AArch64: Optimize SVE exp functions

Improve performance of SVE exps by making better use
of the SVE FEXPA instruction.

Performance improvement on Neoverse V1:
exp2_sve:   21%
exp2f_sve:  24%
exp10f_sve: 23%
expm1_sve:  25%

Reviewed-by: Wilco Dijkstra  <Wilco.Dijkstra@arm.com>

4 files changed, 211 insertions, 155 deletions

diff --git a/sysdeps/aarch64/fpu/exp10f_sve.c b/sysdeps/aarch64/fpu/exp10f_sve.c
index 1a74db2..f3e7f8b 100644
--- a/sysdeps/aarch64/fpu/exp10f_sve.c
+++ b/sysdeps/aarch64/fpu/exp10f_sve.c
@@ -19,26 +19,19 @@
 
 #include "sv_math.h"
 
-/* For x < -Thres, the result is subnormal and not handled correctly by
-   FEXPA.  */
-#define Thres 37.9
+/* For x < -Thres (-log10(2^126)), the result is subnormal and not handled
+   correctly by FEXPA.  */
+#define Thres 0x1.2f702p+5
 
 static const struct data
 {
-  float log2_10_lo, c0, c2, c4;
-  float c1, c3, log10_2;
-  float shift, log2_10_hi, thres;
+  float log10_2, log2_10_hi, log2_10_lo, c1;
+  float c0, shift, thres;
 } data = {
   /* Coefficients generated using Remez algorithm with minimisation of relative
-     error.
-     rel error: 0x1.89dafa3p-24
-     abs error: 0x1.167d55p-23 in [-log10(2)/2, log10(2)/2]
-     maxerr: 0.52 +0.5 ulp.  */
-  .c0 = 0x1.26bb16p+1f,
-  .c1 = 0x1.5350d2p+1f,
-  .c2 = 0x1.04744ap+1f,
-  .c3 = 0x1.2d8176p+0f,
-  .c4 = 0x1.12b41ap-1f,
+     error.  */
+  .c0 = 0x1.26bb62p1,
+  .c1 = 0x1.53524cp1,
   /* 1.5*2^17 + 127, a shift value suitable for FEXPA.  */
   .shift = 0x1.803f8p17f,
   .log10_2 = 0x1.a934fp+1,
@@ -53,28 +46,23 @@ sv_exp10f_inline (svfloat32_t x, const svbool_t pg, const struct data *d)
   /* exp10(x) = 2^(n/N) * 10^r = 2^n * (1 + poly (r)),
      with poly(r) in [1/sqrt(2), sqrt(2)] and
      x = r + n * log10(2) / N, with r in [-log10(2)/2N, log10(2)/2N].  */
-
-  svfloat32_t lane_consts = svld1rq (svptrue_b32 (), &d->log2_10_lo);
+  svfloat32_t lane_consts = svld1rq (svptrue_b32 (), &d->log10_2);
 
   /* n = round(x/(log10(2)/N)).  */
   svfloat32_t shift = sv_f32 (d->shift);
-  svfloat32_t z = svmad_x (pg, sv_f32 (d->log10_2), x, shift);
-  svfloat32_t n = svsub_x (svptrue_b32 (), z, shift);
+  svfloat32_t z = svmla_lane (shift, x, lane_consts, 0);
+  svfloat32_t n = svsub_x (pg, z, shift);
 
   /* r = x - n*log10(2)/N.  */
-  svfloat32_t r = svmsb_x (pg, sv_f32 (d->log2_10_hi), n, x);
-  r = svmls_lane (r, n, lane_consts, 0);
+  svfloat32_t r = x;
+  r = svmls_lane (r, n, lane_consts, 1);
+  r = svmls_lane (r, n, lane_consts, 2);
 
   svfloat32_t scale = svexpa (svreinterpret_u32 (z));
 
