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fastmath.jl
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fastmath.jl
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# This file is a part of Julia. License is MIT: https://julialang.org/license
# Support for @fastmath
# This module provides versions of math functions that may violate
# strict IEEE semantics.
# This allows the following transformations. For more information see
# http://llvm.org/docs/LangRef.html#fast-math-flags:
# nnan: No NaNs - Allow optimizations to assume the arguments and
# result are not NaN. Such optimizations are required to retain
# defined behavior over NaNs, but the value of the result is
# undefined.
# ninf: No Infs - Allow optimizations to assume the arguments and
# result are not +/-Inf. Such optimizations are required to
# retain defined behavior over +/-Inf, but the value of the
# result is undefined.
# nsz: No Signed Zeros - Allow optimizations to treat the sign of a
# zero argument or result as insignificant.
# arcp: Allow Reciprocal - Allow optimizations to use the reciprocal
# of an argument rather than perform division.
# fast: Fast - Allow algebraically equivalent transformations that may
# dramatically change results in floating point (e.g.
# reassociate). This flag implies all the others.
module FastMath
export @fastmath
import Core.Intrinsics: sqrt_llvm_fast, neg_float_fast,
add_float_fast, sub_float_fast, mul_float_fast, div_float_fast, rem_float_fast,
eq_float_fast, ne_float_fast, lt_float_fast, le_float_fast
const fast_op =
Dict(# basic arithmetic
:+ => :add_fast,
:- => :sub_fast,
:* => :mul_fast,
:/ => :div_fast,
:(==) => :eq_fast,
:!= => :ne_fast,
:< => :lt_fast,
:<= => :le_fast,
:abs => :abs_fast,
:abs2 => :abs2_fast,
:cmp => :cmp_fast,
:conj => :conj_fast,
:inv => :inv_fast,
:rem => :rem_fast,
:sign => :sign_fast,
:isfinite => :isfinite_fast,
:isinf => :isinf_fast,
:isnan => :isnan_fast,
:issubnormal => :issubnormal_fast,
# math functions
:^ => :pow_fast,
:acos => :acos_fast,
:acosh => :acosh_fast,
:angle => :angle_fast,
:asin => :asin_fast,
:asinh => :asinh_fast,
:atan => :atan_fast,
:atanh => :atanh_fast,
:cbrt => :cbrt_fast,
:cis => :cis_fast,
:cos => :cos_fast,
:cosh => :cosh_fast,
:exp10 => :exp10_fast,
:exp2 => :exp2_fast,
:exp => :exp_fast,
:expm1 => :expm1_fast,
:hypot => :hypot_fast,
:log10 => :log10_fast,
:log1p => :log1p_fast,
:log2 => :log2_fast,
:log => :log_fast,
:max => :max_fast,
:min => :min_fast,
:minmax => :minmax_fast,
:sin => :sin_fast,
:sincos => :sincos_fast,
:sinh => :sinh_fast,
:sqrt => :sqrt_fast,
:tan => :tan_fast,
:tanh => :tanh_fast)
const rewrite_op =
Dict(:+= => :+,
:-= => :-,
:*= => :*,
:/= => :/,
:^= => :^)
function make_fastmath(expr::Expr)
if expr.head === :quote
return expr
elseif expr.head === :call && expr.args[1] === :^ && expr.args[3] isa Integer
# mimic Julia's literal_pow lowering of literal integer powers
return Expr(:call, :(Base.FastMath.pow_fast), make_fastmath(expr.args[2]), Val{expr.args[3]}())
end
op = get(rewrite_op, expr.head, :nothing)
if op !== :nothing
var = expr.args[1]
rhs = expr.args[2]
if isa(var, Symbol)
# simple assignment
expr = :($var = $op($var, $rhs))
elseif isa(var, Expr) && var.head === :ref
var = var::Expr
# array reference
arr = var.args[1]
inds = var.args[2:end]
arrvar = gensym()
indvars = Any[gensym() for _ in inds]
expr = quote
$(Expr(:(=), arrvar, arr))
$(Expr(:(=), Base.exprarray(:tuple, indvars), Base.exprarray(:tuple, inds)))
$arrvar[$(indvars...)] = $op($arrvar[$(indvars...)], $rhs)
end
end
end
Base.exprarray(make_fastmath(expr.head), Base.mapany(make_fastmath, expr.args))
end
function make_fastmath(symb::Symbol)
fast_symb = get(fast_op, symb, :nothing)
if fast_symb === :nothing
return symb
end
:(Base.FastMath.$fast_symb)
end
make_fastmath(expr) = expr
"""
@fastmath expr
Execute a transformed version of the expression, which calls functions that
may violate strict IEEE semantics. This allows the fastest possible operation,
but results are undefined -- be careful when doing this, as it may change numerical
results.
