math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 10.1s

Alternatives: 17

Speedup: 1.0×

• 49.2% 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

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

FPCore

(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(-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(-im) + exp(im))
end function

Java

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(-im) + math.exp(im))

Julia

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(-im) + exp(im));
end

Wolfram

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. Step-by-step derivation

   1. sub0-neg 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 84.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4.2 \cdot 10^{-7}:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(im \cdot \left(0.5 \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 4.2e-7)
   (* (sin re) (fma 0.5 (* im im) 1.0))
   (if (<= im 1.9e+154)
     (* (* 0.5 re) (+ (exp (- im)) (exp im)))
     (* (sin re) (* im (* 0.5 im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 4.2e-7) {
		tmp = sin(re) * fma(0.5, (im * im), 1.0);
	} else if (im <= 1.9e+154) {
		tmp = (0.5 * re) * (exp(-im) + exp(im));
	} else {
		tmp = sin(re) * (im * (0.5 * im));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 4.2e-7)
		tmp = Float64(sin(re) * fma(0.5, Float64(im * im), 1.0));
	elseif (im <= 1.9e+154)
		tmp = Float64(Float64(0.5 * re) * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(sin(re) * Float64(im * Float64(0.5 * im)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 4.2e-7], N[(N[Sin[re], $MachinePrecision] * N[(0.5 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.9e+154], N[(N[(0.5 * re), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 4.2e-7

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 77.1%

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

   6. Simplified 77.1%

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

   7. Taylor expanded in re around inf 77.1%

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

      1. *-commutative 77.1%

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

      2. associate-*r* 77.1%

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

      3. distribute-rgt1-in 77.1%

         \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2} + 1\right) \cdot \sin re} \]

      4. fma-def 77.1%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, {im}^{2}, 1\right)} \cdot \sin re \]

      5. unpow2 77.1%

         \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{im \cdot im}, 1\right) \cdot \sin re \]

   9. Simplified 77.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, im \cdot im, 1\right) \cdot \sin re} \]

   if 4.2e-7 < im < 1.8999999999999999e154

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 71.9%

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

   6. Simplified 71.9%

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

   if 1.8999999999999999e154 < 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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

      3. distribute-rgt1-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2} + 1\right) \cdot \sin re} \]

      4. fma-def 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, {im}^{2}, 1\right)} \cdot \sin re \]

      5. unpow2 100.0%

         \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{im \cdot im}, 1\right) \cdot \sin re \]

   9. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, im \cdot im, 1\right) \cdot \sin re} \]

   10. Taylor expanded in im around inf 100.0%

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

       1. *-commutative 100.0%

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

       2. associate-*r* 100.0%

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

       3. unpow2 100.0%

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

       4. associate-*r* 100.0%

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

       5. *-commutative 100.0%

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

   12. Simplified 100.0%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.2 \cdot 10^{-7}:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(im \cdot \left(0.5 \cdot im\right)\right)\\ \end{array} \]

Alternative 3: 84.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4.2 \cdot 10^{-7}:\\ \;\;\;\;\sin re + \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(im \cdot \left(0.5 \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 4.2e-7)
   (+ (sin re) (* (* 0.5 (sin re)) (* im im)))
   (if (<= im 1.9e+154)
     (* (* 0.5 re) (+ (exp (- im)) (exp im)))
     (* (sin re) (* im (* 0.5 im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 4.2e-7) {
		tmp = sin(re) + ((0.5 * sin(re)) * (im * im));
	} else if (im <= 1.9e+154) {
		tmp = (0.5 * re) * (exp(-im) + exp(im));
	} else {
		tmp = sin(re) * (im * (0.5 * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 4.2d-7) then
        tmp = sin(re) + ((0.5d0 * sin(re)) * (im * im))
    else if (im <= 1.9d+154) then
        tmp = (0.5d0 * re) * (exp(-im) + exp(im))
    else
        tmp = sin(re) * (im * (0.5d0 * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 4.2e-7) {
		tmp = Math.sin(re) + ((0.5 * Math.sin(re)) * (im * im));
	} else if (im <= 1.9e+154) {
		tmp = (0.5 * re) * (Math.exp(-im) + Math.exp(im));
	} else {
		tmp = Math.sin(re) * (im * (0.5 * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 4.2e-7:
		tmp = math.sin(re) + ((0.5 * math.sin(re)) * (im * im))
	elif im <= 1.9e+154:
		tmp = (0.5 * re) * (math.exp(-im) + math.exp(im))
	else:
		tmp = math.sin(re) * (im * (0.5 * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 4.2e-7)
		tmp = Float64(sin(re) + Float64(Float64(0.5 * sin(re)) * Float64(im * im)));
	elseif (im <= 1.9e+154)
		tmp = Float64(Float64(0.5 * re) * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(sin(re) * Float64(im * Float64(0.5 * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 4.2e-7)
		tmp = sin(re) + ((0.5 * sin(re)) * (im * im));
	elseif (im <= 1.9e+154)
		tmp = (0.5 * re) * (exp(-im) + exp(im));
	else
		tmp = sin(re) * (im * (0.5 * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 4.2e-7], N[(N[Sin[re], $MachinePrecision] + N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.9e+154], N[(N[(0.5 * re), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 4.2e-7

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 77.1%

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

   6. Simplified 77.1%

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

   if 4.2e-7 < im < 1.8999999999999999e154

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 71.9%

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

   6. Simplified 71.9%

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

   if 1.8999999999999999e154 < 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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

      3. distribute-rgt1-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2} + 1\right) \cdot \sin re} \]

      4. fma-def 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, {im}^{2}, 1\right)} \cdot \sin re \]

      5. unpow2 100.0%

         \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{im \cdot im}, 1\right) \cdot \sin re \]

   9. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, im \cdot im, 1\right) \cdot \sin re} \]

   10. Taylor expanded in im around inf 100.0%

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

       1. *-commutative 100.0%

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

       2. associate-*r* 100.0%

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

       3. unpow2 100.0%

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

       4. associate-*r* 100.0%

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

       5. *-commutative 100.0%

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

   12. Simplified 100.0%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.2 \cdot 10^{-7}:\\ \;\;\;\;\sin re + \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(im \cdot \left(0.5 \cdot im\right)\right)\\ \end{array} \]

Alternative 4: 68.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 550:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\left(im \cdot im\right) \cdot \left(0.5 \cdot re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(im \cdot \left(0.5 \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 550.0)
   (sin re)
   (if (<= im 1.9e+154)
     (log1p (expm1 (* (* im im) (* 0.5 re))))
     (* (sin re) (* im (* 0.5 im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 550.0) {
		tmp = sin(re);
	} else if (im <= 1.9e+154) {
		tmp = log1p(expm1(((im * im) * (0.5 * re))));
	} else {
		tmp = sin(re) * (im * (0.5 * im));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 550.0) {
		tmp = Math.sin(re);
	} else if (im <= 1.9e+154) {
		tmp = Math.log1p(Math.expm1(((im * im) * (0.5 * re))));
	} else {
		tmp = Math.sin(re) * (im * (0.5 * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 550.0:
		tmp = math.sin(re)
	elif im <= 1.9e+154:
		tmp = math.log1p(math.expm1(((im * im) * (0.5 * re))))
	else:
		tmp = math.sin(re) * (im * (0.5 * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 550.0)
		tmp = sin(re);
	elseif (im <= 1.9e+154)
		tmp = log1p(expm1(Float64(Float64(im * im) * Float64(0.5 * re))));
	else
		tmp = Float64(sin(re) * Float64(im * Float64(0.5 * im)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 550.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.9e+154], N[Log[1 + N[(Exp[N[(N[(im * im), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 550

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 62.8%

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

   if 550 < im < 1.8999999999999999e154

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 4.3%

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

   6. Simplified 4.3%

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

   7. Taylor expanded in re around 0 21.1%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot {im}^{2}\right) \cdot re} \]
   8. Step-by-step derivation

      1. *-commutative 21.1%

         \[\leadsto \color{blue}{re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)} \]

      2. unpow2 21.1%

         \[\leadsto re \cdot \left(1 + 0.5 \cdot \color{blue}{\left(im \cdot im\right)}\right) \]

   9. Simplified 21.1%

      \[\leadsto \color{blue}{re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)} \]

   10. Taylor expanded in im around inf 21.1%

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

       1. associate-*r* 21.1%

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

       2. unpow2 21.1%

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

   12. Simplified 21.1%

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

       1. log1p-expm1-u 42.7%

          \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(0.5 \cdot re\right) \cdot \left(im \cdot im\right)\right)\right)} \]

       2. *-commutative 42.7%

          \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left(im \cdot im\right) \cdot \left(0.5 \cdot re\right)}\right)\right) \]

   14. Applied egg-rr 42.7%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(im \cdot im\right) \cdot \left(0.5 \cdot re\right)\right)\right)} \]

   if 1.8999999999999999e154 < 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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

      3. distribute-rgt1-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2} + 1\right) \cdot \sin re} \]

      4. fma-def 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, {im}^{2}, 1\right)} \cdot \sin re \]

      5. unpow2 100.0%

         \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{im \cdot im}, 1\right) \cdot \sin re \]

   9. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, im \cdot im, 1\right) \cdot \sin re} \]

   10. Taylor expanded in im around inf 100.0%

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

       1. *-commutative 100.0%

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

       2. associate-*r* 100.0%

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

       3. unpow2 100.0%

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

       4. associate-*r* 100.0%

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

       5. *-commutative 100.0%

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

   12. Simplified 100.0%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 550:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\left(im \cdot im\right) \cdot \left(0.5 \cdot re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(im \cdot \left(0.5 \cdot im\right)\right)\\ \end{array} \]

Alternative 5: 81.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 580:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\left(im \cdot im\right) \cdot \left(0.5 \cdot re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(im \cdot \left(0.5 \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 580.0)
   (* (sin re) (fma 0.5 (* im im) 1.0))
   (if (<= im 1.9e+154)
     (log1p (expm1 (* (* im im) (* 0.5 re))))
     (* (sin re) (* im (* 0.5 im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 580.0) {
		tmp = sin(re) * fma(0.5, (im * im), 1.0);
	} else if (im <= 1.9e+154) {
		tmp = log1p(expm1(((im * im) * (0.5 * re))));
	} else {
		tmp = sin(re) * (im * (0.5 * im));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 580.0)
		tmp = Float64(sin(re) * fma(0.5, Float64(im * im), 1.0));
	elseif (im <= 1.9e+154)
		tmp = log1p(expm1(Float64(Float64(im * im) * Float64(0.5 * re))));
	else
		tmp = Float64(sin(re) * Float64(im * Float64(0.5 * im)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 580.0], N[(N[Sin[re], $MachinePrecision] * N[(0.5 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.9e+154], N[Log[1 + N[(Exp[N[(N[(im * im), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 580

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 77.2%

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

   6. Simplified 77.2%

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

   7. Taylor expanded in re around inf 77.2%

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

      1. *-commutative 77.2%

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

      2. associate-*r* 77.2%

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

      3. distribute-rgt1-in 77.2%

         \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2} + 1\right) \cdot \sin re} \]

      4. fma-def 77.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, {im}^{2}, 1\right)} \cdot \sin re \]

      5. unpow2 77.2%

         \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{im \cdot im}, 1\right) \cdot \sin re \]

