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c++ source #1
Output
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Intel asm syntax
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Filters
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Debug intrinsics
Compiler
6502-c++ 11.1.0
ARM GCC 10.2.0
ARM GCC 10.3.0
ARM GCC 10.4.0
ARM GCC 10.5.0
ARM GCC 11.1.0
ARM GCC 11.2.0
ARM GCC 11.3.0
ARM GCC 11.4.0
ARM GCC 12.1.0
ARM GCC 12.2.0
ARM GCC 12.3.0
ARM GCC 13.1.0
ARM GCC 13.2.0
ARM GCC 13.2.0 (unknown-eabi)
ARM GCC 4.5.4
ARM GCC 4.6.4
ARM GCC 5.4
ARM GCC 6.3.0
ARM GCC 6.4.0
ARM GCC 7.3.0
ARM GCC 7.5.0
ARM GCC 8.2.0
ARM GCC 8.5.0
ARM GCC 9.3.0
ARM GCC 9.4.0
ARM GCC 9.5.0
ARM GCC trunk
ARM gcc 10.2.1 (none)
ARM gcc 10.3.1 (2021.07 none)
ARM gcc 10.3.1 (2021.10 none)
ARM gcc 11.2.1 (none)
ARM gcc 5.4.1 (none)
ARM gcc 7.2.1 (none)
ARM gcc 8.2 (WinCE)
ARM gcc 8.3.1 (none)
ARM gcc 9.2.1 (none)
ARM msvc v19.0 (WINE)
ARM msvc v19.10 (WINE)
ARM msvc v19.14 (WINE)
ARM64 Morello gcc 10.1 Alpha 2
ARM64 gcc 10.2
ARM64 gcc 10.3
ARM64 gcc 10.4
ARM64 gcc 10.5.0
ARM64 gcc 11.1
ARM64 gcc 11.2
ARM64 gcc 11.3
ARM64 gcc 11.4.0
ARM64 gcc 12.1
ARM64 gcc 12.2.0
ARM64 gcc 12.3.0
ARM64 gcc 13.1.0
ARM64 gcc 13.2.0
ARM64 gcc 5.4
ARM64 gcc 6.3
ARM64 gcc 6.4
ARM64 gcc 7.3
ARM64 gcc 7.5
ARM64 gcc 8.2
ARM64 gcc 8.5
ARM64 gcc 9.3
ARM64 gcc 9.4
ARM64 gcc 9.5
ARM64 gcc trunk
ARM64 msvc v19.14 (WINE)
AVR gcc 10.3.0
AVR gcc 11.1.0
AVR gcc 12.1.0
AVR gcc 12.2.0
AVR gcc 12.3.0
AVR gcc 13.1.0
AVR gcc 13.2.0
AVR gcc 4.5.4
AVR gcc 4.6.4
AVR gcc 5.4.0
AVR gcc 9.2.0
AVR gcc 9.3.0
Arduino Mega (1.8.9)
Arduino Uno (1.8.9)
BPF clang (trunk)
BPF clang 13.0.0
BPF clang 14.0.0
BPF clang 15.0.0
BPF clang 16.0.0
BPF clang 17.0.1
BPF clang 18.1.0
BPF gcc 13.1.0
BPF gcc 13.2.0
BPF gcc trunk
EDG (experimental reflection)
EDG 6.5
EDG 6.5 (GNU mode gcc 13)
EDG 6.6
EDG 6.6 (GNU mode gcc 13)
FRC 2019
FRC 2020
FRC 2023
KVX ACB 4.1.0 (GCC 7.5.0)
KVX ACB 4.1.0-cd1 (GCC 7.5.0)
KVX ACB 4.10.0 (GCC 10.3.1)
KVX ACB 4.11.1 (GCC 10.3.1)
KVX ACB 4.12.0 (GCC 11.3.0)
KVX ACB 4.2.0 (GCC 7.5.0)
KVX ACB 4.3.0 (GCC 7.5.0)
KVX ACB 4.4.0 (GCC 7.5.0)
KVX ACB 4.6.0 (GCC 9.4.1)
KVX ACB 4.8.0 (GCC 9.4.1)
KVX ACB 4.9.0 (GCC 9.4.1)
M68K gcc 13.1.0
M68K gcc 13.2.0
M68k clang (trunk)
MRISC32 gcc (trunk)
MSP430 gcc 4.5.3
MSP430 gcc 5.3.0
MSP430 gcc 6.2.1
MinGW clang 14.0.3
MinGW clang 14.0.6
MinGW clang 15.0.7
MinGW clang 16.0.0
MinGW clang 16.0.2
MinGW gcc 11.3.0
MinGW gcc 12.1.0
MinGW gcc 12.2.0
MinGW gcc 13.1.0
RISC-V (32-bits) gcc (trunk)
RISC-V (32-bits) gcc 10.2.0
RISC-V (32-bits) gcc 10.3.0
RISC-V (32-bits) gcc 11.2.0
RISC-V (32-bits) gcc 11.3.0