   /* Polynomial evaluation: poly(r) ~ exp10(r)-1.  */
-  svfloat32_t p12 = svmla_lane (sv_f32 (d->c1), r, lane_consts, 2);
-  svfloat32_t p34 = svmla_lane (sv_f32 (d->c3), r, lane_consts, 3);
-  svfloat32_t r2 = svmul_x (svptrue_b32 (), r, r);
-  svfloat32_t p14 = svmla_x (pg, p12, p34, r2);
-  svfloat32_t p0 = svmul_lane (r, lane_consts, 1);
-  svfloat32_t poly = svmla_x (pg, p0, r2, p14);
-
+  svfloat32_t poly = svmla_lane (sv_f32 (d->c0), r, lane_consts, 3);
+  poly = svmul_x (pg, poly, r);
   return svmla_x (pg, scale, scale, poly);
 }
 
@@ -85,11 +73,10 @@ special_case (svfloat32_t x, svbool_t special, const struct data *d)
 		      special);
 }
 
-/* Single-precision SVE exp10f routine. Implements the same algorithm
-   as AdvSIMD exp10f.
-   Worst case error is 1.02 ULPs.
-   _ZGVsMxv_exp10f(-0x1.040488p-4) got 0x1.ba5f9ep-1
-				  want 0x1.ba5f9cp-1.  */
+/* Single-precision SVE exp10f routine. Based on the FEXPA instruction.
+   Worst case error is 1.10 ULP.
+   _ZGVsMxv_exp10f (0x1.cc76dep+3) got 0x1.be0172p+47
+				  want 0x1.be017p+47.  */
 svfloat32_t SV_NAME_F1 (exp10) (svfloat32_t x, const svbool_t pg)
 {
   const struct data *d = ptr_barrier (&data);
diff --git a/sysdeps/aarch64/fpu/exp2_sve.c b/sysdeps/aarch64/fpu/exp2_sve.c
index 6db8526..c135852 100644
--- a/sysdeps/aarch64/fpu/exp2_sve.c
+++ b/sysdeps/aarch64/fpu/exp2_sve.c
@@ -19,23 +19,21 @@
 
 #include "sv_math.h"
 
-#define N (1 << V_EXP_TABLE_BITS)
-
 #define BigBound 1022
 #define UOFlowBound 1280
 
 static const struct data
 {
-  double c0, c2;
-  double c1, c3;
+  double c2, c4;
+  double c0, c1, c3;
   double shift, big_bound, uoflow_bound;
 } data = {
   /* Coefficients are computed using Remez algorithm with
      minimisation of the absolute error.  */
-  .c0 = 0x1.62e42fefa3686p-1, .c1 = 0x1.ebfbdff82c241p-3,
-  .c2 = 0x1.c6b09b16de99ap-5, .c3 = 0x1.3b2abf5571ad8p-7,
-  .shift = 0x1.8p52 / N,      .uoflow_bound = UOFlowBound,
-  .big_bound = BigBound,
+  .c0 = 0x1.62e42fefa39efp-1,  .c1 = 0x1.ebfbdff82a31bp-3,
+  .c2 = 0x1.c6b08d706c8a5p-5,  .c3 = 0x1.3b2ad2ff7d2f3p-7,
+  .c4 = 0x1.5d8761184beb3p-10, .shift = 0x1.800000000ffc0p+46,
+  .uoflow_bound = UOFlowBound, .big_bound = BigBound,
 };
 
 #define SpecialOffset 0x6000000000000000 /* 0x1p513.  */
@@ -64,50 +62,52 @@ special_case (svbool_t pg, svfloat64_t s, svfloat64_t y, svfloat64_t n,
       svadd_x (pg, svsub_x (pg, svreinterpret_u64 (s), SpecialBias2), b));
 
   /* |n| > 1280 => 2^(n) overflows.  */
-  svbool_t p_cmp = svacgt (pg, n, d->uoflow_bound);
+  svbool_t p_cmp = svacle (pg, n, d->uoflow_bound);
 
   svfloat64_t r1 = svmul_x (svptrue_b64 (), s1, s1);
   svfloat64_t r2 = svmla_x (pg, s2, s2, y);
   svfloat64_t r0 = svmul_x (svptrue_b64 (), r2, s1);
 
