math.cos on complex, real part

Average Error: 0.0 → 0.0

Time: 9.4s

Precision: binary64

Cost: 26304

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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 = ((cos(re) * -exp(im)) - (cos(re) * exp(-im))) * (-0.5d0)
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 ((Math.cos(re) * -Math.exp(im)) - (Math.cos(re) * Math.exp(-im))) * -0.5;
}

Python

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

↓

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

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(cos(re) * Float64(-exp(im))) - Float64(cos(re) * exp(Float64(-im)))) * -0.5)
end

MATLAB

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

↓

function tmp = code(re, im)
	tmp = ((cos(re) * -exp(im)) - (cos(re) * exp(-im))) * -0.5;
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[(N[Cos[re], $MachinePrecision] * (-N[Exp[im], $MachinePrecision])), $MachinePrecision] - N[(N[Cos[re], $MachinePrecision] * N[Exp[(-im)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]

Derivation

1. Initial program 0.0

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

2. Applied egg-rr 0.0

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

3. Applied egg-rr 0.0

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

4. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.0
Cost	19712

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

Alternative 2
Error	0.7
Cost	13312

\[\left(0.5 \cdot \cos re\right) \cdot \left(2 + {im}^{2}\right) \]

Alternative 3
Error	1.1
Cost	6464

\[\cos re \]

Alternative 4
Error	29.5
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023066 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))