RISC-V (32-bits) gcc 11.4.0
RISC-V (32-bits) gcc 12.1.0
RISC-V (32-bits) gcc 12.2.0
RISC-V (32-bits) gcc 12.3.0
RISC-V (32-bits) gcc 13.1.0
RISC-V (32-bits) gcc 13.2.0
RISC-V (32-bits) gcc 8.2.0
RISC-V (32-bits) gcc 8.5.0
RISC-V (32-bits) gcc 9.4.0
RISC-V (64-bits) gcc (trunk)
RISC-V (64-bits) gcc 10.2.0
RISC-V (64-bits) gcc 10.3.0
RISC-V (64-bits) gcc 11.2.0
RISC-V (64-bits) gcc 11.3.0
RISC-V (64-bits) gcc 11.4.0
RISC-V (64-bits) gcc 12.1.0
RISC-V (64-bits) gcc 12.2.0
RISC-V (64-bits) gcc 12.3.0
RISC-V (64-bits) gcc 13.1.0
RISC-V (64-bits) gcc 13.2.0
RISC-V (64-bits) gcc 8.2.0
RISC-V (64-bits) gcc 8.5.0
RISC-V (64-bits) gcc 9.4.0
RISC-V rv32gc clang (trunk)
RISC-V rv32gc clang 10.0.0
RISC-V rv32gc clang 10.0.1
RISC-V rv32gc clang 11.0.0
RISC-V rv32gc clang 11.0.1
RISC-V rv32gc clang 12.0.0
RISC-V rv32gc clang 12.0.1
RISC-V rv32gc clang 13.0.0
RISC-V rv32gc clang 13.0.1
RISC-V rv32gc clang 14.0.0
RISC-V rv32gc clang 15.0.0
RISC-V rv32gc clang 16.0.0
RISC-V rv32gc clang 17.0.1
RISC-V rv32gc clang 18.1.0
RISC-V rv32gc clang 9.0.0
RISC-V rv32gc clang 9.0.1
RISC-V rv64gc clang (trunk)
RISC-V rv64gc clang 10.0.0
RISC-V rv64gc clang 10.0.1
RISC-V rv64gc clang 11.0.0
RISC-V rv64gc clang 11.0.1
RISC-V rv64gc clang 12.0.0
RISC-V rv64gc clang 12.0.1
RISC-V rv64gc clang 13.0.0
RISC-V rv64gc clang 13.0.1
RISC-V rv64gc clang 14.0.0
RISC-V rv64gc clang 15.0.0
RISC-V rv64gc clang 16.0.0
RISC-V rv64gc clang 17.0.1
RISC-V rv64gc clang 18.1.0
RISC-V rv64gc clang 9.0.0
RISC-V rv64gc clang 9.0.1
Raspbian Buster
Raspbian Stretch
SPARC LEON gcc 12.2.0
SPARC LEON gcc 12.3.0
SPARC LEON gcc 13.1.0
SPARC LEON gcc 13.2.0
SPARC gcc 12.2.0
SPARC gcc 12.3.0
SPARC gcc 13.1.0
SPARC gcc 13.2.0
SPARC64 gcc 12.2.0
SPARC64 gcc 12.3.0
SPARC64 gcc 13.1.0
SPARC64 gcc 13.2.0
TI C6x gcc 12.2.0
TI C6x gcc 12.3.0
TI C6x gcc 13.1.0
TI C6x gcc 13.2.0
TI CL430 21.6.1
VAX gcc NetBSDELF 10.4.0
VAX gcc NetBSDELF 10.5.0 (Nov 15 03:50:22 2023)
WebAssembly clang (trunk)
Xtensa ESP32 gcc 11.2.0 (2022r1)
Xtensa ESP32 gcc 12.2.0 (20230208)
Xtensa ESP32 gcc 8.2.0 (2019r2)
Xtensa ESP32 gcc 8.2.0 (2020r1)
Xtensa ESP32 gcc 8.2.0 (2020r2)
Xtensa ESP32 gcc 8.4.0 (2020r3)
Xtensa ESP32 gcc 8.4.0 (2021r1)
Xtensa ESP32 gcc 8.4.0 (2021r2)
Xtensa ESP32-S2 gcc 11.2.0 (2022r1)
Xtensa ESP32-S2 gcc 12.2.0 (20230208)
Xtensa ESP32-S2 gcc 8.2.0 (2019r2)
Xtensa ESP32-S2 gcc 8.2.0 (2020r1)
Xtensa ESP32-S2 gcc 8.2.0 (2020r2)
Xtensa ESP32-S2 gcc 8.4.0 (2020r3)
Xtensa ESP32-S2 gcc 8.4.0 (2021r1)
Xtensa ESP32-S2 gcc 8.4.0 (2021r2)
Xtensa ESP32-S3 gcc 11.2.0 (2022r1)
Xtensa ESP32-S3 gcc 12.2.0 (20230208)
Xtensa ESP32-S3 gcc 8.4.0 (2020r3)
Xtensa ESP32-S3 gcc 8.4.0 (2021r1)
