math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 7.6s

Alternatives: 12

Speedup: 1.0×

• 50.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp((0.0 - 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]

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp((0.0 - 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]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. distribute-rgt-in 100.0%

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

   2. cancel-sign-sub 100.0%

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

   3. distribute-rgt-out-- 100.0%

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

   4. sub-neg 100.0%

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

   5. neg-sub0 100.0%

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

   6. remove-double-neg 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
4. Add Preprocessing

5. Final simplification 100.0%

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

Alternative 2: 84.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.0095:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{2} \cdot \left(0.5 \cdot \sin re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.0095)
   (* (sin re) (* 0.5 (fma im im 2.0)))
   (if (<= im 1.35e+154)
     (* (+ (exp (- im)) (exp im)) (* 0.5 re))
     (* (pow im 2.0) (* 0.5 (sin re))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.0095) {
		tmp = sin(re) * (0.5 * fma(im, im, 2.0));
	} else if (im <= 1.35e+154) {
		tmp = (exp(-im) + exp(im)) * (0.5 * re);
	} else {
		tmp = pow(im, 2.0) * (0.5 * sin(re));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.0095)
		tmp = Float64(sin(re) * Float64(0.5 * fma(im, im, 2.0)));
	elseif (im <= 1.35e+154)
		tmp = Float64(Float64(exp(Float64(-im)) + exp(im)) * Float64(0.5 * re));
	else
		tmp = Float64((im ^ 2.0) * Float64(0.5 * sin(re)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.0095], N[(N[Sin[re], $MachinePrecision] * N[(0.5 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], N[(N[Power[im, 2.0], $MachinePrecision] * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.00949999999999999976

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 83.1%

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

   6. Taylor expanded in re around inf 83.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(\sin re \cdot \left(2 + {im}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 83.1%

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

      2. *-commutative 83.1%

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

      3. +-commutative 83.1%

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

      4. unpow2 83.1%

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

      5. fma-udef 83.1%

         \[\leadsto \left(\sin re \cdot 0.5\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

      6. associate-*r* 83.1%

         \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\right)} \]

   8. Simplified 83.1%

      \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\right)} \]

   if 0.00949999999999999976 < im < 1.35000000000000003e154

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in re around 0 76.4%

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

   if 1.35000000000000003e154 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 100.0%

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

   6. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left({im}^{2} \cdot \sin re\right)} \]
   7. Step-by-step derivation

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

   8. Simplified 100.0%

      \[\leadsto \color{blue}{{im}^{2} \cdot \left(\sin re \cdot 0.5\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.0095:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{2} \cdot \left(0.5 \cdot \sin re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 64.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 850:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{{re}^{-6}}\\ \mathbf{else}:\\ \;\;\;\;{im}^{2} \cdot \left(0.5 \cdot \sin re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 850.0)
   (sin re)
   (if (<= im 1.35e+154)
     (cbrt (pow re -6.0))
     (* (pow im 2.0) (* 0.5 (sin re))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 850.0) {
		tmp = sin(re);
	} else if (im <= 1.35e+154) {
		tmp = cbrt(pow(re, -6.0));
	} else {
		tmp = pow(im, 2.0) * (0.5 * sin(re));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 850.0) {
		tmp = Math.sin(re);
	} else if (im <= 1.35e+154) {
		tmp = Math.cbrt(Math.pow(re, -6.0));
	} else {
		tmp = Math.pow(im, 2.0) * (0.5 * Math.sin(re));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 850.0)
		tmp = sin(re);
	elseif (im <= 1.35e+154)
		tmp = cbrt((re ^ -6.0));
	else
		tmp = Float64((im ^ 2.0) * Float64(0.5 * sin(re)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 850.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.35e+154], N[Power[N[Power[re, -6.0], $MachinePrecision], 1/3], $MachinePrecision], N[(N[Power[im, 2.0], $MachinePrecision] * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 850

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 66.6%

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

   if 850 < im < 1.35000000000000003e154

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 2.7%

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

   7. Applied egg-rr 13.9%

      \[\leadsto \color{blue}{{\sin re}^{-2}} \]

