diff mbox series

[3/4] Fix the inaccuracy of j1f and y1f [BZ #14470 and #14472]

Message ID 20210209052818.3991249-3-Paul.Zimmermann@inria.fr
State Superseded
Headers show
Series [1/4] Add a routine for accurate reduction mod (2pi), for j0f/j1f/y0f/y1f. | expand

Commit Message

Paul Zimmermann Feb. 9, 2021, 5:28 a.m. UTC
For both j1f and y1f, the largest error for all binary32
inputs is reduced to at most 9 ulps for all rounding modes.

The new code is enabled only when there is a cancellation at the very end of
the j1f/y1f computation, or for very large inputs, thus should not give any
visible slowdown on average.  Two different algorithms are used:

* around the first 64 zeros of j1/y1, approximation polynomials of degree 3
  are used, computed using the Sollya tool (https://www.sollya.org/)

* for large inputs, an asymptotic formula from [1] is used

[1] Fast and Accurate Bessel Function Computation,
John Harrison, Proceedings of Arith 19, 2009.

Tested on x86_64 with --disable-multi-arch.
---
 sysdeps/ieee754/flt-32/e_j1f.c | 403 ++++++++++++++++++++++++++++++---
 1 file changed, 373 insertions(+), 30 deletions(-)
diff mbox series

Patch

diff --git a/sysdeps/ieee754/flt-32/e_j1f.c b/sysdeps/ieee754/flt-32/e_j1f.c
index ac5bb76828..92ebbd3a9a 100644
--- a/sysdeps/ieee754/flt-32/e_j1f.c
+++ b/sysdeps/ieee754/flt-32/e_j1f.c
@@ -21,6 +21,7 @@ 
 #include <fenv_private.h>
 #include <math-underflow.h>
 #include <libm-alias-finite.h>
+#include <e_reduce_pi.h>
 
 static float ponef(float), qonef(float);
 
@@ -42,6 +43,184 @@  s05  =  1.2354227016e-11; /* 0x2d59567e */
 
 static const float zero    = 0.0;
 
