Add new log2 implementation

Similar algorithm is used as in log, but there are more operations
(and more error) due to the 1/ln2 multiplier.

There is separate code path when fma instruction is not available for
computing x/c - 1 precisely, for which the table size is doubled,
and to compute (x/c - 1)/ln2 precisely.

The worst case error is 0.547 ULP (0.55 without fma), the read only
global data size is 1168 bytes (2192 without fma).  The non-nearest
rounding error is less than 1 ULP.

Improvements on Cortex-A72 compared to current glibc master:
log latency: 2.04x
log thruput: 1.87x

1 files changed, 144 insertions, 0 deletions

diff --git a/math/log2.c b/math/log2.c
new file mode 100644
index 0000000..bc4b8f8
--- /dev/null
+++ b/math/log2.c
@@ -0,0 +1,144 @@
+/*
+ * Double-precision log2(x) function.
+ *
+ * Copyright (c) 2018, Arm Limited.
+ * SPDX-License-Identifier: Apache-2.0
+ *
+ * Licensed under the Apache License, Version 2.0 (the "License");
+ * you may not use this file except in compliance with the License.
+ * You may obtain a copy of the License at
+ *
+ *     http://www.apache.org/licenses/LICENSE-2.0
+ *
+ * Unless required by applicable law or agreed to in writing, software
+ * distributed under the License is distributed on an "AS IS" BASIS,
+ * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+ * See the License for the specific language governing permissions and
+ * limitations under the License.
+ */
+
+#include <math.h>
+#include <stdint.h>
+#include "math_config.h"
+
+#define T __log2_data.tab
+#define T2 __log2_data.tab2
+#define B __log2_data.poly1
+#define A __log2_data.poly
+#define InvLn2hi __log2_data.invln2hi
+#define InvLn2lo __log2_data.invln2lo
+#define N (1 << LOG2_TABLE_BITS)
+#define OFF 0x3fe6000000000000
+
+static inline uint32_t
+top16 (double x)
+{
+  return asuint64 (x) >> 48;
+}
+
+double
+log2 (double x)
+{
+  /* double_t for better performance on targets with FLT_EVAL_METHOD==2.  */
+  double_t z, r, r2, r4, y, invc, logc, kd, hi, lo, t1, t2, t3, p;
+  uint64_t ix, iz, tmp;
+  uint32_t top;
+  int k, i;
+
+  ix = asuint64 (x);
+  top = top16 (x);
+
+#if LOG2_POLY1_ORDER == 11
+# define LO asuint64 (1.0 - 0x1.5b51p-5)
+# define HI asuint64 (1.0 + 0x1.6ab2p-5)
+#endif
+  if (unlikely (ix - LO < HI - LO))
+    {
+      /* Handle close to 1.0 inputs separately.  */
+      /* Fix sign of zero with downward rounding when x==1.  */
+      if (WANT_ROUNDING && unlikely (ix == asuint64 (1.0)))
+	return 0;
+      r = x - 1.0;
+#if HAVE_FAST_FMA
+      hi = r*InvLn2hi;
+      lo = r*InvLn2lo + fma (r, InvLn2hi, -hi);
+#else
+      double_t rhi, rlo;
+      rhi = asdouble (asuint64 (r) & -1ULL<<32);
+      rlo = r - rhi;
+      hi = rhi*InvLn2hi;
+      lo = rlo*InvLn2hi + r*InvLn2lo;
+#endif
+      r2 = r * r; /* rounding error: 0x1p-62.  */
+      r4 = r2 * r2;
+#if LOG2_POLY1_ORDER == 11
+      /* Worst-case error is less than 0.54 ULP (0.55 ULP without fma).  */
+      p = r2*(B[0] + r*B[1]);
+      y = hi + p;
+      lo += hi - y + p;
+      lo += r4*(B[2] + r*B[3] + r2*(B[4] + r*B[5])
+		+ r4*(B[6] + r*B[7] + r2*(B[8] + r*B[9])));
+      y += lo;
+#endif
+      return y;
+    }
+  if (unlikely (top - 0x0010 >= 0x7ff0 - 0x0010))
+    {
+      /* x < 0x1p-1022 or inf or nan.  */
+      if (ix * 2 == 0)
+	return __math_divzero (1);
+      if (ix == asuint64 (INFINITY)) /* log(inf) == inf.  */
+	return x;
+      if ((top & 0x8000) || (top & 0x7ff0) == 0x7ff0)
+	return __math_invalid (x);
+      /* x is subnormal, normalize it.  */
+      ix = asuint64 (x * 0x1p52);
+      ix -= 52ULL << 52;
+    }
+
+  /* x = 2^k z; where z is in range [OFF,2*OFF) and exact.
+     The range is split into N subintervals.
+     The ith subinterval contains z and c is near its center.  */
+  tmp = ix - OFF;
+  i = (tmp >> (52 - LOG2_TABLE_BITS)) % N;
+  k = (int64_t) tmp >> 52; /* arithmetic shift */
+  iz = ix - (tmp & 0xfffULL << 52);
+  invc = T[i].invc;
+  logc = T[i].logc;
+  z = asdouble (iz);
+  kd = (double_t) k;
+
+  /* log2(x) = log2(z/c) + log2(c) + k.  */
+  /* r ~= z/c - 1, |r| < 1/(2*N).  */
+#if HAVE_FAST_FMA
+  /* rounding error: 0x1p-55/N.  */
+  r = fma (z, invc, -1.0);
+  t1 = r*InvLn2hi;
+  t2 = r*InvLn2lo + fma (r, InvLn2hi, -t1);
+#else
+  double_t rhi, rlo;
+  /* rounding error: 0x1p-55/N + 0x1p-65.  */
+  r = (z - T2[i].chi - T2[i].clo)*invc;
+  rhi = asdouble (asuint64 (r) & -1ULL << 32);
+  rlo = r - rhi;
+  t1 = rhi*InvLn2hi;
+  t2 = rlo*InvLn2hi + r*InvLn2lo;
+#endif
+
+  /* hi + lo = r/ln2 + log2(c) + k.  */
+  t3 = kd + logc;
+  hi = t3 + t1;
+  lo = t3 - hi + t1 + t2;
+
+  /* log2(r+1) = r/ln2 + r^2*poly(r).  */
+  /* Evaluation is optimized assuming superscalar pipelined execution.  */
+  r2 = r * r; /* rounding error: 0x1p-54/N^2.  */
+  r4 = r2 * r2;
+#if LOG2_POLY_ORDER == 7
+  /* Worst-case error if |y| > 0x1p-4: 0.547 ULP (0.550 ULP without fma).
+     ~ 0.5 + 2/N/ln2 + abs-poly-error*0x1p56 ULP (+ 0.003 ULP without fma).  */
+  p = A[0] + r*A[1] + r2*(A[2] + r*A[3]) + r4*(A[4] + r*A[5]);
+  y = lo + r2*p + hi;
+#endif
+  return y;
+}