math.cos on complex, real part

Average Error: 0.0 → 0.1

Time: 13.0s

Precision: binary64

Cost: 32768

Math

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

↓

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

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (* (* (+ (exp (- im)) (exp im)) (/ (cos re) (exp im))) (* 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 ((exp(-im) + exp(im)) * (cos(re) / exp(im))) * (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 = ((exp(-im) + exp(im)) * (cos(re) / exp(im))) * (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 ((Math.exp(-im) + Math.exp(im)) * (Math.cos(re) / Math.exp(im))) * (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 ((math.exp(-im) + math.exp(im)) * (math.cos(re) / math.exp(im))) * (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(exp(Float64(-im)) + exp(im)) * Float64(cos(re) / exp(im))) * 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 = ((exp(-im) + exp(im)) * (cos(re) / exp(im))) * (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[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] / N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 * 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. Applied egg-rr 0.0

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

3. Applied egg-rr 0.1

   \[\leadsto \color{blue}{\frac{\frac{0.5 \cdot \cos re}{e^{-im}}}{\frac{e^{im}}{e^{-im} + e^{im}}}} \]

4. Applied egg-rr 0.1

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

5. Final simplification 0.1

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

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	0.8
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	29.1
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023075 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))