Xtensa ESP32-S3 gcc 8.4.0 (2021r2)
arm64 msvc v19.28 VS16.9
arm64 msvc v19.29 VS16.10
arm64 msvc v19.29 VS16.11
arm64 msvc v19.30
arm64 msvc v19.31
arm64 msvc v19.32
arm64 msvc v19.33
arm64 msvc v19.34
arm64 msvc v19.35
arm64 msvc v19.36
arm64 msvc v19.37
arm64 msvc v19.38
arm64 msvc v19.latest
armv7-a clang (trunk)
armv7-a clang 10.0.0
armv7-a clang 10.0.1
armv7-a clang 11.0.0
armv7-a clang 11.0.1
armv7-a clang 9.0.0
armv7-a clang 9.0.1
armv8-a clang (all architectural features, trunk)
armv8-a clang (trunk)
armv8-a clang 10.0.0
armv8-a clang 10.0.1
armv8-a clang 11.0.0
armv8-a clang 11.0.1
armv8-a clang 12.0.0
armv8-a clang 13.0.0
armv8-a clang 14.0.0
armv8-a clang 15.0.0
armv8-a clang 16.0.0
armv8-a clang 17.0.1
armv8-a clang 18.1.0
armv8-a clang 9.0.0
armv8-a clang 9.0.1
ellcc 0.1.33
ellcc 0.1.34
ellcc 2017-07-16
hexagon-clang 16.0.5
llvm-mos atari2600-3e
llvm-mos atari2600-4k
llvm-mos atari2600-common
llvm-mos atari5200-supercart
llvm-mos atari8-cart-megacart
llvm-mos atari8-cart-std
llvm-mos atari8-cart-xegs
llvm-mos atari8-common
llvm-mos atari8-dos
llvm-mos c128
llvm-mos c64
llvm-mos commodore
llvm-mos cpm65
llvm-mos cx16
llvm-mos dodo
llvm-mos eater
llvm-mos mega65
llvm-mos nes
llvm-mos nes-action53
llvm-mos nes-cnrom
llvm-mos nes-gtrom
llvm-mos nes-mmc1
llvm-mos nes-mmc3
llvm-mos nes-nrom
llvm-mos nes-unrom
llvm-mos nes-unrom-512
llvm-mos osi-c1p
llvm-mos pce
llvm-mos pce-cd
llvm-mos pce-common
llvm-mos pet
llvm-mos rp6502
llvm-mos rpc8e
llvm-mos supervision
llvm-mos vic20
loongarch64 gcc 12.2.0
loongarch64 gcc 12.3.0
loongarch64 gcc 13.1.0
loongarch64 gcc 13.2.0
mips clang 13.0.0
mips clang 14.0.0
mips clang 15.0.0
mips clang 16.0.0
mips clang 17.0.1
mips clang 18.1.0
mips gcc 11.2.0
mips gcc 12.1.0
mips gcc 12.2.0
mips gcc 12.3.0
mips gcc 13.1.0
mips gcc 13.2.0
mips gcc 4.9.4
mips gcc 5.4
mips gcc 5.5.0
mips gcc 9.3.0 (codescape)
mips gcc 9.5.0
mips64 (el) gcc 12.1.0
mips64 (el) gcc 12.2.0
mips64 (el) gcc 12.3.0
mips64 (el) gcc 13.1.0
mips64 (el) gcc 13.2.0
mips64 (el) gcc 4.9.4
mips64 (el) gcc 5.4.0
mips64 (el) gcc 5.5.0
mips64 (el) gcc 9.5.0
mips64 clang 13.0.0
mips64 clang 14.0.0
mips64 clang 15.0.0
mips64 clang 16.0.0
mips64 clang 17.0.1
mips64 clang 18.1.0
mips64 gcc 11.2.0
mips64 gcc 12.1.0
mips64 gcc 12.2.0
mips64 gcc 12.3.0
mips64 gcc 13.1.0
mips64 gcc 13.2.0
mips64 gcc 4.9.4
mips64 gcc 5.4.0
mips64 gcc 5.5.0
mips64 gcc 9.5.0
mips64el clang 13.0.0
mips64el clang 14.0.0
mips64el clang 15.0.0
mips64el clang 16.0.0
mips64el clang 17.0.1
mips64el clang 18.1.0
mipsel clang 13.0.0
mipsel clang 14.0.0
mipsel clang 15.0.0
mipsel clang 16.0.0
mipsel clang 17.0.1
mipsel clang 18.1.0
