pl/math: Add vector/Neon tan

New routine uses a similar technique to the single-precision Neon
routine, but with an extra reduction to pi/8 using the double-angle
formula. It is accurate to 3.5 ULP.

7 files changed, 164 insertions, 0 deletions

diff --git a/pl/math/include/mathlib.h b/pl/math/include/mathlib.h
index 74c3b34..af5f9f9 100644
--- a/pl/math/include/mathlib.h
+++ b/pl/math/include/mathlib.h
@@ -70,6 +70,7 @@ double __s_log10 (double);
 double __s_log1p (double);
 double __s_log2 (double);
 double __s_sinh (double);
+double __s_tan (double);
 double __s_tanh (double);
 
 #if __aarch64__
@@ -113,6 +114,7 @@ __f64x2_t __v_log2 (__f64x2_t);
 __f32x4_t __v_sinhf (__f32x4_t);
 __f64x2_t __v_sinh (__f64x2_t);
 __f32x4_t __v_tanf (__f32x4_t);
+__f64x2_t __v_tan (__f64x2_t);
 __f32x4_t __v_tanhf (__f32x4_t);
 __f64x2_t __v_tanh (__f64x2_t);
 
@@ -149,6 +151,7 @@ __vpcs __f64x2_t __vn_log2 (__f64x2_t);
 __vpcs __f32x4_t __vn_sinhf (__f32x4_t);
 __vpcs __f64x2_t __vn_sinh (__f64x2_t);
 __vpcs __f32x4_t __vn_tanf (__f32x4_t);
+__vpcs __f64x2_t __vn_tan (__f64x2_t);
 __vpcs __f32x4_t __vn_tanhf (__f32x4_t);
 __vpcs __f64x2_t __vn_tanh (__f64x2_t);
 
@@ -182,6 +185,7 @@ __vpcs __f64x2_t _ZGVnN2v_log2 (__f64x2_t);
 __vpcs __f32x4_t _ZGVnN4v_sinhf (__f32x4_t);
 __vpcs __f64x2_t _ZGVnN2v_sinh (__f64x2_t);
 __vpcs __f32x4_t _ZGVnN4v_tanf (__f32x4_t);
+__vpcs __f64x2_t _ZGVnN2v_tan (__f64x2_t);
 __vpcs __f32x4_t _ZGVnN4v_tanhf (__f32x4_t);
 __vpcs __f64x2_t _ZGVnN2v_tanh (__f64x2_t);
 
diff --git a/pl/math/math_config.h b/pl/math/math_config.h
index 03b4ad4..9a7ce96 100644
--- a/pl/math/math_config.h
+++ b/pl/math/math_config.h
@@ -564,4 +564,9 @@ extern const struct cbrt_data
   double table[5];
 } __cbrt_data HIDDEN;
 
