math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 13.0s

Precision: binary64

Cost: 19712

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

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

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

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

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

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

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

• 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

\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

↓

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

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))

↓

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

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
}

↓

double code(double re, double im) {
	return (0.5 * sin(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 * sin(re)) * (exp((0.0d0 - im)) + exp(im))
end function

↓

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp(-im) + exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (Math.exp((0.0 - im)) + Math.exp(im));
}

↓

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (Math.exp(-im) + Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * (math.exp((0.0 - im)) + math.exp(im))

↓

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

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im)))
end

↓

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) + exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
end

↓

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

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[re_, im_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

2. Simplified 100.0%

   \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   Step-by-step derivation

   [Start] 100.0	\[ \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   sub0-neg [=>] 100.0	\[ \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

3. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	93.3%
Cost	13842

\[\begin{array}{l} \mathbf{if}\;im \leq -2.1 \cdot 10^{+154} \lor \neg \left(im \leq -0.62\right) \land \left(im \leq 2.3 \lor \neg \left(im \leq 1.35 \cdot 10^{+154}\right)\right):\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \end{array} \]

Alternative 2
Accuracy	93.3%
Cost	13840

\[\begin{array}{l} t_0 := \left(0.5 \cdot re\right) \cdot \left(e^{-im} + e^{im}\right)\\ t_1 := \sin re \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)\\ \mathbf{if}\;im \leq -2.1 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.55:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 2.3:\\ \;\;\;\;\sin re + 0.5 \cdot \left(im \cdot \left(\sin re \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Accuracy	93.3%
Cost	13840

\[\begin{array}{l} t_0 := \left(0.5 \cdot re\right) \cdot \left(e^{-im} + e^{im}\right)\\ t_1 := \sin re \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)\\ \mathbf{if}\;im \leq -2.1 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.55:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 2.3:\\ \;\;\;\;\sin re + \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Accuracy	85.8%
Cost	13778

\[\begin{array}{l} \mathbf{if}\;im \leq -1.35 \cdot 10^{+154} \lor \neg \left(im \leq -1.85 \cdot 10^{+30}\right) \land \left(im \leq 4000000000 \lor \neg \left(im \leq 1.35 \cdot 10^{+154}\right)\right):\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{re \cdot \left(1 - 0.25 \cdot {im}^{4}\right)}{1 + \left(im \cdot im\right) \cdot -0.5}\\ \end{array} \]

Alternative 5
Accuracy	79.5%
Cost	7952

\[\begin{array}{l} t_0 := 0.5 \cdot \left(im \cdot im\right)\\ t_1 := \frac{re \cdot \left(1 - 0.25 \cdot {im}^{4}\right)}{1 + \left(im \cdot im\right) \cdot -0.5}\\ \mathbf{if}\;im \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;re \cdot t_0\\ \mathbf{elif}\;im \leq -2.35 \cdot 10^{+27}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 35000000000:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + t_0\right)\\ \end{array} \]

Alternative 6
Accuracy	73.2%
Cost	6728

\[\begin{array}{l} t_0 := 0.5 \cdot \left(im \cdot im\right)\\ \mathbf{if}\;im \leq -7.6 \cdot 10^{+28}:\\ \;\;\;\;re \cdot t_0\\ \mathbf{elif}\;im \leq 680:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 7 \cdot 10^{+125}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + t_0\right)\\ \end{array} \]

Alternative 7
Accuracy	48.4%
Cost	1097

\[\begin{array}{l} \mathbf{if}\;im \leq 2.2 \lor \neg \left(im \leq 4.5 \cdot 10^{+121}\right):\\ \;\;\;\;re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\\ \end{array} \]

Alternative 8
Accuracy	40.5%
Cost	844

\[\begin{array}{l} t_0 := im \cdot \left(0.5 \cdot \left(re \cdot im\right)\right)\\ \mathbf{if}\;im \leq -1.45:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.0125:\\ \;\;\;\;re\\ \mathbf{elif}\;im \leq 8.5 \cdot 10^{+187}:\\ \;\;\;\;\frac{1}{re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Accuracy	46.9%
Cost	844

\[\begin{array}{l} t_0 := re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{if}\;im \leq -1.45:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.0125:\\ \;\;\;\;re\\ \mathbf{elif}\;im \leq 6.8 \cdot 10^{+143}:\\ \;\;\;\;0.08333333333333333 + \frac{0.25}{re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Accuracy	47.0%
Cost	841

\[\begin{array}{l} \mathbf{if}\;im \leq 0.0125 \lor \neg \left(im \leq 4 \cdot 10^{+143}\right):\\ \;\;\;\;re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.08333333333333333 + \frac{0.25}{re \cdot re}\\ \end{array} \]

Alternative 11
Accuracy	47.0%
Cost	713

\[\begin{array}{l} \mathbf{if}\;im \leq -1.45 \lor \neg \left(im \leq 0.0125\right):\\ \;\;\;\;re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array} \]

Alternative 12
Accuracy	30.7%
Cost	585

\[\begin{array}{l} \mathbf{if}\;im \leq -3.7 \cdot 10^{+95} \lor \neg \left(im \leq 0.0125\right):\\ \;\;\;\;\frac{0.25}{re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array} \]

Alternative 13
Accuracy	30.7%
Cost	585

\[\begin{array}{l} \mathbf{if}\;im \leq -3.7 \cdot 10^{+95} \lor \neg \left(im \leq 0.0125\right):\\ \;\;\;\;\frac{1}{re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array} \]

Alternative 14
Accuracy	25.5%
Cost	64

\[re \]

Reproduce

herbie shell --seed 2023159 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))