math.cos on complex, real part

Average Accuracy: 100.0% → 100.0%

Time: 13.6s

Precision: binary64

Cost: 19712

• Unsound rule application detected in e-graph. Results may not be sound.

• 49.8% of points produce a very large (infinite) output. You may want to add a precondition.

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 100.0%

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

2. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	28.5%
Cost	14096

\[\begin{array}{l} t_0 := e^{-im} + e^{im}\\ t_1 := 0.5 \cdot \left(im \cdot im\right)\\ t_2 := \cos re \cdot t_1\\ \mathbf{if}\;im \leq 5.9 \cdot 10^{+152}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq 0.55:\\ \;\;\;\;0.5 \cdot t_0\\ \mathbf{elif}\;im \leq 2.3:\\ \;\;\;\;\cos re \cdot \left(t_1 + 1\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;t_0 \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Accuracy	28.6%
Cost	13712

\[\begin{array}{l} t_0 := 0.5 \cdot \left(im \cdot im\right)\\ t_1 := \cos re \cdot t_0\\ t_2 := 0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;im \leq 5.9 \cdot 10^{+152}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 0.55:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq 2.3:\\ \;\;\;\;\cos re \cdot \left(t_0 + 1\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Accuracy	28.6%
Cost	7956

\[\begin{array}{l} t_0 := 0.5 \cdot \left(im \cdot im\right)\\ t_1 := \cos re \cdot t_0\\ t_2 := \frac{{im}^{4} \cdot 0.25 + -1}{t_0 + -1}\\ \mathbf{if}\;im \leq 5.9 \cdot 10^{+152}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 1.2 \cdot 10^{+77}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq 7200:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{elif}\;im \leq 3.3 \cdot 10^{+75}:\\ \;\;\;\;\cos re \cdot \left(t_0 + 1\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Accuracy	28.5%
Cost	7760

\[\begin{array}{l} t_0 := 0.5 \cdot \left(im \cdot im\right)\\ t_1 := \cos re \cdot t_0\\ \mathbf{if}\;im \leq 2.1 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 780:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+22}:\\ \;\;\;\;\cos re \cdot \left(t_0 + 1\right)\\ \mathbf{elif}\;im \leq 6.6 \cdot 10^{+152}:\\ \;\;\;\;\mathsf{fma}\left(0.5, im \cdot im, 1\right) \cdot \left(1 + \left(re \cdot re\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Accuracy	28.5%
Cost	7376

\[\begin{array}{l} t_0 := \cos re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ t_1 := \left(im \cdot im\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{if}\;im \leq 2.1 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 780:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 4.1 \cdot 10^{+14}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 6.6 \cdot 10^{+152}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Accuracy	28.5%
Cost	7376

\[\begin{array}{l} t_0 := 0.5 \cdot \left(im \cdot im\right)\\ t_1 := \cos re \cdot t_0\\ t_2 := \left(im \cdot im\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{if}\;im \leq 2.1 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 2300:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq 6 \cdot 10^{+22}:\\ \;\;\;\;\cos re + t_0\\ \mathbf{elif}\;im \leq 6.6 \cdot 10^{+152}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Accuracy	28.5%
Cost	7376

\[\begin{array}{l} t_0 := 0.5 \cdot \left(im \cdot im\right)\\ t_1 := \cos re \cdot t_0\\ t_2 := \left(im \cdot im\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{if}\;im \leq 2.1 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 780:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+22}:\\ \;\;\;\;\cos re \cdot \left(t_0 + 1\right)\\ \mathbf{elif}\;im \leq 6.6 \cdot 10^{+152}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Accuracy	26.0%
Cost	6728

\[\begin{array}{l} t_0 := \left(im \cdot im\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{if}\;im \leq 780:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 4.1 \cdot 10^{+14}:\\ \;\;\;\;\cos re\\ \mathbf{elif}\;im \leq 2.7 \cdot 10^{+193}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right) + 1\\ \end{array} \]

Alternative 9
Accuracy	26.0%
Cost	1100

\[\begin{array}{l} t_0 := \left(im \cdot im\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{if}\;re \leq 7.8 \cdot 10^{+89}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq 4.6 \cdot 10^{+136}:\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right) + 1\\ \mathbf{elif}\;re \leq 2.36 \cdot 10^{+213}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.25 + \left(re \cdot re\right) \cdot 0.25\\ \end{array} \]

Alternative 10
Accuracy	21.7%
Cost	716

\[\begin{array}{l} t_0 := im \cdot \left(0.5 \cdot im\right)\\ \mathbf{if}\;im \leq 1.85 \cdot 10^{+158}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 220:\\ \;\;\;\;0.25 + \left(re \cdot re\right) \cdot 0.25\\ \mathbf{elif}\;im \leq 1.4:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Accuracy	21.7%
Cost	712

\[\begin{array}{l} \mathbf{if}\;im \leq 1.85 \cdot 10^{+158}:\\ \;\;\;\;im \cdot \left(0.5 \cdot im\right)\\ \mathbf{elif}\;im \leq 250:\\ \;\;\;\;0.25 + \left(re \cdot re\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot im\right) + 1\\ \end{array} \]

Alternative 12
Accuracy	21.7%
Cost	585

\[\begin{array}{l} \mathbf{if}\;im \leq 1.75 \cdot 10^{+19} \lor \neg \left(im \leq 1.3\right):\\ \;\;\;\;im \cdot \left(0.5 \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 13
Accuracy	8.0%
Cost	64

\[0.25 \]

Alternative 14
Accuracy	27.6%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023159 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))