diff --git a/math/auto-libm-test-in b/math/auto-libm-test-in
index 73840b8bef..dbf1dca35c 100644
--- a/math/auto-libm-test-in
+++ b/math/auto-libm-test-in
@@ -5756,8 +5756,9 @@ j0 -0x1.001000001p+593
 j0 0x1p1023
 j0 0x1p16382
 j0 0x1p16383
-# the next value generates larger error bounds on x86_64 (binary32)
+# the next values generate large error bounds on x86_64 (binary32)
 j0 0x2.602774p+0 xfail-rounding:ibm128-libgcc
+j0 0x1.8bde7ep+5
 # the next value exercises the flt-32 code path for x >= 2^127
 j0 0x8.2f4ecp+124
 
diff --git a/math/auto-libm-test-out-j0 b/math/auto-libm-test-out-j0
index cc1fea074e..d659f47632 100644
--- a/math/auto-libm-test-out-j0
+++ b/math/auto-libm-test-out-j0
@@ -1359,6 +1359,31 @@ j0 0x2.602774p+0 xfail-rounding:ibm128-libgcc
 = 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
diff --git a/sysdeps/ieee754/flt-32/e_j0f.c b/sysdeps/ieee754/flt-32/e_j0f.c
index 5d29611eb7..f6efab828c 100644
--- a/sysdeps/ieee754/flt-32/e_j0f.c
+++ b/sysdeps/ieee754/flt-32/e_j0f.c
@@ -37,6 +37,330 @@ S04  =  1.1661400734e-09; /* 0x30a045e8 */
 
 static const float zero = 0.0;
 
+#define FIRST_ZERO 2.404825f
+
+#define SMALL_SIZE 64
+
+/* The following table contains successive zeros of j0 and degree-3 polynomial
+   approximations of j0 around these zeros: P[0] for the first zero (2.404825), P[1]
+   for the second one (5.520078), and so one. 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 P[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 */
+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;
+  /* beta0 = 1 - 16/x^2 + 53/(512*x^4) */
+  double beta0 = 1.0f + y2 * (-0x1p-4f + 0x1.a8p-4 * y2);
+  /* alpha0 = 1/(8*x) - 25/(384*x^3) */
+  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) / (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 = P[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)
 {
@@ -75,12 +399,17 @@ __ieee754_j0f(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>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 */
diff --git a/sysdeps/x86_64/fpu/libm-test-ulps b/sysdeps/x86_64/fpu/libm-test-ulps
index 633d2ab8e4..75deb1462d 100644
--- a/sysdeps/x86_64/fpu/libm-test-ulps
+++ b/sysdeps/x86_64/fpu/libm-test-ulps
@@ -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
