Document the methods used to compute logf. Taken and edited from the

double version, with adaptations for the differences between it and
the float version.

1 files changed, 51 insertions, 0 deletions

diff --git a/lib/msun/src/e_logf.c b/lib/msun/src/e_logf.c
index c3be6ed..77fe479 100644
--- a/lib/msun/src/e_logf.c
+++ b/lib/msun/src/e_logf.c
@@ -19,6 +19,57 @@ __FBSDID("$FreeBSD$");
 #include "math.h"
 #include "math_private.h"
 
+/* __ieee754_log(x)
+ * Return the logrithm of x
+ *
+ * Method :                  
+ *   1. Argument Reduction: find k and f such that 
+ *                      x = 2^k * (1+f), 
+ *         where  sqrt(2)/2 < 1+f < sqrt(2) .
+ *
+ *   2. Approximation of log(1+f).
+ *      Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
+ *               = 2s + 2/3 s**3 + 2/5 s**5 + .....,
+ *               = 2s + s*R
+ *      We use a special Reme algorithm on [0,0.1716] to generate 
+ *      a polynomial of degree 8 to approximate R The maximum error 
+ *      of this polynomial approximation is bounded by 2**-34.24. In
+ *      other words,
+ *                      2      4      6      8
+ *          R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s
+ *      (the values of Lg1 to Lg7 are listed in the program)
+ *      and
+ *          |      2          8           |     -34.24
+ *          | Lg1*s +...+Lg4*s    -  R(z) | <= 2 
+ *          |                             |
+ *      Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
+ *      In order to guarantee error in log below 1ulp, we compute log
+ *      by
+ *              log(1+f) = f - s*(f - R)        (if f is not too large)
+ *              log(1+f) = f - (hfsq - s*(hfsq+R)).     (better accuracy)
+ *
+ *      3. Finally,  log(x) = k*ln2 + log(1+f).  
+ *                          = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
+ *         Here ln2 is split into two floating point number: 
+ *                      ln2_hi + ln2_lo,
+ *         where n*ln2_hi is always exact for |n| < 2000.
+ *
+ * Special cases:
+ *      log(x) is NaN with signal if x < 0 (including -INF) ; 
+ *      log(+INF) is +INF; log(0) is -INF with signal;
+ *      log(NaN) is that NaN with no signal.
+ *
+ * Accuracy:
+ *      according to an error analysis, the error is always less than
+ *      1 ulp (unit in the last place).
+ *
+ * Constants:
+ * The hexadecimal values are the intended ones for the following 
+ * constants. The decimal values may be used, provided that the 
+ * compiler will convert from decimal to binary accurately enough 
+ * to produce the hexadecimal values shown.
+ */
+
 static const float
 ln2_hi =   6.9313812256e-01,	/* 0x3f317180 */
 ln2_lo =   9.0580006145e-06,	/* 0x3717f7d1 */