@@ -5756,8 +5756,8 @@ j0 -0x1.001000001p+593
j0 0x1p1023
j0 0x1p16382
j0 0x1p16383
-# the next value generates larger error bounds on x86_64 (binary32)
-j0 0x2.602774p+0 xfail-rounding:ibm128-libgcc
+# the next value generates large error bounds on x86_64 (binary32)
+j0 0x1.8bde7ep+5
# the next value exercises the flt-32 code path for x >= 2^127
j0 0x8.2f4ecp+124
@@ -8225,8 +8225,8 @@ y0 0x1p-100
y0 0x1p-110
y0 0x1p-600
y0 0x1p-10000
-# the next value generates larger error bounds on x86_64 (binary32)
-y0 0xd.3432bp-4
+# the next value generates large error bounds on x86_64 (binary32)
+y0 0x1.3d4e56p+6
y0 min
y0 min_subnorm
@@ -1334,31 +1334,31 @@ j0 0x1p16383
= j0 tonearest ibm128 0xf.ffffffffffffbffffffffffffcp+1020 : -0xb.a80d0ee91ce259a722e1f01904p-516 : inexact-ok
= j0 towardzero ibm128 0xf.ffffffffffffbffffffffffffcp+1020 : -0xb.a80d0ee91ce259a722e1f019p-516 : inexact-ok
= j0 upward ibm128 0xf.ffffffffffffbffffffffffffcp+1020 : -0xb.a80d0ee91ce259a722e1f019p-516 : inexact-ok
-j0 0x2.602774p+0 xfail-rounding:ibm128-libgcc
-= j0 downward binary32 0x2.602774p+0 : 0x3.e83778p-8 : xfail:ibm128-libgcc inexact-ok
-= j0 tonearest binary32 0x2.602774p+0 : 0x3.e83778p-8 : inexact-ok
-= j0 towardzero binary32 0x2.602774p+0 : 0x3.e83778p-8 : xfail:ibm128-libgcc inexact-ok
-= j0 upward binary32 0x2.602774p+0 : 0x3.e8377cp-8 : xfail:ibm128-libgcc inexact-ok
-= j0 downward binary64 0x2.602774p+0 : 0x3.e83779fe1991p-8 : xfail:ibm128-libgcc inexact-ok
-= j0 tonearest binary64 0x2.602774p+0 : 0x3.e83779fe19912p-8 : inexact-ok
-= j0 towardzero binary64 0x2.602774p+0 : 0x3.e83779fe1991p-8 : xfail:ibm128-libgcc inexact-ok
-= j0 upward binary64 0x2.602774p+0 : 0x3.e83779fe19912p-8 : xfail:ibm128-libgcc inexact-ok
-= j0 downward intel96 0x2.602774p+0 : 0x3.e83779fe19911fa8p-8 : xfail:ibm128-libgcc inexact-ok
-= j0 tonearest intel96 0x2.602774p+0 : 0x3.e83779fe19911fa8p-8 : inexact-ok
-= j0 towardzero intel96 0x2.602774p+0 : 0x3.e83779fe19911fa8p-8 : xfail:ibm128-libgcc inexact-ok
-= j0 upward intel96 0x2.602774p+0 : 0x3.e83779fe19911facp-8 : xfail:ibm128-libgcc inexact-ok
-= j0 downward m68k96 0x2.602774p+0 : 0x3.e83779fe19911fa8p-8 : xfail:ibm128-libgcc inexact-ok
-= j0 tonearest m68k96 0x2.602774p+0 : 0x3.e83779fe19911fa8p-8 : inexact-ok
-= j0 towardzero m68k96 0x2.602774p+0 : 0x3.e83779fe19911fa8p-8 : xfail:ibm128-libgcc inexact-ok
-= j0 upward m68k96 0x2.602774p+0 : 0x3.e83779fe19911facp-8 : xfail:ibm128-libgcc inexact-ok
-= j0 downward binary128 0x2.602774p+0 : 0x3.e83779fe19911fa806cee83ab7p-8 : xfail:ibm128-libgcc inexact-ok
-= j0 tonearest binary128 0x2.602774p+0 : 0x3.e83779fe19911fa806cee83ab7p-8 : inexact-ok
-= j0 towardzero binary128 0x2.602774p+0 : 0x3.e83779fe19911fa806cee83ab7p-8 : xfail:ibm128-libgcc inexact-ok
-= j0 upward binary128 0x2.602774p+0 : 0x3.e83779fe19911fa806cee83ab702p-8 : xfail:ibm128-libgcc inexact-ok
-= j0 downward ibm128 0x2.602774p+0 : 0x3.e83779fe19911fa806cee83ab7p-8 : xfail:ibm128-libgcc inexact-ok
-= j0 tonearest ibm128 0x2.602774p+0 : 0x3.e83779fe19911fa806cee83ab7p-8 : inexact-ok
-= j0 towardzero ibm128 0x2.602774p+0 : 0x3.e83779fe19911fa806cee83ab7p-8 : xfail:ibm128-libgcc inexact-ok
-= j0 upward ibm128 0x2.602774p+0 : 0x3.e83779fe19911fa806cee83ab8p-8 : xfail:ibm128-libgcc inexact-ok
+j0 0x1.8bde7ep+5
+= j0 downward binary32 0x3.17bcfcp+4 : 0x7.a5f028p-16 : inexact-ok
+= j0 tonearest binary32 0x3.17bcfcp+4 : 0x7.a5f03p-16 : inexact-ok
+= j0 towardzero binary32 0x3.17bcfcp+4 : 0x7.a5f028p-16 : inexact-ok
+= j0 upward binary32 0x3.17bcfcp+4 : 0x7.a5f03p-16 : inexact-ok
+= j0 downward binary64 0x3.17bcfcp+4 : 0x7.a5f02c0a5b77cp-16 : inexact-ok
+= j0 tonearest binary64 0x3.17bcfcp+4 : 0x7.a5f02c0a5b77cp-16 : inexact-ok
+= j0 towardzero binary64 0x3.17bcfcp+4 : 0x7.a5f02c0a5b77cp-16 : inexact-ok
+= j0 upward binary64 0x3.17bcfcp+4 : 0x7.a5f02c0a5b78p-16 : inexact-ok
+= j0 downward intel96 0x3.17bcfcp+4 : 0x7.a5f02c0a5b77d0a8p-16 : inexact-ok
+= j0 tonearest intel96 0x3.17bcfcp+4 : 0x7.a5f02c0a5b77d0bp-16 : inexact-ok
+= j0 towardzero intel96 0x3.17bcfcp+4 : 0x7.a5f02c0a5b77d0a8p-16 : inexact-ok
+= j0 upward intel96 0x3.17bcfcp+4 : 0x7.a5f02c0a5b77d0bp-16 : inexact-ok
+= j0 downward m68k96 0x3.17bcfcp+4 : 0x7.a5f02c0a5b77d0a8p-16 : inexact-ok
+= j0 tonearest m68k96 0x3.17bcfcp+4 : 0x7.a5f02c0a5b77d0bp-16 : inexact-ok
+= j0 towardzero m68k96 0x3.17bcfcp+4 : 0x7.a5f02c0a5b77d0a8p-16 : inexact-ok
+= j0 upward m68k96 0x3.17bcfcp+4 : 0x7.a5f02c0a5b77d0bp-16 : inexact-ok
+= j0 downward binary128 0x3.17bcfcp+4 : 0x7.a5f02c0a5b77d0af558d6935f168p-16 : inexact-ok
+= j0 tonearest binary128 0x3.17bcfcp+4 : 0x7.a5f02c0a5b77d0af558d6935f168p-16 : inexact-ok
+= j0 towardzero binary128 0x3.17bcfcp+4 : 0x7.a5f02c0a5b77d0af558d6935f168p-16 : inexact-ok
+= j0 upward binary128 0x3.17bcfcp+4 : 0x7.a5f02c0a5b77d0af558d6935f16cp-16 : inexact-ok
+= j0 downward ibm128 0x3.17bcfcp+4 : 0x7.a5f02c0a5b77d0af558d6935fp-16 : inexact-ok
+= j0 tonearest ibm128 0x3.17bcfcp+4 : 0x7.a5f02c0a5b77d0af558d6935f2p-16 : inexact-ok
+= j0 towardzero ibm128 0x3.17bcfcp+4 : 0x7.a5f02c0a5b77d0af558d6935fp-16 : inexact-ok
+= j0 upward ibm128 0x3.17bcfcp+4 : 0x7.a5f02c0a5b77d0af558d6935f2p-16 : inexact-ok