   8. Taylor expanded in re around 0 13.8%

      \[\leadsto {\color{blue}{re}}^{-2} \]
   9. Step-by-step derivation

      1. metadata-eval 13.8%

         \[\leadsto {re}^{\color{blue}{\left(-2\right)}} \]

      2. pow-flip 13.8%

         \[\leadsto \color{blue}{\frac{1}{{re}^{2}}} \]

      3. add-cbrt-cube 25.2%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{1}{{re}^{2}} \cdot \frac{1}{{re}^{2}}\right) \cdot \frac{1}{{re}^{2}}}} \]

      4. pow3 25.2%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{1}{{re}^{2}}\right)}^{3}}} \]

      5. pow-flip 25.2%

         \[\leadsto \sqrt[3]{{\color{blue}{\left({re}^{\left(-2\right)}\right)}}^{3}} \]

      6. metadata-eval 25.2%

         \[\leadsto \sqrt[3]{{\left({re}^{\color{blue}{-2}}\right)}^{3}} \]

      7. pow-pow 25.2%

         \[\leadsto \sqrt[3]{\color{blue}{{re}^{\left(-2 \cdot 3\right)}}} \]

      8. metadata-eval 25.2%

         \[\leadsto \sqrt[3]{{re}^{\color{blue}{-6}}} \]

   10. Applied egg-rr 25.2%

       \[\leadsto \color{blue}{\sqrt[3]{{re}^{-6}}} \]

   if 1.35000000000000003e154 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 100.0%

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

   6. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left({im}^{2} \cdot \sin re\right)} \]
   7. Step-by-step derivation

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

   8. Simplified 100.0%

      \[\leadsto \color{blue}{{im}^{2} \cdot \left(\sin re \cdot 0.5\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 850:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{{re}^{-6}}\\ \mathbf{else}:\\ \;\;\;\;{im}^{2} \cdot \left(0.5 \cdot \sin re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 980:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{{re}^{-6}}\\ \mathbf{else}:\\ \;\;\;\;{im}^{2} \cdot \left(0.5 \cdot \sin re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 980.0)
   (* (sin re) (* 0.5 (fma im im 2.0)))
   (if (<= im 1.35e+154)
     (cbrt (pow re -6.0))
     (* (pow im 2.0) (* 0.5 (sin re))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 980.0) {
		tmp = sin(re) * (0.5 * fma(im, im, 2.0));
	} else if (im <= 1.35e+154) {
		tmp = cbrt(pow(re, -6.0));
	} else {
		tmp = pow(im, 2.0) * (0.5 * sin(re));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 980.0)
		tmp = Float64(sin(re) * Float64(0.5 * fma(im, im, 2.0)));
	elseif (im <= 1.35e+154)
		tmp = cbrt((re ^ -6.0));
	else
		tmp = Float64((im ^ 2.0) * Float64(0.5 * sin(re)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 980.0], N[(N[Sin[re], $MachinePrecision] * N[(0.5 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.35e+154], N[Power[N[Power[re, -6.0], $MachinePrecision], 1/3], $MachinePrecision], N[(N[Power[im, 2.0], $MachinePrecision] * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 980

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 83.0%

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

   6. Taylor expanded in re around inf 83.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(\sin re \cdot \left(2 + {im}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 83.0%

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

      2. *-commutative 83.0%

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

      3. +-commutative 83.0%

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

      4. unpow2 83.0%

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

      5. fma-udef 83.0%

         \[\leadsto \left(\sin re \cdot 0.5\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

      6. associate-*r* 83.0%

         \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\right)} \]

   8. Simplified 83.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\right)} \]

   if 980 < im < 1.35000000000000003e154

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 2.7%

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

   7. Applied egg-rr 13.9%

      \[\leadsto \color{blue}{{\sin re}^{-2}} \]

   8. Taylor expanded in re around 0 13.8%

      \[\leadsto {\color{blue}{re}}^{-2} \]
   9. Step-by-step derivation

      1. metadata-eval 13.8%

         \[\leadsto {re}^{\color{blue}{\left(-2\right)}} \]

      2. pow-flip 13.8%

         \[\leadsto \color{blue}{\frac{1}{{re}^{2}}} \]