   9. Simplified 77.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, im \cdot im, 1\right) \cdot \sin re} \]

   if 580 < im < 1.8999999999999999e154

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 4.3%

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

   6. Simplified 4.3%

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

   7. Taylor expanded in re around 0 21.1%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot {im}^{2}\right) \cdot re} \]
   8. Step-by-step derivation

      1. *-commutative 21.1%

         \[\leadsto \color{blue}{re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)} \]

      2. unpow2 21.1%

         \[\leadsto re \cdot \left(1 + 0.5 \cdot \color{blue}{\left(im \cdot im\right)}\right) \]

   9. Simplified 21.1%

      \[\leadsto \color{blue}{re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)} \]

   10. Taylor expanded in im around inf 21.1%

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

       1. associate-*r* 21.1%

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

       2. unpow2 21.1%

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

   12. Simplified 21.1%

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

       1. log1p-expm1-u 42.7%

          \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(0.5 \cdot re\right) \cdot \left(im \cdot im\right)\right)\right)} \]

       2. *-commutative 42.7%

          \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left(im \cdot im\right) \cdot \left(0.5 \cdot re\right)}\right)\right) \]

   14. Applied egg-rr 42.7%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(im \cdot im\right) \cdot \left(0.5 \cdot re\right)\right)\right)} \]

   if 1.8999999999999999e154 < 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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

      3. distribute-rgt1-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2} + 1\right) \cdot \sin re} \]

      4. fma-def 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, {im}^{2}, 1\right)} \cdot \sin re \]

      5. unpow2 100.0%

         \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{im \cdot im}, 1\right) \cdot \sin re \]

   9. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, im \cdot im, 1\right) \cdot \sin re} \]

   10. Taylor expanded in im around inf 100.0%

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

       1. *-commutative 100.0%

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

       2. associate-*r* 100.0%

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

       3. unpow2 100.0%

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

       4. associate-*r* 100.0%

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

       5. *-commutative 100.0%

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

   12. Simplified 100.0%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 580:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(0.5, im \cdot im, 1\right)\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\left(im \cdot im\right) \cdot \left(0.5 \cdot re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(im \cdot \left(0.5 \cdot im\right)\right)\\ \end{array} \]

Alternative 6: 69.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(im \cdot im\right) \cdot \left(0.5 \cdot re\right)\\ t_1 := im \cdot \left(0.5 \cdot im\right)\\ \mathbf{if}\;im \leq 22000000:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2.3 \cdot 10^{+56}:\\ \;\;\;\;\sqrt[3]{t_0 \cdot \left(t_0 \cdot t_0\right)}\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;\frac{re \cdot \left(1 - 0.25 \cdot {im}^{4}\right)}{1 - t_1}\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* im im) (* 0.5 re))) (t_1 (* im (* 0.5 im))))
   (if (<= im 22000000.0)
     (sin re)
     (if (<= im 2.3e+56)
       (cbrt (* t_0 (* t_0 t_0)))
       (if (<= im 1.9e+154)
         (/ (* re (- 1.0 (* 0.25 (pow im 4.0)))) (- 1.0 t_1))
         (* (sin re) t_1))))))

C

double code(double re, double im) {
	double t_0 = (im * im) * (0.5 * re);
	double t_1 = im * (0.5 * im);
	double tmp;
	if (im <= 22000000.0) {
		tmp = sin(re);
	} else if (im <= 2.3e+56) {
		tmp = cbrt((t_0 * (t_0 * t_0)));
	} else if (im <= 1.9e+154) {
		tmp = (re * (1.0 - (0.25 * pow(im, 4.0)))) / (1.0 - t_1);
	} else {
		tmp = sin(re) * t_1;
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = (im * im) * (0.5 * re);
	double t_1 = im * (0.5 * im);
	double tmp;
	if (im <= 22000000.0) {
		tmp = Math.sin(re);
	} else if (im <= 2.3e+56) {
		tmp = Math.cbrt((t_0 * (t_0 * t_0)));
	} else if (im <= 1.9e+154) {
		tmp = (re * (1.0 - (0.25 * Math.pow(im, 4.0)))) / (1.0 - t_1);
	} else {
		tmp = Math.sin(re) * t_1;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(im * im) * Float64(0.5 * re))
	t_1 = Float64(im * Float64(0.5 * im))
	tmp = 0.0
	if (im <= 22000000.0)
		tmp = sin(re);
	elseif (im <= 2.3e+56)
		tmp = cbrt(Float64(t_0 * Float64(t_0 * t_0)));
	elseif (im <= 1.9e+154)
		tmp = Float64(Float64(re * Float64(1.0 - Float64(0.25 * (im ^ 4.0)))) / Float64(1.0 - t_1));
	else
		tmp = Float64(sin(re) * t_1);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(im * im), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 22000000.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 2.3e+56], N[Power[N[(t$95$0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], If[LessEqual[im, 1.9e+154], N[(N[(re * N[(1.0 - N[(0.25 * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < 2.2e7

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 62.8%

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

   if 2.2e7 < im < 2.30000000000000015e56

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 3.2%

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

   6. Simplified 3.2%

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

   7. Taylor expanded in re around 0 10.0%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot {im}^{2}\right) \cdot re} \]
   8. Step-by-step derivation

      1. *-commutative 10.0%

         \[\leadsto \color{blue}{re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)} \]

      2. unpow2 10.0%

         \[\leadsto re \cdot \left(1 + 0.5 \cdot \color{blue}{\left(im \cdot im\right)}\right) \]

   9. Simplified 10.0%

      \[\leadsto \color{blue}{re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)} \]

   10. Taylor expanded in im around inf 10.0%

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

       1. associate-*r* 10.0%

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

       2. unpow2 10.0%

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

   12. Simplified 10.0%

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

       1. add-cbrt-cube 32.0%

          \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\left(0.5 \cdot re\right) \cdot \left(im \cdot im\right)\right) \cdot \left(\left(0.5 \cdot re\right) \cdot \left(im \cdot im\right)\right)\right) \cdot \left(\left(0.5 \cdot re\right) \cdot \left(im \cdot im\right)\right)}} \]

       2. *-commutative 32.0%

          \[\leadsto \sqrt[3]{\left(\color{blue}{\left(\left(im \cdot im\right) \cdot \left(0.5 \cdot re\right)\right)} \cdot \left(\left(0.5 \cdot re\right) \cdot \left(im \cdot im\right)\right)\right) \cdot \left(\left(0.5 \cdot re\right) \cdot \left(im \cdot im\right)\right)} \]

       3. *-commutative 32.0%

          \[\leadsto \sqrt[3]{\left(\left(\left(im \cdot im\right) \cdot \left(0.5 \cdot re\right)\right) \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot \left(0.5 \cdot re\right)\right)}\right) \cdot \left(\left(0.5 \cdot re\right) \cdot \left(im \cdot im\right)\right)} \]

       4. *-commutative 32.0%

          \[\leadsto \sqrt[3]{\left(\left(\left(im \cdot im\right) \cdot \left(0.5 \cdot re\right)\right) \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 \cdot re\right)\right)\right) \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot \left(0.5 \cdot re\right)\right)}} \]

   14. Applied egg-rr 32.0%

       \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\left(im \cdot im\right) \cdot \left(0.5 \cdot re\right)\right) \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 \cdot re\right)\right)\right) \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 \cdot re\right)\right)}} \]

   if 2.30000000000000015e56 < im < 1.8999999999999999e154

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 5.1%

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

   6. Simplified 5.1%

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

   7. Taylor expanded in re around 0 29.1%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot {im}^{2}\right) \cdot re} \]
   8. Step-by-step derivation

      1. *-commutative 29.1%

         \[\leadsto \color{blue}{re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)} \]

      2. unpow2 29.1%

         \[\leadsto re \cdot \left(1 + 0.5 \cdot \color{blue}{\left(im \cdot im\right)}\right) \]

   9. Simplified 29.1%

      \[\leadsto \color{blue}{re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative 29.1%

          \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \left(im \cdot im\right)\right) \cdot re} \]

       2. flip-+ 50.5%

          \[\leadsto \color{blue}{\frac{1 \cdot 1 - \left(0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)}{1 - 0.5 \cdot \left(im \cdot im\right)}} \cdot re \]

       3. associate-*l/ 50.5%

          \[\leadsto \color{blue}{\frac{\left(1 \cdot 1 - \left(0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\right) \cdot re}{1 - 0.5 \cdot \left(im \cdot im\right)}} \]

       4. metadata-eval 50.5%

          \[\leadsto \frac{\left(\color{blue}{1} - \left(0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\right) \cdot re}{1 - 0.5 \cdot \left(im \cdot im\right)} \]

       5. swap-sqr 50.5%

          \[\leadsto \frac{\left(1 - \color{blue}{\left(0.5 \cdot 0.5\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)}\right) \cdot re}{1 - 0.5 \cdot \left(im \cdot im\right)} \]

       6. metadata-eval 50.5%

          \[\leadsto \frac{\left(1 - \color{blue}{0.25} \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right) \cdot re}{1 - 0.5 \cdot \left(im \cdot im\right)} \]

       7. pow2 50.5%

          \[\leadsto \frac{\left(1 - 0.25 \cdot \left(\color{blue}{{im}^{2}} \cdot \left(im \cdot im\right)\right)\right) \cdot re}{1 - 0.5 \cdot \left(im \cdot im\right)} \]

       8. pow2 50.5%

          \[\leadsto \frac{\left(1 - 0.25 \cdot \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right) \cdot re}{1 - 0.5 \cdot \left(im \cdot im\right)} \]

       9. pow-prod-up 50.5%

          \[\leadsto \frac{\left(1 - 0.25 \cdot \color{blue}{{im}^{\left(2 + 2\right)}}\right) \cdot re}{1 - 0.5 \cdot \left(im \cdot im\right)} \]

       10. metadata-eval 50.5%

           \[\leadsto \frac{\left(1 - 0.25 \cdot {im}^{\color{blue}{4}}\right) \cdot re}{1 - 0.5 \cdot \left(im \cdot im\right)} \]

       11. associate-*r* 50.5%

           \[\leadsto \frac{\left(1 - 0.25 \cdot {im}^{4}\right) \cdot re}{1 - \color{blue}{\left(0.5 \cdot im\right) \cdot im}} \]

       12. *-commutative 50.5%

           \[\leadsto \frac{\left(1 - 0.25 \cdot {im}^{4}\right) \cdot re}{1 - \color{blue}{im \cdot \left(0.5 \cdot im\right)}} \]

   11. Applied egg-rr 50.5%

       \[\leadsto \color{blue}{\frac{\left(1 - 0.25 \cdot {im}^{4}\right) \cdot re}{1 - im \cdot \left(0.5 \cdot im\right)}} \]

   if 1.8999999999999999e154 < 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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

      3. distribute-rgt1-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2} + 1\right) \cdot \sin re} \]

      4. fma-def 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, {im}^{2}, 1\right)} \cdot \sin re \]

      5. unpow2 100.0%

         \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{im \cdot im}, 1\right) \cdot \sin re \]

   9. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, im \cdot im, 1\right) \cdot \sin re} \]