j0 0x8.2f4ecp+124
= j0 downward binary32 0x8.2f4ecp+124 : 0xd.331efp-68 : inexact-ok
= j0 tonearest binary32 0x8.2f4ecp+124 : 0xd.331efp-68 : inexact-ok
@@ -795,31 +795,31 @@ y0 0x1p-10000
= y0 tonearest binary128 0x1p-10000 : -0x1.13cc92aab9d385d1d0f2693cb631p+12 : inexact-ok
= y0 towardzero binary128 0x1p-10000 : -0x1.13cc92aab9d385d1d0f2693cb631p+12 : inexact-ok
= y0 upward binary128 0x1p-10000 : -0x1.13cc92aab9d385d1d0f2693cb631p+12 : inexact-ok
-y0 0xd.3432bp-4
-= y0 downward binary32 0xd.3432bp-4 : -0xf.fdd88p-8 : inexact-ok
-= y0 tonearest binary32 0xd.3432bp-4 : -0xf.fdd87p-8 : inexact-ok
-= y0 towardzero binary32 0xd.3432bp-4 : -0xf.fdd87p-8 : inexact-ok
-= y0 upward binary32 0xd.3432bp-4 : -0xf.fdd87p-8 : inexact-ok
-= y0 downward binary64 0xd.3432bp-4 : -0xf.fdd871793bc78p-8 : inexact-ok
-= y0 tonearest binary64 0xd.3432bp-4 : -0xf.fdd871793bc7p-8 : inexact-ok
-= y0 towardzero binary64 0xd.3432bp-4 : -0xf.fdd871793bc7p-8 : inexact-ok
-= y0 upward binary64 0xd.3432bp-4 : -0xf.fdd871793bc7p-8 : inexact-ok
-= y0 downward intel96 0xd.3432bp-4 : -0xf.fdd871793bc71fap-8 : inexact-ok
-= y0 tonearest intel96 0xd.3432bp-4 : -0xf.fdd871793bc71f9p-8 : inexact-ok
-= y0 towardzero intel96 0xd.3432bp-4 : -0xf.fdd871793bc71f9p-8 : inexact-ok
-= y0 upward intel96 0xd.3432bp-4 : -0xf.fdd871793bc71f9p-8 : inexact-ok
-= y0 downward m68k96 0xd.3432bp-4 : -0xf.fdd871793bc71fap-8 : inexact-ok
-= y0 tonearest m68k96 0xd.3432bp-4 : -0xf.fdd871793bc71f9p-8 : inexact-ok
-= y0 towardzero m68k96 0xd.3432bp-4 : -0xf.fdd871793bc71f9p-8 : inexact-ok
-= y0 upward m68k96 0xd.3432bp-4 : -0xf.fdd871793bc71f9p-8 : inexact-ok
-= y0 downward binary128 0xd.3432bp-4 : -0xf.fdd871793bc71f92d6b137b44118p-8 : inexact-ok
-= y0 tonearest binary128 0xd.3432bp-4 : -0xf.fdd871793bc71f92d6b137b44118p-8 : inexact-ok
-= y0 towardzero binary128 0xd.3432bp-4 : -0xf.fdd871793bc71f92d6b137b4411p-8 : inexact-ok
-= y0 upward binary128 0xd.3432bp-4 : -0xf.fdd871793bc71f92d6b137b4411p-8 : inexact-ok
-= y0 downward ibm128 0xd.3432bp-4 : -0xf.fdd871793bc71f92d6b137b444p-8 : inexact-ok
-= y0 tonearest ibm128 0xd.3432bp-4 : -0xf.fdd871793bc71f92d6b137b44p-8 : inexact-ok
-= y0 towardzero ibm128 0xd.3432bp-4 : -0xf.fdd871793bc71f92d6b137b44p-8 : inexact-ok
-= y0 upward ibm128 0xd.3432bp-4 : -0xf.fdd871793bc71f92d6b137b44p-8 : inexact-ok
+y0 0x1.3d4e56p+6
+= y0 downward binary32 0x4.f53958p+4 : 0x1.b42862p-16 : inexact-ok
+= y0 tonearest binary32 0x4.f53958p+4 : 0x1.b42864p-16 : inexact-ok
+= y0 towardzero binary32 0x4.f53958p+4 : 0x1.b42862p-16 : inexact-ok
+= y0 upward binary32 0x4.f53958p+4 : 0x1.b42864p-16 : inexact-ok
+= y0 downward binary64 0x4.f53958p+4 : 0x1.b428630651a17p-16 : inexact-ok
+= y0 tonearest binary64 0x4.f53958p+4 : 0x1.b428630651a17p-16 : inexact-ok
+= y0 towardzero binary64 0x4.f53958p+4 : 0x1.b428630651a17p-16 : inexact-ok
+= y0 upward binary64 0x4.f53958p+4 : 0x1.b428630651a18p-16 : inexact-ok
+= y0 downward intel96 0x4.f53958p+4 : 0x1.b428630651a175acp-16 : inexact-ok
+= y0 tonearest intel96 0x4.f53958p+4 : 0x1.b428630651a175acp-16 : inexact-ok
+= y0 towardzero intel96 0x4.f53958p+4 : 0x1.b428630651a175acp-16 : inexact-ok
+= y0 upward intel96 0x4.f53958p+4 : 0x1.b428630651a175aep-16 : inexact-ok
+= y0 downward m68k96 0x4.f53958p+4 : 0x1.b428630651a175acp-16 : inexact-ok
+= y0 tonearest m68k96 0x4.f53958p+4 : 0x1.b428630651a175acp-16 : inexact-ok
+= y0 towardzero m68k96 0x4.f53958p+4 : 0x1.b428630651a175acp-16 : inexact-ok
+= y0 upward m68k96 0x4.f53958p+4 : 0x1.b428630651a175aep-16 : inexact-ok
+= y0 downward binary128 0x4.f53958p+4 : 0x1.b428630651a175ac20abf8104eeap-16 : inexact-ok
+= y0 tonearest binary128 0x4.f53958p+4 : 0x1.b428630651a175ac20abf8104eebp-16 : inexact-ok
+= y0 towardzero binary128 0x4.f53958p+4 : 0x1.b428630651a175ac20abf8104eeap-16 : inexact-ok
+= y0 upward binary128 0x4.f53958p+4 : 0x1.b428630651a175ac20abf8104eebp-16 : inexact-ok
+= y0 downward ibm128 0x4.f53958p+4 : 0x1.b428630651a175ac20abf8104e8p-16 : inexact-ok
+= y0 tonearest ibm128 0x4.f53958p+4 : 0x1.b428630651a175ac20abf8104fp-16 : inexact-ok
+= y0 towardzero ibm128 0x4.f53958p+4 : 0x1.b428630651a175ac20abf8104e8p-16 : inexact-ok
+= y0 upward ibm128 0x4.f53958p+4 : 0x1.b428630651a175ac20abf8104fp-16 : inexact-ok
y0 min
= y0 downward binary32 0x4p-128 : -0x3.7ac89cp+4 : inexact-ok
= y0 tonearest binary32 0x4p-128 : -0x3.7ac89cp+4 : inexact-ok
@@ -37,6 +37,330 @@ S04 = 1.1661400734e-09; /* 0x30a045e8 */
static const float zero = 0.0;
+#define FIRST_ZERO_J0 2.404825f
+
+#define SMALL_SIZE 64
+
+/* The following table contains successive zeros of j0 and degree-3 polynomial
+ approximations of j0 around these zeros: Pj[0] for the first zero (2.404825),
+ Pj[1] for the second one (5.520078), and so on. Each line contains:
+ {x0, xmid, x1, p0, p1, p2, p3}
+ where [x0,x1] is the interval around the zero, xmid is the binary32 number closest
+ to the zero, and p0+p1*x+p2*x^2+p3*x^3 is the approximation polynomial.
+ Each polynomial was generated using Remez' algorithm on the interval [x0,x1]
+ around the corresponding zero where the error is larger than 9 ulps for the
+ default code below. Degree 3 is enough to get an error less than 4 ulps.
