The algorithm is exp(y * log(x)), where log(x) is computed with about
1.3*2^-68 relative error (1.5*2^-68 without fma), returning the result
in two doubles, and the exp part uses the same algorithm (and lookup
tables) as exp, but takes the input as two doubles and a sign (to handle
negative bases with odd integer exponent). The __exp1 internal symbol
is no longer necessary.
There is separate code path when fma is not available but the worst case
error is about 0.54 ULP in both cases. The lookup table and consts for
log are 4168 bytes. The .rodata+.text is decreased by 37908 bytes on
aarch64. The non-nearest rounding error is less than 1 ULP.
Improvements on Cortex-A72 compared to current glibc master:
pow thruput: 2.40x in [0.01 11.1]x[0.01 11.1]
pow latency: 1.84x in [0.01 11.1]x[0.01 11.1]
Tested on
aarch64-linux-gnu (defined __FP_FAST_FMA, TOINT_INTRINSICS) and
arm-linux-gnueabihf (!defined __FP_FAST_FMA, !TOINT_INTRINSICS) and
x86_64-linux-gnu (!defined __FP_FAST_FMA, !TOINT_INTRINSICS) and
powerpc64le-linux-gnu (defined __FP_FAST_FMA, !TOINT_INTRINSICS) targets.
* NEWS: Mention pow improvements.
* math/Makefile (type-double-routines): Add e_pow_log_data.
* sysdeps/generic/math_private.h (__exp1): Remove.
* sysdeps/i386/fpu/e_pow_log_data.c: New file.
* sysdeps/ia64/fpu/e_pow_log_data.c: New file.
* sysdeps/ieee754/dbl-64/Makefile (CFLAGS-e_pow.c): Allow fma
contraction.
* sysdeps/ieee754/dbl-64/e_exp.c (__exp1): Remove.
(exp_inline): Remove.
(__ieee754_exp): Only single double input is handled.
* sysdeps/ieee754/dbl-64/e_pow.c: Rewrite.
* sysdeps/ieee754/dbl-64/e_pow_log_data.c: New file.
* sysdeps/ieee754/dbl-64/math_config.h (issignaling_inline): Define.
(__pow_log_data): Define.
* sysdeps/ieee754/dbl-64/upow.h: Remove.
* sysdeps/ieee754/dbl-64/upow.tbl: Remove.
* sysdeps/m68k/m680x0/fpu/e_pow_log_data.c: New file.
* sysdeps/x86_64/fpu/multiarch/Makefile (CFLAGS-e_pow-fma.c): Allow fma
contraction.
(CFLAGS-e_pow-fma4.c): Likewise.
Use += instead of = to avoid overriding target specific CFLAGS settings.
Ideally the settings in target Makefiles would have precedence, but the
Makefile inclusion order does not allow that, with this fix at least the
target settings are not dropped.