   10. Taylor expanded in im around inf 100.0%

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

       1. *-commutative 100.0%

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

       2. associate-*r* 100.0%

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

       3. unpow2 100.0%

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

       4. associate-*r* 100.0%

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

       5. *-commutative 100.0%

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

   12. Simplified 100.0%

       \[\leadsto \color{blue}{\sin re \cdot \left(im \cdot \left(0.5 \cdot im\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 22000000:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2.3 \cdot 10^{+56}:\\ \;\;\;\;\sqrt[3]{\left(\left(im \cdot im\right) \cdot \left(0.5 \cdot re\right)\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \left(0.5 \cdot re\right)\right) \cdot \left(\left(im \cdot im\right) \cdot \left(0.5 \cdot re\right)\right)\right)}\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;\frac{re \cdot \left(1 - 0.25 \cdot {im}^{4}\right)}{1 - im \cdot \left(0.5 \cdot im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(im \cdot \left(0.5 \cdot im\right)\right)\\ \end{array} \]

Alternative 7: 68.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.016666666666666666 \cdot \left(re \cdot re\right)\\ t_1 := im \cdot \left(0.5 \cdot im\right)\\ \mathbf{if}\;im \leq 560:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2.7 \cdot 10^{+25}:\\ \;\;\;\;0.08333333333333333 + \sqrt[3]{t_0 \cdot \left(t_0 \cdot t_0\right)}\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;\frac{re \cdot \left(1 - 0.25 \cdot {im}^{4}\right)}{1 - t_1}\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.016666666666666666 (* re re))) (t_1 (* im (* 0.5 im))))
   (if (<= im 560.0)
     (sin re)
     (if (<= im 2.7e+25)
       (+ 0.08333333333333333 (cbrt (* t_0 (* t_0 t_0))))
       (if (<= im 1.9e+154)
         (/ (* re (- 1.0 (* 0.25 (pow im 4.0)))) (- 1.0 t_1))
         (* (sin re) t_1))))))

C

double code(double re, double im) {
	double t_0 = 0.016666666666666666 * (re * re);
	double t_1 = im * (0.5 * im);
	double tmp;
	if (im <= 560.0) {
		tmp = sin(re);
	} else if (im <= 2.7e+25) {
		tmp = 0.08333333333333333 + cbrt((t_0 * (t_0 * t_0)));
	} else if (im <= 1.9e+154) {
		tmp = (re * (1.0 - (0.25 * pow(im, 4.0)))) / (1.0 - t_1);
	} else {
		tmp = sin(re) * t_1;
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = 0.016666666666666666 * (re * re);
	double t_1 = im * (0.5 * im);
	double tmp;
	if (im <= 560.0) {
		tmp = Math.sin(re);
	} else if (im <= 2.7e+25) {
		tmp = 0.08333333333333333 + Math.cbrt((t_0 * (t_0 * t_0)));
	} else if (im <= 1.9e+154) {
		tmp = (re * (1.0 - (0.25 * Math.pow(im, 4.0)))) / (1.0 - t_1);
	} else {
		tmp = Math.sin(re) * t_1;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.016666666666666666 * Float64(re * re))
	t_1 = Float64(im * Float64(0.5 * im))
	tmp = 0.0
	if (im <= 560.0)
		tmp = sin(re);
	elseif (im <= 2.7e+25)
		tmp = Float64(0.08333333333333333 + cbrt(Float64(t_0 * Float64(t_0 * t_0))));
	elseif (im <= 1.9e+154)
		tmp = Float64(Float64(re * Float64(1.0 - Float64(0.25 * (im ^ 4.0)))) / Float64(1.0 - t_1));
	else
		tmp = Float64(sin(re) * t_1);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.016666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 560.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 2.7e+25], N[(0.08333333333333333 + N[Power[N[(t$95$0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.9e+154], N[(N[(re * N[(1.0 - N[(0.25 * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < 560

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 62.8%

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

   if 560 < im < 2.7e25

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 4.6%

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

   6. Taylor expanded in re around 0 6.6%

      \[\leadsto \color{blue}{0.08333333333333333 + \left(0.25 \cdot \frac{1}{{re}^{2}} + 0.016666666666666666 \cdot {re}^{2}\right)} \]
   7. Step-by-step derivation

      1. associate-*r/ 6.6%

         \[\leadsto 0.08333333333333333 + \left(\color{blue}{\frac{0.25 \cdot 1}{{re}^{2}}} + 0.016666666666666666 \cdot {re}^{2}\right) \]

      2. metadata-eval 6.6%

         \[\leadsto 0.08333333333333333 + \left(\frac{\color{blue}{0.25}}{{re}^{2}} + 0.016666666666666666 \cdot {re}^{2}\right) \]

      3. unpow2 6.6%

         \[\leadsto 0.08333333333333333 + \left(\frac{0.25}{\color{blue}{re \cdot re}} + 0.016666666666666666 \cdot {re}^{2}\right) \]

      4. *-commutative 6.6%

         \[\leadsto 0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \color{blue}{{re}^{2} \cdot 0.016666666666666666}\right) \]

      5. unpow2 6.6%

         \[\leadsto 0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \color{blue}{\left(re \cdot re\right)} \cdot 0.016666666666666666\right) \]

   8. Simplified 6.6%

      \[\leadsto \color{blue}{0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)} \]

   9. Taylor expanded in re around inf 5.1%

      \[\leadsto \color{blue}{0.08333333333333333 + 0.016666666666666666 \cdot {re}^{2}} \]
   10. Step-by-step derivation

       1. unpow2 5.1%

          \[\leadsto 0.08333333333333333 + 0.016666666666666666 \cdot \color{blue}{\left(re \cdot re\right)} \]

   11. Simplified 5.1%

       \[\leadsto \color{blue}{0.08333333333333333 + 0.016666666666666666 \cdot \left(re \cdot re\right)} \]
   12. Step-by-step derivation

       1. add-cbrt-cube 41.9%

          \[\leadsto 0.08333333333333333 + \color{blue}{\sqrt[3]{\left(\left(0.016666666666666666 \cdot \left(re \cdot re\right)\right) \cdot \left(0.016666666666666666 \cdot \left(re \cdot re\right)\right)\right) \cdot \left(0.016666666666666666 \cdot \left(re \cdot re\right)\right)}} \]

   13. Applied egg-rr 41.9%

       \[\leadsto 0.08333333333333333 + \color{blue}{\sqrt[3]{\left(\left(0.016666666666666666 \cdot \left(re \cdot re\right)\right) \cdot \left(0.016666666666666666 \cdot \left(re \cdot re\right)\right)\right) \cdot \left(0.016666666666666666 \cdot \left(re \cdot re\right)\right)}} \]

   if 2.7e25 < im < 1.8999999999999999e154

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 4.5%

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

   6. Simplified 4.5%

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

   7. Taylor expanded in re around 0 24.8%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot {im}^{2}\right) \cdot re} \]
   8. Step-by-step derivation

      1. *-commutative 24.8%

         \[\leadsto \color{blue}{re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)} \]

      2. unpow2 24.8%

         \[\leadsto re \cdot \left(1 + 0.5 \cdot \color{blue}{\left(im \cdot im\right)}\right) \]

   9. Simplified 24.8%

      \[\leadsto \color{blue}{re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative 24.8%

          \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \left(im \cdot im\right)\right) \cdot re} \]

       2. flip-+ 39.6%

          \[\leadsto \color{blue}{\frac{1 \cdot 1 - \left(0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)}{1 - 0.5 \cdot \left(im \cdot im\right)}} \cdot re \]

       3. associate-*l/ 39.5%

          \[\leadsto \color{blue}{\frac{\left(1 \cdot 1 - \left(0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\right) \cdot re}{1 - 0.5 \cdot \left(im \cdot im\right)}} \]

       4. metadata-eval 39.5%

          \[\leadsto \frac{\left(\color{blue}{1} - \left(0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\right) \cdot re}{1 - 0.5 \cdot \left(im \cdot im\right)} \]

       5. swap-sqr 39.5%

          \[\leadsto \frac{\left(1 - \color{blue}{\left(0.5 \cdot 0.5\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)}\right) \cdot re}{1 - 0.5 \cdot \left(im \cdot im\right)} \]

       6. metadata-eval 39.5%

          \[\leadsto \frac{\left(1 - \color{blue}{0.25} \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right) \cdot re}{1 - 0.5 \cdot \left(im \cdot im\right)} \]

       7. pow2 39.5%

          \[\leadsto \frac{\left(1 - 0.25 \cdot \left(\color{blue}{{im}^{2}} \cdot \left(im \cdot im\right)\right)\right) \cdot re}{1 - 0.5 \cdot \left(im \cdot im\right)} \]

       8. pow2 39.5%

          \[\leadsto \frac{\left(1 - 0.25 \cdot \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right) \cdot re}{1 - 0.5 \cdot \left(im \cdot im\right)} \]

       9. pow-prod-up 39.5%

          \[\leadsto \frac{\left(1 - 0.25 \cdot \color{blue}{{im}^{\left(2 + 2\right)}}\right) \cdot re}{1 - 0.5 \cdot \left(im \cdot im\right)} \]

       10. metadata-eval 39.5%

           \[\leadsto \frac{\left(1 - 0.25 \cdot {im}^{\color{blue}{4}}\right) \cdot re}{1 - 0.5 \cdot \left(im \cdot im\right)} \]

       11. associate-*r* 39.5%

           \[\leadsto \frac{\left(1 - 0.25 \cdot {im}^{4}\right) \cdot re}{1 - \color{blue}{\left(0.5 \cdot im\right) \cdot im}} \]

       12. *-commutative 39.5%

           \[\leadsto \frac{\left(1 - 0.25 \cdot {im}^{4}\right) \cdot re}{1 - \color{blue}{im \cdot \left(0.5 \cdot im\right)}} \]

   11. Applied egg-rr 39.5%

       \[\leadsto \color{blue}{\frac{\left(1 - 0.25 \cdot {im}^{4}\right) \cdot re}{1 - im \cdot \left(0.5 \cdot im\right)}} \]

   if 1.8999999999999999e154 < 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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

      3. distribute-rgt1-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2} + 1\right) \cdot \sin re} \]

      4. fma-def 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, {im}^{2}, 1\right)} \cdot \sin re \]

      5. unpow2 100.0%

         \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{im \cdot im}, 1\right) \cdot \sin re \]

   9. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, im \cdot im, 1\right) \cdot \sin re} \]

   10. Taylor expanded in im around inf 100.0%

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

       1. *-commutative 100.0%

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

       2. associate-*r* 100.0%

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

       3. unpow2 100.0%

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

       4. associate-*r* 100.0%

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

       5. *-commutative 100.0%

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

   12. Simplified 100.0%

       \[\leadsto \color{blue}{\sin re \cdot \left(im \cdot \left(0.5 \cdot im\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 560:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2.7 \cdot 10^{+25}:\\ \;\;\;\;0.08333333333333333 + \sqrt[3]{\left(0.016666666666666666 \cdot \left(re \cdot re\right)\right) \cdot \left(\left(0.016666666666666666 \cdot \left(re \cdot re\right)\right) \cdot \left(0.016666666666666666 \cdot \left(re \cdot re\right)\right)\right)}\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;\frac{re \cdot \left(1 - 0.25 \cdot {im}^{4}\right)}{1 - im \cdot \left(0.5 \cdot im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(im \cdot \left(0.5 \cdot im\right)\right)\\ \end{array} \]