+*/
+static float Pj[SMALL_SIZE][7] = {
+ { 0x2.63cb8cp+0, 0x2.67a2a4p+0, 0x2.6f5d28p+0, 0xf.26247p-28, -0x8.4e6d9p-4, 0x1.ba1c7p-4, 0xe.6c06ap-8 },
+/* The following polynomial was generated by Sollya. */
+ { 0x5.83abe8p+0, 0x5.8523d8p+0, 0x5.8a557p+0, 0x6.9205fp-28, 0x5.71b98p-4, -0x7.e3e798p-8, -0xd.87d1p-8 },
+ { 0x8.a66f1p+0, 0x8.a75abp+0, 0x8.a92a7p+0, 0x1.bcc1cap-24, -0x4.57de6p-4, 0x4.03debp-8, 0xb.44a4ap-8 },
+ { 0xb.c9905p+0, 0xb.caa2p+0, 0xb.ccc6bp+0, -0xf.2976fp-32, 0x3.b827ccp-4, -0x2.85fdb8p-8, -0x9.c5a97p-8 },
+ { 0xe.edb6ap+0, 0xe.ee50ap+0, 0xe.ef864p+0, -0x1.bd67d8p-28, -0x3.4e03ap-4, 0x1.c54b8ep-8, 0x8.bb70dp-8 },
+ { 0x1.2119fp+4, 0x1.212314p+4, 0x1.21375p+4, 0x1.62209cp-28, 0x3.00efecp-4, -0x1.5467fp-8, -0x7.f5a2p-8 },
+ { 0x1.535d28p+4, 0x1.5362dep+4, 0x1.536cbcp+4, -0x2.853f74p-24, -0x2.c5b274p-4, 0x1.0bab0ap-8, 0x7.5c05b8p-8 },
+ { 0x1.859df6p+4, 0x1.85a3bap+4, 0x1.85afcap+4, 0x2.19ed1cp-24, 0x2.96545cp-4, -0xd.995c9p-12, -0x6.e024cp-8 },
+ { 0x1.b7dfe6p+4, 0x1.b7e54ap+4, 0x1.b7ebccp+4, 0xe.959aep-28, -0x2.6f5594p-4, 0xb.55ff1p-12, 0x6.79cf58p-8 },
+ { 0x1.ea22bcp+4, 0x1.ea275ap+4, 0x1.ea2e2ep+4, 0x2.0c3964p-24, 0x2.4e80fcp-4, -0x9.a35abp-12, -0x6.234b08p-8 },
+ { 0x2.1c6638p+4, 0x2.1c69c4p+4, 0x2.1c6d7cp+4, -0x3.642554p-24, -0x2.325e48p-4, 0x8.534f1p-12, 0x5.d9048p-8 },
+ { 0x2.4ea8dcp+4, 0x2.4eac7p+4, 0x2.4eb39cp+4, 0x1.6c015ap-24, 0x2.19e7d8p-4, -0x7.4915f8p-12, -0x5.984698p-8 },
+ { 0x2.80ec2p+4, 0x2.80ef5p+4, 0x2.80f72cp+4, -0x4.b18c9p-28, -0x2.046174p-4, 0x6.720468p-12, 0x5.5f426p-8 },
+ { 0x2.b32e6p+4, 0x2.b33258p+4, 0x2.b33654p+4, -0x1.8b8792p-24, 0x1.f13fbp-4, -0x5.c149dp-12, -0x5.2c935p-8 },
+ { 0x2.e5736p+4, 0x2.e5758p+4, 0x2.e57894p+4, 0x3.a26e0cp-24, -0x1.e018dap-4, 0x5.2df918p-12, 0x4.ff0f68p-8 },
+ { 0x3.17b694p+4, 0x3.17b8c4p+4, 0x3.17bcecp+4, -0x2.18fabcp-24, 0x1.d09b22p-4, -0x4.b1c31p-12, -0x4.d5ecd8p-8 },
+ { 0x3.49f9d8p+4, 0x3.49fc1cp+4, 0x3.4a0084p+4, 0x3.2370e8p-24, -0x1.c28614p-4, 0x4.47bb78p-12, 0x4.b08458p-8 },
+ { 0x3.7c3d78p+4, 0x3.7c3f88p+4, 0x3.7c43ep+4, -0x5.9eae3p-28, 0x1.b5a622p-4, -0x3.ec892cp-12, -0x4.8e4d3p-8 },
+ { 0x3.ae80ap+4, 0x3.ae83p+4, 0x3.ae8528p+4, 0x2.9fa1e8p-24, -0x1.a9d184p-4, 0x3.9d2fa8p-12, 0x4.6edccp-8 },
+ { 0x3.e0c484p+4, 0x3.e0c688p+4, 0x3.e0c8a4p+4, 0x9.9ac67p-28, 0x1.9ee5eep-4, -0x3.57e9ep-12, -0x4.51d1e8p-8 },
+ { 0x4.1308f8p+4, 0x4.130a18p+4, 0x4.130c58p+4, 0xd.6ab94p-28, -0x1.94c6f6p-4, 0x3.1ac03p-12, 0x4.36e4bp-8 },
+ { 0x4.454cp+4, 0x4.454dbp+4, 0x4.45504p+4, -0x4.4cb2d8p-24, 0x1.8b5cccp-4, -0x2.e477ap-12, -0x4.1dd858p-8 },
+ { 0x4.778f6p+4, 0x4.779158p+4, 0x4.779368p+4, -0x4.4aa8c8p-24, -0x1.829356p-4, 0x2.b4726cp-12, 0x4.0676dp-8 },
+ { 0x4.a9d3ep+4, 0x4.a9d5p+4, 0x4.a9d6cp+4, 0x2.077c38p-24, 0x1.7a597ep-4, -0x2.891dbcp-12, -0x3.f09154p-8 },
+ { 0x4.dc175p+4, 0x4.dc18bp+4, 0x4.dc1a08p+4, -0x2.6a6cd8p-24, -0x1.72a09ap-4, 0x2.62315cp-12, 0x3.dc034p-8 },
+ { 0x5.0e5bb8p+4, 0x5.0e5c6p+4, 0x5.0e5dbp+4, -0x5.08287p-24, 0x1.6b5c06p-4, -0x2.3ec48p-12, -0x3.c8a91cp-8 },
+ { 0x5.409ebp+4, 0x5.40a02p+4, 0x5.40a188p+4, -0x3.4598dcp-24, -0x1.6480c4p-4, 0x2.1f1798p-12, 0x3.b667ccp-8 },
+ { 0x5.72e268p+4, 0x5.72e3ep+4, 0x5.72e54p+4, 0x5.4e74bp-24, 0x1.5e0544p-4, -0x2.021254p-12, -0x3.a5248cp-8 },
+ { 0x5.a5263p+4, 0x5.a527ap+4, 0x5.a528d8p+4, -0x2.05751cp-24, -0x1.57e12p-4, 0x1.e7643ep-12, 0x3.94c994p-8 },
+ { 0x5.d76acp+4, 0x5.d76b68p+4, 0x5.d76ccp+4, 0x4.c5e278p-24, 0x1.520ceep-4, -0x1.cf1d4ep-12, -0x3.85428cp-8 },
+ { 0x6.09ae88p+4, 0x6.09af3p+4, 0x6.09b0bp+4, -0x3.50e62cp-24, -0x1.4c822p-4, 0x1.b8ab9ap-12, 0x3.767f34p-8 },
+ { 0x6.3bf21p+4, 0x6.3bf2f8p+4, 0x6.3bf418p+4, -0x1.c39f38p-24, 0x1.473ae6p-4, -0x1.a3dccep-12, -0x3.68700cp-8 },
+ { 0x6.6e362p+4, 0x6.6e36c8p+4, 0x6.6e3818p+4, -0x1.9245b6p-28, -0x1.42320ap-4, 0x1.90d5f2p-12, 0x3.5b0634p-8 },
+ { 0x6.a079dp+4, 0x6.a07a98p+4, 0x6.a07b78p+4, -0x1.0acf8p-24, 0x1.3d62e6p-4, -0x1.7f1e42p-12, -0x3.4e3678p-8 },
+ { 0x6.d2bda8p+4, 0x6.d2be68p+4, 0x6.d2bfb8p+4, 0x4.cd92d8p-24, -0x1.38c94ap-4, 0x1.6e94e2p-12, 0x3.41f4acp-8 },
+ { 0x7.05018p+4, 0x7.05024p+4, 0x7.0503p+4, -0x1.37bf8ap-24, 0x1.34617p-4, -0x1.5f6a22p-12, -0x3.3637c8p-8 },
