math.cos on complex, real part

Average Error: 0.0 → 0.0

Time: 5.7s

Precision: binary64

Math

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

↓

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

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) (* 0.5 (cos re)))))

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) * (0.5 * cos(re)));
}

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) * (0.5d0 * cos(re)))
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) * (0.5 * Math.cos(re)));
}

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) * (0.5 * math.cos(re)))

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

Error

Bits error versus re

Bits error versus im

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

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

3. Taylor expanded in im around inf 0.0

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

4. Simplified 0.0

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

5. Final simplification 0.0

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

Reproduce

herbie shell --seed 2022170 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))