This sets the [LLVM Fast-Math flags](http://llvm.org/docs/LangRef.html#fast-math-flags),
and corresponds to the `-ffast-math` option in clang. See [the notes on performance
annotations](@ref man-performance-annotations) for more details.
# Examples
```jldoctest
julia> @fastmath 1+2
3
julia> @fastmath(sin(3))
0.1411200080598672
```
"""
macro fastmath(expr)
make_fastmath(esc(expr))
end
# Basic arithmetic
const FloatTypes = Union{Float16,Float32,Float64}
sub_fast(x::FloatTypes) = neg_float_fast(x)
add_fast(x::T, y::T) where {T<:FloatTypes} = add_float_fast(x, y)
sub_fast(x::T, y::T) where {T<:FloatTypes} = sub_float_fast(x, y)
mul_fast(x::T, y::T) where {T<:FloatTypes} = mul_float_fast(x, y)
div_fast(x::T, y::T) where {T<:FloatTypes} = div_float_fast(x, y)
rem_fast(x::T, y::T) where {T<:FloatTypes} = rem_float_fast(x, y)
add_fast(x::T, y::T, zs::T...) where {T<:FloatTypes} =
add_fast(add_fast(x, y), zs...)
mul_fast(x::T, y::T, zs::T...) where {T<:FloatTypes} =
mul_fast(mul_fast(x, y), zs...)
@fastmath begin
cmp_fast(x::T, y::T) where {T<:FloatTypes} = ifelse(x==y, 0, ifelse(x<y, -1, +1))
log_fast(b::T, x::T) where {T<:FloatTypes} = log_fast(x)/log_fast(b)
end
eq_fast(x::T, y::T) where {T<:FloatTypes} = eq_float_fast(x, y)
ne_fast(x::T, y::T) where {T<:FloatTypes} = ne_float_fast(x, y)
lt_fast(x::T, y::T) where {T<:FloatTypes} = lt_float_fast(x, y)
le_fast(x::T, y::T) where {T<:FloatTypes} = le_float_fast(x, y)
isinf_fast(x) = false
isfinite_fast(x) = true
isnan_fast(x) = false
issubnormal_fast(x) = false
# complex numbers
ComplexTypes = Union{ComplexF32, ComplexF64}
@fastmath begin
abs_fast(x::ComplexTypes) = hypot(real(x), imag(x))
abs2_fast(x::ComplexTypes) = real(x)*real(x) + imag(x)*imag(x)
conj_fast(x::T) where {T<:ComplexTypes} = T(real(x), -imag(x))
inv_fast(x::ComplexTypes) = conj(x) / abs2(x)
sign_fast(x::ComplexTypes) = x == 0 ? float(zero(x)) : x/abs(x)
add_fast(x::T, y::T) where {T<:ComplexTypes} =
T(real(x)+real(y), imag(x)+imag(y))
add_fast(x::Complex{T}, b::T) where {T<:FloatTypes} =
Complex{T}(real(x)+b, imag(x))
add_fast(a::T, y::Complex{T}) where {T<:FloatTypes} =
Complex{T}(a+real(y), imag(y))
sub_fast(x::T, y::T) where {T<:ComplexTypes} =
T(real(x)-real(y), imag(x)-imag(y))