+ { 0x7.37459p+4, 0x7.374618p+4, 0x7.3747ap+4, -0x1.8f62dep-28, -0x1.3027fp-4, 0x1.51357ap-12, 0x3.2af594p-8 },
+ { 0x7.69892p+4, 0x7.6989fp+4, 0x7.698b98p+4, -0x9.81e79p-28, 0x1.2c19b4p-4, -0x1.43e0aep-12, -0x3.20271p-8 },
+ { 0x7.9bccf8p+4, 0x7.9bcdc8p+4, 0x7.9bceap+4, 0x3.103b3p-24, -0x1.2833eep-4, 0x1.37580ep-12, 0x3.15c484p-8 },
+ { 0x7.ce10dp+4, 0x7.ce11a8p+4, 0x7.ce136p+4, 0x2.07b058p-24, 0x1.24740ap-4, -0x1.2bd334p-12, -0x3.0bc628p-8 },
+ { 0x8.0054cp+4, 0x8.00558p+4, 0x8.00562p+4, 0x3.87576cp-24, -0x1.20d7b6p-4, 0x1.20af6cp-12, 0x3.022738p-8 },
+ { 0x8.32994p+4, 0x8.32996p+4, 0x8.329a4p+4, -0x1.691ecp-24, 0x1.1d5ccap-4, -0x1.167022p-12, -0x2.f8e084p-8 },
+ { 0x8.64dcdp+4, 0x8.64dd4p+4, 0x8.64dd9p+4, 0x9.b406dp-28, -0x1.1a015p-4, 0x1.0cbfd2p-12, 0x2.efee34p-8 },
+ { 0x8.97209p+4, 0x8.97212p+4, 0x8.9721bp+4, -0xf.bfd8fp-28, 0x1.16c37ap-4, -0x1.039356p-12, -0x2.e74a78p-8 },
+ { 0x8.c9649p+4, 0x8.c965p+4, 0x8.c965bp+4, 0x2.6d50c8p-24, -0x1.13a19ep-4, 0xf.ae0ap-16, 0x2.def13p-8 },
+ { 0x8.fba89p+4, 0x8.fba8ep+4, 0x8.fba9dp+4, -0x4.d475c8p-24, 0x1.109a32p-4, -0xf.29e9cp-16, -0x2.d6de4cp-8 },
+ { 0x9.2dec7p+4, 0x9.2deccp+4, 0x9.2dedp+4, 0x8.1982p-24, -0x1.0dabc8p-4, 0xe.ac514p-16, 0x2.cf0e6p-8 },
+ { 0x9.60306p+4, 0x9.6030bp+4, 0x9.60318p+4, 0x4.864518p-24, 0x1.0ad51p-4, -0xe.3d1fbp-16, -0x2.c77d28p-8 },
+ { 0x9.92743p+4, 0x9.92749p+4, 0x9.9274ep+4, 0x6.8917a8p-28, -0x1.0814d4p-4, 0xd.cb25ap-16, 0x2.c0283p-8 },
+ { 0x9.c4b81p+4, 0x9.c4b87p+4, 0x9.c4b8ep+4, -0x5.fa18fp-24, 0x1.0569fp-4, -0xd.5e6d8p-16, -0x2.b90bep-8 },
+ { 0x9.f6fc2p+4, 0x9.f6fc6p+4, 0x9.f6fcep+4, -0x4.0e5c98p-24, -0x1.02d354p-4, 0xc.feb37p-16, 0x2.b2259p-8 },
+ { 0xa.293f8p+4, 0xa.29404p+4, 0xa.29408p+4, -0x2.c3ddbp-24, 0x1.005004p-4, -0xc.9b641p-16, -0x2.ab72fp-8 },
+ { 0xa.5b83bp+4, 0xa.5b843p+4, 0xa.5b852p+4, -0x5.d052p-24, -0xf.ddf16p-8, 0xc.444dcp-16, 0x2.a4f0d4p-8 },
+ { 0xa.8dc7ap+4, 0xa.8dc81p+4, 0xa.8dc87p+4, -0x2.0b97dcp-24, 0xf.b7fafp-8, -0xb.e93a7p-16, -0x2.9e9dccp-8 },
+ { 0xa.c00b4p+4, 0xa.c00cp+4, 0xa.c00c4p+4, -0x5.4aab5p-24, -0xf.930fep-8, 0xb.99b61p-16, 0x2.98774p-8 },
+ { 0xa.f24f5p+4, 0xa.f24fep+4, 0xa.f2509p+4, -0x3.6dadd8p-24, 0xf.6f245p-8, -0xb.45e12p-16, -0x2.927b08p-8 },
+ { 0xb.24939p+4, 0xb.2493dp+4, 0xb.24948p+4, -0x2.d7e038p-24, -0xf.4c2cep-8, 0xa.fd076p-16, 0x2.8ca788p-8 },
+ { 0xb.56d73p+4, 0xb.56d7bp+4, 0xb.56d82p+4, -0x6.977a1p-24, 0xf.2a1fp-8, -0xa.af99ap-16, -0x2.86fb2p-8 },
+ { 0xb.891b3p+4, 0xb.891bap+4, 0xb.891c7p+4, 0x1.3cc95ep-24, -0xf.08f0ap-8, 0xa.6ca59p-16, 0x2.8173b8p-8 },
+ { 0xb.bb5f5p+4, 0xb.bb5f9p+4, 0xb.bb5fdp+4, 0x3.a4921p-24, 0xe.e8986p-8, -0xa.2c5b8p-16, -0x2.7c1024p-8 },
+ { 0xb.eda37p+4, 0xb.eda37p+4, 0xb.eda3bp+4, 0x6.b45a7p-24, -0xe.c90d8p-8, 0x9.e7307p-16, 0x2.76ceacp-8 },
+ { 0xc.1fe71p+4, 0xc.1fe76p+4, 0xc.1fe87p+4, -0x2.8f34a4p-24, 0xe.aa478p-8, -0x9.abd8fp-16, -0x2.71adecp-8 },
+ { 0xc.522aep+4, 0xc.522b5p+4, 0xc.522c4p+4, -0x1.325968p-24, -0xe.8c3eap-8, 0x9.72bf7p-16, 0x2.6cacd4p-8 },
+ { 0xc.846eep+4, 0xc.846f4p+4, 0xc.846fap+4, 0x4.96b808p-24, 0xe.6eeb5p-8, -0x9.3bc53p-16, -0x2.67ca04p-8 }
+};
+
+/* Argument reduction mod 2pi: T[i] ~ 2^i mod (2*pi). */
+static double T[128] = {
+ 0x1p+0,
+ 0x2p+0,
+ -0x2.487ed5110b462p+0,
+ 0x1.b7812aeef4b9fp+0,
+ -0x2.d97c7f3321d24p+0,
+ 0x9.585d6aac7a1a8p-4,
+ 0x1.2b0bad558f435p+0,
+ 0x2.56175aab1e86ap+0,
+ -0x1.9c501fbace38dp+0,
+ 0x3.0fde959b6ed46p+0,
+ -0x2.8c1a9da2d9d3cp-4,
+ -0x5.18353b45b3a78p-4,
+ -0xa.306a768b674fp-4,
+ -0x1.460d4ed16ce9ep+0,
+ -0x2.8c1a9da2d9d3cp+0,
+ 0x1.304999cb579e8p+0,
+ 0x2.60933396af3dp+0,
+ -0x1.87586de3accc3p+0,
+ -0x3.0eb0dbc759986p+0,
+ 0x2.b1d1d8258155p-4,
+ 0x5.63a3b04b02aap-4,
+ 0xa.c74760960554p-4,
+ 0x1.58e8ec12c0aa8p+0,
+ 0x2.b1d1d8258155p+0,
+ -0xe.4db24c6089bf8p-4,
+ -0x1.c9b6498c1137fp+0,
+ 0x2.b51241f8e8d64p+0,
+ -0xd.e5a511f3999bp-4,
+ -0x1.bcb4a23e73336p+0,
+ 0x2.cf15909424df4p+0,
+ -0xa.a53b3e8c1877p-4,
+ -0x1.54a767d1830eep+0,
+ -0x2.a94ecfa3061dcp+0,
+ 0xf.5e135caff0a78p-4,
+ 0x1.ebc26b95fe14fp+0,
+ -0x2.70f9fde50f1c4p+0,
+ 0x1.668ad946ed0dap+0,
+ 0x2.cd15b28dda1b4p+0,
+ -0xa.e536ff5570fbp-4,
+ -0x1.5ca6dfeaae1f6p+0,
+ -0x2.b94dbfd55c3ecp+0,
+ 0xd.5e3556652c89p-4,
+ 0x1.abc6aacca5912p+0,