+#define FIRST_ZERO_J1 3.831705f /* First positive zero of j1. */
+
+#define SMALL_SIZE 64
+
+/* The following table contains successive zeros of j1 and degree-3 polynomial
+   approximations of j1 around these zeros: Pj[0] for the first positive zero
+   (3.831705), Pj[1] for the second one (7.015586), 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 Sollya on the interval
+   [x0,x1] around the corresponding zero where the error is larger than 9 ulps
+   for the alternate code.  Degree 3 is enough to get an error at most 9 ulps,
+   except around the first zero.
+*/
+static const float Pj[SMALL_SIZE][7] = {
+  /* For index 0, we use a degree-4 polynomial generated by Sollya, with the
+     coefficient of degree 4 hard-coded in j1f_near_root(). */
+  { 0x1.e09e5ep+1, 0x1.ea7558p+1, 0x1.ef7352p+1, -0x8.4f069p-28, -0x6.71b3d8p-4, 0xd.744a2p-8, 0xd.acd9p-8/*, -0x1.3e51aap-8*/ }, /* 0 */
+  { 0x1.bdb4c2p+2, 0x1.c0ff6p+2, 0x1.c27a8cp+2, 0xe.c2858p-28, 0x4.cd464p-4, -0x5.79b71p-8, -0xc.11124p-8 }, /* 1 */
+  { 0x1.43b214p+3, 0x1.458d0ep+3, 0x1.460ccep+3, -0x1.e7acecp-24, -0x3.feca9p-4, 0x3.2470f8p-8, 0xa.625b7p-8 }, /* 2 */
+  { 0x1.a9c98p+3, 0x1.aa5bbp+3, 0x1.aaa492p+3, 0x1.698158p-24, 0x3.7e666cp-4, -0x2.1900ap-8, -0x9.2755p-8 }, /* 3 */
+  { 0x1.073be4p+4, 0x1.0787b4p+4, 0x1.07aed8p+4, -0x1.f5f658p-24, -0x3.24b8ep-4, 0x1.86e35cp-8, 0x8.4e4bbp-8 }, /* 4 */
+  { 0x1.39ae2ap+4, 0x1.39da8ep+4, 0x1.39f3dap+4, -0x1.4e744p-24, 0x2.e18a24p-4, -0x1.2ccd16p-8, -0x7.a27ep-8 }, /* 5 */
+  { 0x1.6bfa46p+4, 0x1.6c294ep+4, 0x1.6c412p+4, 0xa.3fb7fp-28, -0x2.acc9c4p-4, 0xf.0b783p-12, 0x7.1c0d3p-8 }, /* 6 */
+  { 0x1.9e44a2p+4, 0x1.9e757p+4, 0x1.9e876ep+4, -0x2.29f6f4p-24, 0x2.81f21p-4, -0xc.641bp-12, -0x6.a7ea58p-8 }, /* 7 */
+  { 0x1.d08a3ep+4, 0x1.d0bfdp+4, 0x1.d0cd3cp+4, -0x1.b5d196p-24, -0x2.5e40e4p-4, 0xa.7059fp-12, 0x6.4d6bfp-8 }, /* 8 */
+  { 0x1.017794p+5, 0x1.018476p+5, 0x1.018b8cp+5, -0x4.0e001p-24, 0x2.3febep-4, -0x8.f23aap-12, -0x6.0102cp-8 }, /* 9 */
+  { 0x1.1a9e78p+5, 0x1.1aa89p+5, 0x1.1aaf88p+5, 0x3.b26f2p-24, -0x2.25babp-4, 0x7.c6d948p-12, 0x5.a1d988p-8 }, /* 10 */
+  { 0x1.33bddep+5, 0x1.33cc52p+5, 0x1.33d2e4p+5, -0xf.c8cdap-28, 0x2.0ed05p-4, -0x6.d97dbp-12, -0x5.8da498p-8 }, /* 11 */
+  { 0x1.4ce7cp+5, 0x1.4cefdp+5, 0x1.4cf7d4p+5, -0x3.9940e4p-24, -0x1.fa8b4p-4, 0x6.16108p-12, 0x5.4355e8p-8 }, /* 12 */
+  { 0x1.6603e8p+5, 0x1.661316p+5, 0x1.66173p+5, 0x9.da15dp-28, 0x1.e8727ep-4, -0x5.742468p-12, -0x5.117c28p-8 }, /* 13 */
+  { 0x1.7f2ebcp+5, 0x1.7f3632p+5, 0x1.7f3a7ep+5, -0x3.39b218p-24, -0x1.d8293ap-4, 0x4.ee3348p-12, 0x4.f9bep-8 }, /* 14 */
+  { 0x1.9850e6p+5, 0x1.985928p+5, 0x1.985d9ep+5, -0x3.7b5108p-24, 0x1.c96702p-4, -0x4.7b0d08p-12, -0x4.c784a8p-8 }, /* 15 */
+  { 0x1.b172e8p+5, 0x1.b17c04p+5, 0x1.b1805cp+5, -0x1.91e43ep-24, -0x1.bbf246p-4, 0x4.18ad78p-12, 0x4.9bfae8p-8 }, /* 16 */
+  { 0x1.ca955ap+5, 0x1.ca9ec6p+5, 0x1.caa2a4p+5, 0x1.28453cp-24, 0x1.af9cb4p-4, -0x3.c3a494p-12, -0x4.78b69p-8 }, /* 17 */
+  { 0x1.e3bc94p+5, 0x1.e3c174p+5, 0x1.e3c64p+5, -0x2.e7fef4p-24, -0x1.a4407ep-4, 0x3.79b228p-12, 0x4.874f7p-8 }, /* 18 */
+  { 0x1.fcdf16p+5, 0x1.fce40ep+5, 0x1.fce71p+5, -0x3.23b2fcp-24, 0x1.99be76p-4, -0x3.39ad7cp-12, -0x4.92a55p-8 }, /* 19 */
+  { 0x1.0afe34p+6, 0x1.0b034ep+6, 0x1.0b0532p+6, -0xd.19e93p-28, -0x1.8ffc9cp-4, 0x2.fee7f8p-12, 0x4.2d33b8p-8 }, /* 20 */
+  { 0x1.179344p+6, 0x1.17948ep+6, 0x1.1795a2p+6, 0x1.c2ac48p-24, 0x1.86e51cp-4, -0x2.cc5abp-12, -0x4.866d08p-8 }, /* 21 */
+  { 0x1.24231ep+6, 0x1.2425c8p+6, 0x1.2426e2p+6, -0xd.31027p-28, -0x1.7e656ep-4, 0x2.9db23cp-12, 0x3.cc63c8p-8 }, /* 22 */
+  { 0x1.30b5a8p+6, 0x1.30b6fep+6, 0x1.30b84ep+6, 0x5.b5e53p-24, 0x1.766dc2p-4, -0x2.754cfcp-12, -0x3.c39bb4p-8 }, /* 23 */
+  { 0x1.3d46ccp+6, 0x1.3d482ep+6, 0x1.3d495ep+6, -0x1.340a8ap-24, -0x1.6ef07ep-4, 0x2.4ff9d4p-12, 0x3.9b36e4p-8 }, /* 24 */
+  { 0x1.49d688p+6, 0x1.49d95ap+6, 0x1.49dabep+6, -0x3.ba66p-24, 0x1.67e1dcp-4, -0x2.2f32b8p-12, -0x3.e2aaf4p-8 }, /* 25 */
+  { 0x1.566916p+6, 0x1.566a84p+6, 0x1.566bcp+6, 0x6.47ca5p-28, -0x1.61379ap-4, 0x2.1096acp-12, 0x4.2d0968p-8 }, /* 26 */
+  { 0x1.62f8dap+6, 0x1.62fbaap+6, 0x1.62fc9cp+6, -0x2.12affp-24, 0x1.5ae8c4p-4, -0x1.f32444p-12, -0x3.9e592p-8 }, /* 27 */
