math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 7.2s

Alternatives: 17

Speedup: 1.0×

• 50.1% 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. distribute-rgt-in 100.0%

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

   2. cancel-sign-sub 100.0%

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

   3. distribute-rgt-out-- 100.0%

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

   4. sub-neg 100.0%

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

   5. neg-sub0 100.0%

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

   6. remove-double-neg 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 71.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.1 \cdot 10^{-11}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.1e-11)
   (sin re)
   (if (<= im 1.32e+154)
     (* (* 0.5 re) (+ (exp (- im)) (exp im)))
     (* (sin re) (* 0.5 (pow im 2.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.1e-11) {
		tmp = sin(re);
	} else if (im <= 1.32e+154) {
		tmp = (0.5 * re) * (exp(-im) + exp(im));
	} else {
		tmp = sin(re) * (0.5 * pow(im, 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.1d-11) then
        tmp = sin(re)
    else if (im <= 1.32d+154) then
        tmp = (0.5d0 * re) * (exp(-im) + exp(im))
    else
        tmp = sin(re) * (0.5d0 * (im ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.1e-11) {
		tmp = Math.sin(re);
	} else if (im <= 1.32e+154) {
		tmp = (0.5 * re) * (Math.exp(-im) + Math.exp(im));
	} else {
		tmp = Math.sin(re) * (0.5 * Math.pow(im, 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.1e-11:
		tmp = math.sin(re)
	elif im <= 1.32e+154:
		tmp = (0.5 * re) * (math.exp(-im) + math.exp(im))
	else:
		tmp = math.sin(re) * (0.5 * math.pow(im, 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.1e-11)
		tmp = sin(re);
	elseif (im <= 1.32e+154)
		tmp = Float64(Float64(0.5 * re) * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(sin(re) * Float64(0.5 * (im ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.1e-11)
		tmp = sin(re);
	elseif (im <= 1.32e+154)
		tmp = (0.5 * re) * (exp(-im) + exp(im));
	else
		tmp = sin(re) * (0.5 * (im ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.1e-11], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.32e+154], N[(N[(0.5 * re), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.1000000000000001e-11

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 67.6%

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

   if 1.1000000000000001e-11 < im < 1.31999999999999998e154

   1. Initial program 99.9%

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

      1. distribute-rgt-in 99.9%

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

      2. cancel-sign-sub 99.9%

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

      3. distribute-rgt-out-- 99.9%

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

      4. sub-neg 99.9%

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

      5. neg-sub0 99.9%

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

      6. remove-double-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in re around 0 86.8%

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

   7. Simplified 86.8%

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

   if 1.31999999999999998e154 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.1 \cdot 10^{-11}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 65.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 680:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{0.5}{re}\right)}^{6}}\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 680.0)
   (sin re)
   (if (<= im 1.32e+154)
     (cbrt (pow (/ 0.5 re) 6.0))
     (* (sin re) (* 0.5 (pow im 2.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 680.0) {
		tmp = sin(re);
	} else if (im <= 1.32e+154) {
		tmp = cbrt(pow((0.5 / re), 6.0));
	} else {
		tmp = sin(re) * (0.5 * pow(im, 2.0));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 680.0) {
		tmp = Math.sin(re);
	} else if (im <= 1.32e+154) {
		tmp = Math.cbrt(Math.pow((0.5 / re), 6.0));
	} else {
		tmp = Math.sin(re) * (0.5 * Math.pow(im, 2.0));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 680.0)
		tmp = sin(re);
	elseif (im <= 1.32e+154)
		tmp = cbrt((Float64(0.5 / re) ^ 6.0));
	else
		tmp = Float64(sin(re) * Float64(0.5 * (im ^ 2.0)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 680.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.32e+154], N[Power[N[Power[N[(0.5 / re), $MachinePrecision], 6.0], $MachinePrecision], 1/3], $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 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. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 66.9%

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

   if 680 < im < 1.31999999999999998e154

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 27.3%

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

   7. Taylor expanded in re around 0 27.1%

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

      1. add-sqr-sqrt 27.1%

         \[\leadsto \color{blue}{\sqrt{\frac{0.25}{{re}^{2}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}}} \]

      2. sqrt-div 27.1%

         \[\leadsto \color{blue}{\frac{\sqrt{0.25}}{\sqrt{{re}^{2}}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

      3. metadata-eval 27.1%

         \[\leadsto \frac{\color{blue}{0.5}}{\sqrt{{re}^{2}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

      4. unpow2 27.1%

         \[\leadsto \frac{0.5}{\sqrt{\color{blue}{re \cdot re}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

