math.cos on complex, real part

Average Error: 0.0 → 0.0

Time: 6.6s

Precision: binary64

Cost: 19712

Math

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

↓

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

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)) (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) + 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) + 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.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.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(0.5 * cos(re)) * Float64(exp(Float64(-im)) + 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) + 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[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.0

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

2. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.3
Cost	19908

\[\begin{array}{l} \mathbf{if}\;\cos re \leq 0.999995:\\ \;\;\;\;\cos re + \cos re \cdot \left(im \cdot \left(0.5 \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot \left(1 + \left(re \cdot re\right) \cdot -0.5\right)\\ \end{array} \]

Alternative 2
Error	0.5
Cost	13508

\[\begin{array}{l} \mathbf{if}\;\cos re \leq 0.999995:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot \left(1 + \left(re \cdot re\right) \cdot -0.5\right)\\ \end{array} \]

Alternative 3
Error	0.3
Cost	13508

\[\begin{array}{l} \mathbf{if}\;\cos re \leq 0.999995:\\ \;\;\;\;\cos re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot \left(1 + \left(re \cdot re\right) \cdot -0.5\right)\\ \end{array} \]

Alternative 4
Error	0.6
Cost	12996

\[\begin{array}{l} \mathbf{if}\;\cos re \leq 0.999995:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\cosh im\\ \end{array} \]

Alternative 5
Error	1.0
Cost	6464

\[\cos re \]

Alternative 6
Error	28.8
Cost	448

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

Alternative 7
Error	29.0
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022354 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))