+  { 0x1.6f89e6p+6, 0x1.6f8ccep+6, 0x1.6f8e34p+6, -0x7.4853ap-28, -0x1.54ed76p-4, 0x1.db004ap-12, 0x3.907034p-8 }, /* 28 */
+  { 0x1.7c1c6ap+6, 0x1.7c1deep+6, 0x1.7c1f1ap+6, -0x4.f0a998p-24, 0x1.4f3ebcp-4, -0x1.c26808p-12, -0x2.da8df8p-8 }, /* 29 */
+  { 0x1.88adaep+6, 0x1.88af0ep+6, 0x1.88afc4p+6, -0x1.80c246p-24, -0x1.49d668p-4, 0x1.aebc26p-12, 0x3.af7b5cp-8 }, /* 30 */
+  { 0x1.953d7p+6, 0x1.95402ap+6, 0x1.9540a6p+6, -0x2.22aff8p-24, 0x1.44aefap-4, -0x1.99f25p-12, -0x3.5e9198p-8 }, /* 31 */
+  { 0x1.a1d01ep+6, 0x1.a1d146p+6, 0x1.a1d202p+6, -0x3.aad6d4p-24, -0x1.3fc386p-4, 0x1.892858p-12, 0x3.fe0184p-8 }, /* 32 */
+  { 0x1.ae60ecp+6, 0x1.ae625ep+6, 0x1.ae6322p+6, -0x2.010be4p-24, 0x1.3b0fa4p-4, -0x1.7539ap-12, -0x2.b2c9bp-8 }, /* 33 */
+  { 0x1.baf234p+6, 0x1.baf376p+6, 0x1.baf442p+6, -0xd.4fd17p-32, -0x1.368f5cp-4, 0x1.6734e4p-12, 0x3.59f514p-8 }, /* 34 */
+  { 0x1.c782e6p+6, 0x1.c7848cp+6, 0x1.c78516p+6, -0xa.d662dp-28, 0x1.323f18p-4, -0x1.571c02p-12, -0x3.2be5bp-8 }, /* 35 */
+  { 0x1.d4144ep+6, 0x1.d415ap+6, 0x1.d41622p+6, 0x4.9f217p-24, -0x1.2e1b9ap-4, 0x1.4a2edap-12, 0x3.a4e96p-8 }, /* 36 */
+  { 0x1.e0a60cp+6, 0x1.e0a6b4p+6, 0x1.e0a788p+6, -0x2.d167p-24, 0x1.2a21eep-4, -0x1.3c4b46p-12, -0x4.9e0978p-8 }, /* 37 */
+  { 0x1.ed36eep+6, 0x1.ed37c8p+6, 0x1.ed3892p+6, -0x4.15a83p-24, -0x1.264f66p-4, 0x1.31dea4p-12, 0x3.d125ecp-8 }, /* 38 */
+  { 0x1.f9c77p+6, 0x1.f9c8d8p+6, 0x1.f9c9acp+6, -0x2.a5bbbp-24, 0x1.22a192p-4, -0x1.25e59ep-12, -0x2.ef6934p-8 }, /* 39 */
+  { 0x1.032c54p+7, 0x1.032cf4p+7, 0x1.032d66p+7, 0x4.e828bp-24, -0x1.1f1634p-4, 0x1.1c2394p-12, 0x3.6d744cp-8 }, /* 40 */
+  { 0x1.09750cp+7, 0x1.09757cp+7, 0x1.0975b2p+7, -0x3.28a3bcp-24, 0x1.1bab3ep-4, -0x1.1569cep-12, -0x5.84da7p-8 }, /* 41 */
+  { 0x1.0fbd9ap+7, 0x1.0fbe04p+7, 0x1.0fbe5ep+7, -0x2.2f667p-24, -0x1.185eccp-4, 0x1.07f42cp-12, 0x2.87896cp-8 }, /* 42 */
+  { 0x1.160628p+7, 0x1.16068ap+7, 0x1.1606cep+7, -0x6.9097dp-24, 0x1.152f28p-4, -0x1.0227fep-12, -0x5.da6e6p-8 }, /* 43 */
+  { 0x1.1c4e9ap+7, 0x1.1c4f12p+7, 0x1.1c4f7cp+7, -0x5.1b408p-24, -0x1.121abp-4, 0xf.6efcp-16, 0x2.c5e954p-8 }, /* 44 */
+  { 0x1.2296aap+7, 0x1.229798p+7, 0x1.2297cap+7, 0x2.70d0dp-24, 0x1.0f1ffp-4, -0xf.523f5p-16, -0x3.5c0568p-8 }, /* 45 */
+  { 0x1.28dfa4p+7, 0x1.28e01ep+7, 0x1.28e054p+7, -0x2.7c176p-24, -0x1.0c3d8ap-4, 0xe.8329ap-16, 0x3.5eb34p-8 }, /* 46 */
+  { 0x1.2f282ap+7, 0x1.2f28a4p+7, 0x1.2f28dep+7, 0x4.fd6368p-24, 0x1.097236p-4, -0xe.17299p-16, -0x3.73a2e4p-8 }, /* 47 */
+  { 0x1.3570bp+7, 0x1.357128p+7, 0x1.35715p+7, 0x6.b05f68p-24, -0x1.06bccap-4, 0xd.527b8p-16, 0x2.b8bf9cp-8 }, /* 48 */
+  { 0x1.3bb932p+7, 0x1.3bb9aep+7, 0x1.3bb9eap+7, 0x4.0d622p-28, 0x1.041c28p-4, -0xd.0ac11p-16, -0x1.65f2ccp-8 }, /* 49 */
+  { 0x1.4201b6p+7, 0x1.420232p+7, 0x1.42027p+7, 0x7.0d98cp-24, -0x1.018f52p-4, 0xc.c4d8ep-16, 0x2.8f250cp-8 }, /* 50 */
+  { 0x1.484a78p+7, 0x1.484ab8p+7, 0x1.484ad8p+7, 0x3.93d10cp-24, 0xf.f154fp-8, -0xc.7b7fep-16, -0x3.6b6e4cp-8 }, /* 51 */
+  { 0x1.4e92c8p+7, 0x1.4e933cp+7, 0x1.4e9368p+7, 0xd.88185p-32, -0xf.cad3fp-8, 0xc.1462p-16, 0x2.bd66p-8 }, /* 52 */
+  { 0x1.54db84p+7, 0x1.54dbcp+7, 0x1.54dbf4p+7, -0x1.fe6b92p-24, 0xf.a564cp-8, -0xb.c4e6cp-16, -0x3.d51decp-8 }, /* 53 */
+  { 0x1.5b23c4p+7, 0x1.5b2444p+7, 0x1.5b2486p+7, 0x2.6137f4p-24, -0xf.80faep-8, 0xb.5199ep-16, 0x1.9ca85ap-8 }, /* 54 */
+  { 0x1.616c88p+7, 0x1.616cc8p+7, 0x1.616d0ap+7, -0x1.55468p-24, 0xf.5d8acp-8, -0xb.21d16p-16, -0x1.b8809ap-8 }, /* 55 */
+  { 0x1.67b4fp+7, 0x1.67b54cp+7, 0x1.67b588p+7, -0x1.08c6bep-24, -0xf.3b096p-8, 0xa.e65efp-16, 0x3.642304p-8 }, /* 56 */
+  { 0x1.6dfd8ep+7, 0x1.6dfddp+7, 0x1.6dfdfap+7, 0x4.9ebfa8p-24, 0xf.196c7p-8, -0xa.ba8c8p-16, -0x5.ad6a2p-8 }, /* 57 */
+  { 0x1.74461p+7, 0x1.744652p+7, 0x1.744692p+7, 0x5.a4017p-24, -0xe.f8aa5p-8, 0xa.49748p-16, 0x2.a86498p-8 }, /* 58 */
+  { 0x1.7a8e5ep+7, 0x1.7a8ed6p+7, 0x1.7a8ef8p+7, 0x3.bcb2a8p-28, 0xe.d8b9dp-8, -0x9.c43bep-16, -0x1.e7124ap-8 }, /* 59 */
+  { 0x1.80d714p+7, 0x1.80d75ap+7, 0x1.80d78ap+7, -0x7.1091fp-24, -0xe.b9925p-8, 0x9.c43dap-16, 0x1.aba86p-8 }, /* 60 */
+  { 0x1.871f58p+7, 0x1.871fdcp+7, 0x1.87201ap+7, 0x2.ca1cf4p-28, 0xe.9b2bep-8, -0x9.843b3p-16, -0x2.093e68p-8 }, /* 61 */
+  { 0x1.8d6836p+7, 0x1.8d685ep+7, 0x1.8d6882p+7, 0x5.aa8908p-24, -0xe.7d7ecp-8, 0x9.501a8p-16, 0x2.54a754p-8 }, /* 62 */