Alternative 8: 69.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(0.5 \cdot im\right)\\ \mathbf{if}\;im \leq 710:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 5.4 \cdot 10^{+70}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + re \cdot \left(re \cdot 0.016666666666666666\right)\right)\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;\frac{re \cdot \left(1 - 0.25 \cdot {im}^{4}\right)}{1 - t_0}\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (* 0.5 im))))
   (if (<= im 710.0)
     (sin re)
     (if (<= im 5.4e+70)
       (+
        0.08333333333333333
        (+ (/ 0.25 (* re re)) (* re (* re 0.016666666666666666))))
       (if (<= im 1.9e+154)
         (/ (* re (- 1.0 (* 0.25 (pow im 4.0)))) (- 1.0 t_0))
         (* (sin re) t_0))))))

C

double code(double re, double im) {
	double t_0 = im * (0.5 * im);
	double tmp;
	if (im <= 710.0) {
		tmp = sin(re);
	} else if (im <= 5.4e+70) {
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + (re * (re * 0.016666666666666666)));
	} else if (im <= 1.9e+154) {
		tmp = (re * (1.0 - (0.25 * pow(im, 4.0)))) / (1.0 - t_0);
	} else {
		tmp = sin(re) * t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = im * (0.5d0 * im)
    if (im <= 710.0d0) then
        tmp = sin(re)
    else if (im <= 5.4d+70) then
        tmp = 0.08333333333333333d0 + ((0.25d0 / (re * re)) + (re * (re * 0.016666666666666666d0)))
    else if (im <= 1.9d+154) then
        tmp = (re * (1.0d0 - (0.25d0 * (im ** 4.0d0)))) / (1.0d0 - t_0)
    else
        tmp = sin(re) * t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * (0.5 * im);
	double tmp;
	if (im <= 710.0) {
		tmp = Math.sin(re);
	} else if (im <= 5.4e+70) {
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + (re * (re * 0.016666666666666666)));
	} else if (im <= 1.9e+154) {
		tmp = (re * (1.0 - (0.25 * Math.pow(im, 4.0)))) / (1.0 - t_0);
	} else {
		tmp = Math.sin(re) * t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (0.5 * im)
	tmp = 0
	if im <= 710.0:
		tmp = math.sin(re)
	elif im <= 5.4e+70:
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + (re * (re * 0.016666666666666666)))
	elif im <= 1.9e+154:
		tmp = (re * (1.0 - (0.25 * math.pow(im, 4.0)))) / (1.0 - t_0)
	else:
		tmp = math.sin(re) * t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(0.5 * im))
	tmp = 0.0
	if (im <= 710.0)
		tmp = sin(re);
	elseif (im <= 5.4e+70)
		tmp = Float64(0.08333333333333333 + Float64(Float64(0.25 / Float64(re * re)) + Float64(re * Float64(re * 0.016666666666666666))));
	elseif (im <= 1.9e+154)
		tmp = Float64(Float64(re * Float64(1.0 - Float64(0.25 * (im ^ 4.0)))) / Float64(1.0 - t_0));
	else
		tmp = Float64(sin(re) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (0.5 * im);
	tmp = 0.0;
	if (im <= 710.0)
		tmp = sin(re);
	elseif (im <= 5.4e+70)
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + (re * (re * 0.016666666666666666)));
	elseif (im <= 1.9e+154)
		tmp = (re * (1.0 - (0.25 * (im ^ 4.0)))) / (1.0 - t_0);
	else
		tmp = sin(re) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 710.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 5.4e+70], N[(0.08333333333333333 + N[(N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision] + N[(re * N[(re * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.9e+154], N[(N[(re * N[(1.0 - N[(0.25 * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if im < 710

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 62.8%

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

   if 710 < im < 5.3999999999999999e70

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 13.9%

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

   6. Taylor expanded in re around 0 14.8%

      \[\leadsto \color{blue}{0.08333333333333333 + \left(0.25 \cdot \frac{1}{{re}^{2}} + 0.016666666666666666 \cdot {re}^{2}\right)} \]
   7. Step-by-step derivation

      1. associate-*r/ 14.8%

         \[\leadsto 0.08333333333333333 + \left(\color{blue}{\frac{0.25 \cdot 1}{{re}^{2}}} + 0.016666666666666666 \cdot {re}^{2}\right) \]

      2. metadata-eval 14.8%

         \[\leadsto 0.08333333333333333 + \left(\frac{\color{blue}{0.25}}{{re}^{2}} + 0.016666666666666666 \cdot {re}^{2}\right) \]

      3. unpow2 14.8%

         \[\leadsto 0.08333333333333333 + \left(\frac{0.25}{\color{blue}{re \cdot re}} + 0.016666666666666666 \cdot {re}^{2}\right) \]

      4. *-commutative 14.8%

         \[\leadsto 0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \color{blue}{{re}^{2} \cdot 0.016666666666666666}\right) \]

      5. unpow2 14.8%

         \[\leadsto 0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \color{blue}{\left(re \cdot re\right)} \cdot 0.016666666666666666\right) \]

   8. Simplified 14.8%

      \[\leadsto \color{blue}{0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)} \]

   9. Taylor expanded in re around 0 14.8%

      \[\leadsto 0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \color{blue}{0.016666666666666666 \cdot {re}^{2}}\right) \]
   10. Step-by-step derivation

       1. unpow2 14.8%

          \[\leadsto 0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + 0.016666666666666666 \cdot \color{blue}{\left(re \cdot re\right)}\right) \]

       2. *-commutative 14.8%

          \[\leadsto 0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \color{blue}{\left(re \cdot re\right) \cdot 0.016666666666666666}\right) \]

       3. associate-*r* 14.8%

          \[\leadsto 0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \color{blue}{re \cdot \left(re \cdot 0.016666666666666666\right)}\right) \]

   11. Simplified 14.8%

       \[\leadsto 0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \color{blue}{re \cdot \left(re \cdot 0.016666666666666666\right)}\right) \]

   if 5.3999999999999999e70 < im < 1.8999999999999999e154

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 5.7%

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

   6. Simplified 5.7%

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

   7. Taylor expanded in re around 0 32.0%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot {im}^{2}\right) \cdot re} \]
   8. Step-by-step derivation

      1. *-commutative 32.0%

         \[\leadsto \color{blue}{re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)} \]

      2. unpow2 32.0%

         \[\leadsto re \cdot \left(1 + 0.5 \cdot \color{blue}{\left(im \cdot im\right)}\right) \]

   9. Simplified 32.0%

      \[\leadsto \color{blue}{re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative 32.0%

          \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \left(im \cdot im\right)\right) \cdot re} \]

       2. flip-+ 61.5%

          \[\leadsto \color{blue}{\frac{1 \cdot 1 - \left(0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)}{1 - 0.5 \cdot \left(im \cdot im\right)}} \cdot re \]

       3. associate-*l/ 61.5%

          \[\leadsto \color{blue}{\frac{\left(1 \cdot 1 - \left(0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\right) \cdot re}{1 - 0.5 \cdot \left(im \cdot im\right)}} \]

       4. metadata-eval 61.5%

          \[\leadsto \frac{\left(\color{blue}{1} - \left(0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\right) \cdot re}{1 - 0.5 \cdot \left(im \cdot im\right)} \]

       5. swap-sqr 61.5%

          \[\leadsto \frac{\left(1 - \color{blue}{\left(0.5 \cdot 0.5\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)}\right) \cdot re}{1 - 0.5 \cdot \left(im \cdot im\right)} \]

       6. metadata-eval 61.5%

          \[\leadsto \frac{\left(1 - \color{blue}{0.25} \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right) \cdot re}{1 - 0.5 \cdot \left(im \cdot im\right)} \]

       7. pow2 61.5%

          \[\leadsto \frac{\left(1 - 0.25 \cdot \left(\color{blue}{{im}^{2}} \cdot \left(im \cdot im\right)\right)\right) \cdot re}{1 - 0.5 \cdot \left(im \cdot im\right)} \]

       8. pow2 61.5%

          \[\leadsto \frac{\left(1 - 0.25 \cdot \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right) \cdot re}{1 - 0.5 \cdot \left(im \cdot im\right)} \]

       9. pow-prod-up 61.5%

          \[\leadsto \frac{\left(1 - 0.25 \cdot \color{blue}{{im}^{\left(2 + 2\right)}}\right) \cdot re}{1 - 0.5 \cdot \left(im \cdot im\right)} \]

       10. metadata-eval 61.5%

           \[\leadsto \frac{\left(1 - 0.25 \cdot {im}^{\color{blue}{4}}\right) \cdot re}{1 - 0.5 \cdot \left(im \cdot im\right)} \]

       11. associate-*r* 61.5%

           \[\leadsto \frac{\left(1 - 0.25 \cdot {im}^{4}\right) \cdot re}{1 - \color{blue}{\left(0.5 \cdot im\right) \cdot im}} \]

       12. *-commutative 61.5%

           \[\leadsto \frac{\left(1 - 0.25 \cdot {im}^{4}\right) \cdot re}{1 - \color{blue}{im \cdot \left(0.5 \cdot im\right)}} \]

   11. Applied egg-rr 61.5%

       \[\leadsto \color{blue}{\frac{\left(1 - 0.25 \cdot {im}^{4}\right) \cdot re}{1 - im \cdot \left(0.5 \cdot im\right)}} \]

   if 1.8999999999999999e154 < 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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

      3. distribute-rgt1-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2} + 1\right) \cdot \sin re} \]

      4. fma-def 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, {im}^{2}, 1\right)} \cdot \sin re \]

      5. unpow2 100.0%

         \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{im \cdot im}, 1\right) \cdot \sin re \]

   9. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, im \cdot im, 1\right) \cdot \sin re} \]

   10. Taylor expanded in im around inf 100.0%

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

       1. *-commutative 100.0%

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

       2. associate-*r* 100.0%

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

       3. unpow2 100.0%

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

       4. associate-*r* 100.0%

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

       5. *-commutative 100.0%

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

   12. Simplified 100.0%

       \[\leadsto \color{blue}{\sin re \cdot \left(im \cdot \left(0.5 \cdot im\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 710:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 5.4 \cdot 10^{+70}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + re \cdot \left(re \cdot 0.016666666666666666\right)\right)\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;\frac{re \cdot \left(1 - 0.25 \cdot {im}^{4}\right)}{1 - im \cdot \left(0.5 \cdot im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(im \cdot \left(0.5 \cdot im\right)\right)\\ \end{array} \]

Alternative 9: 69.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(0.5 \cdot im\right)\\ \mathbf{if}\;im \leq 500:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+72}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + re \cdot \left(re \cdot 0.016666666666666666\right)\right)\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;\frac{{im}^{4} \cdot \left(re \cdot -0.25\right)}{1 - t_0}\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (* 0.5 im))))
   (if (<= im 500.0)
     (sin re)
     (if (<= im 1.4e+72)
       (+
        0.08333333333333333
        (+ (/ 0.25 (* re re)) (* re (* re 0.016666666666666666))))
       (if (<= im 1.9e+154)
         (/ (* (pow im 4.0) (* re -0.25)) (- 1.0 t_0))
         (* (sin re) t_0))))))