      5. sqrt-prod 26.8%

         \[\leadsto \frac{0.5}{\color{blue}{\sqrt{re} \cdot \sqrt{re}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

      6. add-sqr-sqrt 42.4%

         \[\leadsto \frac{0.5}{\color{blue}{re}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

      7. sqrt-div 42.4%

         \[\leadsto \frac{0.5}{re} \cdot \color{blue}{\frac{\sqrt{0.25}}{\sqrt{{re}^{2}}}} \]

      8. metadata-eval 42.4%

         \[\leadsto \frac{0.5}{re} \cdot \frac{\color{blue}{0.5}}{\sqrt{{re}^{2}}} \]

      9. unpow2 42.4%

         \[\leadsto \frac{0.5}{re} \cdot \frac{0.5}{\sqrt{\color{blue}{re \cdot re}}} \]

      10. sqrt-prod 26.8%

          \[\leadsto \frac{0.5}{re} \cdot \frac{0.5}{\color{blue}{\sqrt{re} \cdot \sqrt{re}}} \]

      11. add-sqr-sqrt 27.1%

          \[\leadsto \frac{0.5}{re} \cdot \frac{0.5}{\color{blue}{re}} \]

   9. Applied egg-rr 27.1%

      \[\leadsto \color{blue}{\frac{0.5}{re} \cdot \frac{0.5}{re}} \]
   10. Step-by-step derivation

       1. add-cbrt-cube 34.0%

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

       2. pow1/3 34.0%

          \[\leadsto \color{blue}{{\left(\left(\left(\frac{0.5}{re} \cdot \frac{0.5}{re}\right) \cdot \left(\frac{0.5}{re} \cdot \frac{0.5}{re}\right)\right) \cdot \left(\frac{0.5}{re} \cdot \frac{0.5}{re}\right)\right)}^{0.3333333333333333}} \]

       3. pow3 34.0%

          \[\leadsto {\color{blue}{\left({\left(\frac{0.5}{re} \cdot \frac{0.5}{re}\right)}^{3}\right)}}^{0.3333333333333333} \]

       4. pow2 34.0%

          \[\leadsto {\left({\color{blue}{\left({\left(\frac{0.5}{re}\right)}^{2}\right)}}^{3}\right)}^{0.3333333333333333} \]

       5. pow-pow 34.0%

          \[\leadsto {\color{blue}{\left({\left(\frac{0.5}{re}\right)}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} \]

       6. metadata-eval 34.0%

          \[\leadsto {\left({\left(\frac{0.5}{re}\right)}^{\color{blue}{6}}\right)}^{0.3333333333333333} \]

   11. Applied egg-rr 34.0%

       \[\leadsto \color{blue}{{\left({\left(\frac{0.5}{re}\right)}^{6}\right)}^{0.3333333333333333}} \]
   12. Step-by-step derivation

       1. unpow1/3 34.0%

          \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{0.5}{re}\right)}^{6}}} \]

   13. Simplified 34.0%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{0.5}{re}\right)}^{6}}} \]

   if 1.31999999999999998e154 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 680:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{0.5}{re}\right)}^{6}}\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 61.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 750:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{+142}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{0.5}{re}\right)}^{6}}\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot {im}^{2} + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 750.0)
   (sin re)
   (if (<= im 2.5e+142)
     (cbrt (pow (/ 0.5 re) 6.0))
     (* re (+ (* 0.5 (pow im 2.0)) 1.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 750.0) {
		tmp = sin(re);
	} else if (im <= 2.5e+142) {
		tmp = cbrt(pow((0.5 / re), 6.0));
	} else {
		tmp = re * ((0.5 * pow(im, 2.0)) + 1.0);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 750.0) {
		tmp = Math.sin(re);
	} else if (im <= 2.5e+142) {
		tmp = Math.cbrt(Math.pow((0.5 / re), 6.0));
	} else {
		tmp = re * ((0.5 * Math.pow(im, 2.0)) + 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 750.0)
		tmp = sin(re);
	elseif (im <= 2.5e+142)
		tmp = cbrt((Float64(0.5 / re) ^ 6.0));
	else
		tmp = Float64(re * Float64(Float64(0.5 * (im ^ 2.0)) + 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 750.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 2.5e+142], N[Power[N[Power[N[(0.5 / re), $MachinePrecision], 6.0], $MachinePrecision], 1/3], $MachinePrecision], N[(re * N[(N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 750