mipsel gcc 12.1.0
mipsel gcc 12.2.0
mipsel gcc 12.3.0
mipsel gcc 13.1.0
mipsel gcc 13.2.0
mipsel gcc 4.9.4
mipsel gcc 5.4.0
mipsel gcc 5.5.0
mipsel gcc 9.5.0
nanoMIPS gcc 6.3.0 (mtk)
power gcc 11.2.0
power gcc 12.1.0
power gcc 12.2.0
power gcc 12.3.0
power gcc 13.1.0
power gcc 13.2.0
power gcc 4.8.5
power64 AT12.0 (gcc8)
power64 AT13.0 (gcc9)
power64 gcc 11.2.0
power64 gcc 12.1.0
power64 gcc 12.2.0
power64 gcc 12.3.0
power64 gcc 13.1.0
power64 gcc 13.2.0
power64 gcc trunk
power64le AT12.0 (gcc8)
power64le AT13.0 (gcc9)
power64le clang (trunk)
power64le gcc 11.2.0
power64le gcc 12.1.0
power64le gcc 12.2.0
power64le gcc 12.3.0
power64le gcc 13.1.0
power64le gcc 13.2.0
power64le gcc 6.3.0
power64le gcc trunk
powerpc64 clang (trunk)
s390x gcc 11.2.0
s390x gcc 12.1.0
s390x gcc 12.2.0
s390x gcc 12.3.0
s390x gcc 13.1.0
s390x gcc 13.2.0
sh gcc 12.2.0
sh gcc 12.3.0
sh gcc 13.1.0
sh gcc 13.2.0
sh gcc 4.9.4
sh gcc 9.5.0
vast (trunk)
x64 msvc v19.0 (WINE)
x64 msvc v19.10 (WINE)
x64 msvc v19.14
x64 msvc v19.14 (WINE)
x64 msvc v19.15
x64 msvc v19.16
x64 msvc v19.20
x64 msvc v19.21
x64 msvc v19.22
x64 msvc v19.23
x64 msvc v19.24
x64 msvc v19.25
x64 msvc v19.26
x64 msvc v19.27
x64 msvc v19.28
x64 msvc v19.28 VS16.9
x64 msvc v19.29 VS16.10
x64 msvc v19.29 VS16.11
x64 msvc v19.30
x64 msvc v19.31
x64 msvc v19.32
x64 msvc v19.33
x64 msvc v19.34
x64 msvc v19.35
x64 msvc v19.36
x64 msvc v19.37
x64 msvc v19.38
x64 msvc v19.latest
x86 djgpp 4.9.4
x86 djgpp 5.5.0
x86 djgpp 6.4.0
x86 djgpp 7.2.0
x86 msvc v19.0 (WINE)
x86 msvc v19.10 (WINE)
x86 msvc v19.14
x86 msvc v19.14 (WINE)
x86 msvc v19.15
x86 msvc v19.16
x86 msvc v19.20
x86 msvc v19.21
x86 msvc v19.22
x86 msvc v19.23
x86 msvc v19.24
x86 msvc v19.25
x86 msvc v19.26
x86 msvc v19.27
x86 msvc v19.28
x86 msvc v19.28 VS16.9
x86 msvc v19.29 VS16.10
x86 msvc v19.29 VS16.11
x86 msvc v19.30
x86 msvc v19.31
x86 msvc v19.32
x86 msvc v19.33
x86 msvc v19.34
x86 msvc v19.35
x86 msvc v19.36
x86 msvc v19.37
x86 msvc v19.38
x86 msvc v19.latest
x86 nvc++ 22.11
x86 nvc++ 22.7
x86 nvc++ 22.9
x86 nvc++ 23.1
x86 nvc++ 23.11
x86 nvc++ 23.3
x86 nvc++ 23.5
x86 nvc++ 23.7
x86 nvc++ 23.9
x86 nvc++ 24.1
x86 nvc++ 24.3
x86-64 Zapcc 190308
x86-64 clang (amd-stg-open)
x86-64 clang (assertions trunk)
x86-64 clang (clangir)
x86-64 clang (experimental -Wlifetime)
x86-64 clang (experimental P1061)
x86-64 clang (experimental P1144)
x86-64 clang (experimental P1221)
x86-64 clang (experimental P2996)
x86-64 clang (experimental metaprogramming - P2632)
x86-64 clang (experimental pattern matching)
x86-64 clang (old concepts branch)
x86-64 clang (reflection)
x86-64 clang (resugar)
x86-64 clang (thephd.dev)