C

double code(double re, double im) {
	double t_0 = im * (0.5 * im);
	double tmp;
	if (im <= 500.0) {
		tmp = sin(re);
	} else if (im <= 1.4e+72) {
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + (re * (re * 0.016666666666666666)));
	} else if (im <= 1.9e+154) {
		tmp = (pow(im, 4.0) * (re * -0.25)) / (1.0 - t_0);
	} else {
		tmp = sin(re) * t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = im * (0.5d0 * im)
    if (im <= 500.0d0) then
        tmp = sin(re)
    else if (im <= 1.4d+72) then
        tmp = 0.08333333333333333d0 + ((0.25d0 / (re * re)) + (re * (re * 0.016666666666666666d0)))
    else if (im <= 1.9d+154) then
        tmp = ((im ** 4.0d0) * (re * (-0.25d0))) / (1.0d0 - t_0)
    else
        tmp = sin(re) * t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * (0.5 * im);
	double tmp;
	if (im <= 500.0) {
		tmp = Math.sin(re);
	} else if (im <= 1.4e+72) {
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + (re * (re * 0.016666666666666666)));
	} else if (im <= 1.9e+154) {
		tmp = (Math.pow(im, 4.0) * (re * -0.25)) / (1.0 - t_0);
	} else {
		tmp = Math.sin(re) * t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (0.5 * im)
	tmp = 0
	if im <= 500.0:
		tmp = math.sin(re)
	elif im <= 1.4e+72:
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + (re * (re * 0.016666666666666666)))
	elif im <= 1.9e+154:
		tmp = (math.pow(im, 4.0) * (re * -0.25)) / (1.0 - t_0)
	else:
		tmp = math.sin(re) * t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(0.5 * im))
	tmp = 0.0
	if (im <= 500.0)
		tmp = sin(re);
	elseif (im <= 1.4e+72)
		tmp = Float64(0.08333333333333333 + Float64(Float64(0.25 / Float64(re * re)) + Float64(re * Float64(re * 0.016666666666666666))));
	elseif (im <= 1.9e+154)
		tmp = Float64(Float64((im ^ 4.0) * Float64(re * -0.25)) / Float64(1.0 - t_0));
	else
		tmp = Float64(sin(re) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (0.5 * im);
	tmp = 0.0;
	if (im <= 500.0)
		tmp = sin(re);
	elseif (im <= 1.4e+72)
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + (re * (re * 0.016666666666666666)));
	elseif (im <= 1.9e+154)
		tmp = ((im ^ 4.0) * (re * -0.25)) / (1.0 - t_0);
	else
		tmp = sin(re) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 500.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.4e+72], N[(0.08333333333333333 + N[(N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision] + N[(re * N[(re * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.9e+154], N[(N[(N[Power[im, 4.0], $MachinePrecision] * N[(re * -0.25), $MachinePrecision]), $MachinePrecision] / N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if im < 500

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 62.8%

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

   if 500 < im < 1.4e72

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 13.9%

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

   6. Taylor expanded in re around 0 14.8%

      \[\leadsto \color{blue}{0.08333333333333333 + \left(0.25 \cdot \frac{1}{{re}^{2}} + 0.016666666666666666 \cdot {re}^{2}\right)} \]
   7. Step-by-step derivation

      1. associate-*r/ 14.8%

         \[\leadsto 0.08333333333333333 + \left(\color{blue}{\frac{0.25 \cdot 1}{{re}^{2}}} + 0.016666666666666666 \cdot {re}^{2}\right) \]

      2. metadata-eval 14.8%

         \[\leadsto 0.08333333333333333 + \left(\frac{\color{blue}{0.25}}{{re}^{2}} + 0.016666666666666666 \cdot {re}^{2}\right) \]

      3. unpow2 14.8%

         \[\leadsto 0.08333333333333333 + \left(\frac{0.25}{\color{blue}{re \cdot re}} + 0.016666666666666666 \cdot {re}^{2}\right) \]

      4. *-commutative 14.8%

         \[\leadsto 0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \color{blue}{{re}^{2} \cdot 0.016666666666666666}\right) \]

      5. unpow2 14.8%

         \[\leadsto 0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \color{blue}{\left(re \cdot re\right)} \cdot 0.016666666666666666\right) \]

   8. Simplified 14.8%

      \[\leadsto \color{blue}{0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)} \]

   9. Taylor expanded in re around 0 14.8%

      \[\leadsto 0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \color{blue}{0.016666666666666666 \cdot {re}^{2}}\right) \]
   10. Step-by-step derivation

       1. unpow2 14.8%

          \[\leadsto 0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + 0.016666666666666666 \cdot \color{blue}{\left(re \cdot re\right)}\right) \]

       2. *-commutative 14.8%

          \[\leadsto 0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \color{blue}{\left(re \cdot re\right) \cdot 0.016666666666666666}\right) \]

       3. associate-*r* 14.8%

          \[\leadsto 0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \color{blue}{re \cdot \left(re \cdot 0.016666666666666666\right)}\right) \]

   11. Simplified 14.8%

       \[\leadsto 0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \color{blue}{re \cdot \left(re \cdot 0.016666666666666666\right)}\right) \]

   if 1.4e72 < im < 1.8999999999999999e154

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 5.7%

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

   6. Simplified 5.7%

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

   7. Taylor expanded in re around 0 32.0%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot {im}^{2}\right) \cdot re} \]
   8. Step-by-step derivation

      1. *-commutative 32.0%

         \[\leadsto \color{blue}{re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)} \]

      2. unpow2 32.0%

         \[\leadsto re \cdot \left(1 + 0.5 \cdot \color{blue}{\left(im \cdot im\right)}\right) \]

   9. Simplified 32.0%

      \[\leadsto \color{blue}{re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative 32.0%

          \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \left(im \cdot im\right)\right) \cdot re} \]

       2. flip-+ 61.5%

          \[\leadsto \color{blue}{\frac{1 \cdot 1 - \left(0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)}{1 - 0.5 \cdot \left(im \cdot im\right)}} \cdot re \]

       3. associate-*l/ 61.5%

          \[\leadsto \color{blue}{\frac{\left(1 \cdot 1 - \left(0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\right) \cdot re}{1 - 0.5 \cdot \left(im \cdot im\right)}} \]

       4. metadata-eval 61.5%

          \[\leadsto \frac{\left(\color{blue}{1} - \left(0.5 \cdot \left(im \cdot im\right)\right) \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\right) \cdot re}{1 - 0.5 \cdot \left(im \cdot im\right)} \]

       5. swap-sqr 61.5%

          \[\leadsto \frac{\left(1 - \color{blue}{\left(0.5 \cdot 0.5\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)}\right) \cdot re}{1 - 0.5 \cdot \left(im \cdot im\right)} \]

       6. metadata-eval 61.5%

          \[\leadsto \frac{\left(1 - \color{blue}{0.25} \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right) \cdot re}{1 - 0.5 \cdot \left(im \cdot im\right)} \]

       7. pow2 61.5%

          \[\leadsto \frac{\left(1 - 0.25 \cdot \left(\color{blue}{{im}^{2}} \cdot \left(im \cdot im\right)\right)\right) \cdot re}{1 - 0.5 \cdot \left(im \cdot im\right)} \]

       8. pow2 61.5%

          \[\leadsto \frac{\left(1 - 0.25 \cdot \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right) \cdot re}{1 - 0.5 \cdot \left(im \cdot im\right)} \]

       9. pow-prod-up 61.5%

          \[\leadsto \frac{\left(1 - 0.25 \cdot \color{blue}{{im}^{\left(2 + 2\right)}}\right) \cdot re}{1 - 0.5 \cdot \left(im \cdot im\right)} \]

       10. metadata-eval 61.5%

           \[\leadsto \frac{\left(1 - 0.25 \cdot {im}^{\color{blue}{4}}\right) \cdot re}{1 - 0.5 \cdot \left(im \cdot im\right)} \]

       11. associate-*r* 61.5%

           \[\leadsto \frac{\left(1 - 0.25 \cdot {im}^{4}\right) \cdot re}{1 - \color{blue}{\left(0.5 \cdot im\right) \cdot im}} \]

       12. *-commutative 61.5%

           \[\leadsto \frac{\left(1 - 0.25 \cdot {im}^{4}\right) \cdot re}{1 - \color{blue}{im \cdot \left(0.5 \cdot im\right)}} \]

   11. Applied egg-rr 61.5%

       \[\leadsto \color{blue}{\frac{\left(1 - 0.25 \cdot {im}^{4}\right) \cdot re}{1 - im \cdot \left(0.5 \cdot im\right)}} \]

   12. Taylor expanded in im around inf 61.5%

       \[\leadsto \frac{\color{blue}{-0.25 \cdot \left(re \cdot {im}^{4}\right)}}{1 - im \cdot \left(0.5 \cdot im\right)} \]
   13. Step-by-step derivation

       1. associate-*r* 61.5%

          \[\leadsto \frac{\color{blue}{\left(-0.25 \cdot re\right) \cdot {im}^{4}}}{1 - im \cdot \left(0.5 \cdot im\right)} \]

   14. Simplified 61.5%

       \[\leadsto \frac{\color{blue}{\left(-0.25 \cdot re\right) \cdot {im}^{4}}}{1 - im \cdot \left(0.5 \cdot im\right)} \]

   if 1.8999999999999999e154 < 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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

      3. distribute-rgt1-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2} + 1\right) \cdot \sin re} \]

      4. fma-def 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, {im}^{2}, 1\right)} \cdot \sin re \]

      5. unpow2 100.0%

         \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{im \cdot im}, 1\right) \cdot \sin re \]

   9. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, im \cdot im, 1\right) \cdot \sin re} \]

   10. Taylor expanded in im around inf 100.0%

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

       1. *-commutative 100.0%

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

       2. associate-*r* 100.0%

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

       3. unpow2 100.0%

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

       4. associate-*r* 100.0%

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

       5. *-commutative 100.0%

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

   12. Simplified 100.0%

       \[\leadsto \color{blue}{\sin re \cdot \left(im \cdot \left(0.5 \cdot im\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 500:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+72}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + re \cdot \left(re \cdot 0.016666666666666666\right)\right)\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;\frac{{im}^{4} \cdot \left(re \cdot -0.25\right)}{1 - im \cdot \left(0.5 \cdot im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(im \cdot \left(0.5 \cdot im\right)\right)\\ \end{array} \]

Alternative 10: 66.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(0.5 \cdot im\right)\\ t_1 := re \cdot t_0\\ \mathbf{if}\;im \leq 680:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.55 \cdot 10^{+96}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + re \cdot \left(re \cdot 0.016666666666666666\right)\right)\\ \mathbf{elif}\;im \leq 1.85 \cdot 10^{+140}:\\ \;\;\;\;re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;\frac{re \cdot \left(re - \left(0.5 \cdot im\right) \cdot \left(im \cdot t_1\right)\right)}{re - t_1}\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (* 0.5 im))) (t_1 (* re t_0)))
   (if (<= im 680.0)
     (sin re)
     (if (<= im 1.55e+96)
       (+
        0.08333333333333333
        (+ (/ 0.25 (* re re)) (* re (* re 0.016666666666666666))))
       (if (<= im 1.85e+140)
         (* re (+ 1.0 (* 0.5 (* im im))))
         (if (<= im 1.9e+154)
           (/ (* re (- re (* (* 0.5 im) (* im t_1)))) (- re t_1))
           (* (sin re) t_0)))))))

