math.exp on complex, imaginary part

Average Error: 0.04% → 0.04%

Time: 9.2s

Precision: binary64

Cost: 12992

Math

\[e^{re} \cdot \sin im \]

↓

\[e^{re} \cdot \sin im \]

FPCore

(FPCore (re im) :precision binary64 (* (exp re) (sin im)))

↓

(FPCore (re im) :precision binary64 (* (exp re) (sin im)))

C

double code(double re, double im) {
	return exp(re) * sin(im);
}

↓

double code(double re, double im) {
	return exp(re) * sin(im);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = exp(re) * sin(im)
end function

↓

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = exp(re) * sin(im)
end function

Java

public static double code(double re, double im) {
	return Math.exp(re) * Math.sin(im);
}

↓

public static double code(double re, double im) {
	return Math.exp(re) * Math.sin(im);
}

Python

def code(re, im):
	return math.exp(re) * math.sin(im)

↓

def code(re, im):
	return math.exp(re) * math.sin(im)

Julia

function code(re, im)
	return Float64(exp(re) * sin(im))
end

↓

function code(re, im)
	return Float64(exp(re) * sin(im))
end

MATLAB

function tmp = code(re, im)
	tmp = exp(re) * sin(im);
end

↓

function tmp = code(re, im)
	tmp = exp(re) * sin(im);
end

Wolfram

code[re_, im_] := N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]

↓

code[re_, im_] := N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.04

   \[e^{re} \cdot \sin im \]

2. Final simplification 0.04

   \[\leadsto e^{re} \cdot \sin im \]

Alternatives

Alternative 1
Error	0.95%
Cost	13636

\[\begin{array}{l} \mathbf{if}\;e^{re} \leq 0.99998:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]

Alternative 2
Error	1.14%
Cost	13252

\[\begin{array}{l} \mathbf{if}\;e^{re} \leq 0.99998:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \]

Alternative 3
Error	1.78%
Cost	13124

\[\begin{array}{l} \mathbf{if}\;e^{re} \leq 0.99998:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \]

Alternative 4
Error	2.04%
Cost	6596

\[\begin{array}{l} \mathbf{if}\;re \leq -61:\\ \;\;\;\;\left(re + 1\right) \cdot 0\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \]

Alternative 5
Error	33.56%
Cost	964

\[\begin{array}{l} \mathbf{if}\;re \leq -85:\\ \;\;\;\;\left(re + 1\right) \cdot 0\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(1 + re \cdot \left(re \cdot 0.5\right)\right) + re \cdot im\\ \end{array} \]

Alternative 6
Error	33.57%
Cost	836

\[\begin{array}{l} \mathbf{if}\;re \leq -80:\\ \;\;\;\;\left(re + 1\right) \cdot 0\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]

Alternative 7
Error	33.71%
Cost	452

\[\begin{array}{l} \mathbf{if}\;re \leq -1:\\ \;\;\;\;\left(re + 1\right) \cdot 0\\ \mathbf{else}:\\ \;\;\;\;im + re \cdot im\\ \end{array} \]

Alternative 8
Error	65.53%
Cost	64

\[im \]

Reproduce

herbie shell --seed 2023089 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))