      3. add-cbrt-cube 25.2%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{1}{{re}^{2}} \cdot \frac{1}{{re}^{2}}\right) \cdot \frac{1}{{re}^{2}}}} \]

      4. pow3 25.2%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{1}{{re}^{2}}\right)}^{3}}} \]

      5. pow-flip 25.2%

         \[\leadsto \sqrt[3]{{\color{blue}{\left({re}^{\left(-2\right)}\right)}}^{3}} \]

      6. metadata-eval 25.2%

         \[\leadsto \sqrt[3]{{\left({re}^{\color{blue}{-2}}\right)}^{3}} \]

      7. pow-pow 25.2%

         \[\leadsto \sqrt[3]{\color{blue}{{re}^{\left(-2 \cdot 3\right)}}} \]

      8. metadata-eval 25.2%

         \[\leadsto \sqrt[3]{{re}^{\color{blue}{-6}}} \]

   10. Applied egg-rr 25.2%

       \[\leadsto \color{blue}{\sqrt[3]{{re}^{-6}}} \]

   if 1.35000000000000003e154 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 100.0%

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

   6. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left({im}^{2} \cdot \sin re\right)} \]
   7. Step-by-step derivation

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

   8. Simplified 100.0%

      \[\leadsto \color{blue}{{im}^{2} \cdot \left(\sin re \cdot 0.5\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 980:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{{re}^{-6}}\\ \mathbf{else}:\\ \;\;\;\;{im}^{2} \cdot \left(0.5 \cdot \sin re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 60.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 780:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{+162}:\\ \;\;\;\;\sqrt[3]{{re}^{-6}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 780.0)
   (sin re)
   (if (<= im 2.5e+162) (cbrt (pow re -6.0)) (* (fma im im 2.0) (* 0.5 re)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 780.0) {
		tmp = sin(re);
	} else if (im <= 2.5e+162) {
		tmp = cbrt(pow(re, -6.0));
	} else {
		tmp = fma(im, im, 2.0) * (0.5 * re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 780.0)
		tmp = sin(re);
	elseif (im <= 2.5e+162)
		tmp = cbrt((re ^ -6.0));
	else
		tmp = Float64(fma(im, im, 2.0) * Float64(0.5 * re));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 780.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 2.5e+162], N[Power[N[Power[re, -6.0], $MachinePrecision], 1/3], $MachinePrecision], N[(N[(im * im + 2.0), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 780

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 66.6%

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

   if 780 < im < 2.4999999999999998e162

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 2.7%

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

   7. Applied egg-rr 13.5%

      \[\leadsto \color{blue}{{\sin re}^{-2}} \]

   8. Taylor expanded in re around 0 13.5%

      \[\leadsto {\color{blue}{re}}^{-2} \]
   9. Step-by-step derivation

      1. metadata-eval 13.5%

         \[\leadsto {re}^{\color{blue}{\left(-2\right)}} \]

      2. pow-flip 13.5%

         \[\leadsto \color{blue}{\frac{1}{{re}^{2}}} \]

      3. add-cbrt-cube 24.5%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{1}{{re}^{2}} \cdot \frac{1}{{re}^{2}}\right) \cdot \frac{1}{{re}^{2}}}} \]

      4. pow3 24.5%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{1}{{re}^{2}}\right)}^{3}}} \]

      5. pow-flip 24.5%

         \[\leadsto \sqrt[3]{{\color{blue}{\left({re}^{\left(-2\right)}\right)}}^{3}} \]

      6. metadata-eval 24.5%

         \[\leadsto \sqrt[3]{{\left({re}^{\color{blue}{-2}}\right)}^{3}} \]

      7. pow-pow 24.5%

         \[\leadsto \sqrt[3]{\color{blue}{{re}^{\left(-2 \cdot 3\right)}}} \]

      8. metadata-eval 24.5%

         \[\leadsto \sqrt[3]{{re}^{\color{blue}{-6}}} \]

   10. Applied egg-rr 24.5%

       \[\leadsto \color{blue}{\sqrt[3]{{re}^{-6}}} \]

   if 2.4999999999999998e162 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 100.0%