+ -0x2.f0f17f77c023ep+0,
+ 0x6.69bd6218afe64p-4,
+ 0xc.d37ac4315fcc8p-4,
+ 0x1.9a6f58862bf99p+0,
+ -0x3.13a02404b352ep+0,
+ 0x2.13e8d07a4a046p-4,
+ 0x4.27d1a0f49408cp-4,
+ 0x8.4fa341e928118p-4,
+ 0x1.09f4683d25023p+0,
+ 0x2.13e8d07a4a046p+0,
+ -0x2.20ad341c773d4p+0,
+ 0x2.07246cd81ccb8p+0,
+ -0x2.3a35fb60d1af2p+0,
+ 0x1.d412de4f67e7ep+0,
+ -0x2.a05918723b764p+0,
+ 0x1.07cca42c94599p+0,
+ 0x2.0f99485928b32p+0,
+ -0x2.294c445eb9dfep+0,
+ 0x1.f5e64c5397865p+0,
+ -0x2.5cb23c69dc396p+0,
+ 0x1.8f1a5c3d52d34p+0,
+ 0x3.1e34b87aa5a68p+0,
+ -0xc.15641bbff90bp-8,
+ -0x1.82ac8377ff216p-4,
+ -0x3.055906effe42cp-4,
+ -0x6.0ab20ddffc858p-4,
+ -0xc.15641bbff90bp-4,
+ -0x1.82ac8377ff216p+0,
+ -0x3.055906effe42cp+0,
+ 0x3.dccc7310ec09ap-4,
+ 0x7.b998e621d8134p-4,
+ 0xf.7331cc43b0268p-4,
+ 0x1.ee6639887604dp+0,
+ -0x2.6bb262001f3c6p+0,
+ 0x1.711a1110cccd4p+0,
+ 0x2.e2342221999a8p+0,
+ -0x8.41690cdd81128p-4,
+ -0x1.082d219bb0225p+0,
+ -0x2.105a43376044ap+0,
+ 0x2.27ca4ea24abcep+0,
+ -0x1.f8ea37cc75cc6p+0,
+ 0x2.56aa65781fad6p+0,
+ -0x1.9b2a0a20cbeb6p+0,
+ 0x3.122ac0cf736f6p+0,
+ -0x2.4295372246764p-4,
+ -0x4.852a6e448cec8p-4,
+ -0x9.0a54dc8919d9p-4,
+ -0x1.214a9b91233b2p+0,
+ -0x2.4295372246764p+0,
+ 0x1.c35466cc7e59ap+0,
+ -0x2.c1d607780e92ep+0,
+ 0xc.4d2c620ee206p-4,
+ 0x1.89a58c41dc40cp+0,
+ 0x3.134b1883b8818p+0,
+ -0x2.1e8a4099a4324p-4,
+ -0x4.3d14813348648p-4,
+ -0x8.7a29026690c9p-4,
+ -0x1.0f45204cd2192p+0,
+ -0x2.1e8a4099a4324p+0,
+ 0x2.0b6a53ddc2e18p+0,
+ -0x2.31aa2d558583p+0,
+ 0x1.e52a7a6600402p+0,
+ -0x2.7e29e0450ac5cp+0,
+ 0x1.4c2b1486f5ba8p+0,
+ 0x2.9856290deb75p+0,
+ -0x1.17d282f5345bfp+0,
+ -0x2.2fa505ea68b7ep+0,
+ 0x1.e934c93c39d64p+0,
+ -0x2.761542989799ap+0,
+ 0x1.5c544fdfdc12fp+0,
+ 0x2.b8a89fbfb825ep+0,
+ -0xd.72d95919afa58p-4,
+ -0x1.ae5b2b2335f4bp+0,
+ 0x2.ebc87eca9f5cap+0,
+ -0x7.0edd77bcc8ccp-4,
+ -0xe.1dbaef799198p-4,
+ -0x1.c3b75def3233p+0,
+ 0x2.c1101932a6e02p+0,
+ -0xc.65ea2abbd85fp-4,
+ -0x1.8cbd45577b0bep+0,
+ -0x3.197a8aaef617cp+0,
+ 0x1.589bfb31f1687p-4,
+ 0x2.b137f663e2d0ep-4,
+ 0x5.626fecc7c5a1cp-4,
+ 0xa.c4dfd98f8b438p-4
+};
+
+/* Return h such that x - pi/4 mod (2*pi) ~ h. */
+static double
+reduce_mod_twopi (double x)
+{
+ double sign = 1, h;
+ if (x < 0)
+ {
+ x = -x;
+ sign = -1;
+ }
+ h = 0; /* Invariant is x+h mod (2*pi). */
+ while (x >= 4)
+ {
+ int i = ilogb (x);
+ x -= ldexp (1.0f, i);
+ h += T[i];
+ }
+ /* Add the remainder x. */
+ h += x;
+ /* Subtract pi/4. */
+ double piover4 = 0xc.90fdaa22168cp-4;
+ h -= piover4;
+ /* Reduce mod 2*pi. */
+ double twopi = 0x6.487ed5110b46p+0;
+ while (fabs (h) > twopi * 0.5)
+ {
+ if (h > 0)
+ h -= twopi;
+ else
+ h += twopi;
+ }
+ return sign * h;
+}
+
+/* Formula page 5 of https://www.cl.cam.ac.uk/~jrh13/papers/bessel.pdf:
+ j0f(x) ~ sqrt(2/(pi*x))*beta0(x)*cos(x-pi/4-alpha0(x))
+ where beta0(x) = 1 - 16/x^2 + 53/(512*x^4)
+ and alpha0(x) = 1/(8*x) - 25/(384*x^3). */
+static float
+j0f_asympt (float x)
+{
+ /* The following code fails to give an error <= 9 ulps in only one case,
+ for which we tabulate the result. */
+ if (x == 0x1.4665d2p+24f)
+ return 0xa.50206p-52f;
+ double y = 1.0 / (double) x;
+ double y2 = y * y;
+ double beta0 = 1.0f + y2 * (-0x1p-4f + 0x1.a8p-4 * y2);
+ double alpha0 = y * (0x2p-4 - 0x1.0aaaaap-4 * y2);
+ double h;
+ h = reduce_mod_twopi ((double) x);
+ /* Subtract alpha0. */
+ h -= alpha0;
+ /* Now reduce mod pi/2. */
+ double piover2 = 0x1.921fb54442d18p+0;
+ int n = 0;
+ while (fabs (h) > piover2 / 2)
+ {
+ if (h > 0)
+ {
+ h -= piover2;
+ n ++;
+ }
+ else
+ {
+ h += piover2;
+ n --;
+ }
+ }
+ /* Now x - pi/4 - alpha0 = h + n*pi/2 mod (2*pi). */
+ float xr = (float) h;
+ n = n & 3;
+ float cst = 0xc.c422ap-4; /* sqrt(2/pi) rounded to nearest */
+ float t = cst / sqrtf (x) * (float) beta0;
+ if (n == 0)
+ return t * cosf (xr);
+ else if (n == 2) /* cos(x+pi) = -cos(x) */
+ return -t * cosf (xr);
+ else if (n == 1) /* cos(x+pi/2) = -sin(x) */
+ return -t * sinf (xr);
+ else /* cos(x+3pi/2) = sin(x) */
+ return t * sinf (xr);
+}
+
+/* Special code for x near a root of j0.
+ z is the value computed by the generic code.
+ For small x, we use a polynomial approximating j0 around its root.