sub_fast(x::Complex{T}, b::T) where {T<:FloatTypes} =
Complex{T}(real(x)-b, imag(x))
sub_fast(a::T, y::Complex{T}) where {T<:FloatTypes} =
Complex{T}(a-real(y), -imag(y))
mul_fast(x::T, y::T) where {T<:ComplexTypes} =
T(real(x)*real(y) - imag(x)*imag(y),
real(x)*imag(y) + imag(x)*real(y))
mul_fast(x::Complex{T}, b::T) where {T<:FloatTypes} =
Complex{T}(real(x)*b, imag(x)*b)
mul_fast(a::T, y::Complex{T}) where {T<:FloatTypes} =
Complex{T}(a*real(y), a*imag(y))
@inline div_fast(x::T, y::T) where {T<:ComplexTypes} =
T(real(x)*real(y) + imag(x)*imag(y),
imag(x)*real(y) - real(x)*imag(y)) / abs2(y)
div_fast(x::Complex{T}, b::T) where {T<:FloatTypes} =
Complex{T}(real(x)/b, imag(x)/b)
div_fast(a::T, y::Complex{T}) where {T<:FloatTypes} =
Complex{T}(a*real(y), -a*imag(y)) / abs2(y)
eq_fast(x::T, y::T) where {T<:ComplexTypes} =
(real(x)==real(y)) & (imag(x)==imag(y))
eq_fast(x::Complex{T}, b::T) where {T<:FloatTypes} =
(real(x)==b) & (imag(x)==T(0))
eq_fast(a::T, y::Complex{T}) where {T<:FloatTypes} =
(a==real(y)) & (T(0)==imag(y))
ne_fast(x::T, y::T) where {T<:ComplexTypes} = !(x==y)
# Note: we use the same comparison for min, max, and minmax, so
# that the compiler can convert between them
max_fast(x::T, y::T) where {T<:FloatTypes} = ifelse(y > x, y, x)
min_fast(x::T, y::T) where {T<:FloatTypes} = ifelse(y > x, x, y)
minmax_fast(x::T, y::T) where {T<:FloatTypes} = ifelse(y > x, (x,y), (y,x))
max_fast(x::T, y::T, z::T...) where {T<:FloatTypes} = max_fast(max_fast(x, y), z...)
min_fast(x::T, y::T, z::T...) where {T<:FloatTypes} = min_fast(min_fast(x, y), z...)
end
# fall-back implementations and type promotion
for op in (:abs, :abs2, :conj, :inv, :sign)
op_fast = fast_op[op]
@eval begin
# fall-back implementation for non-numeric types
$op_fast(xs...) = $op(xs...)
end
end
for op in (:+, :-, :*, :/, :(==), :!=, :<, :<=, :cmp, :rem, :min, :max, :minmax)
op_fast = fast_op[op]
@eval begin
# fall-back implementation for non-numeric types
$op_fast(xs...) = $op(xs...)
# type promotion
$op_fast(x::Number, y::Number, zs::Number...) =
$op_fast(promote(x,y,zs...)...)
# fall-back implementation that applies after promotion
$op_fast(x::T,ys::T...) where {T<:Number} = $op(x,ys...)