+  { 0x1.93b09cp+7, 0x1.93b0e2p+7, 0x1.93b10ep+7, 0x3.d9cd9cp-24, 0xe.6083ap-8, -0x9.45dadp-16, -0x5.112908p-8 }, /* 63 */
+};
+
+/* Reduce h mod (pi/2), and add k to n if k*pi/2 was subtracted from h. */
+static double
+reduce_mod_piover2 (double h, int *n)
+{
+  double piover2 = 0x1.921fb54442d18p+0;
+  while (fabs (h) > piover2 / 2)
+    {
+      if (h > 0)
+        {
+          h -= piover2;
+          (*n) ++;
+        }
+      else
+        {
+          h += piover2;
+          (*n) --;
+        }
+    }
+  return h;
+}
+
+/* Formula page 5 of https://www.cl.cam.ac.uk/~jrh13/papers/bessel.pdf:
+   j1f(x) ~ sqrt(2/(pi*x))*beta1(x)*cos(x-3pi/4-alpha1(x))
+   where beta1(x) = 1 + 3/(16*x^2) - 99/(512*x^4)
+   and alpha1(x) = -3/(8*x) + 21/(128*x^3) - 1899/(5120*x^5). */
+static float
+j1f_asympt (float x)
+{
+  float cst = 0xc.c422ap-4; /* sqrt(2/pi) rounded to nearest */
+  if (x < 0)
+    {
+      x = -x;
+      cst = -cst;
+    }
+  double y = 1.0 / (double) x;
+  double y2 = y * y;
+  double beta1 = 1.0f + y2 * (0x3p-4 - 0x3.18p-4 * y2);
+  double alpha1 = y * (-0x6p-4 + y2 * (0x2.ap-4 - 0x5.ef33333333334p-4 * y2));
+  double h;
+  h = __math_reduce_mod_twopi ((double) x);
+  int n = -1; /* Accounts for -pi/2 (pi/4 was subtracted in
+                 __math_reduce_mod_twopi). */
+  /* Now reduce mod pi/2. */
+  h = reduce_mod_piover2 (h, &n);
+  /* Subtract alpha1. */
+  h -= alpha1;
+  /* Reduce again mod pi/2 if needed. */
+  h = reduce_mod_piover2 (h, &n);
+  /* Now x - 3pi/4 - alpha1 = h + n*pi/2 mod (2*pi). */
+  float xr = (float) h;
+  n = n & 3;
+  float t = cst / sqrtf (x) * (float) beta1;
+  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 j1.
+   z is the value computed by the generic code.
+   For small x, we use a polynomial approximating j1 around its root.
+   For large x, we use an asymptotic formula (j1f_asympt). */
+static float
+j1f_near_root (float x, float z)
+{
+  float index_f, sign = 1.0f;
+  int index;
+
+  if (x < 0)
+    {
+      x = -x;
+      sign = -1.0f;
+    }
+  index_f = roundf ((x - FIRST_ZERO_J1) / (float) M_PI);
+  if (index_f >= SMALL_SIZE)
+    return sign * j1f_asympt (x);
+  index = (int) index_f;
+  const float *p = Pj[index];
+  float x0 = p[0];
+  float x1 = p[2];
+  /* If not in the interval [x0,x1] around xmid, return the value z. */
+  if (! (x0 <= x && x <= x1))
+    return z;
+  float xmid = p[1];
+  float y = x - xmid;
+  float p6 = (index > 0) ? p[6] : p[6] + y * -0x1.3e51aap-8f;
+  return sign * (p[3] + y * (p[4] + y * (p[5] + y * p6)));
+}
+
 float
 __ieee754_j1f(float x)
 {
@@ -53,25 +232,32 @@  __ieee754_j1f(float x)
 	if(__builtin_expect(ix>=0x7f800000, 0)) return one/x;
 	y = fabsf(x);
 	if(ix >= 0x40000000) {	/* |x| >= 2.0 */
+                SET_RESTORE_ROUNDF (FE_TONEAREST);
 		__sincosf (y, &s, &c);
 		ss = -s-c;
 		cc = s-c;
-		if(ix<0x7f000000) {  /* make sure y+y not overflow */
-		    z = __cosf(y+y);
-		    if ((s*c)>zero) cc = z/ss;
-		    else	    ss = z/cc;
-		}
+                if (ix >= 0x7f000000) /* x >= 2^127: use asymptotic expansion. */
+                  return j1f_asympt (x);
+                /* Now we are sure that x+x cannot overflow. */
+                z = __cosf(y+y);
+                if ((s*c)>zero) cc = z/ss;
+                else	        ss = z/cc;
 	/*
 	 * j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x)
 	 * y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x)
 	 */
-		if(ix>0x5c000000) z = (invsqrtpi*cc)/sqrtf(y);
-		else {
+		if (ix <= 0x5c000000)
+                  {
 		    u = ponef(y); v = qonef(y);
-		    z = invsqrtpi*(u*cc-v*ss)/sqrtf(y);
-		}
-		if(hx<0) return -z;
-		else	 return  z;
+		    cc = u*cc-v*ss;
+                  }
+                z = (invsqrtpi * cc) / sqrtf(y);
+                /* Adjust sign of z. */
+                z = (hx < 0) ? -z : z;
+                if (fabsf (cc) > 0x1p-3) /* Most likely. */
+                  return z;
+                else /* |cc| <= 2^-3 */
+                  return j1f_near_root (x, z);
 	}
 	if(__builtin_expect(ix<0x32000000, 0)) {	/* |x|<2**-27 */
 	    if(huge+x>one) {		/* inexact if x!=0 necessary */
@@ -105,6 +291,157 @@  static const float V0[5] = {
   1.6655924903e-11, /* 0x2d9281cf */
 };
 