+ For large x, we use an asymptotic formula (j0f_asympt). */
+static float
+j0f_near_root (float x, float z)
+{
+ float index_f;
+ int index;
+ index_f = roundf ((x - FIRST_ZERO_J0) / (float) M_PI);
+ /* j0f_asympt fails to give an error <= 9 ulps for x=0x1.324e92p+7 (index 48)
+ thus we can't reduce SMALL_SIZE below 49. */
+ if (index_f >= SMALL_SIZE)
+ return j0f_asympt (x);
+ index = (int) index_f;
+ float *p = Pj[index];
+ float x0 = p[0];
+ float x1 = p[2];
+ /* If we are not in the interval [x0,x1] around xmid, we return the value z. */
+ if (! (x0 <= x && x <= x1))
+ return z;
+ float xmid = p[1];
+ float y = x - xmid;
+ return p[3] + y * (p[4] + y * (p[5] + y * p[6]));
+}
+
float
__ieee754_j0f(float x)
{
@@ -51,36 +375,27 @@ __ieee754_j0f(float x)
__sincosf (x, &s, &c);
ss = s-c;
cc = s+c;
- if(ix<0x7f000000) { /* make sure x+x not overflow */
- z = -__cosf(x+x);
- if ((s*c)<zero) cc = z/ss;
- else ss = z/cc;
- } else {
- /* We subtract (exactly) a value x0 such that
- cos(x0)+sin(x0) is very near to 0, and use the identity
- sin(x-x0) = sin(x)*cos(x0)-cos(x)*sin(x0) to get
- sin(x) + cos(x) with extra accuracy. */
- float x0 = 0xe.d4108p+124f;
- float y = x - x0; /* exact */
- /* sin(y) = sin(x)*cos(x0)-cos(x)*sin(x0) */
- z = __sinf (y);
- float eps = 0x1.5f263ep-24f;
- /* cos(x0) ~ -sin(x0) + eps */
- z += eps * __cosf (x);
- /* now z ~ (sin(x)-cos(x))*cos(x0) */
- float cosx0 = -0xb.504f3p-4f;
- cc = z / cosx0;
- }
+ if (ix >= 0x7f000000) /* x >= 2^127: use asymptotic expansion. */
+ return j0f_asympt (x);
+ /* Now we are sure that x+x cannot overflow. */
+ z = -__cosf(x+x);
+ if ((s*c)<zero) cc = z/ss;
+ else ss = z/cc;
/*
* j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
* y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
*/
- if(ix>0x5c000000) z = (invsqrtpi*cc)/sqrtf(x);
- else {
- u = pzerof(x); v = qzerof(x);
- z = invsqrtpi*(u*cc-v*ss)/sqrtf(x);
- }
- return z;
+ if (ix <= 0x5c000000)
+ {
+ u = pzerof(x); v = qzerof(x);
+ cc = u*cc-v*ss;
+ }
+ z = (invsqrtpi * cc) / sqrtf(x);
+ float magic = 0x1.8p+20; /* 2^20 + 2^19 */
+ if (magic + cc != magic) /* Most likely. */
+ return z;
+ else /* |cc| <= 2^-4 */
+ return j0f_near_root (x, z);
}
if(ix<0x39000000) { /* |x| < 2**-13 */
math_force_eval(huge+x); /* raise inexact if x != 0 */
@@ -112,6 +427,168 @@ v02 = 7.6006865129e-05, /* 0x389f65e0 */
v03 = 2.5915085189e-07, /* 0x348b216c */
v04 = 4.4111031494e-10; /* 0x2ff280c2 */
+#define FIRST_ZERO_Y0 0.893576f
+
+#define SMALL_SIZE 64
+
+/* The following table contains successive zeros of y0 and degree-3 polynomial
+ approximations of y0 around these zeros: Py[0] for the first zero (0.893576),
+ Py[1] for the second one (3.957678), and so on. Each line contains:
+ {x0, xmid, x1, p0, p1, p2, p3}
+ where [x0,x1] is the interval around the zero, xmid is the binary32 number closest
+ to the zero, and p0+p1*x+p2*x^2+p3*x^3 is the approximation polynomial.
+ Each polynomial was generated using Remez' algorithm on the interval [x0,x1]
+ around the corresponding zero where the error is larger than 9 ulps for the
+ default code below. Degree 3 is enough to get an error less than 4 ulps.
+*/
+static float Py[SMALL_SIZE][7] = {
+ /* For the first root we use a degree-4 polynomial since degree 3 is not enough,
+ where the coefficient of degree 4 is hard-coded in y0f_near_root(). */
+ { 0xd.bd613p-4, 0xe.4c176p-4, 0xe.e0897p-4, 0x3.274468p-28, 0xe.121b7p-4, -0x7.df8eap-4, 0x3.88cc2p-4 },
+ { 0x3.f2af8cp+0, 0x3.f52a68p+0, 0x3.fa1fa4p+0, 0xa.f1f83p-28, -0x6.70d098p-4, 0xd.04dc4p-8, 0xe.f2a5p-8 },
+ { 0x7.1464ap+0, 0x7.16077p+0, 0x7.191dap+0, -0x5.e2a51p-28, 0x4.cd3328p-4, -0x5.6bbb5p-8, -0xc.48cfbp-8 },
+ { 0xa.37ec2p+0, 0xa.38ebap+0, 0xa.3ad94p+0, -0x1.4b0aeep-24, -0x3.fec6b8p-4, 0x3.206ccp-8, 0xa.72401p-8 },
+ { 0xd.5bd7dp+0, 0xd.5c70ep+0, 0xd.5d94ap+0, -0x8.179d7p-28, 0x3.7e6544p-4, -0x2.178554p-8, -0x9.35f5bp-8 },
+ { 0x1.07f9aap+4, 0x1.0803c8p+4, 0x1.08170cp+4, -0x2.5b8078p-24, -0x3.24b868p-4, 0x1.86265ep-8, 0x8.51de2p-8 },
+ { 0x1.3a3d44p+4, 0x1.3a42cep+4, 0x1.3a4d8ap+4, 0x1.cd304ap-28, 0x2.e189ecp-4, -0x1.2c673ap-8, -0x7.a4726p-8 },
+ { 0x1.6c7d5ep+4, 0x1.6c833p+4, 0x1.6c99fp+4, -0x6.c63f1p-28, -0x2.acc9a8p-4, 0xf.077a1p-12, 0x7.1aba98p-8 },
+ { 0x1.9ebec4p+4, 0x1.9ec47p+4, 0x1.9ed016p+4, 0x1.e53838p-24, 0x2.81f2p-4, -0xc.61ccp-12, -0x6.aaa99p-8 },
+ { 0x1.d1008ep+4, 0x1.d10644p+4, 0x1.d10cf2p+4, 0x1.636feep-24, -0x2.5e40dcp-4, 0xa.6dfp-12, 0x6.4cd5a8p-8 },
+ { 0x2.0344f8p+4, 0x2.034884p+4, 0x2.034d4p+4, -0x4.04e1bp-28, 0x2.3febd8p-4, -0x8.f0ff9p-12, -0x5.fcd088p-8 },
+ { 0x2.358778p+4, 0x2.358b1p+4, 0x2.359224p+4, 0x3.6063d8p-24, -0x2.25baacp-4, 0x7.c6a088p-12, 0x5.b78ff8p-8 },
+ { 0x2.67ca1p+4, 0x2.67cdd8p+4, 0x2.67d434p+4, -0x3.f39ebcp-24, 0x2.0ed04cp-4, -0x6.d7eaf8p-12, -0x5.7aeaa8p-8 },
+ { 0x2.9a0d0cp+4, 0x2.9a10dp+4, 0x2.9a1828p+4, -0x4.ea23p-28, -0x1.fa8b4p-4, 0x6.158438p-12, 0x5.45324p-8 },
+ { 0x2.cc51ecp+4, 0x2.cc53e8p+4, 0x2.cc580cp+4, -0x3.5df0c8p-24, 0x1.e8727ep-4, -0x5.7460d8p-12, -0x5.1536p-8 },
+ { 0x2.fe94f8p+4, 0x2.fe972p+4, 0x2.fe9b18p+4, 0x1.1ef09ep-24, -0x1.d8293ap-4, 0x4.ed6058p-12, 0x4.e9fcc8p-8 },