C

double code(double re, double im) {
	double t_0 = im * (0.5 * im);
	double t_1 = re * t_0;
	double tmp;
	if (im <= 680.0) {
		tmp = sin(re);
	} else if (im <= 1.55e+96) {
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + (re * (re * 0.016666666666666666)));
	} else if (im <= 1.85e+140) {
		tmp = re * (1.0 + (0.5 * (im * im)));
	} else if (im <= 1.9e+154) {
		tmp = (re * (re - ((0.5 * im) * (im * t_1)))) / (re - t_1);
	} else {
		tmp = sin(re) * t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = im * (0.5d0 * im)
    t_1 = re * t_0
    if (im <= 680.0d0) then
        tmp = sin(re)
    else if (im <= 1.55d+96) then
        tmp = 0.08333333333333333d0 + ((0.25d0 / (re * re)) + (re * (re * 0.016666666666666666d0)))
    else if (im <= 1.85d+140) then
        tmp = re * (1.0d0 + (0.5d0 * (im * im)))
    else if (im <= 1.9d+154) then
        tmp = (re * (re - ((0.5d0 * im) * (im * t_1)))) / (re - t_1)
    else
        tmp = sin(re) * t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * (0.5 * im);
	double t_1 = re * t_0;
	double tmp;
	if (im <= 680.0) {
		tmp = Math.sin(re);
	} else if (im <= 1.55e+96) {
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + (re * (re * 0.016666666666666666)));
	} else if (im <= 1.85e+140) {
		tmp = re * (1.0 + (0.5 * (im * im)));
	} else if (im <= 1.9e+154) {
		tmp = (re * (re - ((0.5 * im) * (im * t_1)))) / (re - t_1);
	} else {
		tmp = Math.sin(re) * t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (0.5 * im)
	t_1 = re * t_0
	tmp = 0
	if im <= 680.0:
		tmp = math.sin(re)
	elif im <= 1.55e+96:
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + (re * (re * 0.016666666666666666)))
	elif im <= 1.85e+140:
		tmp = re * (1.0 + (0.5 * (im * im)))
	elif im <= 1.9e+154:
		tmp = (re * (re - ((0.5 * im) * (im * t_1)))) / (re - t_1)
	else:
		tmp = math.sin(re) * t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(0.5 * im))
	t_1 = Float64(re * t_0)
	tmp = 0.0
	if (im <= 680.0)
		tmp = sin(re);
	elseif (im <= 1.55e+96)
		tmp = Float64(0.08333333333333333 + Float64(Float64(0.25 / Float64(re * re)) + Float64(re * Float64(re * 0.016666666666666666))));
	elseif (im <= 1.85e+140)
		tmp = Float64(re * Float64(1.0 + Float64(0.5 * Float64(im * im))));
	elseif (im <= 1.9e+154)
		tmp = Float64(Float64(re * Float64(re - Float64(Float64(0.5 * im) * Float64(im * t_1)))) / Float64(re - t_1));
	else
		tmp = Float64(sin(re) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (0.5 * im);
	t_1 = re * t_0;
	tmp = 0.0;
	if (im <= 680.0)
		tmp = sin(re);
	elseif (im <= 1.55e+96)
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + (re * (re * 0.016666666666666666)));
	elseif (im <= 1.85e+140)
		tmp = re * (1.0 + (0.5 * (im * im)));
	elseif (im <= 1.9e+154)
		tmp = (re * (re - ((0.5 * im) * (im * t_1)))) / (re - t_1);
	else
		tmp = sin(re) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(re * t$95$0), $MachinePrecision]}, If[LessEqual[im, 680.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.55e+96], N[(0.08333333333333333 + N[(N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision] + N[(re * N[(re * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.85e+140], N[(re * N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.9e+154], N[(N[(re * N[(re - N[(N[(0.5 * im), $MachinePrecision] * N[(im * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(re - t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * t$95$0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if im < 680

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 62.8%

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

   if 680 < im < 1.5499999999999999e96

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 12.1%

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

   6. Taylor expanded in re around 0 17.5%

      \[\leadsto \color{blue}{0.08333333333333333 + \left(0.25 \cdot \frac{1}{{re}^{2}} + 0.016666666666666666 \cdot {re}^{2}\right)} \]
   7. Step-by-step derivation

      1. associate-*r/ 17.5%

         \[\leadsto 0.08333333333333333 + \left(\color{blue}{\frac{0.25 \cdot 1}{{re}^{2}}} + 0.016666666666666666 \cdot {re}^{2}\right) \]

      2. metadata-eval 17.5%

         \[\leadsto 0.08333333333333333 + \left(\frac{\color{blue}{0.25}}{{re}^{2}} + 0.016666666666666666 \cdot {re}^{2}\right) \]

      3. unpow2 17.5%

         \[\leadsto 0.08333333333333333 + \left(\frac{0.25}{\color{blue}{re \cdot re}} + 0.016666666666666666 \cdot {re}^{2}\right) \]

      4. *-commutative 17.5%

         \[\leadsto 0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \color{blue}{{re}^{2} \cdot 0.016666666666666666}\right) \]

      5. unpow2 17.5%

         \[\leadsto 0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \color{blue}{\left(re \cdot re\right)} \cdot 0.016666666666666666\right) \]

   8. Simplified 17.5%

      \[\leadsto \color{blue}{0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)} \]

   9. Taylor expanded in re around 0 17.5%

      \[\leadsto 0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \color{blue}{0.016666666666666666 \cdot {re}^{2}}\right) \]
   10. Step-by-step derivation

       1. unpow2 17.5%

          \[\leadsto 0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + 0.016666666666666666 \cdot \color{blue}{\left(re \cdot re\right)}\right) \]

       2. *-commutative 17.5%

          \[\leadsto 0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \color{blue}{\left(re \cdot re\right) \cdot 0.016666666666666666}\right) \]

       3. associate-*r* 17.5%

          \[\leadsto 0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \color{blue}{re \cdot \left(re \cdot 0.016666666666666666\right)}\right) \]

   11. Simplified 17.5%

       \[\leadsto 0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \color{blue}{re \cdot \left(re \cdot 0.016666666666666666\right)}\right) \]

   if 1.5499999999999999e96 < im < 1.85000000000000001e140

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 6.7%

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

   6. Simplified 6.7%

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

   7. Taylor expanded in re around 0 50.0%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot {im}^{2}\right) \cdot re} \]
   8. Step-by-step derivation

      1. *-commutative 50.0%

         \[\leadsto \color{blue}{re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)} \]

      2. unpow2 50.0%

         \[\leadsto re \cdot \left(1 + 0.5 \cdot \color{blue}{\left(im \cdot im\right)}\right) \]

   9. Simplified 50.0%

      \[\leadsto \color{blue}{re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)} \]

   if 1.85000000000000001e140 < im < 1.8999999999999999e154

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 5.1%

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

   6. Simplified 5.1%

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

   7. Taylor expanded in re around 0 5.1%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot {im}^{2}\right) \cdot re} \]
   8. Step-by-step derivation

      1. *-commutative 5.1%

         \[\leadsto \color{blue}{re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)} \]

      2. unpow2 5.1%

         \[\leadsto re \cdot \left(1 + 0.5 \cdot \color{blue}{\left(im \cdot im\right)}\right) \]

   9. Simplified 5.1%

      \[\leadsto \color{blue}{re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)} \]
   10. Step-by-step derivation

       1. distribute-rgt-in 5.1%

          \[\leadsto \color{blue}{1 \cdot re + \left(0.5 \cdot \left(im \cdot im\right)\right) \cdot re} \]

       2. *-un-lft-identity 5.1%

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

       3. flip-+ 100.0%

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

       4. *-commutative 100.0%

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

       5. associate-*r* 100.0%

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

       6. *-commutative 100.0%

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

       7. *-commutative 100.0%

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

       8. associate-*r* 100.0%

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

       9. *-commutative 100.0%

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

       10. *-commutative 100.0%

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

       11. associate-*r* 100.0%

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

       12. *-commutative 100.0%

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

   11. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\frac{re \cdot re - \left(re \cdot \left(im \cdot \left(0.5 \cdot im\right)\right)\right) \cdot \left(re \cdot \left(im \cdot \left(0.5 \cdot im\right)\right)\right)}{re - re \cdot \left(im \cdot \left(0.5 \cdot im\right)\right)}} \]
   12. Step-by-step derivation

       1. associate-*l* 100.0%

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

       2. distribute-lft-out-- 100.0%

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

       3. *-commutative 100.0%

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

       4. associate-*l* 100.0%

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

   13. Simplified 100.0%

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

   if 1.8999999999999999e154 < 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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

      3. distribute-rgt1-in 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot {im}^{2} + 1\right) \cdot \sin re} \]

      4. fma-def 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, {im}^{2}, 1\right)} \cdot \sin re \]

      5. unpow2 100.0%

         \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{im \cdot im}, 1\right) \cdot \sin re \]

   9. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, im \cdot im, 1\right) \cdot \sin re} \]

   10. Taylor expanded in im around inf 100.0%

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

       1. *-commutative 100.0%

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

       2. associate-*r* 100.0%

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

       3. unpow2 100.0%

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

       4. associate-*r* 100.0%

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

       5. *-commutative 100.0%

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

   12. Simplified 100.0%

       \[\leadsto \color{blue}{\sin re \cdot \left(im \cdot \left(0.5 \cdot im\right)\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 680:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.55 \cdot 10^{+96}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + re \cdot \left(re \cdot 0.016666666666666666\right)\right)\\ \mathbf{elif}\;im \leq 1.85 \cdot 10^{+140}:\\ \;\;\;\;re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;\frac{re \cdot \left(re - \left(0.5 \cdot im\right) \cdot \left(im \cdot \left(re \cdot \left(im \cdot \left(0.5 \cdot im\right)\right)\right)\right)\right)}{re - re \cdot \left(im \cdot \left(0.5 \cdot im\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(im \cdot \left(0.5 \cdot im\right)\right)\\ \end{array} \]

Alternative 11: 63.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(im \cdot \left(0.5 \cdot im\right)\right)\\ \mathbf{if}\;im \leq 550:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.65 \cdot 10^{+97}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + re \cdot \left(re \cdot 0.016666666666666666\right)\right)\\ \mathbf{elif}\;im \leq 5.8 \cdot 10^{+141} \lor \neg \left(im \leq 4.4 \cdot 10^{+153}\right):\\ \;\;\;\;re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{re \cdot \left(re - \left(0.5 \cdot im\right) \cdot \left(im \cdot t_0\right)\right)}{re - t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (* im (* 0.5 im)))))
   (if (<= im 550.0)
     (sin re)
     (if (<= im 1.65e+97)
       (+
        0.08333333333333333
        (+ (/ 0.25 (* re re)) (* re (* re 0.016666666666666666))))
       (if (or (<= im 5.8e+141) (not (<= im 4.4e+153)))
         (* re (+ 1.0 (* 0.5 (* im im))))
         (/ (* re (- re (* (* 0.5 im) (* im t_0)))) (- re t_0)))))))