x86-64 clang (trunk)
x86-64 clang (variadic friends - P2893)
x86-64 clang (widberg)
x86-64 clang 10.0.0
x86-64 clang 10.0.0 (assertions)
x86-64 clang 10.0.1
x86-64 clang 11.0.0
x86-64 clang 11.0.0 (assertions)
x86-64 clang 11.0.1
x86-64 clang 12.0.0
x86-64 clang 12.0.0 (assertions)
x86-64 clang 12.0.1
x86-64 clang 13.0.0
x86-64 clang 13.0.0 (assertions)
x86-64 clang 13.0.1
x86-64 clang 14.0.0
x86-64 clang 14.0.0 (assertions)
x86-64 clang 15.0.0
x86-64 clang 15.0.0 (assertions)
x86-64 clang 16.0.0
x86-64 clang 16.0.0 (assertions)
x86-64 clang 17.0.1
x86-64 clang 17.0.1 (assertions)
x86-64 clang 18.1.0
x86-64 clang 18.1.0 (assertions)
x86-64 clang 2.6.0 (assertions)
x86-64 clang 2.7.0 (assertions)
x86-64 clang 2.8.0 (assertions)
x86-64 clang 2.9.0 (assertions)
x86-64 clang 3.0.0
x86-64 clang 3.0.0 (assertions)
x86-64 clang 3.1
x86-64 clang 3.1 (assertions)
x86-64 clang 3.2
x86-64 clang 3.2 (assertions)
x86-64 clang 3.3
x86-64 clang 3.3 (assertions)
x86-64 clang 3.4 (assertions)
x86-64 clang 3.4.1
x86-64 clang 3.5
x86-64 clang 3.5 (assertions)
x86-64 clang 3.5.1
x86-64 clang 3.5.2
x86-64 clang 3.6
x86-64 clang 3.6 (assertions)
x86-64 clang 3.7
x86-64 clang 3.7 (assertions)
x86-64 clang 3.7.1
x86-64 clang 3.8
x86-64 clang 3.8 (assertions)
x86-64 clang 3.8.1
x86-64 clang 3.9.0
x86-64 clang 3.9.0 (assertions)
x86-64 clang 3.9.1
x86-64 clang 4.0.0
x86-64 clang 4.0.0 (assertions)
x86-64 clang 4.0.1
x86-64 clang 5.0.0
x86-64 clang 5.0.0 (assertions)
x86-64 clang 5.0.1
x86-64 clang 5.0.2
x86-64 clang 6.0.0
x86-64 clang 6.0.0 (assertions)
x86-64 clang 6.0.1
x86-64 clang 7.0.0
x86-64 clang 7.0.0 (assertions)
x86-64 clang 7.0.1
x86-64 clang 7.1.0
x86-64 clang 8.0.0
x86-64 clang 8.0.0 (assertions)
x86-64 clang 8.0.1
x86-64 clang 9.0.0
x86-64 clang 9.0.0 (assertions)
x86-64 clang 9.0.1
x86-64 clang rocm-4.5.2
x86-64 clang rocm-5.0.2
x86-64 clang rocm-5.1.3
x86-64 clang rocm-5.2.3
x86-64 clang rocm-5.3.3
x86-64 clang rocm-5.7.0
x86-64 gcc (contract labels)
x86-64 gcc (contracts natural syntax)
x86-64 gcc (contracts)
x86-64 gcc (coroutines)
x86-64 gcc (modules)
x86-64 gcc (trunk)
x86-64 gcc 10.1
x86-64 gcc 10.2
x86-64 gcc 10.3
x86-64 gcc 10.4
x86-64 gcc 10.5
x86-64 gcc 11.1
x86-64 gcc 11.2
x86-64 gcc 11.3
x86-64 gcc 11.4
x86-64 gcc 12.1
x86-64 gcc 12.2
x86-64 gcc 12.3
x86-64 gcc 13.1
x86-64 gcc 13.2
x86-64 gcc 4.1.2
x86-64 gcc 4.4.7
x86-64 gcc 4.5.3
x86-64 gcc 4.6.4
x86-64 gcc 4.7.1
x86-64 gcc 4.7.2
x86-64 gcc 4.7.3
x86-64 gcc 4.7.4
x86-64 gcc 4.8.1
x86-64 gcc 4.8.2
x86-64 gcc 4.8.3
x86-64 gcc 4.8.4
x86-64 gcc 4.8.5
x86-64 gcc 4.9.0
x86-64 gcc 4.9.1
x86-64 gcc 4.9.2
x86-64 gcc 4.9.3
x86-64 gcc 4.9.4
