[PATCH] Document the log table generation method

[Szabolcs Nagy <szabolcs dot nagy at arm dot com>]

documentation update following https://github.com/ARM-software/optimized-routines

>From 73f901c13d64a615406346b0b9d698ce6996e4c5 Mon Sep 17 00:00:00 2001
From: Szabolcs Nagy <szabolcs.nagy@arm.com>
Date: Wed, 5 Sep 2018 12:15:55 +0100
Subject: [PATCH] Document the log table generation method

Add comments with enough detail so the log lookup tables can be recreated.
---
 newlib/libm/common/log2_data.c    | 26 ++++++++++++++++++++++++++
 newlib/libm/common/log_data.c     | 26 ++++++++++++++++++++++++++
 newlib/libm/common/pow_log_data.c | 22 ++++++++++++++++++++++
 3 files changed, 74 insertions(+)

diff --git a/newlib/libm/common/log2_data.c b/newlib/libm/common/log2_data.c
index ee9efcc2a..f13e0c2f5 100644
--- a/newlib/libm/common/log2_data.c
+++ b/newlib/libm/common/log2_data.c
@@ -66,6 +66,32 @@ const struct log2_data __log2_data = {
 0x1.a6225e117f92ep-3,
 #endif
 },
+/* Algorithm:
+
+	x = 2^k z
+	log2(x) = k + log2(c) + log2(z/c)
+	log2(z/c) = poly(z/c - 1)
+
+where z is in [1.6p-1; 1.6p0] which is split into N subintervals and z falls
+into the ith one, then table entries are computed as
+
+	tab[i].invc = 1/c
+	tab[i].logc = (double)log2(c)
+	tab2[i].chi = (double)c
+	tab2[i].clo = (double)(c - (double)c)
+
+where c is near the center of the subinterval and is chosen by trying +-2^29
+floating point invc candidates around 1/center and selecting one for which
+
+	1) the rounding error in 0x1.8p10 + logc is 0,
+	2) the rounding error in z - chi - clo is < 0x1p-64 and
+	3) the rounding error in (double)log2(c) is minimized (< 0x1p-68).
+
+Note: 1) ensures that k + logc can be computed without rounding error, 2)
+ensures that z/c - 1 can be computed as (z - chi - clo)*invc with close to a
+single rounding error when there is no fast fma for z*invc - 1, 3) ensures
+that logc + poly(z/c - 1) has small error, however near x == 1 when
+|log2(x)| < 0x1p-4, this is not enough so that is special cased.  */
 .tab = {
 #if N == 64
 {0x1.724286bb1acf8p+0, -0x1.1095feecdb000p-1},
diff --git a/newlib/libm/common/log_data.c b/newlib/libm/common/log_data.c
index 26b9c3f44..27753c2cf 100644
--- a/newlib/libm/common/log_data.c
+++ b/newlib/libm/common/log_data.c
@@ -110,6 +110,32 @@ const struct log_data __log_data = {
 0x1.2493c29331a5cp-3,
 #endif
 },
+/* Algorithm:
+
+	x = 2^k z
+	log(x) = k ln2 + log(c) + log(z/c)
+	log(z/c) = poly(z/c - 1)
+
+where z is in [1.6p-1; 1.6p0] which is split into N subintervals and z falls
+into the ith one, then table entries are computed as
+
+	tab[i].invc = 1/c
+	tab[i].logc = (double)log(c)
+	tab2[i].chi = (double)c
+	tab2[i].clo = (double)(c - (double)c)
+
+where c is near the center of the subinterval and is chosen by trying +-2^29
+floating point invc candidates around 1/center and selecting one for which
+
+	1) the rounding error in 0x1.8p9 + logc is 0,
+	2) the rounding error in z - chi - clo is < 0x1p-66 and
+	3) the rounding error in (double)log(c) is minimized (< 0x1p-66).
+
+Note: 1) ensures that k*ln2hi + logc can be computed without rounding error,
+2) ensures that z/c - 1 can be computed as (z - chi - clo)*invc with close to
+a single rounding error when there is no fast fma for z*invc - 1, 3) ensures
+that logc + poly(z/c - 1) has small error, however near x == 1 when
+|log(x)| < 0x1p-4, this is not enough so that is special cased.  */
 .tab = {
 #if N == 64
 {0x1.7242886495cd8p+0, -0x1.79e267bdfe000p-2},
diff --git a/newlib/libm/common/pow_log_data.c b/newlib/libm/common/pow_log_data.c
index 9b94f5e4b..b5d78e065 100644
--- a/newlib/libm/common/pow_log_data.c
+++ b/newlib/libm/common/pow_log_data.c
@@ -50,6 +50,28 @@ const struct pow_log_data __pow_log_data = {
 -0x1.0002b8b263fc3p-3 * -8,
 #endif
 },
+/* Algorithm:
+
+	x = 2^k z
+	log(x) = k ln2 + log(c) + log(z/c)
+	log(z/c) = poly(z/c - 1)
+
+where z is in [0x1.69555p-1; 0x1.69555p0] which is split into N subintervals
+and z falls into the ith one, then table entries are computed as
+
+	tab[i].invc = 1/c
+	tab[i].logc = round(0x1p43*log(c))/0x1p43
+	tab[i].logctail = (double)(log(c) - logc)
+
+where c is chosen near the center of the subinterval such that 1/c has only a
+few precision bits so z/c - 1 is exactly representible as double:
+
+	1/c = center < 1 ? round(N/center)/N : round(2*N/center)/N/2
+
+Note: |z/c - 1| < 1/N for the chosen c, |log(c) - logc - logctail| < 0x1p-97,
+the last few bits of logc are rounded away so k*ln2hi + logc has no rounding
+error and the interval for z is selected such that near x == 1, where log(x)
+is tiny, large cancellation error is avoided in logc + poly(z/c - 1).  */
 .tab = {
 #if N == 128
 #define A(a,b,c) {a,0,b,c},
-- 
2.14.1