s + * 2 -s+c -c-s + * 3 s+c c-s + */ + long double s; + long double c; + __sincosl (x, &s, &c); + switch (n & 3) + { + case 0: + temp = c + s; + break; + case 1: + temp = -c + s; + break; + case 2: + temp = -c - s; + break; + case 3: + temp = c - s; + break; + } + b = invsqrtpi * temp / __ieee754_sqrtl (x); + } + else + { + a = __ieee754_j0l (x); + b = __ieee754_j1l (x); + for (i = 1; i < n; i++) + { + temp = b; + b = b * ((long double) (i + i) / x) - a; /* avoid underflow */ + a = temp; + } + } + } + else + { + if (ix < 0x3fde) + { /* x < 2**-33 */ + /* x is tiny, return the first Taylor expansion of J(n,x) + * J(n,x) = 1/n!*(x/2)^n - ... + */ + if (n >= 400) /* underflow, result < 10^-4952 */ + b = zero; + else + { + temp = x * 0.5; + b = temp; + for (a = one, i = 2; i <= n; i++) + { + a *= (long double) i; /* a = n! */ + b *= temp; /* b = (x/2)^n */ + } + b = b / a; + } + } + else + { + /* use backward recurrence */ + /* x x^2 x^2 + * J(n,x)/J(n-1,x) = ---- ------ ------ ..... + * 2n - 2(n+1) - 2(n+2) + * + * 1 1 1 + * (for large x) = ---- ------ ------ ..... + * 2n 2(n+1) 2(n+2) + * -- - ------ - ------ - + * x x x + * + * Let w = 2n/x and h=2/x, then the above quotient + * is equal to the continued fraction: + * 1 + * = ----------------------- + * 1 + * w - ----------------- + * 1 + * w+h - --------- + * w+2h - ... + * + * To determine how many terms needed, let + * Q(0) = w, Q(1) = w(w+h) - 1, + * Q(k) = (w+k*h)*Q(k-1) - Q(k-2), + * When Q(k) > 1e4 good for single + * When Q(k) > 1e9 good for double + * When Q(k) > 1e17 good for quadruple + */ + /* determine k */ + long double t, v; + long double q0, q1, h, tmp; + int32_t k, m; + w = (n + n) / (long double) x; + h = 2.0L / (long double) x; + q0 = w; + z = w + h; + q1 = w * z - 1.0L; + k = 1; + while (q1 < 1.0e11L) + { + k += 1; + z += h; + tmp = z * q1 - q0; + q0 = q1; + q1 = tmp; + } + m = n + n; + for (t = zero, i = 2 * (n + k); i >= m; i -= 2) + t = one / (i / x - t); + a = t; + b = one; + /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n) + * Hence, if n*(log(2n/x)) > ... + * single 8.8722839355e+01 + * double 7.09782712893383973096e+02 + * long double 1.1356523406294143949491931077970765006170e+04 + * then recurrent value may overflow and the result is + * likely underflow to zero + */ + tmp = n; + v = two / x; + tmp = tmp * __ieee754_logl (fabsl (v * tmp)); - if (tmp < 1.1356523406294143949491931077970765006170e+04L) - { - for (i = n - 1, di = (long double) (i + i); i > 0; i--) - { - temp = b; - b *= di; - b = b / x - a; - a = temp; - di -= two; - } - } - else - { - for (i = n - 1, di = (long double) (i + i); i > 0; i--) - { - temp = b; - b *= di; - b = b / x - a; - a = temp; - di -= two; - /* scale b to avoid spurious overflow */ - if (b > 1e100L) - { - a /= b; - t /= b; - b = one; - } - } - } - /* j0() and j1() suffer enormous loss of precision at and - * near zero; however, we know that their zero points never - * coincide, so just choose the one further away from zero. - */ - z = __ieee754_j0l (x); - w = __ieee754_j1l (x); - if (fabsl (z) >= fabsl (w)) - b = (t * z / b); - else - b = (t * w / a); - } + if (tmp < 1.1356523406294143949491931077970765006170e+04L) + { + for (i = n - 1, di = (long double) (i + i); i > 0; i--) + { + temp = b; + b *= di; + b = b / x - a; + a = temp; + di -= two; + } + } + else + { + for (i = n - 1, di = (long double) (i + i); i > 0; i--) + { + temp = b; + b *= di; + b = b / x - a; + a = temp; + di -= two; + /* scale b to avoid spurious overflow */ + if (b > 1e100L) + { + a /= b; + t /= b; + b = one; + } + } + } + /* j0() and j1() suffer enormous loss of precision at and + * near zero; however, we know that their zero points never + * coincide, so