e6226e6b72
Reduce the number of subintervals that need lookup table and optimize the evaluation steps. Currently, `exp2f` is computed by reducing to `2^hi * 2^mid * 2^lo` where `-16/32 <= mid <= 15/32` and `-1/64 <= lo <= 1/64`, and `2^lo` is then approximated by a degree 6 polynomial. Experiment with Sollya showed that by using a degree 6 polynomial, we can approximate `2^lo` for a bigger range with reasonable errors: ``` > P = fpminimax((2^x - 1)/x, 5, [|D...|], [-1/64, 1/64]); > dirtyinfnorm(2^x - 1 - x*P, [-1/64, 1/64]); 0x1.e18a1bc09114def49eb851655e2e5c4dd08075ac2p-63 > P = fpminimax((2^x - 1)/x, 5, [|D...|], [-1/32, 1/32]); > dirtyinfnorm(2^x - 1 - x*P, [-1/32, 1/32]); 0x1.05627b6ed48ca417fe53e3495f7df4baf84a05e2ap-56 ``` So we can optimize the implementation a bit with: # Reduce the range to `mid = i/16` for `i = 0..15` and `-1/32 <= lo <= 1/32` # Store the table `2^mid` in bits, and add `hi` directly to its exponent field to compute `2^hi * 2^mid` # Rearrange the order of evaluating the polynomial approximating `2^lo`. Performance benchmark using perf tool from the CORE-MATH project on Ryzen 1700: ``` $ CORE_MATH_PERF_MODE="rdtsc" ./perf.sh exp2f GNU libc version: 2.35 GNU libc release: stable CORE-MATH reciprocal throughput : 9.534 System LIBC reciprocal throughput : 6.229 BEFORE: LIBC reciprocal throughput : 21.405 LIBC reciprocal throughput : 15.241 (with `-msse4.2` flag) LIBC reciprocal throughput : 11.111 (with `-mfma` flag) AFTER: LIBC reciprocal throughput : 18.617 LIBC reciprocal throughput : 12.852 (with `-msse4.2` flag) LIBC reciprocal throughput : 9.253 (with `-mfma` flag) $ CORE_MATH_PERF_MODE="rdtsc" ./perf.sh exp2f --latency GNU libc version: 2.35 GNU libc release: stable CORE-MATH latency : 40.869 System LIBC latency : 30.580 BEFORE LIBC latency : 64.888 LIBC latency : 61.027 (with `-msse4.2` flag) LIBC latency : 48.778 (with `-mfma` flag) AFTER LIBC latency : 48.803 LIBC latency : 45.047 (with `-msse4.2` flag) LIBC latency : 37.487 (with `-mfma` flag) ``` Reviewed By: sivachandra, orex Differential Revision: https://reviews.llvm.org/D133870
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