+extern const struct v_tan_data
+{
+  double neg_half_pi_hi, neg_half_pi_lo;
+  double poly[9];
+} __v_tan_data HIDDEN;
 #endif
diff --git a/pl/math/s_tan_3u5.c b/pl/math/s_tan_3u5.c
new file mode 100644
index 0000000..adb807c
--- /dev/null
+++ b/pl/math/s_tan_3u5.c
@@ -0,0 +1,6 @@
+/*
+ * Copyright (c) 2023, Arm Limited.
+ * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
+ */
+#define SCALAR 1
+#include "v_tan_3u5.c"
diff --git a/pl/math/tools/tan.sollya b/pl/math/tools/tan.sollya
new file mode 100644
index 0000000..bb0bb28
--- /dev/null
+++ b/pl/math/tools/tan.sollya
@@ -0,0 +1,20 @@
+// polynomial for approximating double precision tan(x)
+//
+// Copyright (c) 2023, Arm Limited.
+// SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
+
+deg = 8;
+
+// interval bounds
+a = 0x1.0p-126;
+b = pi / 8;
+
+display = hexadecimal;
+
+f = (tan(sqrt(x))-sqrt(x))/x^(3/2);
+poly = fpminimax(f, deg, [|double ...|], [a*a;b*b]);
+
+//print("rel error:", accurateinfnorm(1-poly(x)/f(x), [a;b], 30));
+print("in [",a,b,"]");
+print("coeffs:");
+for i from 0 to deg do coeff(poly,i);
diff --git a/pl/math/v_tan_3u5.c b/pl/math/v_tan_3u5.c
new file mode 100644
index 0000000..f87bacc
--- /dev/null
+++ b/pl/math/v_tan_3u5.c
@@ -0,0 +1,102 @@
+/*
+ * Double-precision vector tan(x) function.
+ *
+ * Copyright (c) 2023, Arm Limited.
+ * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
+ */
+
+#include "v_math.h"
+#include "estrin.h"
+#include "pl_sig.h"
+#include "pl_test.h"
+
+#if V_SUPPORTED
+
+#define MHalfPiHi v_f64 (__v_tan_data.neg_half_pi_hi)
+#define MHalfPiLo v_f64 (__v_tan_data.neg_half_pi_lo)
+#define TwoOverPi v_f64 (0x1.45f306dc9c883p-1)
+#define Shift v_f64 (0x1.8p52)
+#define AbsMask 0x7fffffffffffffff
+#define RangeVal 0x4160000000000000  /* asuint64(2^23).  */
+#define TinyBound 0x3e50000000000000 /* asuint64(2^-26).  */
+#define C(i) v_f64 (__v_tan_data.poly[i])
+
+/* Special cases (fall back to scalar calls).  */
+VPCS_ATTR
+NOINLINE static v_f64_t
+specialcase (v_f64_t x)
+{
+  return v_call_f64 (tan, x, x, v_u64 (-1));
+}
+
+/* Vector approximation for double-precision tan.
+   Maximum measured error is 3.48 ULP:
+   __v_tan(0x1.4457047ef78d8p+20) got -0x1.f6ccd8ecf7dedp+37
+				 want -0x1.f6ccd8ecf7deap+37.   */
+VPCS_ATTR
+v_f64_t V_NAME (tan) (v_f64_t x)
+{
+  v_u64_t iax = v_as_u64_f64 (x) & AbsMask;
+
+  /* Our argument reduction cannot calculate q with sufficient accuracy for very
+     large inputs. Fall back to scalar routine for all lanes if any are too
+     large, or Inf/NaN. If fenv exceptions are expected, also fall back for tiny
+     input to avoid underflow. Note pl does not supply a scalar double-precision
+     tan, so the fallback will be statically linked from the system libm.  */
+#if WANT_SIMD_EXCEPT
+  if (unlikely (v_any_u64 (iax - TinyBound > RangeVal - TinyBound)))
+#else
+  if (unlikely (v_any_u64 (iax > RangeVal)))
+#endif
+    return specialcase (x);
+
+  /* q = nearest integer to 2 * x / pi.  */
+  v_f64_t q = v_fma_f64 (x, TwoOverPi, Shift) - Shift;
+  v_s64_t qi = v_to_s64_f64 (q);