+ { 0x3.30d8b8p+4, 0x3.30da7p+4, 0x3.30debcp+4, 0x1.b70cecp-24, 0x1.c967p-4, -0x4.7ad838p-12, -0x4.c2c9d8p-8 },
+ { 0x3.631b94p+4, 0x3.631ddp+4, 0x3.632244p+4, 0x1.abaadcp-24, -0x1.bbf246p-4, 0x4.187ba8p-12, 0x4.9f09f8p-8 },
+ { 0x3.955f7cp+4, 0x3.956144p+4, 0x3.9565fcp+4, 0x1.63989ep-24, 0x1.af9cb4p-4, -0x3.c397f8p-12, -0x4.7e4038p-8 },
+ { 0x3.c7a2bp+4, 0x3.c7a4c4p+4, 0x3.c7a878p+4, -0x1.68a8ecp-24, -0x1.a4407ep-4, 0x3.797fdcp-12, 0x4.600d3p-8 },
+ { 0x3.f9e62cp+4, 0x3.f9e85p+4, 0x3.f9ea7p+4, 0x1.e1bb5p-24, 0x1.99be74p-4, -0x3.38739cp-12, -0x4.441c5p-8 },
+ { 0x4.2c2a1p+4, 0x4.2c2be8p+4, 0x4.2c2e7p+4, -0x5.5bbcfp-24, -0x1.8ffc9ap-4, 0x2.ff0f5cp-12, 0x4.2a266p-8 },
+ { 0x4.5e6d98p+4, 0x4.5e6f8p+4, 0x4.5e71c8p+4, -0x4.9e34a8p-24, 0x1.86e51cp-4, -0x2.cba284p-12, -0x4.11f568p-8 },
+ { 0x4.90b17p+4, 0x4.90b328p+4, 0x4.90b5ap+4, 0x1.966706p-24, -0x1.7e657p-4, 0x2.9e0d44p-12, 0x3.fb56a4p-8 },
+ { 0x4.c2f59p+4, 0x4.c2f6d8p+4, 0x4.c2fac8p+4, 0x3.4b3b68p-24, 0x1.766dc2p-4, -0x2.752fbp-12, -0x3.e61fcp-8 },
+ { 0x4.f53968p+4, 0x4.f53a88p+4, 0x4.f53cb8p+4, 0x6.a99008p-28, -0x1.6ef07ep-4, 0x2.501294p-12, 0x3.d230ep-8 },
+ { 0x5.277dp+4, 0x5.277e4p+4, 0x5.278108p+4, -0x7.a9fa6p-32, 0x1.67e1dap-4, -0x2.2e9388p-12, -0x3.bf663cp-8 },
+ { 0x5.59c0e8p+4, 0x5.59c2p+4, 0x5.59c398p+4, -0x5.026808p-24, -0x1.613798p-4, 0x2.104558p-12, 0x3.ada76cp-8 },
+ { 0x5.8c0498p+4, 0x5.8c05cp+4, 0x5.8c0898p+4, 0x4.46aa2p-24, 0x1.5ae8c2p-4, -0x1.f474eep-12, -0x3.9cda1cp-8 },
+ { 0x5.be48dp+4, 0x5.be498p+4, 0x5.be4aap+4, 0x1.5cfccp-24, -0x1.54ed76p-4, 0x1.dad812p-12, 0x3.8cec8p-8 },
+ { 0x5.f08c08p+4, 0x5.f08d48p+4, 0x5.f08ecp+4, -0xb.4dc4cp-28, 0x1.4f3ebcp-4, -0x1.c38338p-12, -0x3.7dc9dp-8 },
+ { 0x6.22d05p+4, 0x6.22d11p+4, 0x6.22d428p+4, 0x3.f5343p-24, -0x1.49d668p-4, 0x1.ade97cp-12, 0x3.6f610cp-8 },
+ { 0x6.55137p+4, 0x6.5514ep+4, 0x6.551638p+4, -0x6.e6f32p-28, 0x1.44aefap-4, -0x1.9a2d3ep-12, -0x3.61a778p-8 },
+ { 0x6.8757e8p+4, 0x6.8758bp+4, 0x6.8759c8p+4, 0x1.f359c2p-28, -0x1.3fc386p-4, 0x1.87d25cp-12, 0x3.548be4p-8 },
+ { 0x6.b99bp+4, 0x6.b99c8p+4, 0x6.b99e2p+4, -0x2.9de748p-24, 0x1.3b0fa4p-4, -0x1.76b5aap-12, -0x3.48048p-8 },
+ { 0x6.ebdfb8p+4, 0x6.ebe058p+4, 0x6.ebe1bp+4, -0x2.24ccc8p-24, -0x1.368f5cp-4, 0x1.67061p-12, 0x3.3c0608p-8 },
+ { 0x7.1e2368p+4, 0x7.1e243p+4, 0x7.1e25bp+4, 0x4.7dcea8p-24, 0x1.323f16p-4, -0x1.5858c4p-12, -0x3.3087acp-8 },
+ { 0x7.50673p+4, 0x7.506808p+4, 0x7.5069ap+4, -0x4.b538p-24, -0x1.2e1b98p-4, 0x1.4a9624p-12, 0x3.258078p-8 },
+ { 0x7.82ab38p+4, 0x7.82abep+4, 0x7.82ad78p+4, 0x3.09dc4cp-24, 0x1.2a21ecp-4, -0x1.3da94p-12, -0x3.1ae88cp-8 },
+ { 0x7.b4eeep+4, 0x7.b4efb8p+4, 0x7.b4f158p+4, 0x4.d5a58p-28, -0x1.264f66p-4, 0x1.317f8cp-12, 0x3.10b8fcp-8 },
+ { 0x7.e732cp+4, 0x7.e73398p+4, 0x7.e73548p+4, 0x3.f4c44cp-24, 0x1.22a192p-4, -0x1.265128p-12, -0x3.06eb08p-8 },
+ { 0x8.1976bp+4, 0x8.19777p+4, 0x8.19783p+4, 0x2.4beae8p-24, -0x1.1f1634p-4, 0x1.1b7d2ap-12, 0x2.fd7934p-8 },
+ { 0x8.4bbbp+4, 0x8.4bbb5p+4, 0x8.4bbcep+4, -0xd.a581ep-28, 0x1.1bab3cp-4, -0x1.1186d6p-12, -0x2.f45cep-8 },
+ { 0x8.7dfe8p+4, 0x8.7dff3p+4, 0x8.7dffbp+4, 0xa.7c0bdp-28, -0x1.185eccp-4, 0x1.0819c2p-12, 0x2.eb92ccp-8 },
+ { 0x8.b042bp+4, 0x8.b0431p+4, 0x8.b043dp+4, -0x1.9452ecp-24, 0x1.152f26p-4, -0xf.f2b59p-16, -0x2.e314bp-8 },
+ { 0x8.e2868p+4, 0x8.e286fp+4, 0x8.e287cp+4, 0x3.83b7b8p-24, -0x1.121ab2p-4, 0xf.6b21p-16, 0x2.dadf34p-8 },
+ { 0x9.14ca8p+4, 0x9.14cadp+4, 0x9.14cbbp+4, -0x6.5ca3a8p-24, 0x1.0f1ff2p-4, -0xe.ea544p-16, -0x2.d2ee28p-8 },
+ { 0x9.470e6p+4, 0x9.470ecp+4, 0x9.470fp+4, -0x6.bb61ap-24, -0x1.0c3d8ap-4, 0xe.7833fp-16, 0x2.cb3e2cp-8 },
+ { 0x9.79524p+4, 0x9.7952ap+4, 0x9.79539p+4, 0x2.2438p-24, 0x1.097236p-4, -0xe.03747p-16, -0x2.c3cb48p-8 },
+ { 0x9.ab95cp+4, 0x9.ab968p+4, 0x9.ab971p+4, 0x3.1e0054p-24, -0x1.06bcc8p-4, 0xd.94272p-16, 0x2.bc9328p-8 },
+ { 0x9.ddd9fp+4, 0x9.ddda7p+4, 0x9.dddb5p+4, 0x7.46c908p-24, 0x1.041c28p-4, -0xd.320e5p-16, -0x2.b5920cp-8 },
+ { 0xa.101d7p+4, 0xa.101e5p+4, 0xa.101f3p+4, -0xb.4f158p-28, -0x1.018f52p-4, 0xc.cc7dfp-16, 0x2.aec5f4p-8 },
+ { 0xa.4261cp+4, 0xa.42623p+4, 0xa.4262bp+4, -0x6.5a89c8p-24, 0xf.f155p-8, -0xc.6b5cbp-16, -0x2.a82be4p-8 },
+ { 0xa.74a5cp+4, 0xa.74a62p+4, 0xa.74a6ap+4, -0x1.ef16c8p-24, -0xf.cad3fp-8, 0xc.16478p-16, 0x2.a1c1ap-8 },
+ { 0xa.a6e9bp+4, 0xa.a6eap+4, 0xa.a6ea9p+4, -0x6.1e7b68p-24, 0xf.a564cp-8, -0xb.bd1eap-16, -0x2.9b84f8p-8 },
+ { 0xa.d92d7p+4, 0xa.d92dfp+4, 0xa.d92eap+4, -0xf.8c858p-28, -0xf.80faep-8, 0xb.6f5dbp-16, 0x2.9573dcp-8 },
+ { 0xb.0b71ap+4, 0xb.0b71ep+4, 0xb.0b727p+4, 0x7.75d858p-24, 0xf.5d8abp-8, -0xb.24e88p-16, -0x2.8f8c4cp-8 },
+ { 0xb.3db58p+4, 0xb.3db5cp+4, 0xb.3db68p+4, 0x1.d78632p-24, -0xf.3b096p-8, 0xa.d5ef1p-16, 0x2.89cc8p-8 },
+ { 0xb.6ff96p+4, 0xb.6ff9bp+4, 0xb.6ffaap+4, 0x3.b24794p-24, 0xf.196c7p-8, -0xa.918e2p-16, -0x2.8432cp-8 },
+ { 0xb.a23d1p+4, 0xb.a23d9p+4, 0xb.a23e5p+4, 0x6.39cbc8p-24, -0xe.f8aa5p-8, 0xa.486fap-16, 0x2.7ebd9p-8 },
+ { 0xb.d481p+4, 0xb.d4818p+4, 0xb.d481cp+4, -0x1.820e3ap-24, 0xe.d8b9dp-8, -0xa.0973fp-16, -0x2.796b4cp-8 },
+ { 0xc.06c4fp+4, 0xc.06c57p+4, 0xc.06c5fp+4, -0x2.c7e038p-24, -0xe.b9925p-8, 0x9.cce94p-16, 0x2.743a5cp-8 },
+ { 0xc.3908ep+4, 0xc.39096p+4, 0xc.390a2p+4, 0x6.ab31c8p-24, 0xe.9b2bep-8, -0x9.92ad2p-16, -0x2.6f2994p-8 },