just choose the one further away from zero. + */ + z = __ieee754_j0l (x); + w = __ieee754_j1l (x); + if (fabsl (z) >= fabsl (w)) + b = (t * z / b); + else + b = (t * w / a); + } + } + if (sgn == 1) + ret = -b; + else + ret = b; + } + if (ret == 0) + ret = __copysignl (LDBL_MIN, ret) * LDBL_MIN; + else if (fabsl (ret) < LDBL_MIN) + { + long double force_underflow = ret * ret; + math_force_eval (force_underflow); } - if (sgn == 1) - return -b; - else - return b; + return ret; } strong_alias (__ieee754_jnl, __jnl_finite) @@ -293,7 +305,7 @@ __ieee754_ynl (int n, long double x) u_int32_t se, i0, i1; int32_t i, ix; int32_t sign; - long double a, b, temp; + long double a, b, temp, ret; GET_LDOUBLE_WORDS (se, i0, i1, x); @@ -314,69 +326,79 @@ __ieee754_ynl (int n, long double x) } if (n == 0) return (__ieee754_y0l (x)); - if (n == 1) - return (sign * __ieee754_y1l (x)); - if (__builtin_expect (ix == 0x7fff, 0)) - return zero; - if (ix >= 0x412D) - { /* x > 2**302 */ + { + SET_RESTORE_ROUNDL (FE_TONEAREST); + if (n == 1) + { + ret = sign * __ieee754_y1l (x); + goto out; + } + if (__glibc_unlikely (ix == 0x7fff)) + return zero; + if (ix >= 0x412D) + { /* x > 2**302 */ - /* ??? See comment above on the possible futility of this. */ + /* ??? See comment above on the possible futility of this. */ - /* (x >> n**2) - * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) - * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) - * Let s=sin(x), c=cos(x), - * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then - * - * n sin(xn)*sqt2 cos(xn)*sqt2 - * ---------------------------------- - * 0 s-c c+s - * 1 -s-c -c+s - * 2 -s+c -c-s - * 3 s+c c-s - */ - long double s; - long double c; - __sincosl (x, &s, &c); - switch (n & 3) - { - case 0: - temp = s - c; - break; - case 1: - temp = -s - c; - break; - case 2: - temp = -s + c; - break; - case 3: - temp = s + c; - break; - } - b = invsqrtpi * temp / __ieee754_sqrtl (x); - } - else - { - a = __ieee754_y0l (x); - b = __ieee754_y1l (x); - /* quit if b is -inf */ - GET_LDOUBLE_WORDS (se, i0, i1, b); - /* Use 0xffffffff since GET_LDOUBLE_WORDS sign-extends SE. */ - for (i = 1; i < n && se != 0xffffffff; i++) - { - temp = b; - b = ((long double) (i + i) / x) * b - a; - GET_LDOUBLE_WORDS (se, i0, i1, b); - a = temp; - } - } - /* If B is +-Inf, set up errno accordingly. */ - if (! __finitel (b)) - __set_errno (ERANGE); - if (sign > 0) - return b; - else - return -b; + /* (x >> n**2) + * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) + * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) + * Let s=sin(x), c=cos(x), + * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then + * + * n sin(xn)*sqt2 cos(xn)*sqt2 + * ---------------------------------- + * 0 s-c c+s + * 1 -s-c -c+s + * 2 -s+c -c-s + * 3 s+c c-s + */ + long double s; + long double c; + __sincosl (x, &s, &c); + switch (n & 3) + { + case 0: + temp = s - c; + break; + case 1: + temp = -s - c; + break; + case 2: + temp = -s + c; + break; + case 3: + temp = s + c; + break; + } + b = invsqrtpi * temp / __ieee754_sqrtl (x); + } + else + { + a = __ieee754_y0l (x); + b = __ieee754_y1l (x); + /* quit if b is -inf */ + GET_LDOUBLE_WORDS (se, i0, i1, b); + /* Use 0xffffffff since GET_LDOUBLE_WORDS sign-extends SE. */ + for (i = 1; i < n && se != 0xffffffff; i++) + { + temp = b; + b = ((long double) (i + i) / x) * b - a; + GET_LDOUBLE_WORDS (se, i0, i1, b); + a = temp; + } + } + /* If B is +-Inf, set up errno accordingly. */ + if (! isfinite (b)) + __set_errno (ERANGE); + if (sign > 0) + ret = b; + else + ret = -b; + } + out: + if (isinf (ret)) + ret = __copysignl (LDBL_MAX, ret) * LDBL_MAX; + return ret; } strong_alias (__ieee754_ynl, __ynl_finite) |