+
+  /* Use q to reduce x to r in [-pi/4, pi/4], by:
+     r = x - q * pi/2, in extended precision.  */
+  v_f64_t r = x;
+  r = v_fma_f64 (q, MHalfPiHi, r);
+  r = v_fma_f64 (q, MHalfPiLo, r);
+  /* Further reduce r to [-pi/8, pi/8], to be reconstructed using double angle
+     formula.  */
+  r = r * 0.5;
+
+  /* Approximate tan(r) using order 8 polynomial.
+     tan(x) is odd, so polynomial has the form:
+     tan(x) ~= x + C0 * x^3 + C1 * x^5 + C3 * x^7 + ...
+     Hence we first approximate P(r) = C1 + C2 * r^2 + C3 * r^4 + ...
+     Then compute the approximation by:
+     tan(r) ~= r + r^3 * (C0 + r^2 * P(r)).  */
+  v_f64_t r2 = r * r, r4 = r2 * r2, r8 = r4 * r4;
+  /* Use offset version of Estrin wrapper to evaluate from C1 onwards.  */
+  v_f64_t p = ESTRIN_7_ (r2, r4, r8, C, 1);
+  p = v_fma_f64 (p, r2, C (0));
+  p = v_fma_f64 (r2, p * r, r);
+
+  /* Recombination uses double-angle formula:
+     tan(2x) = 2 * tan(x) / (1 - (tan(x))^2)
+     and reciprocity around pi/2:
+     tan(x) = 1 / (tan(pi/2 - x))
+     to assemble result using change-of-sign and conditional selection of
+     numerator/denominator, dependent on odd/even-ness of q (hence quadrant). */
+  v_f64_t n = v_fma_f64 (p, p, v_f64 (-1));
+  v_f64_t d = p * 2;
+
+  v_u64_t use_recip = v_cond_u64 ((v_as_u64_s64 (qi) & 1) == 0);
+
+  return v_sel_f64 (use_recip, -d, n) / v_sel_f64 (use_recip, n, d);
+}
+VPCS_ALIAS
+
+PL_SIG (V, D, 1, tan, -3.1, 3.1)
+PL_TEST_ULP (V_NAME (tan), 2.99)
+PL_TEST_EXPECT_FENV (V_NAME (tan), WANT_SIMD_EXCEPT)
+PL_TEST_INTERVAL (V_NAME (tan), 0, TinyBound, 5000)
+PL_TEST_INTERVAL (V_NAME (tan), TinyBound, RangeVal, 100000)
+PL_TEST_INTERVAL (V_NAME (tan), RangeVal, inf, 5000)
+PL_TEST_INTERVAL (V_NAME (tan), -0, -TinyBound, 5000)
+PL_TEST_INTERVAL (V_NAME (tan), -TinyBound, -RangeVal, 100000)
+PL_TEST_INTERVAL (V_NAME (tan), -RangeVal, -inf, 5000)
+#endif
diff --git a/pl/math/v_tan_data.c b/pl/math/v_tan_data.c
new file mode 100644
index 0000000..04e2516
--- /dev/null
+++ b/pl/math/v_tan_data.c
@@ -0,0 +1,15 @@
+/*
+ * Coefficients and helpers for double-precision vector tan(x) function.
+ *
+ * Copyright (c) 2023, Arm Limited.
+ * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
+ */
+#include "math_config.h"
+
+const struct v_tan_data __v_tan_data
+  = {.neg_half_pi_hi = -0x1.921fb54442d18p0,
+     .neg_half_pi_lo = -0x1.1a62633145c07p-54,
+     .poly
+     = {0x1.5555555555556p-2, 0x1.1111111110a63p-3, 0x1.ba1ba1bb46414p-5,
+	0x1.664f47e5b5445p-6, 0x1.226e5e5ecdfa3p-7, 0x1.d6c7ddbf87047p-9,
+	0x1.7ea75d05b583ep-10, 0x1.289f22964a03cp-11, 0x1.4e4fd14147622p-12}};
diff --git a/pl/math/vn_tan_3u5.c b/pl/math/vn_tan_3u5.c
new file mode 100644
index 0000000..a4efb06
--- /dev/null
+++ b/pl/math/vn_tan_3u5.c
@@ -0,0 +1,12 @@
+/*
+ * AdvSIMD vector PCS variant of __v_tan.
+ *
+ * Copyright (c) 2023, Arm Limited.
+ * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
+ */
+#include "include/mathlib.h"
+#ifdef __vpcs
+#define VPCS 1
+#define VPCS_ALIAS PL_ALIAS (__vn_tan, _ZGVnN2v_tan)
+#include "v_tan_3u5.c"
+#endif