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

   6. Taylor expanded in re around 0 75.7%

      \[\leadsto \color{blue}{0.5 \cdot \left(re \cdot \left(2 + {im}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative 75.7%

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

      2. *-commutative 75.7%

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

      3. associate-*l* 75.7%

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

      4. +-commutative 75.7%

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

      5. unpow2 75.7%

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

      6. fma-udef 75.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \cdot \left(re \cdot 0.5\right) \]

   8. Simplified 75.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(im, im, 2\right) \cdot \left(re \cdot 0.5\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 780:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{+162}:\\ \;\;\;\;\sqrt[3]{{re}^{-6}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 650:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{+162}:\\ \;\;\;\;0.3333333333333333 + {re}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 650.0)
   (sin re)
   (if (<= im 2.5e+162)
     (+ 0.3333333333333333 (pow re -2.0))
     (* (fma im im 2.0) (* 0.5 re)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 650.0) {
		tmp = sin(re);
	} else if (im <= 2.5e+162) {
		tmp = 0.3333333333333333 + pow(re, -2.0);
	} else {
		tmp = fma(im, im, 2.0) * (0.5 * re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 650.0)
		tmp = sin(re);
	elseif (im <= 2.5e+162)
		tmp = Float64(0.3333333333333333 + (re ^ -2.0));
	else
		tmp = Float64(fma(im, im, 2.0) * Float64(0.5 * re));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 650.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 2.5e+162], N[(0.3333333333333333 + N[Power[re, -2.0], $MachinePrecision]), $MachinePrecision], N[(N[(im * im + 2.0), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 650

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 66.6%

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

   if 650 < im < 2.4999999999999998e162

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 2.7%

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

   7. Applied egg-rr 13.5%

      \[\leadsto \color{blue}{{\sin re}^{-2}} \]

   8. Taylor expanded in re around 0 13.5%

      \[\leadsto \color{blue}{0.3333333333333333 + \frac{1}{{re}^{2}}} \]
   9. Step-by-step derivation

      1. +-commutative 13.5%

         \[\leadsto \color{blue}{\frac{1}{{re}^{2}} + 0.3333333333333333} \]

   10. Simplified 13.5%

       \[\leadsto \color{blue}{\frac{1}{{re}^{2}} + 0.3333333333333333} \]

   11. Taylor expanded in re around 0 13.5%

       \[\leadsto \color{blue}{0.3333333333333333 + \frac{1}{{re}^{2}}} \]
   12. Step-by-step derivation

       1. unpow2 13.5%

          \[\leadsto 0.3333333333333333 + \frac{1}{\color{blue}{re \cdot re}} \]

       2. associate-/l/ 13.5%

          \[\leadsto 0.3333333333333333 + \color{blue}{\frac{\frac{1}{re}}{re}} \]

       3. *-rgt-identity 13.5%

          \[\leadsto 0.3333333333333333 + \frac{\color{blue}{\frac{1}{re} \cdot 1}}{re} \]

       4. associate-*r/ 13.5%

          \[\leadsto 0.3333333333333333 + \color{blue}{\frac{1}{re} \cdot \frac{1}{re}} \]

       5. unpow-1 13.5%

          \[\leadsto 0.3333333333333333 + \color{blue}{{re}^{-1}} \cdot \frac{1}{re} \]

       6. unpow-1 13.5%

          \[\leadsto 0.3333333333333333 + {re}^{-1} \cdot \color{blue}{{re}^{-1}} \]

       7. pow-sqr 13.5%

          \[\leadsto 0.3333333333333333 + \color{blue}{{re}^{\left(2 \cdot -1\right)}} \]

       8. metadata-eval 13.5%

          \[\leadsto 0.3333333333333333 + {re}^{\color{blue}{-2}} \]

   13. Simplified 13.5%

       \[\leadsto \color{blue}{0.3333333333333333 + {re}^{-2}} \]

   if 2.4999999999999998e162 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 100.0%

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

   6. Taylor expanded in re around 0 75.7%

      \[\leadsto \color{blue}{0.5 \cdot \left(re \cdot \left(2 + {im}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative 75.7%