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 66.9%

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

   if 750 < im < 2.5000000000000001e142

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 27.3%

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

   7. Taylor expanded in re around 0 27.1%

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

      1. add-sqr-sqrt 27.1%

         \[\leadsto \color{blue}{\sqrt{\frac{0.25}{{re}^{2}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}}} \]

      2. sqrt-div 27.1%

         \[\leadsto \color{blue}{\frac{\sqrt{0.25}}{\sqrt{{re}^{2}}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

      3. metadata-eval 27.1%

         \[\leadsto \frac{\color{blue}{0.5}}{\sqrt{{re}^{2}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

      4. unpow2 27.1%

         \[\leadsto \frac{0.5}{\sqrt{\color{blue}{re \cdot re}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

      5. sqrt-prod 26.8%

         \[\leadsto \frac{0.5}{\color{blue}{\sqrt{re} \cdot \sqrt{re}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

      6. add-sqr-sqrt 40.8%

         \[\leadsto \frac{0.5}{\color{blue}{re}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

      7. sqrt-div 40.8%

         \[\leadsto \frac{0.5}{re} \cdot \color{blue}{\frac{\sqrt{0.25}}{\sqrt{{re}^{2}}}} \]

      8. metadata-eval 40.8%

         \[\leadsto \frac{0.5}{re} \cdot \frac{\color{blue}{0.5}}{\sqrt{{re}^{2}}} \]

      9. unpow2 40.8%

         \[\leadsto \frac{0.5}{re} \cdot \frac{0.5}{\sqrt{\color{blue}{re \cdot re}}} \]

      10. sqrt-prod 26.8%

          \[\leadsto \frac{0.5}{re} \cdot \frac{0.5}{\color{blue}{\sqrt{re} \cdot \sqrt{re}}} \]

      11. add-sqr-sqrt 27.1%

          \[\leadsto \frac{0.5}{re} \cdot \frac{0.5}{\color{blue}{re}} \]

   9. Applied egg-rr 27.1%

      \[\leadsto \color{blue}{\frac{0.5}{re} \cdot \frac{0.5}{re}} \]
   10. Step-by-step derivation

       1. add-cbrt-cube 31.1%

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

       2. pow1/3 31.1%

          \[\leadsto \color{blue}{{\left(\left(\left(\frac{0.5}{re} \cdot \frac{0.5}{re}\right) \cdot \left(\frac{0.5}{re} \cdot \frac{0.5}{re}\right)\right) \cdot \left(\frac{0.5}{re} \cdot \frac{0.5}{re}\right)\right)}^{0.3333333333333333}} \]

       3. pow3 31.1%

          \[\leadsto {\color{blue}{\left({\left(\frac{0.5}{re} \cdot \frac{0.5}{re}\right)}^{3}\right)}}^{0.3333333333333333} \]

       4. pow2 31.1%

          \[\leadsto {\left({\color{blue}{\left({\left(\frac{0.5}{re}\right)}^{2}\right)}}^{3}\right)}^{0.3333333333333333} \]

       5. pow-pow 31.1%

          \[\leadsto {\color{blue}{\left({\left(\frac{0.5}{re}\right)}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} \]

       6. metadata-eval 31.1%

          \[\leadsto {\left({\left(\frac{0.5}{re}\right)}^{\color{blue}{6}}\right)}^{0.3333333333333333} \]

   11. Applied egg-rr 31.1%

       \[\leadsto \color{blue}{{\left({\left(\frac{0.5}{re}\right)}^{6}\right)}^{0.3333333333333333}} \]
   12. Step-by-step derivation

       1. unpow1/3 31.1%

          \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{0.5}{re}\right)}^{6}}} \]

   13. Simplified 31.1%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{0.5}{re}\right)}^{6}}} \]

   if 2.5000000000000001e142 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 88.9%