-  return svsel (p_cmp, r1, r0);
+  return svsel (p_cmp, r0, r1);
 }
 
 /* Fast vector implementation of exp2.
-   Maximum measured error is 1.65 ulp.
-   _ZGVsMxv_exp2(-0x1.4c264ab5b559bp-6) got 0x1.f8db0d4df721fp-1
-				       want 0x1.f8db0d4df721dp-1.  */
+   Maximum measured error is 0.52 + 0.5 ulp.
+   _ZGVsMxv_exp2 (0x1.3b72ad5b701bfp-1) got 0x1.8861641b49e08p+0
+				       want 0x1.8861641b49e07p+0.  */
 svfloat64_t SV_NAME_D1 (exp2) (svfloat64_t x, svbool_t pg)
 {
   const struct data *d = ptr_barrier (&data);
-  svbool_t no_big_scale = svacle (pg, x, d->big_bound);
-  svbool_t special = svnot_z (pg, no_big_scale);
-
-  /* Reduce x to k/N + r, where k is integer and r in [-1/2N, 1/2N].  */
-  svfloat64_t shift = sv_f64 (d->shift);
-  svfloat64_t kd = svadd_x (pg, x, shift);
-  svuint64_t ki = svreinterpret_u64 (kd);
-  /* kd = k/N.  */
-  kd = svsub_x (pg, kd, shift);
-  svfloat64_t r = svsub_x (pg, x, kd);
-
-  /* scale ~= 2^(k/N).  */
-  svuint64_t idx = svand_x (pg, ki, N - 1);
-  svuint64_t sbits = svld1_gather_index (pg, __v_exp_data, idx);
-  /* This is only a valid scale when -1023*N < k < 1024*N.  */
-  svuint64_t top = svlsl_x (pg, ki, 52 - V_EXP_TABLE_BITS);
-  svfloat64_t scale = svreinterpret_f64 (svadd_x (pg, sbits, top));
-
-  svfloat64_t c13 = svld1rq (svptrue_b64 (), &d->c1);
-  /* Approximate exp2(r) using polynomial.  */
-  /* y = exp2(r) - 1 ~= C0 r + C1 r^2 + C2 r^3 + C3 r^4.  */
+  svbool_t special = svacge (pg, x, d->big_bound);
+
+  svfloat64_t z = svadd_x (svptrue_b64 (), x, d->shift);
+  svfloat64_t n = svsub_x (svptrue_b64 (), z, d->shift);
+  svfloat64_t r = svsub_x (svptrue_b64 (), x, n);
+
+  svfloat64_t scale = svexpa (svreinterpret_u64 (z));
+
   svfloat64_t r2 = svmul_x (svptrue_b64 (), r, r);
-  svfloat64_t p01 = svmla_lane (sv_f64 (d->c0), r, c13, 0);
-  svfloat64_t p23 = svmla_lane (sv_f64 (d->c2), r, c13, 1);
-  svfloat64_t p = svmla_x (pg, p01, p23, r2);
+  svfloat64_t c24 = svld1rq (svptrue_b64 (), &d->c2);
+
+  /* Approximate exp2(r) using polynomial.  */
+  /* y = exp2(r) - 1 ~= r * (C0 + C1 r + C2 r^2 + C3 r^3 + C4 r^4).  */
+  svfloat64_t p12 = svmla_lane (sv_f64 (d->c1), r, c24, 0);
+  svfloat64_t p34 = svmla_lane (sv_f64 (d->c3), r, c24, 1);
+  svfloat64_t p = svmla_x (pg, p12, p34, r2);
+  p = svmad_x (pg, p, r, d->c0);
   svfloat64_t y = svmul_x (svptrue_b64 (), r, p);
+
   /* Assemble exp2(x) = exp2(r) * scale.  */
   if (__glibc_unlikely (svptest_any (pg, special)))
-    return special_case (pg, scale, y, kd, d);
+    {
+      /* FEXPA zeroes the sign bit, however the sign is meaningful to the
+          special case function so needs to be copied.
+          e = sign bit of u << 46.  */
+      svuint64_t e = svand_x (pg, svlsl_x (pg, svreinterpret_u64 (z), 46),
+            0x8000000000000000);
+      scale = svreinterpret_f64 (svadd_x (pg, e, svreinterpret_u64 (scale)));
+      return special_case (pg, scale, y, n, d);
+    }
+
   return svmla_x (pg, scale, scale, y);
 }
diff --git a/sysdeps/aarch64/fpu/exp2f_sve.c b/sysdeps/aarch64/fpu/exp2f_sve.c
index fcd7830..989cefb 100644
--- a/sysdeps/aarch64/fpu/exp2f_sve.c
+++ b/sysdeps/aarch64/fpu/exp2f_sve.c
@@ -18,21 +18,17 @@
    <https://www.gnu.org/licenses/>.  */
 