+ { 0xc.6b4cdp+4, 0xc.6b4d4p+4, 0xc.6b4d8p+4, 0x4.4ef25p-24, -0xe.7d7ecp-8, 0x9.5360fp-16, 0x2.6a37dp-8 }
+};
+
+/* Formula page 5 of https://www.cl.cam.ac.uk/~jrh13/papers/bessel.pdf:
+ y0(x) ~ sqrt(2/(pi*x))*beta0(x)*sin(x-pi/4-alpha0(x))
+ where beta0(x) = 1 - 16/x^2 + 53/(512*x^4)
+ and alpha0(x) = 1/(8*x) - 25/(384*x^3). */
+static float
+y0f_asympt (float x)
+{
+ /* The following code fails to give an error <= 9 ulps in only two cases,
+ for which we tabulate the correctly-rounded result. */
+ if (x == 0x1.bfad96p+7f)
+ return -0x7.f32bdp-32f;
+ if (x == 0x1.2e2a42p+17f)
+ return 0x1.a48974p-40f;
+ double y = 1.0 / (double) x;
+ double y2 = y * y;
+ double beta0 = 1.0f + y2 * (-0x1p-4f + 0x1.a8p-4 * y2);
+ double alpha0 = y * (0x2p-4 - 0x1.0aaaaap-4 * y2);
+ double h;
+ h = reduce_mod_twopi ((double) x);
+ /* Subtract alpha0. */
+ h -= alpha0;
+ /* Now reduce mod pi/2. */
+ double piover2 = 0x1.921fb54442d18p+0;
+ int n = 0;
+ while (fabs (h) > piover2 / 2)
+ {
+ if (h > 0)
+ {
+ h -= piover2;
+ n ++;
+ }
+ else
+ {
+ h += piover2;
+ n --;
+ }
+ }
+ /* Now x - pi/4 - alpha0 = h + n*pi/2 mod (2*pi). */
+ float xr = (float) h;
+ n = n & 3;
+ float cst = 0xc.c422ap-4; /* sqrt(2/pi) rounded to nearest */
+ float t = cst / sqrtf (x) * (float) beta0;
+ if (n == 0)
+ return t * sinf (xr);
+ else if (n == 2) /* sin(x+pi) = -sin(x) */
+ return -t * sinf (xr);
+ else if (n == 1) /* sin(x+pi/2) = cos(x) */
+ return t * cosf (xr);
+ else /* sin(x+3pi/2) = -cos(x) */
+ return -t * cosf (xr);
+}
+
+/* Special code for x near a root of y0.
+ z is the value computed by the generic code.
+ For small x, we use a polynomial approximating y0 around its root.
+ For large x, we use an asymptotic formula (y0f_asympt). */
+static float
+y0f_near_root (float x, float z)
+{
+ float index_f;
+ int index;
+ index_f = roundf ((x - FIRST_ZERO_Y0) / (float) M_PI);
+ if (index_f >= SMALL_SIZE)
+ return y0f_asympt (x);
+ index = (int) index_f;
+ float *p = Py[index];
+ float x0 = p[0];
+ float x1 = p[2];
+ /* If we are not in the interval [x0,x1] around xmid, we return the value z. */
+ if (! (x0 <= x && x <= x1))
+ return z;
+ float xmid = p[1];
+ float y = x - xmid;
+ /* For degree 0 we use a degree-4 polynomial, where the coefficient of degree 4
+ is hard-coded here. */
+ float p6 = (index > 0) ? p[6] : p[6] + y * -0x3.9ce95p-4;
+ return p[3] + y * (p[4] + y * (p[5] + y * p6));
+}
+
float
__ieee754_y0f(float x)
{
@@ -124,7 +601,8 @@ __ieee754_y0f(float x)
if(ix>=0x7f800000) return one/(x+x*x);
if(ix==0) return -1/zero; /* -inf and divide by zero exception. */
if(hx<0) return zero/(zero*x);
- if(ix >= 0x40000000) { /* |x| >= 2.0 */
+ if(ix >= 0x40000000 || (0x3f5bd613 <= ix && ix <= 0x3f6e0897)) {
+ /* |x| >= 2.0 or 0x1.b7ac26p-1 <= |x| <= 0x1.dc112ep-1 (around 1st zero) */
/* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0))
* where x0 = x-pi/4
* Better formula:
@@ -143,17 +621,23 @@ __ieee754_y0f(float x)
* j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
* y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
*/
- if(ix<0x7f000000) { /* make sure x+x not overflow */
- z = -__cosf(x+x);
- if ((s*c)<zero) cc = z/ss;
- else ss = z/cc;
- }
- if(ix>0x5c000000) z = (invsqrtpi*ss)/sqrtf(x);
- else {
- u = pzerof(x); v = qzerof(x);
- z = invsqrtpi*(u*ss+v*cc)/sqrtf(x);
- }
- return z;
+ if (ix >= 0x7f000000) /* x >= 2^127: use asymptotic expansion. */
+ return y0f_asympt (x);
+ /* Now we are sure that x+x cannot overflow. */
+ z = -__cosf(x+x);
+ if ((s*c)<zero) cc = z/ss;
+ else ss = z/cc;
+ if (ix <= 0x5c000000)
+ {
+ u = pzerof(x); v = qzerof(x);
+ ss = u*ss+v*cc;
+ }
+ z = (invsqrtpi*ss)/sqrtf(x);
+ float magic = 0x1.8p+20; /* 2^20 + 2^19 */
+ if (magic + ss != magic) /* Most likely. */
+ return z;
+ else /* |cc| <= 2^-4 */
+ return y0f_near_root (x, z);
}
if(ix<=0x39800000) { /* x < 2**-13 */
return(u00 + tpi*__ieee754_logf(x));
@@ -165,7 +649,7 @@ __ieee754_y0f(float x)
}
libm_alias_finite (__ieee754_y0f, __y0f)
-/* The asymptotic expansions of pzero is
+/* The asymptotic expansion of pzero is
* 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x.
* For x >= 2, We approximate pzero by
* pzero(x) = 1 + (R/S)
@@ -257,7 +741,7 @@ pzerof(float x)
}
-/* For x >= 8, the asymptotic expansions of qzero is
+/* For x >= 8, the asymptotic expansion of qzero is
* -1/8 s + 75/1024 s^3 - ..., where s = 1/x.
* We approximate pzero by
* qzero(x) = s*(-1.25 + (R/S))
@@ -1312,22 +1312,22 @@ float128: 1
ldouble: 1
Function: "j0":
-double: 2
-float: 8
-float128: 2
+double: 5
+float: 9
+float128: 4
ldouble: 2
Function: "j0_downward":
double: 2
-float: 4
+float: 9
float128: 4
ldouble: 6
Function: "j0_towardzero":
-double: 5
-float: 6
+double: 9
+float: 7
float128: 4
-ldouble: 6
+ldouble: 9
Function: "j0_upward":
double: 4
@@ -1748,10 +1748,10 @@ float128: 4
ldouble: 5
Function: "y0":
-double: 3
-float: 8
-float128: 3
-ldouble: 1
+double: 8
+float: 9
+float128: 4
+ldouble: 2
Function: "y0_downward":
double: 3
@@ -1760,15 +1760,15 @@ float128: 4
ldouble: 5
Function: "y0_towardzero":
-double: 3
+double: 5
float: 3
float128: 3
-ldouble: 6
+ldouble: 7
Function: "y0_upward":
double: 3
float: 6
-float128: 3
+float128: 7
ldouble: 5
Function: "y1":