+#define FIRST_ZERO_Y1 2.197141f /* First zero of y1. */
+
+/* The following table contains successive zeros of y1 and degree-3 polynomial
+   approximations of y1 around these zeros: Py[0] for the first positive zero
+   (2.197141), Py[1] for the second one (5.429681), 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 Sollya on the interval
+   [x0,x1] around the corresponding zero where the error is larger than 9 ulps
+   for the alternate code.  Degree 3 is enough (except around the first roots).
+*/
+static const float Py[SMALL_SIZE][7] = {
+  /* For index 0, we use a degree-5 polynomial generated by Sollya, with the
+     coefficients of degree 4 and 5 hard-coded in y1f_near_root(). */
+  { 0x1.f7f16ap+0, 0x1.193beep+1, 0x1.2105dcp+1, 0xb.96749p-28, 0x8.55241p-4, -0x1.e570bp-4, -0x8.68b61p-8/*, -0x1.28043p-8, 0x2.50e83p-8*/ }, /* 0 */
+  /* For index 1, we use a degree-4 polynomial generated by Sollya, with the
+     coefficient of degree 4 hard-coded in y1f_near_root(). */
+  { 0x1.55c6d2p+2, 0x1.5b7fe4p+2, 0x1.5cf8cap+2, 0x1.3c7822p-24, -0x5.71f158p-4, 0x8.05cb4p-8, 0xd.0b15p-8/*, -0xf.ff6b8p-12*/ }, /* 1 */
+  { 0x1.113c6p+3, 0x1.13127ap+3, 0x1.138718p+3, -0x1.f3ad8ep-24, 0x4.57e66p-4, -0x4.0afb58p-8, -0xb.29207p-8 }, /* 2 */
+  { 0x1.76e7dep+3, 0x1.77f914p+3, 0x1.786a6ap+3, -0xd.5608fp-28, -0x3.b829d4p-4, 0x2.8852cp-8, 0x9.b70e3p-8 }, /* 3 */
+  { 0x1.dc2794p+3, 0x1.dcb7d8p+3, 0x1.dd032p+3, -0xe.a7c04p-28, 0x3.4e0458p-4, -0x1.c64b18p-8, -0x8.b0e7fp-8 }, /* 4 */
+  { 0x1.20874p+4, 0x1.20b1c6p+4, 0x1.20c71p+4, 0x1.c2626p-24, -0x3.00f03cp-4, 0x1.54f806p-8, 0x7.f9cf9p-8 }, /* 5 */
+  { 0x1.52d89ap+4, 0x1.530254p+4, 0x1.531962p+4, -0x1.9503ecp-24, 0x2.c5b29cp-4, -0x1.0bf28p-8, -0x7.562e58p-8 }, /* 6 */
+  { 0x1.851e64p+4, 0x1.854fa4p+4, 0x1.85679p+4, -0x2.8d40fcp-24, -0x2.96547p-4, 0xd.9c38bp-12, 0x6.dcbf8p-8 }, /* 7 */
+  { 0x1.b7808ep+4, 0x1.b79acep+4, 0x1.b7b2a8p+4, -0x2.36df5cp-24, 0x2.6f55ap-4, -0xb.57f9fp-12, -0x6.82569p-8 }, /* 8 */
+  { 0x1.e9c916p+4, 0x1.e9e48p+4, 0x1.e9f24p+4, 0xd.e2eb7p-28, -0x2.4e8104p-4, 0x9.a4be2p-12, 0x6.2541fp-8 }, /* 9 */
+  { 0x1.0e0808p+5, 0x1.0e169p+5, 0x1.0e1d92p+5, -0x2.3070f4p-24, 0x2.325e4cp-4, -0x8.53604p-12, -0x5.ca03a8p-8 }, /* 10 */
+  { 0x1.273318p+5, 0x1.273a7cp+5, 0x1.2741fcp+5, -0x3.525508p-24, -0x2.19e7dcp-4, 0x7.49d1dp-12, 0x5.9cb02p-8 }, /* 11 */
+  { 0x1.405122p+5, 0x1.405e18p+5, 0x1.4065cep+5, -0xe.6e158p-28, 0x2.046174p-4, -0x6.71b3dp-12, -0x5.4c3c8p-8 }, /* 12 */
+  { 0x1.5971dcp+5, 0x1.598178p+5, 0x1.598592p+5, 0x1.e72698p-24, -0x1.f13fb2p-4, 0x5.c0f938p-12, 0x5.28ca78p-8 }, /* 13 */
+  { 0x1.729c4ep+5, 0x1.72a4a8p+5, 0x1.72a8eap+5, -0x1.5bed9cp-24, 0x1.e018dcp-4, -0x5.2f11e8p-12, -0x5.16ce48p-8 }, /* 14 */
+  { 0x1.8bbf4ep+5, 0x1.8bc7b2p+5, 0x1.8bcc1p+5, -0x3.6b654cp-24, -0x1.d09b2p-4, 0x4.b1747p-12, 0x4.bd22fp-8 }, /* 15 */
+  { 0x1.a4e272p+5, 0x1.a4ea9ap+5, 0x1.a4eef4p+5, 0x1.6f11bp-24, 0x1.c28612p-4, -0x4.47462p-12, -0x4.947c5p-8 }, /* 16 */
+  { 0x1.be08bep+5, 0x1.be0d68p+5, 0x1.be1088p+5, -0x2.0bc074p-24, -0x1.b5a622p-4, 0x3.ed52d4p-12, 0x4.b76fc8p-8 }, /* 17 */
+  { 0x1.d7272ap+5, 0x1.d7301ep+5, 0x1.d734aep+5, -0x2.87dd4p-24, 0x1.a9d184p-4, -0x3.9cf494p-12, -0x4.6303ep-8 }, /* 18 */
+  { 0x1.f0499ap+5, 0x1.f052c4p+5, 0x1.f05758p+5, -0x2.fb964p-24, -0x1.9ee5eep-4, 0x3.5800dp-12, 0x4.4e9f9p-8 }, /* 19 */
+  { 0x1.04b63ap+6, 0x1.04baacp+6, 0x1.04bc92p+6, 0x2.cf5adp-24, 0x1.94c6f4p-4, -0x3.1a83e4p-12, -0x4.2311fp-8 }, /* 20 */
+  { 0x1.1146dp+6, 0x1.114beep+6, 0x1.114e12p+6, 0x3.6766fp-24, -0x1.8b5cccp-4, 0x2.e4a4e4p-12, 0x4.20bf9p-8 }, /* 21 */
+  { 0x1.1dda8cp+6, 0x1.1ddd2cp+6, 0x1.1dde7ap+6, 0x3.501424p-24, 0x1.829356p-4, -0x2.b47524p-12, -0x4.04bf18p-8 }, /* 22 */
+  { 0x1.2a6bcp+6, 0x1.2a6e64p+6, 0x1.2a6faap+6, -0x5.c05808p-24, -0x1.7a597ep-4, 0x2.8a0498p-12, 0x4.187258p-8 }, /* 23 */
+  { 0x1.36fcd6p+6, 0x1.36ff96p+6, 0x1.3700f6p+6, 0x7.1e1478p-28, 0x1.72a09ap-4, -0x2.61a7fp-12, -0x3.c0b54p-8 }, /* 24 */
+  { 0x1.438f46p+6, 0x1.4390c4p+6, 0x1.4391e2p+6, 0x3.e36e6cp-24, -0x1.6b5c06p-4, 0x2.3f612p-12, 0x4.18f868p-8 }, /* 25 */
+  { 0x1.501f4cp+6, 0x1.5021fp+6, 0x1.50235p+6, 0x1.3f9e5ap-24, 0x1.6480c4p-4, -0x2.1f28fcp-12, -0x3.bb4e3cp-8 }, /* 26 */