C

double code(double re, double im) {
	double t_0 = re * (im * (0.5 * im));
	double tmp;
	if (im <= 550.0) {
		tmp = sin(re);
	} else if (im <= 1.65e+97) {
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + (re * (re * 0.016666666666666666)));
	} else if ((im <= 5.8e+141) || !(im <= 4.4e+153)) {
		tmp = re * (1.0 + (0.5 * (im * im)));
	} else {
		tmp = (re * (re - ((0.5 * im) * (im * t_0)))) / (re - t_0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = re * (im * (0.5d0 * im))
    if (im <= 550.0d0) then
        tmp = sin(re)
    else if (im <= 1.65d+97) then
        tmp = 0.08333333333333333d0 + ((0.25d0 / (re * re)) + (re * (re * 0.016666666666666666d0)))
    else if ((im <= 5.8d+141) .or. (.not. (im <= 4.4d+153))) then
        tmp = re * (1.0d0 + (0.5d0 * (im * im)))
    else
        tmp = (re * (re - ((0.5d0 * im) * (im * t_0)))) / (re - t_0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re * (im * (0.5 * im));
	double tmp;
	if (im <= 550.0) {
		tmp = Math.sin(re);
	} else if (im <= 1.65e+97) {
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + (re * (re * 0.016666666666666666)));
	} else if ((im <= 5.8e+141) || !(im <= 4.4e+153)) {
		tmp = re * (1.0 + (0.5 * (im * im)));
	} else {
		tmp = (re * (re - ((0.5 * im) * (im * t_0)))) / (re - t_0);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * (im * (0.5 * im))
	tmp = 0
	if im <= 550.0:
		tmp = math.sin(re)
	elif im <= 1.65e+97:
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + (re * (re * 0.016666666666666666)))
	elif (im <= 5.8e+141) or not (im <= 4.4e+153):
		tmp = re * (1.0 + (0.5 * (im * im)))
	else:
		tmp = (re * (re - ((0.5 * im) * (im * t_0)))) / (re - t_0)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(im * Float64(0.5 * im)))
	tmp = 0.0
	if (im <= 550.0)
		tmp = sin(re);
	elseif (im <= 1.65e+97)
		tmp = Float64(0.08333333333333333 + Float64(Float64(0.25 / Float64(re * re)) + Float64(re * Float64(re * 0.016666666666666666))));
	elseif ((im <= 5.8e+141) || !(im <= 4.4e+153))
		tmp = Float64(re * Float64(1.0 + Float64(0.5 * Float64(im * im))));
	else
		tmp = Float64(Float64(re * Float64(re - Float64(Float64(0.5 * im) * Float64(im * t_0)))) / Float64(re - t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * (im * (0.5 * im));
	tmp = 0.0;
	if (im <= 550.0)
		tmp = sin(re);
	elseif (im <= 1.65e+97)
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + (re * (re * 0.016666666666666666)));
	elseif ((im <= 5.8e+141) || ~((im <= 4.4e+153)))
		tmp = re * (1.0 + (0.5 * (im * im)));
	else
		tmp = (re * (re - ((0.5 * im) * (im * t_0)))) / (re - t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 550.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.65e+97], N[(0.08333333333333333 + N[(N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision] + N[(re * N[(re * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[im, 5.8e+141], N[Not[LessEqual[im, 4.4e+153]], $MachinePrecision]], N[(re * N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(re * N[(re - N[(N[(0.5 * im), $MachinePrecision] * N[(im * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(re - t$95$0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if im < 550

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 62.8%

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

   if 550 < im < 1.6500000000000001e97

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 12.1%

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

   6. Taylor expanded in re around 0 17.5%

      \[\leadsto \color{blue}{0.08333333333333333 + \left(0.25 \cdot \frac{1}{{re}^{2}} + 0.016666666666666666 \cdot {re}^{2}\right)} \]
   7. Step-by-step derivation

      1. associate-*r/ 17.5%

         \[\leadsto 0.08333333333333333 + \left(\color{blue}{\frac{0.25 \cdot 1}{{re}^{2}}} + 0.016666666666666666 \cdot {re}^{2}\right) \]

      2. metadata-eval 17.5%

         \[\leadsto 0.08333333333333333 + \left(\frac{\color{blue}{0.25}}{{re}^{2}} + 0.016666666666666666 \cdot {re}^{2}\right) \]

      3. unpow2 17.5%

         \[\leadsto 0.08333333333333333 + \left(\frac{0.25}{\color{blue}{re \cdot re}} + 0.016666666666666666 \cdot {re}^{2}\right) \]

      4. *-commutative 17.5%

         \[\leadsto 0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \color{blue}{{re}^{2} \cdot 0.016666666666666666}\right) \]

      5. unpow2 17.5%

         \[\leadsto 0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \color{blue}{\left(re \cdot re\right)} \cdot 0.016666666666666666\right) \]

   8. Simplified 17.5%

      \[\leadsto \color{blue}{0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)} \]

   9. Taylor expanded in re around 0 17.5%

      \[\leadsto 0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \color{blue}{0.016666666666666666 \cdot {re}^{2}}\right) \]
   10. Step-by-step derivation

       1. unpow2 17.5%

          \[\leadsto 0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + 0.016666666666666666 \cdot \color{blue}{\left(re \cdot re\right)}\right) \]

       2. *-commutative 17.5%

          \[\leadsto 0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \color{blue}{\left(re \cdot re\right) \cdot 0.016666666666666666}\right) \]

       3. associate-*r* 17.5%

          \[\leadsto 0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \color{blue}{re \cdot \left(re \cdot 0.016666666666666666\right)}\right) \]

   11. Simplified 17.5%

       \[\leadsto 0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \color{blue}{re \cdot \left(re \cdot 0.016666666666666666\right)}\right) \]

   if 1.6500000000000001e97 < im < 5.80000000000000013e141 or 4.3999999999999999e153 < 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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 83.0%

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

   6. Simplified 83.0%

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

   7. Taylor expanded in re around 0 72.7%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot {im}^{2}\right) \cdot re} \]
   8. Step-by-step derivation

      1. *-commutative 72.7%

         \[\leadsto \color{blue}{re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)} \]

      2. unpow2 72.7%

         \[\leadsto re \cdot \left(1 + 0.5 \cdot \color{blue}{\left(im \cdot im\right)}\right) \]

   9. Simplified 72.7%

      \[\leadsto \color{blue}{re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)} \]

   if 5.80000000000000013e141 < im < 4.3999999999999999e153

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 5.1%

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

   6. Simplified 5.1%

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

   7. Taylor expanded in re around 0 5.1%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot {im}^{2}\right) \cdot re} \]
   8. Step-by-step derivation

      1. *-commutative 5.1%

         \[\leadsto \color{blue}{re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)} \]

      2. unpow2 5.1%

         \[\leadsto re \cdot \left(1 + 0.5 \cdot \color{blue}{\left(im \cdot im\right)}\right) \]

   9. Simplified 5.1%

      \[\leadsto \color{blue}{re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)} \]
   10. Step-by-step derivation

       1. distribute-rgt-in 5.1%

          \[\leadsto \color{blue}{1 \cdot re + \left(0.5 \cdot \left(im \cdot im\right)\right) \cdot re} \]

       2. *-un-lft-identity 5.1%

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

       3. flip-+ 100.0%

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

       4. *-commutative 100.0%

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

       5. associate-*r* 100.0%

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

       6. *-commutative 100.0%

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

       7. *-commutative 100.0%

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

       8. associate-*r* 100.0%

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

       9. *-commutative 100.0%

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

       10. *-commutative 100.0%

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

       11. associate-*r* 100.0%

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

       12. *-commutative 100.0%

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

   11. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\frac{re \cdot re - \left(re \cdot \left(im \cdot \left(0.5 \cdot im\right)\right)\right) \cdot \left(re \cdot \left(im \cdot \left(0.5 \cdot im\right)\right)\right)}{re - re \cdot \left(im \cdot \left(0.5 \cdot im\right)\right)}} \]
   12. Step-by-step derivation

       1. associate-*l* 100.0%

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

       2. distribute-lft-out-- 100.0%

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

       3. *-commutative 100.0%

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

       4. associate-*l* 100.0%

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

   13. Simplified 100.0%

       \[\leadsto \color{blue}{\frac{re \cdot \left(re - \left(0.5 \cdot im\right) \cdot \left(im \cdot \left(re \cdot \left(im \cdot \left(0.5 \cdot im\right)\right)\right)\right)\right)}{re - re \cdot \left(im \cdot \left(0.5 \cdot im\right)\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 550:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.65 \cdot 10^{+97}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + re \cdot \left(re \cdot 0.016666666666666666\right)\right)\\ \mathbf{elif}\;im \leq 5.8 \cdot 10^{+141} \lor \neg \left(im \leq 4.4 \cdot 10^{+153}\right):\\ \;\;\;\;re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{re \cdot \left(re - \left(0.5 \cdot im\right) \cdot \left(im \cdot \left(re \cdot \left(im \cdot \left(0.5 \cdot im\right)\right)\right)\right)\right)}{re - re \cdot \left(im \cdot \left(0.5 \cdot im\right)\right)}\\ \end{array} \]

Alternative 12: 37.1% accurate, 27.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 390:\\ \;\;\;\;re\\ \mathbf{elif}\;im \leq 3.1 \cdot 10^{+25}:\\ \;\;\;\;0.08333333333333333 + 0.016666666666666666 \cdot \left(re \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 390.0)
   re
   (if (<= im 3.1e+25)
     (+ 0.08333333333333333 (* 0.016666666666666666 (* re re)))
     (* (* im im) (* 0.5 re)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 390.0) {
		tmp = re;
	} else if (im <= 3.1e+25) {
		tmp = 0.08333333333333333 + (0.016666666666666666 * (re * re));
	} else {
		tmp = (im * im) * (0.5 * 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 <= 390.0d0) then
        tmp = re
    else if (im <= 3.1d+25) then
        tmp = 0.08333333333333333d0 + (0.016666666666666666d0 * (re * re))
    else
        tmp = (im * im) * (0.5d0 * re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 390.0) {
		tmp = re;
	} else if (im <= 3.1e+25) {
		tmp = 0.08333333333333333 + (0.016666666666666666 * (re * re));
	} else {
		tmp = (im * im) * (0.5 * re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 390.0:
		tmp = re
	elif im <= 3.1e+25:
		tmp = 0.08333333333333333 + (0.016666666666666666 * (re * re))
	else:
		tmp = (im * im) * (0.5 * re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 390.0)
		tmp = re;
	elseif (im <= 3.1e+25)
		tmp = Float64(0.08333333333333333 + Float64(0.016666666666666666 * Float64(re * re)));
	else
		tmp = Float64(Float64(im * im) * Float64(0.5 * re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 390.0)
		tmp = re;
	elseif (im <= 3.1e+25)
		tmp = 0.08333333333333333 + (0.016666666666666666 * (re * re));
	else
		tmp = (im * im) * (0.5 * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 390.0], re, If[LessEqual[im, 3.1e+25], N[(0.08333333333333333 + N[(0.016666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im * im), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 390

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 77.2%

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

   6. Simplified 77.2%

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

   7. Taylor expanded in re around 0 49.4%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot {im}^{2}\right) \cdot re} \]
   8. Step-by-step derivation

      1. *-commutative 49.4%

         \[\leadsto \color{blue}{re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)} \]

      2. unpow2 49.4%

         \[\leadsto re \cdot \left(1 + 0.5 \cdot \color{blue}{\left(im \cdot im\right)}\right) \]