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

      2. *-commutative 75.7%

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

      3. associate-*l* 75.7%

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

      4. +-commutative 75.7%

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

      5. unpow2 75.7%

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

      6. fma-udef 75.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \cdot \left(re \cdot 0.5\right) \]

   8. Simplified 75.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(im, im, 2\right) \cdot \left(re \cdot 0.5\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 650:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{+162}:\\ \;\;\;\;0.3333333333333333 + {re}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 53.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 720:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 + {re}^{-2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 720.0) (sin re) (+ 0.3333333333333333 (pow re -2.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 720.0) {
		tmp = sin(re);
	} else {
		tmp = 0.3333333333333333 + pow(re, -2.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 720.0d0) then
        tmp = sin(re)
    else
        tmp = 0.3333333333333333d0 + (re ** (-2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 720.0) {
		tmp = Math.sin(re);
	} else {
		tmp = 0.3333333333333333 + Math.pow(re, -2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 720.0:
		tmp = math.sin(re)
	else:
		tmp = 0.3333333333333333 + math.pow(re, -2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 720.0)
		tmp = sin(re);
	else
		tmp = Float64(0.3333333333333333 + (re ^ -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 720.0)
		tmp = sin(re);
	else
		tmp = 0.3333333333333333 + (re ^ -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 720.0], N[Sin[re], $MachinePrecision], N[(0.3333333333333333 + N[Power[re, -2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 720

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 66.6%

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

   if 720 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 2.7%

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

   7. Applied egg-rr 14.3%

      \[\leadsto \color{blue}{{\sin re}^{-2}} \]

   8. Taylor expanded in re around 0 14.3%

      \[\leadsto \color{blue}{0.3333333333333333 + \frac{1}{{re}^{2}}} \]
   9. Step-by-step derivation

      1. +-commutative 14.3%

         \[\leadsto \color{blue}{\frac{1}{{re}^{2}} + 0.3333333333333333} \]

   10. Simplified 14.3%

       \[\leadsto \color{blue}{\frac{1}{{re}^{2}} + 0.3333333333333333} \]

   11. Taylor expanded in re around 0 14.3%

       \[\leadsto \color{blue}{0.3333333333333333 + \frac{1}{{re}^{2}}} \]
   12. Step-by-step derivation

       1. unpow2 14.3%

          \[\leadsto 0.3333333333333333 + \frac{1}{\color{blue}{re \cdot re}} \]

       2. associate-/l/ 14.3%

          \[\leadsto 0.3333333333333333 + \color{blue}{\frac{\frac{1}{re}}{re}} \]

       3. *-rgt-identity 14.3%

          \[\leadsto 0.3333333333333333 + \frac{\color{blue}{\frac{1}{re} \cdot 1}}{re} \]

       4. associate-*r/ 14.3%

          \[\leadsto 0.3333333333333333 + \color{blue}{\frac{1}{re} \cdot \frac{1}{re}} \]

       5. unpow-1 14.3%

          \[\leadsto 0.3333333333333333 + \color{blue}{{re}^{-1}} \cdot \frac{1}{re} \]

       6. unpow-1 14.3%

          \[\leadsto 0.3333333333333333 + {re}^{-1} \cdot \color{blue}{{re}^{-1}} \]

       7. pow-sqr 14.3%

          \[\leadsto 0.3333333333333333 + \color{blue}{{re}^{\left(2 \cdot -1\right)}} \]

       8. metadata-eval 14.3%

          \[\leadsto 0.3333333333333333 + {re}^{\color{blue}{-2}} \]

   13. Simplified 14.3%

       \[\leadsto \color{blue}{0.3333333333333333 + {re}^{-2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 720:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 + {re}^{-2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 53.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 720:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{re}}{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 720.0) (sin re) (/ (/ 1.0 re) re)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 720.0) {
		tmp = sin(re);
	} else {
		tmp = (1.0 / re) / re;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 720.0d0) then
        tmp = sin(re)
    else
        tmp = (1.0d0 / re) / re
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 720.0) {
		tmp = Math.sin(re);
	} else {
		tmp = (1.0 / re) / re;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 720.0:
		tmp = math.sin(re)
	else:
		tmp = (1.0 / re) / re
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 720.0)
		tmp = sin(re);
	else
		tmp = Float64(Float64(1.0 / re) / re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 720.0)
		tmp = sin(re);
	else
		tmp = (1.0 / re) / re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 720.0], N[Sin[re], $MachinePrecision], N[(N[(1.0 / re), $MachinePrecision] / re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 720