x86-64 gcc 5.1
x86-64 gcc 5.2
x86-64 gcc 5.3
x86-64 gcc 5.4
x86-64 gcc 5.5
x86-64 gcc 6.1
x86-64 gcc 6.2
x86-64 gcc 6.3
x86-64 gcc 6.4
x86-64 gcc 7.1
x86-64 gcc 7.2
x86-64 gcc 7.3
x86-64 gcc 7.4
x86-64 gcc 7.5
x86-64 gcc 8.1
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#include <stdio.h> #include <stdlib.h> #include <stdint.h> #include <math.h> #define SINCOS_NJ (1) #define SINCOS_OLLYW (2) #define VARIANT (SINCOS_NJ) /* a single instruction in many 32-bit architectures */ uint32_t umul32hi (uint32_t a, uint32_t b) { return (uint32_t)(((uint64_t)a * b) >> 32); } /* a single instruction in many 32-bit architectures */ int32_t mul32hi (int32_t a, int32_t b) { return (int32_t)(uint32_t)((uint64_t)((int64_t)a * b) >> 32); } /* compute sine and cosine of argument in [0, PI/2] input and output in S8.23 format max err sine = 9.86237533e-8 max err cosine = 1.02729891e-7 rms err sine = 4.11141973e-8 rms err cosine = 4.11752018e-8 sin correctly rounded: 10961278 (83.19%) cos correctly rounded: 11070113 (84.01%) */ void sincos_fixed_nj (int32_t x, int32_t *sine, int32_t *cosine) { // minimax polynomial approximation for sine on [0, PI/4] const uint32_t s0 = (uint32_t)(1.9510998390614986e-4 * (1LL << 32) + 0.5); const uint32_t s1 = (uint32_t)(8.3322080317884684e-3 * (1LL << 32) + 0.5); const uint32_t s2 = (uint32_t)(1.6666648373939097e-1 * (1LL << 32) + 0.5); const uint32_t s3 = (uint32_t)(9.9999991734512150e-1 * (1LL << 32) + 0.5); // minimax polynomial approximation for cosine on [0, PI/4] const uint32_t c0 = (uint32_t)(1.3578890357166529e-3 * (1LL << 32) + 0.5); const uint32_t c1 = (uint32_t)(4.1654359549283981e-2 * (1LL << 32) + 0.5); const uint32_t c2 = (uint32_t)(4.9999838648363948e-1 * (1LL << 32) + 0.5); const uint32_t c3 = (uint32_t)(9.9999997159466147e-1 * (1LL << 32) + 0.5); // auxilliary constants const int32_t hpi_p23 = (int32_t)(3.141592653590 / 2 * (1LL << 23) + 0.5); const int32_t qpi_p23 = (int32_t)(3.141592653590 / 4 * (1LL << 23) + 0.5); const int32_t one_p23 = (int32_t)(1.0000000000000e+0 * (1LL << 23) + 0.5); uint32_t a, s, q, h, l, t, sn, cs; /* reduce range from [0, PI/2] to [0, PI/4] */ t = (x > qpi_p23) ? (hpi_p23 - x) : x; // S8.23 /* scale up argument for maximum precision in intermediate computation */ a = t << 9; // U0.32 /* pre-compute a**2 and a**4 */ s = umul32hi (a, a); // U0.32 q = umul32hi (s, s); // U0.32 /* approximate sine on [0, PI/4] */ h = s3 - umul32hi (s2, s); // U0.32 l = umul32hi (s1 - umul32hi (s0, s), q); // U0.32 sn = umul32hi (h + l, a); // U0.32 /* approximate cosine on [0, PI/4] */ h = c3 - umul32hi (c2, s); // U0.32 l = umul32hi (c1 - umul32hi (c0, s), q); // U0.32 cs = h + l; // U0.32 /* round results to target precision */ sn = ((sn + 256) >> 9); // S8.23 cs = ((cs + 256) >> 9); // S8.23 /* cosine result overflows U0.32 format for small arguments */ cs = (t < 0xb50) ? one_p23 : cs; // S8.23 /* map sine/cosine approximations based on quadrant */ *sine = (t != x) ? cs : sn; // S8.23 *cosine = (t != x) ? sn : cs; // S8.23 } /* compute sine and cosine of argument in [0, PI/2] input and output in S8.23 format max err sine = 1.13173883e-7 max err cosine = 1.13158773e-7 rms err sine = 4.30955921e-8 rms err cosine = 4.31472191e-8 sin correctly rounded: 10844170 (82.30%) cos correctly rounded: 10855609 (82.38%) Based on an approach by OllyW (http://www.olliw.eu/2014/fast-functions/, retrieved 10/23/2020). We transform a = 2/PI*x-1/2, then we approximate sin ((2*PI*a + PI)/4 and cos ((2*PI*a + PI)/4. Except for sign flipping in the odd-power terms of the expansions the two series expansions match: https://www.wolframalpha.com/input/?i=series++sin+%28%282*pi*a+%2B+pi%29%2F4%29 https://www.wolframalpha.com/input/?i=series++cos+%28%282*pi*a+%2B+pi%29%2F4%29 This means we can sum the odd-power and the even-power terms seperately, then compute the sum and difference of those sums giving sine and cosine. */ void sincos_fixed_ollyw (int32_t x, int32_t *sine, int32_t *cosine) { // minimax polynomial approximation for sin ((2*PI*a + PI)/4 on [-0.5, 0.5] const uint32_t c0 = (uint32_t)(7.0710676768794656e-1 * (1LL << 32) + 0.5); const uint32_t c1 = (uint32_t)((1.110721191857 -.25) * (1LL << 32) + 0.5); const uint32_t c2 = (uint32_t)(8.7235601339489222e-1 * (1LL << 32) + 0.5); const uint32_t c3 = (uint32_t)(4.5677902549505234e-1 * (1LL << 32) + 0.5); const uint32_t c4 = (uint32_t)(1.7932640877552330e-1 * (1LL << 32) + 0.5); const uint32_t c5 = (uint32_t)(5.6449491763487458e-2 * (1LL << 32) + 0.5); const uint32_t c6 = (uint32_t)(1.4444266213104129e-2 * (1LL << 32) + 0.5); const uint32_t c7 = (uint32_t)(3.4931597765535116e-3 * (1LL << 32) + 0.5); // auxiliary constants const uint32_t twoopi = (uint32_t)(2/3.1415926535898 * (1LL << 32) + 0.5); const uint32_t half_p31 = (uint32_t)(0.5000000000000 * (1LL << 31) + 0.5); const uint32_t quarter_p30 = (uint32_t)(0.2500000000 * (1LL << 30) + 0.5); uint32_t s, t, q, h, l; int32_t a, o, e, sn, cs; /* scale up argument for maximum precision in intermediate computation */ t = (uint32_t)x << 8; // U1.31 /* a = 2*PI*x - 0.5 */ a = umul32hi (twoopi, t) - half_p31; // S0.31 /* precompute a**2 and a**4 */ s = (uint32_t)mul32hi (a, a) << 2; // U0.32 q = umul32hi (s, s); // U0.32 /* sum odd power terms; add