+  { 0x1.5cb07cp+6, 0x1.5cb318p+6, 0x1.5cb464p+6, -0x2.39e41cp-24, -0x1.5e0544p-4, 0x2.0189f4p-12, 0x3.8b55acp-8 }, /* 27 */
+  { 0x1.694166p+6, 0x1.69443cp+6, 0x1.694594p+6, -0x2.912f84p-24, 0x1.57e12p-4, -0x1.e6fabep-12, -0x3.850174p-8 }, /* 28 */
+  { 0x1.75d27cp+6, 0x1.75d55ep+6, 0x1.75d642p+6, 0x3.d5b00cp-24, -0x1.520ceep-4, 0x1.d0286ep-12, 0x3.8e7d1p-8 }, /* 29 */
+  { 0x1.82653ep+6, 0x1.82667ep+6, 0x1.82674p+6, -0x3.1726ecp-24, 0x1.4c8222p-4, -0x1.b98206p-12, -0x3.f34978p-8 }, /* 30 */
+  { 0x1.8ef4c8p+6, 0x1.8ef79cp+6, 0x1.8ef888p+6, 0x1.949e22p-24, -0x1.473ae6p-4, 0x1.a47388p-12, 0x3.69eefcp-8 }, /* 31 */
+  { 0x1.9b8728p+6, 0x1.9b88b8p+6, 0x1.9b896cp+6, -0x5.5553bp-28, 0x1.42320ap-4, -0x1.90f0b8p-12, -0x3.6565p-8 }, /* 32 */
+  { 0x1.a8183cp+6, 0x1.a819d2p+6, 0x1.a81aecp+6, 0x3.2df7ecp-28, -0x1.3d62e4p-4, 0x1.7dae28p-12, 0x2.9eb128p-8 }, /* 33 */
+  { 0x1.b4aa1cp+6, 0x1.b4aaeap+6, 0x1.b4abb8p+6, -0x1.e13fcep-24, 0x1.38c948p-4, -0x1.6eb0ecp-12, -0x1.f9ddf8p-8 }, /* 34 */
+  { 0x1.c13a94p+6, 0x1.c13c02p+6, 0x1.c13cbp+6, -0x3.ad9974p-24, -0x1.34616ep-4, 0x1.5e36ecp-12, 0x2.a9fc5p-8 }, /* 35 */
+  { 0x1.cdcb76p+6, 0x1.cdcd16p+6, 0x1.cdcde4p+6, -0x3.6977e8p-24, 0x1.3027fp-4, -0x1.4f703p-12, -0x2.9817d4p-8 }, /* 36 */
+  { 0x1.da5cf2p+6, 0x1.da5e2ap+6, 0x1.da5efp+6, 0x4.654cbp-24, -0x1.2c19b6p-4, 0x1.455982p-12, 0x3.f1c564p-8 }, /* 37 */
+  { 0x1.e6edccp+6, 0x1.e6ef3ep+6, 0x1.e6f00ap+6, 0x8.825c8p-32, 0x1.2833eep-4, -0x1.39097p-12, -0x3.b2646p-8 }, /* 38 */
+  { 0x1.f37f86p+6, 0x1.f3805p+6, 0x1.f3812ap+6, -0x2.0d11d8p-28, -0x1.24740ap-4, 0x1.2c16p-12, 0x1.fc3804p-8 }, /* 39 */
+  { 0x1.000842p+7, 0x1.0008bp+7, 0x1.000908p+7, -0x4.4e495p-24, 0x1.20d7b6p-4, -0x1.20816p-12, -0x2.d1ebe8p-8 }, /* 40 */
+  { 0x1.06505cp+7, 0x1.065138p+7, 0x1.06518p+7, 0x4.81c1c8p-24, -0x1.1d5ccap-4, 0x1.17ad5ap-12, 0x2.fda33p-8 }, /* 41 */
+  { 0x1.0c98dap+7, 0x1.0c99cp+7, 0x1.0c9a28p+7, -0xe.99386p-28, 0x1.1a015p-4, -0x1.0bd50ap-12, -0x2.9dfb68p-8 }, /* 42 */
+  { 0x1.12e212p+7, 0x1.12e248p+7, 0x1.12e29p+7, -0x6.16f1c8p-24, -0x1.16c37ap-4, 0x1.0303dcp-12, 0x4.34316p-8 }, /* 43 */
+  { 0x1.192a6ep+7, 0x1.192acep+7, 0x1.192b02p+7, -0x1.129336p-24, 0x1.13a19ep-4, -0xf.bd247p-16, -0x3.851d18p-8 }, /* 44 */
+  { 0x1.1f727p+7, 0x1.1f7354p+7, 0x1.1f73ap+7, 0x5.19c09p-24, -0x1.109a32p-4, 0xf.09644p-16, 0x2.d78194p-8 }, /* 45 */
+  { 0x1.25bb8p+7, 0x1.25bbdap+7, 0x1.25bc12p+7, -0x6.497dp-24, 0x1.0dabc8p-4, -0xe.a1d25p-16, -0x2.3378bp-8 }, /* 46 */
+  { 0x1.2c04p+7, 0x1.2c046p+7, 0x1.2c04ap+7, 0x4.e4f338p-24, -0x1.0ad512p-4, 0xe.52d84p-16, 0x4.3bfa08p-8 }, /* 47 */
+  { 0x1.324cbp+7, 0x1.324ce6p+7, 0x1.324d4p+7, -0x1.287c58p-24, 0x1.0814d4p-4, -0xe.03a95p-16, 0x3.9930ap-12 }, /* 48 */
+  { 0x1.3894f6p+7, 0x1.38956cp+7, 0x1.38959ap+7, -0x4.b594ep-24, -0x1.0569fp-4, 0xd.6787ep-16, 0x4.0a5148p-8 }, /* 49 */
+  { 0x1.3edd98p+7, 0x1.3eddfp+7, 0x1.3ede2ap+7, -0x3.a8f164p-24, 0x1.02d354p-4, -0xd.0309dp-16, -0x3.2ebfb4p-8 }, /* 50 */
+  { 0x1.452638p+7, 0x1.452676p+7, 0x1.4526b4p+7, -0x6.12505p-24, -0x1.005004p-4, 0xc.a0045p-16, 0x4.87c67p-8 }, /* 51 */
+  { 0x1.4b6e8p+7, 0x1.4b6efap+7, 0x1.4b6f34p+7, 0x1.8acf4ep-24, 0xf.ddf16p-8, -0xc.2d207p-16, -0x1.da6c36p-8 }, /* 52 */
+  { 0x1.51b742p+7, 0x1.51b77ep+7, 0x1.51b7b2p+7, 0x1.39cf86p-24, -0xf.b7faep-8, 0xb.db598p-16, -0x8.945b1p-12 }, /* 53 */
+  { 0x1.57ffc4p+7, 0x1.580002p+7, 0x1.58003cp+7, -0x2.5f8de8p-24, 0xf.930fep-8, -0xb.91889p-16, -0xa.30df9p-12 }, /* 54 */
+  { 0x1.5e483p+7, 0x1.5e4886p+7, 0x1.5e48c8p+7, 0x2.073d64p-24, -0xf.6f245p-8, 0xb.4085fp-16, 0x2.128188p-8 }, /* 55 */
+  { 0x1.64908cp+7, 0x1.64910ap+7, 0x1.64912ap+7, -0x4.ed26ep-28, 0xf.4c2cep-8, -0xa.fe719p-16, -0x2.9374b8p-8 }, /* 56 */
+  { 0x1.6ad91ep+7, 0x1.6ad98ep+7, 0x1.6ad9cep+7, -0x2.ae5204p-24, -0xf.2a1efp-8, 0xa.aa585p-16, 0x2.1c0834p-8 }, /* 57 */
+  { 0x1.7121cep+7, 0x1.712212p+7, 0x1.712238p+7, 0x6.d72168p-24, 0xf.08f09p-8, -0xa.7da49p-16, -0x3.4f5f1cp-8 }, /* 58 */
+  { 0x1.776a0cp+7, 0x1.776a94p+7, 0x1.776accp+7, 0x2.d3f294p-24, -0xe.e8986p-8, 0xa.23ccdp-16, 0x2.2a6678p-8 }, /* 59 */
+  { 0x1.7db2e8p+7, 0x1.7db318p+7, 0x1.7db35ap+7, 0x3.88c0fp-24, 0xe.c90d7p-8, -0x9.eaeap-16, -0x2.86438cp-8 }, /* 60 */
+  { 0x1.83fb56p+7, 0x1.83fb9ap+7, 0x1.83fbep+7, 0x3.d94d34p-24, -0xe.aa478p-8, 0x9.abac7p-16, 0x1.ac2d84p-8 }, /* 61 */