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 66.6%

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

   if 720 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 2.7%

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

   7. Applied egg-rr 14.3%

      \[\leadsto \color{blue}{{\sin re}^{-2}} \]

   8. Taylor expanded in re around 0 14.3%

      \[\leadsto {\color{blue}{re}}^{-2} \]
   9. Step-by-step derivation

      1. metadata-eval 14.3%

         \[\leadsto {re}^{\color{blue}{\left(-2\right)}} \]

      2. pow-flip 14.3%

         \[\leadsto \color{blue}{\frac{1}{{re}^{2}}} \]

      3. unpow2 14.3%

         \[\leadsto \frac{1}{\color{blue}{re \cdot re}} \]

      4. associate-/r* 14.3%

         \[\leadsto \color{blue}{\frac{\frac{1}{re}}{re}} \]

   10. Applied egg-rr 14.3%

       \[\leadsto \color{blue}{\frac{\frac{1}{re}}{re}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 720:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{re}}{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 29.4% accurate, 43.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1080:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{re}}{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 1080.0) re (/ (/ 1.0 re) re)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1080.0) {
		tmp = re;
	} else {
		tmp = (1.0 / re) / re;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1080.0d0) then
        tmp = re
    else
        tmp = (1.0d0 / re) / re
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1080.0) {
		tmp = re;
	} else {
		tmp = (1.0 / re) / re;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1080.0:
		tmp = re
	else:
		tmp = (1.0 / re) / re
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1080.0)
		tmp = re;
	else
		tmp = Float64(Float64(1.0 / re) / re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1080.0)
		tmp = re;
	else
		tmp = (1.0 / re) / re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1080.0], re, N[(N[(1.0 / re), $MachinePrecision] / re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1080

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 66.6%

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

   6. Taylor expanded in re around 0 29.7%

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

   if 1080 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 2.7%

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

   7. Applied egg-rr 14.3%

      \[\leadsto \color{blue}{{\sin re}^{-2}} \]

   8. Taylor expanded in re around 0 14.3%

      \[\leadsto {\color{blue}{re}}^{-2} \]
   9. Step-by-step derivation

      1. metadata-eval 14.3%

         \[\leadsto {re}^{\color{blue}{\left(-2\right)}} \]

      2. pow-flip 14.3%

         \[\leadsto \color{blue}{\frac{1}{{re}^{2}}} \]

      3. unpow2 14.3%

         \[\leadsto \frac{1}{\color{blue}{re \cdot re}} \]

      4. associate-/r* 14.3%

         \[\leadsto \color{blue}{\frac{\frac{1}{re}}{re}} \]

   10. Applied egg-rr 14.3%

       \[\leadsto \color{blue}{\frac{\frac{1}{re}}{re}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 25.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1080:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{re}}{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 27.2% accurate, 100.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 3.55 \cdot 10^{-10}:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 3.55e-10) re 0.3333333333333333))

C

double code(double re, double im) {
	double tmp;
	if (re <= 3.55e-10) {
		tmp = re;
	} else {
		tmp = 0.3333333333333333;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 3.55d-10) then
        tmp = re
    else
        tmp = 0.3333333333333333d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 3.55e-10) {
		tmp = re;
	} else {
		tmp = 0.3333333333333333;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 3.55e-10:
		tmp = re
	else:
		tmp = 0.3333333333333333
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 3.55e-10)
		tmp = re;
	else
		tmp = 0.3333333333333333;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 3.55e-10)
		tmp = re;
	else
		tmp = 0.3333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 3.55e-10], re, 0.3333333333333333]

Derivation

1. Split input into 2 regimes

2. if re < 3.5500000000000001e-10

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 49.5%

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

   6. Taylor expanded in re around 0 29.9%

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

   if 3.5500000000000001e-10 < re

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 47.5%

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

   7. Applied egg-rr 7.0%

      \[\leadsto \color{blue}{{\sin re}^{-2}} \]