 #include "sv_math.h"
-#include "poly_sve_f32.h"
 
 #define Thres 0x1.5d5e2ap+6f
 
 static const struct data
 {
-  float c0, c2, c4, c1, c3;
-  float shift, thres;
+  float c0, c1, shift, thres;
 } data = {
-  /* Coefficients copied from the polynomial in AdvSIMD variant.  */
-  .c0 = 0x1.62e422p-1f,
-  .c1 = 0x1.ebf9bcp-3f,
-  .c2 = 0x1.c6bd32p-5f,
-  .c3 = 0x1.3ce9e4p-7f,
-  .c4 = 0x1.59977ap-10f,
+  /* Coefficients generated using Remez algorithm with minimisation of relative
+     error.  */
+  .c0 = 0x1.62e485p-1,
+  .c1 = 0x1.ebfbe0p-3,
   /* 1.5*2^17 + 127.  */
   .shift = 0x1.803f8p17f,
   /* Roughly 87.3. For x < -Thres, the result is subnormal and not handled
@@ -51,16 +47,8 @@ sv_exp2f_inline (svfloat32_t x, const svbool_t pg, const struct data *d)
 
   svfloat32_t scale = svexpa (svreinterpret_u32 (z));
 
-  /* Polynomial evaluation: poly(r) ~ exp2(r)-1.
-     Evaluate polynomial use hybrid scheme - offset ESTRIN by 1 for
-     coefficients 1 to 4, and apply most significant coefficient directly.  */
-  svfloat32_t even_coeffs = svld1rq (svptrue_b32 (), &d->c0);
-  svfloat32_t r2 = svmul_x (svptrue_b32 (), r, r);
-  svfloat32_t p12 = svmla_lane (sv_f32 (d->c1), r, even_coeffs, 1);
-  svfloat32_t p34 = svmla_lane (sv_f32 (d->c3), r, even_coeffs, 2);
-  svfloat32_t p14 = svmla_x (pg, p12, r2, p34);
-  svfloat32_t p0 = svmul_lane (r, even_coeffs, 0);
-  svfloat32_t poly = svmla_x (pg, p0, r2, p14);
+  svfloat32_t poly = svmla_x (pg, sv_f32 (d->c0), r, sv_f32 (d->c1));
+  poly = svmul_x (svptrue_b32 (), poly, r);
 
   return svmla_x (pg, scale, scale, poly);
 }
@@ -72,11 +60,10 @@ special_case (svfloat32_t x, svbool_t special, const struct data *d)
 		      special);
 }
 
-/* Single-precision SVE exp2f routine. Implements the same algorithm
-   as AdvSIMD exp2f.
-   Worst case error is 1.04 ULPs.
-   _ZGVsMxv_exp2f(-0x1.af994ap-3) got 0x1.ba6a66p-1
-				 want 0x1.ba6a64p-1.  */
+/* Single-precision SVE exp2f routine, based on the FEXPA instruction.
+   Worst case error is 1.09 ULPs.
+   _ZGVsMxv_exp2f (0x1.9a2a94p-1) got 0x1.be1054p+0
+				 want 0x1.be1052p+0.  */
 svfloat32_t SV_NAME_F1 (exp2) (svfloat32_t x, const svbool_t pg)
 {
   const struct data *d = ptr_barrier (&data);
diff --git a/sysdeps/aarch64/fpu/expm1_sve.c b/sysdeps/aarch64/fpu/expm1_sve.c
index d4ba8cc..b1d940b 100644
--- a/sysdeps/aarch64/fpu/expm1_sve.c
+++ b/sysdeps/aarch64/fpu/expm1_sve.c
@@ -18,82 +18,164 @@
    <https://www.gnu.org/licenses/>.  */
 
 #include "sv_math.h"
-#include "poly_sve_f64.h"
 