   9. Simplified 49.4%

      \[\leadsto \color{blue}{re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)} \]

   10. Taylor expanded in im around 0 36.2%

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

   if 390 < im < 3.0999999999999998e25

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 4.6%

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

   6. Taylor expanded in re around 0 6.6%

      \[\leadsto \color{blue}{0.08333333333333333 + \left(0.25 \cdot \frac{1}{{re}^{2}} + 0.016666666666666666 \cdot {re}^{2}\right)} \]
   7. Step-by-step derivation

      1. associate-*r/ 6.6%

         \[\leadsto 0.08333333333333333 + \left(\color{blue}{\frac{0.25 \cdot 1}{{re}^{2}}} + 0.016666666666666666 \cdot {re}^{2}\right) \]

      2. metadata-eval 6.6%

         \[\leadsto 0.08333333333333333 + \left(\frac{\color{blue}{0.25}}{{re}^{2}} + 0.016666666666666666 \cdot {re}^{2}\right) \]

      3. unpow2 6.6%

         \[\leadsto 0.08333333333333333 + \left(\frac{0.25}{\color{blue}{re \cdot re}} + 0.016666666666666666 \cdot {re}^{2}\right) \]

      4. *-commutative 6.6%

         \[\leadsto 0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \color{blue}{{re}^{2} \cdot 0.016666666666666666}\right) \]

      5. unpow2 6.6%

         \[\leadsto 0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \color{blue}{\left(re \cdot re\right)} \cdot 0.016666666666666666\right) \]

   8. Simplified 6.6%

      \[\leadsto \color{blue}{0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)} \]

   9. Taylor expanded in re around inf 5.1%

      \[\leadsto \color{blue}{0.08333333333333333 + 0.016666666666666666 \cdot {re}^{2}} \]
   10. Step-by-step derivation

       1. unpow2 5.1%

          \[\leadsto 0.08333333333333333 + 0.016666666666666666 \cdot \color{blue}{\left(re \cdot re\right)} \]

   11. Simplified 5.1%

       \[\leadsto \color{blue}{0.08333333333333333 + 0.016666666666666666 \cdot \left(re \cdot re\right)} \]

   if 3.0999999999999998e25 < 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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 60.0%

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

   6. Simplified 60.0%

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

   7. Taylor expanded in re around 0 55.5%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot {im}^{2}\right) \cdot re} \]
   8. Step-by-step derivation

      1. *-commutative 55.5%

         \[\leadsto \color{blue}{re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)} \]

      2. unpow2 55.5%

         \[\leadsto re \cdot \left(1 + 0.5 \cdot \color{blue}{\left(im \cdot im\right)}\right) \]

   9. Simplified 55.5%

      \[\leadsto \color{blue}{re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)} \]

   10. Taylor expanded in im around inf 55.5%

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

       1. associate-*r* 55.5%

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

       2. unpow2 55.5%

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

   12. Simplified 55.5%

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

4. Final simplification 40.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 390:\\ \;\;\;\;re\\ \mathbf{elif}\;im \leq 3.1 \cdot 10^{+25}:\\ \;\;\;\;0.08333333333333333 + 0.016666666666666666 \cdot \left(re \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \]

Alternative 13: 37.0% accurate, 34.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.4:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.4) re (* (* im im) (* 0.5 re))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.4) {
		tmp = re;
	} else {
		tmp = (im * im) * (0.5 * 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 <= 1.4d0) then
        tmp = re
    else
        tmp = (im * im) * (0.5d0 * re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.4) {
		tmp = re;
	} else {
		tmp = (im * im) * (0.5 * re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.4:
		tmp = re
	else:
		tmp = (im * im) * (0.5 * re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.4)
		tmp = re;
	else
		tmp = Float64(Float64(im * im) * Float64(0.5 * re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.4)
		tmp = re;
	else
		tmp = (im * im) * (0.5 * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.4], re, N[(N[(im * im), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.3999999999999999

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 77.2%

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

   6. Simplified 77.2%

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

   7. Taylor expanded in re around 0 49.4%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot {im}^{2}\right) \cdot re} \]
   8. Step-by-step derivation

      1. *-commutative 49.4%

         \[\leadsto \color{blue}{re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)} \]

      2. unpow2 49.4%

         \[\leadsto re \cdot \left(1 + 0.5 \cdot \color{blue}{\left(im \cdot im\right)}\right) \]

   9. Simplified 49.4%

      \[\leadsto \color{blue}{re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)} \]

   10. Taylor expanded in im around 0 36.2%

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

   if 1.3999999999999999 < 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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 55.7%

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

   6. Simplified 55.7%

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

   7. Taylor expanded in re around 0 51.6%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot {im}^{2}\right) \cdot re} \]
   8. Step-by-step derivation

      1. *-commutative 51.6%

         \[\leadsto \color{blue}{re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)} \]

      2. unpow2 51.6%

         \[\leadsto re \cdot \left(1 + 0.5 \cdot \color{blue}{\left(im \cdot im\right)}\right) \]

   9. Simplified 51.6%

      \[\leadsto \color{blue}{re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)} \]

   10. Taylor expanded in im around inf 51.6%

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

       1. associate-*r* 51.6%

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

       2. unpow2 51.6%

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

   12. Simplified 51.6%

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

4. Final simplification 40.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.4:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \]

Alternative 14: 47.9% accurate, 34.3× speedup

Math

\[\begin{array}{l} \\ re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* re (+ 1.0 (* 0.5 (* im im)))))

C

double code(double re, double im) {
	return re * (1.0 + (0.5 * (im * im)));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = re * (1.0d0 + (0.5d0 * (im * im)))
end function

Java

public static double code(double re, double im) {
	return re * (1.0 + (0.5 * (im * im)));
}

Python

def code(re, im):
	return re * (1.0 + (0.5 * (im * im)))

Julia

function code(re, im)
	return Float64(re * Float64(1.0 + Float64(0.5 * Float64(im * im))))
end

MATLAB

function tmp = code(re, im)
	tmp = re * (1.0 + (0.5 * (im * im)));
end

Wolfram

code[re_, im_] := N[(re * N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $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. sub0-neg 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in im around 0 71.6%

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

6. Simplified 71.6%

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

7. Taylor expanded in re around 0 49.9%

   \[\leadsto \color{blue}{\left(1 + 0.5 \cdot {im}^{2}\right) \cdot re} \]
8. Step-by-step derivation

   1. *-commutative 49.9%

      \[\leadsto \color{blue}{re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)} \]

   2. unpow2 49.9%

      \[\leadsto re \cdot \left(1 + 0.5 \cdot \color{blue}{\left(im \cdot im\right)}\right) \]

9. Simplified 49.9%

   \[\leadsto \color{blue}{re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)} \]

10. Final simplification 49.9%

    \[\leadsto re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right) \]

Alternative 15: 28.9% accurate, 43.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 5.4 \cdot 10^{+35}:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{re \cdot re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 5.4e+35) re (/ 0.25 (* re re))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 5.4e+35) {
		tmp = re;
	} else {
		tmp = 0.25 / (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 <= 5.4d+35) then
        tmp = re
    else
        tmp = 0.25d0 / (re * re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 5.4e+35) {
		tmp = re;
	} else {
		tmp = 0.25 / (re * re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 5.4e+35:
		tmp = re
	else:
		tmp = 0.25 / (re * re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 5.4e+35)
		tmp = re;
	else
		tmp = Float64(0.25 / Float64(re * re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 5.4e+35)
		tmp = re;
	else
		tmp = 0.25 / (re * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 5.4e+35], re, N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 5.40000000000000005e35

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 74.5%

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

   6. Simplified 74.5%

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

   7. Taylor expanded in re around 0 47.7%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot {im}^{2}\right) \cdot re} \]
   8. Step-by-step derivation

      1. *-commutative 47.7%

         \[\leadsto \color{blue}{re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)} \]

      2. unpow2 47.7%

         \[\leadsto re \cdot \left(1 + 0.5 \cdot \color{blue}{\left(im \cdot im\right)}\right) \]

   9. Simplified 47.7%

      \[\leadsto \color{blue}{re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)} \]

   10. Taylor expanded in im around 0 35.0%

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

   if 5.40000000000000005e35 < 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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 11.6%

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

   6. Taylor expanded in re around 0 11.6%

      \[\leadsto \color{blue}{\frac{0.25}{{re}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 11.6%

         \[\leadsto \frac{0.25}{\color{blue}{re \cdot re}} \]

   8. Simplified 11.6%

      \[\leadsto \color{blue}{\frac{0.25}{re \cdot re}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 5.4 \cdot 10^{+35}:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{re \cdot re}\\ \end{array} \]

Alternative 16: 4.2% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ 0.08333333333333333 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.08333333333333333)

C

double code(double re, double im) {
	return 0.08333333333333333;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.08333333333333333d0
end function

Java

public static double code(double re, double im) {
	return 0.08333333333333333;
}

Python

def code(re, im):
	return 0.08333333333333333

Julia

function code(re, im)
	return 0.08333333333333333
end

MATLAB

function tmp = code(re, im)
	tmp = 0.08333333333333333;
end

Wolfram

code[re_, im_] := 0.08333333333333333

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. sub0-neg 100.0%

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

3. Simplified 100.0%

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

5. Applied egg-rr 9.4%

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

6. Taylor expanded in re around 0 9.2%

   \[\leadsto \color{blue}{0.08333333333333333 + 0.25 \cdot \frac{1}{{re}^{2}}} \]
7. Step-by-step derivation

   1. associate-*r/ 9.2%

      \[\leadsto 0.08333333333333333 + \color{blue}{\frac{0.25 \cdot 1}{{re}^{2}}} \]

   2. metadata-eval 9.2%

      \[\leadsto 0.08333333333333333 + \frac{\color{blue}{0.25}}{{re}^{2}} \]

   3. unpow2 9.2%

      \[\leadsto 0.08333333333333333 + \frac{0.25}{\color{blue}{re \cdot re}} \]

8. Simplified 9.2%

   \[\leadsto \color{blue}{0.08333333333333333 + \frac{0.25}{re \cdot re}} \]

9. Taylor expanded in re around inf 4.2%

   \[\leadsto \color{blue}{0.08333333333333333} \]

10. Final simplification 4.2%

    \[\leadsto 0.08333333333333333 \]

Alternative 17: 26.6% accurate, 309.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 re)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return re

Julia

function code(re, im)
	return re
end

MATLAB

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

Wolfram

code[re_, im_] := re

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. sub0-neg 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in im around 0 71.6%

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

6. Simplified 71.6%

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

7. Taylor expanded in re around 0 49.9%

   \[\leadsto \color{blue}{\left(1 + 0.5 \cdot {im}^{2}\right) \cdot re} \]
8. Step-by-step derivation

   1. *-commutative 49.9%

      \[\leadsto \color{blue}{re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)} \]

   2. unpow2 49.9%

      \[\leadsto re \cdot \left(1 + 0.5 \cdot \color{blue}{\left(im \cdot im\right)}\right) \]

9. Simplified 49.9%

   \[\leadsto \color{blue}{re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)} \]

10. Taylor expanded in im around 0 27.5%

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

11. Final simplification 27.5%

    \[\leadsto re \]

Reproduce

herbie shell --seed 2023257 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))