in second portion of c1 (= 0.25) at the end */ h = c1 - umul32hi (c3, s); // U0.32 l = umul32hi ((c5 - umul32hi (c7, s)), q); // U0.32 o = ((h + l) >> 2) + quarter_p30; // S1.30 o = mul32hi (o, a); // S2.29 /* sum even power terms */ h = c0 - umul32hi (c2, s); // U0.32 l = umul32hi ((c4 - umul32hi (c6, s)), q); // U0.32 e = (h + l) >> 3; // S2.29 /* compute sine and cosine as sum and difference of odd / even terms */ sn = e + o; // S2.29 sum -> sine cs = e - o; // S2.29 difference -> cosine /* round results to target precision */ sn = (sn + 32) >> 6; // S8.23 cs = (cs + 32) >> 6; // S8.23 *sine = sn; *cosine = cs; } double s8p23_to_double (int32_t a) { return (double)a / (1LL << 23); } int32_t double_to_s8p23 (double a) { return (int32_t)(a * (1LL << 23) + 0.5); } /* exhaustive test of S8.23 fixed-point sincos on [0,PI/2] */ int main (void) { double errc, errs, maxerrs, maxerrc, errsqs, errsqc; int32_t arg, sin_correctly_rounded, cos_correctly_rounded; #if VARIANT == SINCOS_OLLYW printf ("S8.23 fixed-point sincos OllyW variant\n"); #elif VARIANT == SINCOS_NJ printf ("S8.23 fixed-point sincos NJ variant\n"); #else // VARIANT #error unsupported VARIANT #endif // VARIANT maxerrs = 0; maxerrc = 0; errsqs = 0; errsqc = 0; sin_correctly_rounded = 0; cos_correctly_rounded = 0; for (arg = 0; arg <= double_to_s8p23 (3.14159265358979 / 2); arg++) { double argf, refs, refc; int32_t sine, cosine, refsi, refci; #if VARIANT == SINCOS_OLLYW sincos_fixed_ollyw (arg, &sine, &cosine); #elif VARIANT == SINCOS_NJ sincos_fixed_nj (arg, &sine, &cosine); #endif // VARIANT argf = s8p23_to_double (arg); refs = sin (argf); refc = cos (argf); refsi = double_to_s8p23 (refs); refci = double_to_s8p23 (refc); /* print function values near endpoints of interval */ if ((arg < 5) || (arg > 0xc90fd5)) { printf ("arg=%08x sin=%08x cos=%08x\n", arg, sine, cosine); } if (sine == refsi) sin_correctly_rounded++; if (cosine == refci) cos_correctly_rounded++; errs = fabs (s8p23_to_double (sine) - refs); errc = fabs (s8p23_to_double (cosine) - refc); errsqs += errs * errs; errsqc += errc * errc; if (errs > maxerrs) maxerrs = errs; if (errc > maxerrc) maxerrc = errc; } printf ("max err sine = %15.8e max err cosine = %15.8e\n", maxerrs, maxerrc); printf ("rms err sine = %15.8e rms err cosine = %15.8e\n", sqrt (errsqs / arg), sqrt (errsqc / arg)); printf ("sin correctly rounded: %d (%.2f%%) cos correctly rounded: %d (%.2f%%)\n", sin_correctly_rounded, 1.0 * sin_correctly_rounded / arg * 100, cos_correctly_rounded, 1.0 * cos_correctly_rounded / arg * 100); return EXIT_SUCCESS; }
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