-#define SpecialBound 0x1.62b7d369a5aa9p+9
-#define ExponentBias 0x3ff0000000000000
+#define FexpaBound 0x1.4cb5ecef28adap-3 /* 15*ln2/64.  */
+#define SpecialBound 0x1.628c2855bfaddp+9 /* ln(2^(1023 + 1/128)).  */
 
 static const struct data
 {
-  double poly[11];
-  double shift, inv_ln2, special_bound;
-  /* To be loaded in one quad-word.  */
+  double c2, c4;
+  double inv_ln2;
   double ln2_hi, ln2_lo;
+  double c0, c1, c3;
+  double shift, thres;
+  uint64_t expm1_data[32];
 } data = {
-  /* Generated using fpminimax.  */
-  .poly = { 0x1p-1, 0x1.5555555555559p-3, 0x1.555555555554bp-5,
-            0x1.111111110f663p-7, 0x1.6c16c16c1b5f3p-10, 0x1.a01a01affa35dp-13,
-            0x1.a01a018b4ecbbp-16, 0x1.71ddf82db5bb4p-19, 0x1.27e517fc0d54bp-22,
-            0x1.af5eedae67435p-26, 0x1.1f143d060a28ap-29, },
-
-  .special_bound = SpecialBound,
-  .inv_ln2 = 0x1.71547652b82fep0,
-  .ln2_hi = 0x1.62e42fefa39efp-1,
-  .ln2_lo = 0x1.abc9e3b39803fp-56,
-  .shift = 0x1.8p52,
+  /* Table emulating FEXPA - 1, for values of FEXPA close to 1.
+  The table holds values of 2^(i/64) - 1, computed in arbitrary precision.
+  The first half of the table stores values associated to i from 0 to 15.
+  The second half of the table stores values associated to i from 0 to -15.  */
+  .expm1_data = {
+      0x0000000000000000, 0x3f864d1f3bc03077, 0x3f966c34c5615d0f, 0x3fa0e8a30eb37901,
+      0x3fa6ab0d9f3121ec, 0x3fac7d865a7a3440, 0x3fb1301d0125b50a, 0x3fb429aaea92ddfb,
+      0x3fb72b83c7d517ae, 0x3fba35beb6fcb754, 0x3fbd4873168b9aa8, 0x3fc031dc431466b2,
+		  0x3fc1c3d373ab11c3, 0x3fc35a2b2f13e6e9, 0x3fc4f4efa8fef709, 0x3fc6942d3720185a,
+      0x0000000000000000, 0xbfc331751ec3a814, 0xbfc20224341286e4, 0xbfc0cf85bed0f8b7,
+      0xbfbf332113d56b1f, 0xbfbcc0768d4175a6, 0xbfba46f918837cb7, 0xbfb7c695afc3b424,
+		  0xbfb53f391822dbc7, 0xbfb2b0cfe1266bd4, 0xbfb01b466423250a, 0xbfaafd11874c009e,
+      0xbfa5b505d5b6f268, 0xbfa05e4119ea5d89, 0xbf95f134923757f3, 0xbf860f9f985bc9f4,
+    },
+
+  /* Generated using Remez, in [-log(2)/128, log(2)/128].  */
+  .c0 = 0x1p-1,
+  .c1 = 0x1.55555555548f9p-3,
+  .c2 = 0x1.5555555554c22p-5,
+  .c3 = 0x1.111123aaa2fb2p-7,
+  .c4 = 0x1.6c16d77d98e5bp-10,
+  .ln2_hi = 0x1.62e42fefa3800p-1,
+  .ln2_lo = 0x1.ef35793c76730p-45,
+  .inv_ln2 = 0x1.71547652b82fep+0,
+  .shift = 0x1.800000000ffc0p+46, /* 1.5*2^46+1023.  */
+  .thres = SpecialBound,
 };
 