end
end
# Math functions
exp2_fast(x::Union{Float32,Float64}) = Base.Math.exp2_fast(x)
exp_fast(x::Union{Float32,Float64}) = Base.Math.exp_fast(x)
exp10_fast(x::Union{Float32,Float64}) = Base.Math.exp10_fast(x)
# builtins
pow_fast(x::Float32, y::Integer) = ccall("llvm.powi.f32", llvmcall, Float32, (Float32, Int32), x, y)
pow_fast(x::Float64, y::Integer) = ccall("llvm.powi.f64", llvmcall, Float64, (Float64, Int32), x, y)
pow_fast(x::FloatTypes, ::Val{p}) where {p} = pow_fast(x, p) # inlines already via llvm.powi
@inline pow_fast(x, v::Val) = Base.literal_pow(^, x, v)
sqrt_fast(x::FloatTypes) = sqrt_llvm_fast(x)
sincos_fast(v::FloatTypes) = sincos(v)
@inline function sincos_fast(v::Float16)
s, c = sincos_fast(Float32(v))
return Float16(s), Float16(c)
end
sincos_fast(v::AbstractFloat) = (sin_fast(v), cos_fast(v))
sincos_fast(v::Real) = sincos_fast(float(v)::AbstractFloat)
sincos_fast(v) = (sin_fast(v), cos_fast(v))
@fastmath begin
hypot_fast(x::T, y::T) where {T<:FloatTypes} = sqrt(x*x + y*y)
# complex numbers
function cis_fast(x::T) where {T<:FloatTypes}
s, c = sincos_fast(x)
Complex{T}(c, s)
end
# See <http://en.cppreference.com/w/cpp/numeric/complex>
pow_fast(x::T, y::T) where {T<:ComplexTypes} = exp(y*log(x))
pow_fast(x::T, y::Complex{T}) where {T<:FloatTypes} = exp(y*log(x))
pow_fast(x::Complex{T}, y::T) where {T<:FloatTypes} = exp(y*log(x))
acos_fast(x::T) where {T<:ComplexTypes} =
convert(T,π)/2 + im*log(im*x + sqrt(1-x*x))
acosh_fast(x::ComplexTypes) = log(x + sqrt(x+1) * sqrt(x-1))
angle_fast(x::ComplexTypes) = atan(imag(x), real(x))
asin_fast(x::ComplexTypes) = -im*asinh(im*x)
asinh_fast(x::ComplexTypes) = log(x + sqrt(1+x*x))
atan_fast(x::ComplexTypes) = -im*atanh(im*x)
atanh_fast(x::T) where {T<:ComplexTypes} = convert(T,1)/2*(log(1+x) - log(1-x))
cis_fast(x::ComplexTypes) = exp(-imag(x)) * cis(real(x))
cos_fast(x::ComplexTypes) = cosh(im*x)
cosh_fast(x::T) where {T<:ComplexTypes} = convert(T,1)/2*(exp(x) + exp(-x))
exp10_fast(x::T) where {T<:ComplexTypes} =
exp10(real(x)) * cis(imag(x)*log(convert(T,10)))
exp2_fast(x::T) where {T<:ComplexTypes} =
exp2(real(x)) * cis(imag(x)*log(convert(T,2)))
exp_fast(x::ComplexTypes) = exp(real(x)) * cis(imag(x))
expm1_fast(x::ComplexTypes) = exp(x)-1
log10_fast(x::T) where {T<:ComplexTypes} = log(x) / log(convert(T,10))
log1p_fast(x::ComplexTypes) = log(1+x)
log2_fast(x::T) where {T<:ComplexTypes} = log(x) / log(convert(T,2))
log_fast(x::T) where {T<:ComplexTypes} = T(log(abs2(x))/2, angle(x))
log_fast(b::T, x::T) where {T<:ComplexTypes} = T(log(x)/log(b))
sin_fast(x::ComplexTypes) = -im*sinh(im*x)
sinh_fast(x::T) where {T<:ComplexTypes} = convert(T,1)/2*(exp(x) - exp(-x))
sqrt_fast(x::ComplexTypes) = sqrt(abs(x)) * cis(angle(x)/2)
tan_fast(x::ComplexTypes) = -im*tanh(im*x)
tanh_fast(x::ComplexTypes) = (a=exp(x); b=exp(-x); (a-b)/(a+b))
end
# fall-back implementations and type promotion
for f in (:acos, :acosh, :angle, :asin, :asinh, :atan, :atanh, :cbrt,
:cis, :cos, :cosh, :exp10, :exp2, :exp, :expm1,
:log10, :log1p, :log2, :log, :sin, :sinh, :sqrt, :tan,
:tanh)
f_fast = fast_op[f]
@eval begin
$f_fast(x) = $f(x)
end
end
for f in (:^, :atan, :hypot, :log)
f_fast = fast_op[f]
@eval begin
# fall-back implementation for non-numeric types
$f_fast(x, y) = $f(x, y)
# type promotion
$f_fast(x::Number, y::Number) = $f_fast(promote(x, y)...)
# fall-back implementation that applies after promotion
$f_fast(x::T, y::T) where {T<:Number} = $f(x, y)
end
end
end