[PATCH 2/3 v2] powf: Fixed 2 bugs in the computation /* t_h=ax+bp[k] High */. (FreeBSD)

[Thomas Wucher]

On 26.09.24 04:11, Keith Packard wrote:
> From 3dd0fce84efd8a45e3501175c05086f8a1cbd832 Mon Sep 17 00:00:00 2001
> From: Keith Packard <keithp@keithp.com>
> Date: Wed, 25 Sep 2024 18:34:28 -0700
> Subject: [PATCH] libm: Improve powf accuracy
> 
> One of the constants, 'cp', was not correctly converted to extended
> precision (using two floats) leading to reduced precision in the
> computation of log2(x).
> 
> The new values use a larger value for cp_h which is closer to the
> actual value.  That leaves a smaller magnitude for cp_l resulting in
> increased precision for the result. These values were computed
> according to the Veltkamp/Dekker method.
> 
> 2/(3*ln2): 0x1.ec709dc3a03fd748p-1
> Old cp_h:  0x1.ec7p-1
> New cp_h:  0x1.ec8p-1
> 
> The desired value is much closer to the higher value than the lower.
> 
> Signed-off-by: Keith Packard <keithp@keithp.com>
> ---
>   newlib/libm/math/ef_pow.c | 4 ++--
>   1 file changed, 2 insertions(+), 2 deletions(-)
> 
> diff --git a/newlib/libm/math/ef_pow.c b/newlib/libm/math/ef_pow.c
> index a2aba1054..d69f1ae48 100644
> --- a/newlib/libm/math/ef_pow.c
> +++ b/newlib/libm/math/ef_pow.c
> @@ -47,8 +47,8 @@ lg2_h  =  6.93145752e-01, /* 0x3f317200 */
>   lg2_l  =  1.42860654e-06, /* 0x35bfbe8c */
>   ovt =  4.2995665694e-08, /* -(128-log2(ovfl+.5ulp)) */
>   cp    =  9.6179670095e-01, /* 0x3f76384f =2/(3ln2) */
> -cp_h  =  9.6179199219e-01, /* 0x3f763800 =head of cp */
> -cp_l  =  4.7017383622e-06, /* 0x369dc3a0 =tail of cp_h */
> +cp_h  =  9.61914062e-01,   /* 0x3f764000 =head of cp */
> +cp_l  = -1.17368574e-04,   /* 0xb8f623c6 =tail of cp */
>   ivln2    =  1.4426950216e+00, /* 0x3fb8aa3b =1/ln2 */
>   ivln2_h  =  1.4426879883e+00, /* 0x3fb8aa00 =16b 1/ln2*/
>   ivln2_l  =  7.0526075433e-06; /* 0x36eca570 =1/ln2 tail*/

This looks correct, thanks Keith! A very similar change has also been 
done in FreeBSD:

https://github.com/freebsd/freebsd-src/commit/b4437c3d322a0f6d23d12b6f76d2fc72d2ff0ec2