math.cos on complex, real part

Average Error: 0.02% → 0.02%

Time: 10.1s

Precision: binary64

Cost: 26304

Math

\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]

↓

\[\frac{0.5 \cdot \cos re}{e^{im}} + \cos re \cdot \left(0.5 \cdot e^{im}\right) \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))

↓

(FPCore (re im)
 :precision binary64
 (+ (/ (* 0.5 (cos re)) (exp im)) (* (cos re) (* 0.5 (exp im)))))

C

double code(double re, double im) {
	return (0.5 * cos(re)) * (exp(-im) + exp(im));
}

↓

double code(double re, double im) {
	return ((0.5 * cos(re)) / exp(im)) + (cos(re) * (0.5 * exp(im)));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * cos(re)) * (exp(-im) + exp(im))
end function

↓

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = ((0.5d0 * cos(re)) / exp(im)) + (cos(re) * (0.5d0 * exp(im)))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.cos(re)) * (Math.exp(-im) + Math.exp(im));
}

↓

public static double code(double re, double im) {
	return ((0.5 * Math.cos(re)) / Math.exp(im)) + (Math.cos(re) * (0.5 * Math.exp(im)));
}

Python

def code(re, im):
	return (0.5 * math.cos(re)) * (math.exp(-im) + math.exp(im))

↓

def code(re, im):
	return ((0.5 * math.cos(re)) / math.exp(im)) + (math.cos(re) * (0.5 * math.exp(im)))

Julia

function code(re, im)
	return Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) + exp(im)))
end

↓

function code(re, im)
	return Float64(Float64(Float64(0.5 * cos(re)) / exp(im)) + Float64(cos(re) * Float64(0.5 * exp(im))))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * cos(re)) * (exp(-im) + exp(im));
end

↓

function tmp = code(re, im)
	tmp = ((0.5 * cos(re)) / exp(im)) + (cos(re) * (0.5 * exp(im)));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[re_, im_] := N[(N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] / N[Exp[im], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[re], $MachinePrecision] * N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.02

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]

2. Simplified 0.02

   \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]
   Proof

   [Start] 0.02	\[ \left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   *-commutative [=>] 0.02	\[ \color{blue}{\left(\cos re \cdot 0.5\right)} \cdot \left(e^{-im} + e^{im}\right) \]
   associate-*l* [=>] 0.02	\[ \color{blue}{\cos re \cdot \left(0.5 \cdot \left(e^{-im} + e^{im}\right)\right)} \]
   +-commutative [=>] 0.02	\[ \cos re \cdot \left(0.5 \cdot \color{blue}{\left(e^{im} + e^{-im}\right)}\right) \]
   distribute-lft-in [=>] 0.02	\[ \cos re \cdot \color{blue}{\left(0.5 \cdot e^{im} + 0.5 \cdot e^{-im}\right)} \]
   fma-def [=>] 0.02	\[ \cos re \cdot \color{blue}{\mathsf{fma}\left(0.5, e^{im}, 0.5 \cdot e^{-im}\right)} \]
   exp-neg [=>] 0.02	\[ \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, 0.5 \cdot \color{blue}{\frac{1}{e^{im}}}\right) \]
   associate-*r/ [=>] 0.02	\[ \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{0.5 \cdot 1}{e^{im}}}\right) \]
   metadata-eval [=>] 0.02	\[ \cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{\color{blue}{0.5}}{e^{im}}\right) \]

3. Applied egg-rr 0.02

   \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{e^{im}} + \cos re \cdot \left(0.5 \cdot e^{im}\right)} \]

4. Final simplification 0.02

   \[\leadsto \frac{0.5 \cdot \cos re}{e^{im}} + \cos re \cdot \left(0.5 \cdot e^{im}\right) \]

Alternatives

Alternative 1
Error	0.02%
Cost	26048

\[\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right) \]

Alternative 2
Error	0.02%
Cost	19712

\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{im} + e^{-im}\right) \]

Alternative 3
Error	1.09%
Cost	13760

\[\cos re + \cos re \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right) \]

Alternative 4
Error	1.09%
Cost	13696

\[\cos re \cdot \left(0.5 \cdot \left(im \cdot im\right) + \left(0.041666666666666664 \cdot {im}^{4} + 1\right)\right) \]

Alternative 5
Error	1.3%
Cost	13376

\[\cos re + 0.5 \cdot \left(\cos re \cdot \left(im \cdot im\right)\right) \]

Alternative 6
Error	1.3%
Cost	13248

\[\left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

Alternative 7
Error	1.3%
Cost	6976

\[\cos re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right) \]

Alternative 8
Error	1.6%
Cost	6848

\[\cos re + im \cdot \left(0.5 \cdot im\right) \]

Alternative 9
Error	1.87%
Cost	6464

\[\cos re \]

Alternative 10
Error	46.78%
Cost	448

\[0.5 \cdot \left(im \cdot im\right) + 1 \]

Alternative 11
Error	47.07%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023090 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))