math.exp on complex, real part

Average Error: 0.01% → 0.01%

Time: 6.9s

Precision: binary64

Cost: 12992

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Fortran

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

↓

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

Java

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

↓

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 0.01

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

2. Final simplification 0.01

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

Alternatives

Alternative 1
Error	0.7%
Cost	13892

\[\begin{array}{l} \mathbf{if}\;e^{re} \leq 0.98:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + \left(re \cdot re\right) \cdot \left(re \cdot 0.16666666666666666 + 0.5\right)\right)\\ \end{array} \]

Alternative 2
Error	0.8%
Cost	13636

\[\begin{array}{l} \mathbf{if}\;e^{re} \leq 0.98:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(\left(re + 1\right) + re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 3
Error	1.02%
Cost	13252

\[\begin{array}{l} \mathbf{if}\;e^{re} \leq 0.98:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im \cdot \left(re + 1\right)\\ \end{array} \]

Alternative 4
Error	1.62%
Cost	12996

\[\begin{array}{l} \mathbf{if}\;e^{re} \leq 0.9999995:\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\cos im\\ \end{array} \]

Alternative 5
Error	26.55%
Cost	6596

\[\begin{array}{l} \mathbf{if}\;re \leq -540:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\cos im\\ \end{array} \]

Alternative 6
Error	55.88%
Cost	452

\[\begin{array}{l} \mathbf{if}\;re \leq -2.1:\\ \;\;\;\;-0.5 \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;re + 1\\ \end{array} \]

Alternative 7
Error	63.65%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023115 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))