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

   7. Simplified 88.9%

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

   8. Taylor expanded in re around 0 71.3%

      \[\leadsto \color{blue}{re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 750:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{+142}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{0.5}{re}\right)}^{6}}\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot {im}^{2} + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 61.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 490:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+121}:\\ \;\;\;\;0.08333333333333333 + \frac{\frac{0.25}{re}}{re}\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot {im}^{2} + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 490.0)
   (sin re)
   (if (<= im 1.35e+121)
     (+ 0.08333333333333333 (/ (/ 0.25 re) re))
     (* re (+ (* 0.5 (pow im 2.0)) 1.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 490.0) {
		tmp = sin(re);
	} else if (im <= 1.35e+121) {
		tmp = 0.08333333333333333 + ((0.25 / re) / re);
	} else {
		tmp = re * ((0.5 * pow(im, 2.0)) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 490.0d0) then
        tmp = sin(re)
    else if (im <= 1.35d+121) then
        tmp = 0.08333333333333333d0 + ((0.25d0 / re) / re)
    else
        tmp = re * ((0.5d0 * (im ** 2.0d0)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 490.0) {
		tmp = Math.sin(re);
	} else if (im <= 1.35e+121) {
		tmp = 0.08333333333333333 + ((0.25 / re) / re);
	} else {
		tmp = re * ((0.5 * Math.pow(im, 2.0)) + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 490.0:
		tmp = math.sin(re)
	elif im <= 1.35e+121:
		tmp = 0.08333333333333333 + ((0.25 / re) / re)
	else:
		tmp = re * ((0.5 * math.pow(im, 2.0)) + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 490.0)
		tmp = sin(re);
	elseif (im <= 1.35e+121)
		tmp = Float64(0.08333333333333333 + Float64(Float64(0.25 / re) / re));
	else
		tmp = Float64(re * Float64(Float64(0.5 * (im ^ 2.0)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 490.0)
		tmp = sin(re);
	elseif (im <= 1.35e+121)
		tmp = 0.08333333333333333 + ((0.25 / re) / re);
	else
		tmp = re * ((0.5 * (im ^ 2.0)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 490.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.35e+121], N[(0.08333333333333333 + N[(N[(0.25 / re), $MachinePrecision] / re), $MachinePrecision]), $MachinePrecision], N[(re * N[(N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 490

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 66.9%

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

   if 490 < im < 1.3500000000000001e121

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 28.7%

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

   7. Taylor expanded in re around 0 28.7%

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

      1. associate-*r/ 28.7%

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

      2. metadata-eval 28.7%

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

   9. Simplified 28.7%

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

       1. add-sqr-sqrt 28.5%

          \[\leadsto \color{blue}{\sqrt{\frac{0.25}{{re}^{2}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}}} \]

       2. sqrt-div 28.5%

          \[\leadsto \color{blue}{\frac{\sqrt{0.25}}{\sqrt{{re}^{2}}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

       3. metadata-eval 28.5%

          \[\leadsto \frac{\color{blue}{0.5}}{\sqrt{{re}^{2}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

       4. unpow2 28.5%

          \[\leadsto \frac{0.5}{\sqrt{\color{blue}{re \cdot re}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

       5. sqrt-prod 28.2%

          \[\leadsto \frac{0.5}{\color{blue}{\sqrt{re} \cdot \sqrt{re}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

       6. add-sqr-sqrt 45.8%

          \[\leadsto \frac{0.5}{\color{blue}{re}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

       7. sqrt-div 45.8%

          \[\leadsto \frac{0.5}{re} \cdot \color{blue}{\frac{\sqrt{0.25}}{\sqrt{{re}^{2}}}} \]

       8. metadata-eval 45.8%

          \[\leadsto \frac{0.5}{re} \cdot \frac{\color{blue}{0.5}}{\sqrt{{re}^{2}}} \]

       9. unpow2 45.8%

          \[\leadsto \frac{0.5}{re} \cdot \frac{0.5}{\sqrt{\color{blue}{re \cdot re}}} \]

       10. sqrt-prod 28.2%

           \[\leadsto \frac{0.5}{re} \cdot \frac{0.5}{\color{blue}{\sqrt{re} \cdot \sqrt{re}}} \]

       11. add-sqr-sqrt 28.5%

           \[\leadsto \frac{0.5}{re} \cdot \frac{0.5}{\color{blue}{re}} \]

   11. Applied egg-rr 28.7%

       \[\leadsto 0.08333333333333333 + \color{blue}{\frac{0.5}{re} \cdot \frac{0.5}{re}} \]
   12. Step-by-step derivation

       1. associate-*l/ 28.5%

          \[\leadsto \color{blue}{\frac{0.5 \cdot \frac{0.5}{re}}{re}} \]

       2. associate-*r/ 28.5%

          \[\leadsto \frac{\color{blue}{\frac{0.5 \cdot 0.5}{re}}}{re} \]

       3. metadata-eval 28.5%

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

   13. Applied egg-rr 28.7%

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

   if 1.3500000000000001e121 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 78.3%