-static svfloat64_t NOINLINE
-special_case (svfloat64_t x, svfloat64_t y, svbool_t pg)
+#define SpecialOffset 0x6000000000000000 /* 0x1p513.  */
+/* SpecialBias1 + SpecialBias1 = asuint(1.0).  */
+#define SpecialBias1 0x7000000000000000 /* 0x1p769.  */
+#define SpecialBias2 0x3010000000000000 /* 0x1p-254.  */
+
+static NOINLINE svfloat64_t
+special_case (svbool_t pg, svfloat64_t y, svfloat64_t s, svfloat64_t p,
+	      svfloat64_t n)
 {
-  return sv_call_f64 (expm1, x, y, pg);
+  /* s=2^n may overflow, break it up into s=s1*s2,
+     such that exp = s + s*y can be computed as s1*(s2+s2*y)
+     and s1*s1 overflows only if n>0.  */
+
+  /* If n<=0 then set b to 0x6, 0 otherwise.  */
+  svbool_t p_sign = svcmple (pg, n, 0.0); /* n <= 0.  */
+  svuint64_t b
+      = svdup_u64_z (p_sign, SpecialOffset); /* Inactive lanes set to 0.  */
+
+  /* Set s1 to generate overflow depending on sign of exponent n,
+     ie. s1 = 0x70...0 - b.  */
+  svfloat64_t s1 = svreinterpret_f64 (svsubr_x (pg, b, SpecialBias1));
+  /* Offset s to avoid overflow in final result if n is below threshold.
+     ie. s2 = as_u64 (s) - 0x3010...0 + b.  */
+  svfloat64_t s2 = svreinterpret_f64 (
+      svadd_x (pg, svsub_x (pg, svreinterpret_u64 (s), SpecialBias2), b));
+
+  /* |n| > 1280 => 2^(n) overflows.  */
+  svbool_t p_cmp = svacgt (pg, n, 1280.0);
+
+  svfloat64_t r1 = svmul_x (svptrue_b64 (), s1, s1);
+  svfloat64_t r2 = svmla_x (pg, s2, s2, p);
+  svfloat64_t r0 = svmul_x (svptrue_b64 (), r2, s1);
+
+  svbool_t is_safe = svacle (pg, n, 1023); /* Only correct special lanes.  */
+  return svsel (is_safe, y, svsub_x (pg, svsel (p_cmp, r1, r0), 1.0));
 }
 
-/* Double-precision vector exp(x) - 1 function.
-   The maximum error observed error is 2.18 ULP:
-   _ZGVsMxv_expm1(0x1.634ba0c237d7bp-2) got 0x1.a8b9ea8d66e22p-2
-				       want 0x1.a8b9ea8d66e2p-2.  */
+/* FEXPA based SVE expm1 algorithm.
+   Maximum measured error is 2.81 + 0.5 ULP:
+   _ZGVsMxv_expm1 (0x1.974060e619bfp-3) got 0x1.c290e5858bb53p-3
+				       want 0x1.c290e5858bb5p-3.  */
 svfloat64_t SV_NAME_D1 (expm1) (svfloat64_t x, svbool_t pg)
 {
   const struct data *d = ptr_barrier (&data);
 
-  /* Large, Nan/Inf.  */
-  svbool_t special = svnot_z (pg, svaclt (pg, x, d->special_bound));
-
-  /* Reduce argument to smaller range:
-     Let i = round(x / ln2)
-     and f = x - i * ln2, then f is in [-ln2/2, ln2/2].
-     exp(x) - 1 = 2^i * (expm1(f) + 1) - 1
-     where 2^i is exact because i is an integer.  */
-  svfloat64_t shift = sv_f64 (d->shift);
-  svfloat64_t n = svsub_x (pg, svmla_x (pg, shift, x, d->inv_ln2), shift);
-  svint64_t i = svcvt_s64_x (pg, n);
-  svfloat64_t ln2 = svld1rq (svptrue_b64 (), &d->ln2_hi);
-  svfloat64_t f = svmls_lane (x, n, ln2, 0);
-  f = svmls_lane (f, n, ln2, 1);
-
-  /* Approximate expm1(f) using polynomial.
-     Taylor expansion for expm1(x) has the form:
-	 x + ax^2 + bx^3 + cx^4 ....
-     So we calculate the polynomial P(f) = a + bf + cf^2 + ...
-     and assemble the approximation expm1(f) ~= f + f^2 * P(f).  */
-  svfloat64_t f2 = svmul_x (pg, f, f);
-  svfloat64_t f4 = svmul_x (pg, f2, f2);
-  svfloat64_t f8 = svmul_x (pg, f4, f4);
-  svfloat64_t p
-      = svmla_x (pg, f, f2, sv_estrin_10_f64_x (pg, f, f2, f4, f8, d->poly));
-
-  /* Assemble the result.
-   expm1(x) ~= 2^i * (p + 1) - 1
-   Let t = 2^i.  */
-  svint64_t u = svadd_x (pg, svlsl_x (pg, i, 52), ExponentBias);
-  svfloat64_t t = svreinterpret_f64 (u);
-
-  /* expm1(x) ~= p * t + (t - 1).  */
-  svfloat64_t y = svmla_x (pg, svsub_x (pg, t, 1), p, t);
+  svbool_t special = svacgt (pg, x, d->thres);
 
