math.cos on complex, real part

Average Accuracy: 49.2% → 50.0%

Time: 8.7s

Precision: binary64

Cost: 6464

Math

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

↓

\[\cos re \]

FPCore

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

↓

(FPCore (re im) :precision binary64 (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 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 = 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 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 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 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 = 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[Cos[re], $MachinePrecision]

Derivation

1. Initial program 53.1%

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

2. Taylor expanded in im around 0 54.1%

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

3. Simplified 54.1%

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

4. Taylor expanded in im around 0 54.2%

   \[\leadsto \color{blue}{\cos re} \]

5. Final simplification 54.2%

   \[\leadsto \cos re \]

Alternatives

Alternative 1
Accuracy	3.0%
Cost	320

\[im \cdot \left(im \cdot 0.25\right) \]

Reproduce

herbie shell --seed 2023157 -o generate:proofs
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))