diff options
Diffstat (limited to 'libquadmath/math/jnq.c')
| -rw-r--r-- | libquadmath/math/jnq.c | 531 |
1 files changed, 276 insertions, 255 deletions
diff --git a/libquadmath/math/jnq.c b/libquadmath/math/jnq.c index 56a183604c1..ae318ac36b7 100644 --- a/libquadmath/math/jnq.c +++ b/libquadmath/math/jnq.c @@ -27,8 +27,8 @@ Lesser General Public License for more details. You should have received a copy of the GNU Lesser General Public - License along with this library; if not, write to the Free Software - Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA */ + License along with this library; if not, see + <http://www.gnu.org/licenses/>. */ /* * __ieee754_jn(n, x), __ieee754_yn(n, x) @@ -56,14 +56,13 @@ * */ -#include <errno.h> #include "quadmath-imp.h" static const __float128 invsqrtpi = 5.6418958354775628694807945156077258584405E-1Q, - two = 2.0e0Q, - one = 1.0e0Q, - zero = 0.0Q; + two = 2, + one = 1, + zero = 0; __float128 @@ -71,7 +70,7 @@ jnq (int n, __float128 x) { uint32_t se; int32_t i, ix, sgn; - __float128 a, b, temp, di; + __float128 a, b, temp, di, ret; __float128 z, w; ieee854_float128 u; @@ -104,201 +103,213 @@ jnq (int n, __float128 x) sgn = (n & 1) & (se >> 31); /* even n -- 0, odd n -- sign(x) */ x = fabsq (x); - if (x == 0.0Q || ix >= 0x7fff0000) /* if x is 0 or inf */ - b = zero; - else if ((__float128) n <= x) - { - /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */ - if (ix >= 0x412D0000) - { /* x > 2**302 */ + { + SET_RESTORE_ROUNDF128 (FE_TONEAREST); + if (x == 0 || ix >= 0x7fff0000) /* if x is 0 or inf */ + return sgn == 1 ? -zero : zero; + else if ((__float128) n <= x) + { + /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */ + if (ix >= 0x412D0000) + { /* x > 2**302 */ - /* ??? Could use an expansion for large x here. */ + /* ??? Could use an expansion for large x here. */ - /* (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 - */ - __float128 s; - __float128 c; - sincosq (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 / sqrtq (x); - } - else - { - a = j0q (x); - b = j1q (x); - for (i = 1; i < n; i++) - { - temp = b; - b = b * ((__float128) (i + i) / x) - a; /* avoid underflow */ - a = temp; - } - } - } - else - { - if (ix < 0x3fc60000) - { /* x < 2**-57 */ - /* 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 *= (__float128) 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 */ - __float128 t, v; - __float128 q0, q1, h, tmp; - int32_t k, m; - w = (n + n) / (__float128) x; - h = 2.0Q / (__float128) x; - q0 = w; - z = w + h; - q1 = w * z - 1.0Q; - k = 1; - while (q1 < 1.0e17Q) - { - 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 - * __float128 1.1356523406294143949491931077970765006170e+04 - * then recurrent value may overflow and the result is - * likely underflow to zero - */ - tmp = n; - v = two / x; - tmp = tmp * logq (fabsq (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 + */ + __float128 s; + __float128 c; + sincosq (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 / sqrtq (x); + } + else + { + a = j0q (x); + b = j1q (x); + for (i = 1; i < n; i++) + { + temp = b; + b = b * ((__float128) (i + i) / x) - a; /* avoid underflow */ + a = temp; + } + } + } + else + { + if (ix < 0x3fc60000) + { /* x < 2**-57 */ + /* 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 *= (__float128) 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 */ + __float128 t, v; + __float128 q0, q1, h, tmp; + int32_t k, m; + w = (n + n) / (__float128) x; + h = 2 / (__float128) x; + q0 = w; + z = w + h; + q1 = w * z - 1; + k = 1; + while (q1 < 1.0e17Q) + { + 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 * logq (fabsq (v * tmp)); - if (tmp < 1.1356523406294143949491931077970765006170e+04Q) - { - for (i = n - 1, di = (__float128) (i + i); i > 0; i--) - { - temp = b; - b *= di; - b = b / x - a; - a = temp; - di -= two; - } - } - else - { - for (i = n - 1, di = (__float128) (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 > 1e100Q) - { - 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 = j0q (x); - w = j1q (x); - if (fabsq (z) >= fabsq (w)) - b = (t * z / b); - else - b = (t * w / a); - } + if (tmp < 1.1356523406294143949491931077970765006170e+04Q) + { + for (i = n - 1, di = (__float128) (i + i); i > 0; i--) + { + temp = b; + b *= di; + b = b / x - a; + a = temp; + di -= two; + } + } + else + { + for (i = n - 1, di = (__float128) (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 > 1e100Q) + { + 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 = j0q (x); + w = j1q (x); + if (fabsq (z) >= fabsq (w)) + b = (t * z / b); + else + b = (t * w / a); + } + } + if (sgn == 1) + ret = -b; + else + ret = b; + } + if (ret == 0) + { + ret = copysignq (FLT128_MIN, ret) * FLT128_MIN; + errno = ERANGE; } - if (sgn == 1) - return -b; else - return b; + math_check_force_underflow (ret); + return ret; } + __float128 ynq (int n, __float128 x) { uint32_t se; int32_t i, ix; int32_t sign; - __float128 a, b, temp; + __float128 a, b, temp, ret; ieee854_float128 u; u.value = x; @@ -311,10 +322,10 @@ ynq (int n, __float128 x) if ((u.words32.w0 & 0xffff) | u.words32.w1 | u.words32.w2 | u.words32.w3) return x + x; } - if (x <= 0.0Q) + if (x <= 0) { - if (x == 0.0Q) - return -HUGE_VALQ + x; + if (x == 0) + return ((n < 0 && (n & 1) != 0) ? 1 : -1) / 0.0Q; if (se & 0x80000000) return zero / (zero * x); } @@ -326,69 +337,79 @@ ynq (int n, __float128 x) } if (n == 0) return (y0q (x)); - if (n == 1) - return (sign * y1q (x)); - if (ix >= 0x7fff0000) - return zero; - if (ix >= 0x412D0000) - { /* x > 2**302 */ + { + SET_RESTORE_ROUNDF128 (FE_TONEAREST); + if (n == 1) + { + ret = sign * y1q (x); + goto out; + } + if (ix >= 0x7fff0000) + return zero; + if (ix >= 0x412D0000) + { /* 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 - */ - __float128 s; - __float128 c; - sincosq (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 / sqrtq (x); - } - else - { - a = y0q (x); - b = y1q (x); - /* quit if b is -inf */ - u.value = b; - se = u.words32.w0 & 0xffff0000; - for (i = 1; i < n && se != 0xffff0000; i++) - { - temp = b; - b = ((__float128) (i + i) / x) * b - a; - u.value = b; - se = u.words32.w0 & 0xffff0000; - a = temp; - } - } - /* If B is +-Inf, set up errno accordingly. */ - if (! finiteq (b)) - 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 + */ + __float128 s; + __float128 c; + sincosq (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 / sqrtq (x); + } + else + { + a = y0q (x); + b = y1q (x); + /* quit if b is -inf */ + u.value = b; + se = u.words32.w0 & 0xffff0000; + for (i = 1; i < n && se != 0xffff0000; i++) + { + temp = b; + b = ((__float128) (i + i) / x) * b - a; + u.value = b; + se = u.words32.w0 & 0xffff0000; + a = temp; + } + } + /* If B is +-Inf, set up errno accordingly. */ + if (! finiteq (b)) + errno = ERANGE; + if (sign > 0) + ret = b; + else + ret = -b; + } + out: + if (isinfq (ret)) + ret = copysignq (FLT128_MAX, ret) * FLT128_MAX; + return ret; } |