-  if (__glibc_unlikely (svptest_any (pg, special)))
-    return special_case (x, y, special);
+  svfloat64_t z = svmla_x (pg, sv_f64 (d->shift), x, d->inv_ln2);
+  svuint64_t u = svreinterpret_u64 (z);
+  svfloat64_t n = svsub_x (pg, z, d->shift);
 
+  /* r = x - n * ln2, r is in [-ln2/128, ln2/128].  */
+  svfloat64_t ln2 = svld1rq (svptrue_b64 (), &d->ln2_hi);
+  svfloat64_t r = x;
+  r = svmls_lane (r, n, ln2, 0);
+  r = svmls_lane (r, n, ln2, 1);
+
+  /* y = exp(r) - 1 ~= r + C0 r^2 + C1 r^3 + C2 r^4 + C3 r^5 + C4 r^6.  */
+  svfloat64_t r2 = svmul_x (svptrue_b64 (), r, r);
+  svfloat64_t c24 = svld1rq (svptrue_b64 (), &d->c2);
+
+  svfloat64_t p;
+  svfloat64_t c12 = svmla_lane (sv_f64 (d->c1), r, c24, 0);
+  svfloat64_t c34 = svmla_lane (sv_f64 (d->c3), r, c24, 1);
+  p = svmad_x (pg, c34, r2, c12);
+  p = svmad_x (pg, p, r, sv_f64 (d->c0));
+  p = svmad_x (pg, p, r2, r);
+
+  svfloat64_t scale = svexpa (u);
+  svfloat64_t scalem1 = svsub_x (pg, scale, sv_f64 (1.0));
+
+  /* We want to construct expm1(x) = (scale - 1) + scale * poly.
+     However, for values of scale close to 1, scale-1 causes large ULP errors
+     due to cancellation.
+
+     This can be circumvented by using a small lookup for scale-1
+     when our input is below a certain bound, otherwise we can use FEXPA.
+
+     This bound is based upon the table size:
+	   Bound = (TableSize-1/64) * ln2.
+     The current bound is based upon a table size of 16.  */
+  svbool_t is_small = svaclt (pg, x, FexpaBound);
+
+  if (svptest_any (pg, is_small))
+    {
+      /* Index via the input of FEXPA, but we only care about the lower 4 bits.
+       */
+      svuint64_t base_idx = svand_x (pg, u, 0xf);
+
+      /* We can use the sign of x as a fifth bit to account for the asymmetry
+	 of e^x around 0.  */
+      svuint64_t signBit
+	  = svlsl_x (pg, svlsr_x (pg, svreinterpret_u64 (x), 63), 4);
+      svuint64_t idx = svorr_x (pg, base_idx, signBit);
+
+      /* Lookup values for scale - 1 for small x.  */
+      svfloat64_t lookup = svreinterpret_f64 (
+	  svld1_gather_index (is_small, d->expm1_data, idx));
+
+      /* Select the appropriate scale - 1 value based on x.  */
+      scalem1 = svsel (is_small, lookup, scalem1);
+    }
+
+  svfloat64_t y = svmla_x (pg, scalem1, scale, p);
+
+  /* FEXPA returns nan for large inputs so we special case those.  */
+  if (__glibc_unlikely (svptest_any (pg, special)))
+    {
+      /* FEXPA zeroes the sign bit, however the sign is meaningful to the
+          special case function so needs to be copied.
+          e = sign bit of u << 46.  */
+      svuint64_t e = svand_x (pg, svlsl_x (pg, u, 46), 0x8000000000000000);
+      /* Copy sign to s.  */
+      scale = svreinterpret_f64 (svadd_x (pg, e, svreinterpret_u64 (scale)));
+      return special_case (pg, y, scale, p, n);
+    }
+
+  /* return expm1 = (scale - 1) + (scale * poly).  */
   return y;
 }