+  { 0x1.8a43eep+7, 0x1.8a441ep+7, 0x1.8a446p+7, 0x4.66b7ep-24, 0xe.8c3e9p-8, -0x9.87682p-16, -0x7.9ab4a8p-12 }, /* 62 */
+  { 0x1.908c6p+7, 0x1.908cap+7, 0x1.908cd6p+7, 0xf.f7ac9p-28, -0xe.6eeb6p-8, 0x9.4423p-16, 0x4.54c4d8p-8 }, /* 63 */
+};
+
+/* Formula page 5 of https://www.cl.cam.ac.uk/~jrh13/papers/bessel.pdf:
+   y1f(x) ~ sqrt(2/(pi*x))*beta1(x)*sin(x-3pi/4-alpha1(x))
+   where beta1(x) = 1 + 3/(16*x^2) - 99/(512*x^4)
+   and alpha1(x) = -3/(8*x) + 21/(128*x^3) - 1899/(5120*x^5). */
+static float
+y1f_asympt (float x)
+{
+  float cst = 0xc.c422ap-4; /* sqrt(2/pi) rounded to nearest */
+  double y = 1.0 / (double) x;
+  double y2 = y * y;
+  double beta1 = 1.0f + y2 * (0x3p-4 - 0x3.18p-4 * y2);
+  double alpha1 = y * (-0x6p-4 + y2 * (0x2.ap-4 - 0x5.ef33333333334p-4 * y2));
+  double h;
+  h = __math_reduce_mod_twopi ((double) x);
+  int n = -1; /* Accounts for -pi/2 (pi/4 was subtracted in
+                 __math_reduce_mod_twopi). */
+  /* Now reduce mod pi/2. */
+  h = reduce_mod_piover2 (h, &n);
+  /* Subtract alpha1. */
+  h -= alpha1;
+  /* Reduce again mod pi/2 if needed. */
+  h = reduce_mod_piover2 (h, &n);
+  /* Now x - 3pi/4 - alpha1 = h + n*pi/2 mod (2*pi). */
+  float xr = (float) h;
+  n = n & 3;
+  float t = cst / sqrtf (x) * (float) beta1;
+  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 y1.
+   z is the value computed by the generic code.
+   For small x, we use a polynomial approximating y1 around its root.
+   For large x, we use an asymptotic formula (y1f_asympt). */
+static float
+y1f_near_root (float x, float z)
+{
+  float index_f;
+  int index;
+
+  index_f = roundf ((x - FIRST_ZERO_Y1) / (float) M_PI);
+  if (index_f >= SMALL_SIZE)
+    return y1f_asympt (x);
+  index = (int) index_f;
+  const float *p = Py[index];
+  float x0 = p[0];
+  float x1 = p[2];
+  /* If not in the interval [x0,x1] around xmid, return the value z. */
+  if (! (x0 <= x && x <= x1))
+    return z;
+  float xmid = p[1];
+  float y = x - xmid, p6;
+  if (index == 0)
+    p6 = p[6] + y * (-0x1.28043p-8 + y * 0x2.50e83p-8);
+  else if (index == 1)
+    p6 = p[6] + y * -0xf.ff6b8p-12;
+  else
+    p6 = p[6];
+  return p[3] + y * (p[4] + y * (p[5] + y * p6));
+}
+
 float
 __ieee754_y1f(float x)
 {
@@ -118,16 +455,17 @@  __ieee754_y1f(float x)
 	if(__builtin_expect(ix==0, 0))
 		return -1/zero; /* -inf and divide by zero exception.  */
 	if(__builtin_expect(hx<0, 0)) return zero/(zero*x);
-	if(ix >= 0x40000000) {  /* |x| >= 2.0 */
+        if (ix >= 0x3fe0dfbc) { /* |x| >= 0x1.c1bf78p+0 */
 		SET_RESTORE_ROUNDF (FE_TONEAREST);
 		__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;
-		}
+                if (ix >= 0x7f000000) /* x >= 2^127: use asymptotic expansion. */
+                  return y1f_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;
 	/* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0))
 	 * where x0 = x-3pi/4
 	 *      Better formula:
@@ -139,12 +477,17 @@  __ieee754_y1f(float x)
 	 *              sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
 	 * to compute the worse one.
 	 */
-		if(ix>0x5c000000) z = (invsqrtpi*ss)/sqrtf(x);
-		else {
-		    u = ponef(x); v = qonef(x);
-		    z = invsqrtpi*(u*ss+v*cc)/sqrtf(x);
-		}
-		return z;
+                if (ix <= 0x5c000000)
+                  {
+                    u = ponef(x); v = qonef(x);
+                    ss = u*ss+v*cc;
+                  }
+                z = (invsqrtpi * ss) / sqrtf(x);
+                /* The following value is attained for x=0x1.f7f16ap+0. */
+                if (fabsf (ss) > 0x1.3e014cp-2)
+                  return z;
+                else /* |cc| <= 2^-2 */
+                  return y1f_near_root (x, z);
 	}
 	if(__builtin_expect(ix<=0x33000000, 0)) {    /* x < 2**-25 */
 	    z = -tpi / x;
@@ -152,6 +495,7 @@  __ieee754_y1f(float x)
 		__set_errno (ERANGE);
 	    return z;
 	}
+        /* Now 2**-25 <= x < 0x1.c1bf78p+0. */
 	z = x*x;
 	u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4])));
 	v = one+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4]))));
@@ -159,7 +503,7 @@  __ieee754_y1f(float x)
 }
 libm_alias_finite (__ieee754_y1f, __y1f)
 