   8. Taylor expanded in re around 0 6.4%

      \[\leadsto \color{blue}{0.3333333333333333 + \frac{1}{{re}^{2}}} \]
   9. Step-by-step derivation

      1. +-commutative 6.4%

         \[\leadsto \color{blue}{\frac{1}{{re}^{2}} + 0.3333333333333333} \]

   10. Simplified 6.4%

       \[\leadsto \color{blue}{\frac{1}{{re}^{2}} + 0.3333333333333333} \]

   11. Taylor expanded in re around inf 6.4%

       \[\leadsto \color{blue}{0.3333333333333333} \]
3. Recombined 2 regimes into one program.

4. Final simplification 23.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 3.55 \cdot 10^{-10}:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 4.3% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ -0.16666666666666666 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 -0.16666666666666666)

C

double code(double re, double im) {
	return -0.16666666666666666;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = -0.16666666666666666d0
end function

Java

public static double code(double re, double im) {
	return -0.16666666666666666;
}

Python

def code(re, im):
	return -0.16666666666666666

Julia

function code(re, im)
	return -0.16666666666666666
end

MATLAB

function tmp = code(re, im)
	tmp = -0.16666666666666666;
end

Wolfram

code[re_, im_] := -0.16666666666666666

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Step-by-step derivation

   1. distribute-rgt-in 100.0%

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

   2. cancel-sign-sub 100.0%

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

   3. distribute-rgt-out-- 100.0%

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

   4. sub-neg 100.0%

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

   5. neg-sub0 100.0%

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

   6. remove-double-neg 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
4. Add Preprocessing

5. Taylor expanded in im around 0 48.9%

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

6. Taylor expanded in re around 0 30.3%

   \[\leadsto \color{blue}{re + -0.16666666666666666 \cdot {re}^{3}} \]

8. Simplified 30.3%

   \[\leadsto \color{blue}{re + {re}^{3} \cdot -0.16666666666666666} \]

10. Applied egg-rr 3.1%

    \[\leadsto re + \color{blue}{-0.16666666666666666} \]

11. Taylor expanded in re around 0 4.3%

    \[\leadsto \color{blue}{-0.16666666666666666} \]

12. Final simplification 4.3%

    \[\leadsto -0.16666666666666666 \]
13. Add Preprocessing

Alternative 12: 4.4% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.3333333333333333)

C

double code(double re, double im) {
	return 0.3333333333333333;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.3333333333333333d0
end function

Java

public static double code(double re, double im) {
	return 0.3333333333333333;
}

Python

def code(re, im):
	return 0.3333333333333333

Julia

function code(re, im)
	return 0.3333333333333333
end

MATLAB

function tmp = code(re, im)
	tmp = 0.3333333333333333;
end

Wolfram

code[re_, im_] := 0.3333333333333333

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Step-by-step derivation

   1. distribute-rgt-in 100.0%

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

   2. cancel-sign-sub 100.0%

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

   3. distribute-rgt-out-- 100.0%

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

   4. sub-neg 100.0%

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

   5. neg-sub0 100.0%

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

   6. remove-double-neg 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
4. Add Preprocessing

5. Taylor expanded in im around 0 48.9%

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

7. Applied egg-rr 11.3%

   \[\leadsto \color{blue}{{\sin re}^{-2}} \]

8. Taylor expanded in re around 0 11.0%

   \[\leadsto \color{blue}{0.3333333333333333 + \frac{1}{{re}^{2}}} \]
9. Step-by-step derivation

   1. +-commutative 11.0%

      \[\leadsto \color{blue}{\frac{1}{{re}^{2}} + 0.3333333333333333} \]

10. Simplified 11.0%

    \[\leadsto \color{blue}{\frac{1}{{re}^{2}} + 0.3333333333333333} \]

11. Taylor expanded in re around inf 4.8%

    \[\leadsto \color{blue}{0.3333333333333333} \]

12. Final simplification 4.8%

    \[\leadsto 0.3333333333333333 \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024010 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))