-/* For x >= 8, the asymptotic expansions of pone is
+/* For x >= 8, the asymptotic expansion of pone is
  *	1 + 15/128 s^2 - 4725/2^15 s^4 - ...,	where s = 1/x.
  * We approximate pone by
  *	pone(x) = 1 + (R/S)
@@ -252,8 +596,7 @@  ponef(float x)
 	return one+ r/s;
 }
 
-
-/* For x >= 8, the asymptotic expansions of qone is
+/* For x >= 8, the asymptotic expansion of qone is
  *	3/8 s - 105/1024 s^3 - ..., where s = 1/x.
  * We approximate pone by
  *	qone(x) = s*(0.375 + (R/S))
@@ -340,10 +683,10 @@  qonef(float x)
 	GET_FLOAT_WORD(ix,x);
 	ix &= 0x7fffffff;
 	/* ix >= 0x40000000 for all calls to this function.  */
-	if(ix>=0x40200000)     {p = qr8; q= qs8;}
-	else if(ix>=0x40f71c58){p = qr5; q= qs5;}
-	else if(ix>=0x4036db68){p = qr3; q= qs3;}
-	else {p = qr2; q= qs2;}
+	if(ix>=0x41000000)     {p = qr8; q= qs8;} /* x >= 8 */
+	else if(ix>=0x40f71c58){p = qr5; q= qs5;} /* x >= 7.722209930e+00 */
+	else if(ix>=0x4036db68){p = qr3; q= qs3;} /* x >= 2.857141495e+00 */
+	else {p = qr2; q= qs2;}                   /* x >= 2 */
 	z = one/(x*x);
 	r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
 	s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));