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

   7. Simplified 78.3%

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

   8. Taylor expanded in re around 0 67.5%

      \[\leadsto \color{blue}{re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 490:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+121}:\\ \;\;\;\;0.08333333333333333 + \frac{\frac{0.25}{re}}{re}\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot {im}^{2} + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 61.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 490:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+121}:\\ \;\;\;\;0.08333333333333333 + \frac{\frac{0.25}{re}}{re}\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 490.0)
   (sin re)
   (if (<= im 1.35e+121)
     (+ 0.08333333333333333 (/ (/ 0.25 re) re))
     (* re (* 0.5 (pow im 2.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 490.0) {
		tmp = sin(re);
	} else if (im <= 1.35e+121) {
		tmp = 0.08333333333333333 + ((0.25 / re) / re);
	} else {
		tmp = re * (0.5 * pow(im, 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 490.0d0) then
        tmp = sin(re)
    else if (im <= 1.35d+121) then
        tmp = 0.08333333333333333d0 + ((0.25d0 / re) / re)
    else
        tmp = re * (0.5d0 * (im ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 490.0) {
		tmp = Math.sin(re);
	} else if (im <= 1.35e+121) {
		tmp = 0.08333333333333333 + ((0.25 / re) / re);
	} else {
		tmp = re * (0.5 * Math.pow(im, 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 490.0:
		tmp = math.sin(re)
	elif im <= 1.35e+121:
		tmp = 0.08333333333333333 + ((0.25 / re) / re)
	else:
		tmp = re * (0.5 * math.pow(im, 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 490.0)
		tmp = sin(re);
	elseif (im <= 1.35e+121)
		tmp = Float64(0.08333333333333333 + Float64(Float64(0.25 / re) / re));
	else
		tmp = Float64(re * Float64(0.5 * (im ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 490.0)
		tmp = sin(re);
	elseif (im <= 1.35e+121)
		tmp = 0.08333333333333333 + ((0.25 / re) / re);
	else
		tmp = re * (0.5 * (im ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 490.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.35e+121], N[(0.08333333333333333 + N[(N[(0.25 / re), $MachinePrecision] / re), $MachinePrecision]), $MachinePrecision], N[(re * N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 490

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 66.9%

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

   if 490 < im < 1.3500000000000001e121

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 28.7%

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

   7. Taylor expanded in re around 0 28.7%

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

      1. associate-*r/ 28.7%

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

      2. metadata-eval 28.7%

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

   9. Simplified 28.7%

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

       1. add-sqr-sqrt 28.5%

          \[\leadsto \color{blue}{\sqrt{\frac{0.25}{{re}^{2}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}}} \]

       2. sqrt-div 28.5%

          \[\leadsto \color{blue}{\frac{\sqrt{0.25}}{\sqrt{{re}^{2}}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

       3. metadata-eval 28.5%

          \[\leadsto \frac{\color{blue}{0.5}}{\sqrt{{re}^{2}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

       4. unpow2 28.5%

          \[\leadsto \frac{0.5}{\sqrt{\color{blue}{re \cdot re}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

       5. sqrt-prod 28.2%

          \[\leadsto \frac{0.5}{\color{blue}{\sqrt{re} \cdot \sqrt{re}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

       6. add-sqr-sqrt 45.8%

          \[\leadsto \frac{0.5}{\color{blue}{re}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

       7. sqrt-div 45.8%

          \[\leadsto \frac{0.5}{re} \cdot \color{blue}{\frac{\sqrt{0.25}}{\sqrt{{re}^{2}}}} \]

       8. metadata-eval 45.8%

          \[\leadsto \frac{0.5}{re} \cdot \frac{\color{blue}{0.5}}{\sqrt{{re}^{2}}} \]

       9. unpow2 45.8%

          \[\leadsto \frac{0.5}{re} \cdot \frac{0.5}{\sqrt{\color{blue}{re \cdot re}}} \]

       10. sqrt-prod 28.2%

           \[\leadsto \frac{0.5}{re} \cdot \frac{0.5}{\color{blue}{\sqrt{re} \cdot \sqrt{re}}} \]

       11. add-sqr-sqrt 28.5%

           \[\leadsto \frac{0.5}{re} \cdot \frac{0.5}{\color{blue}{re}} \]

   11. Applied egg-rr 28.7%

       \[\leadsto 0.08333333333333333 + \color{blue}{\frac{0.5}{re} \cdot \frac{0.5}{re}} \]
   12. Step-by-step derivation

       1. associate-*l/ 28.5%

          \[\leadsto \color{blue}{\frac{0.5 \cdot \frac{0.5}{re}}{re}} \]

       2. associate-*r/ 28.5%

          \[\leadsto \frac{\color{blue}{\frac{0.5 \cdot 0.5}{re}}}{re} \]

       3. metadata-eval 28.5%

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

   13. Applied egg-rr 28.7%

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

   if 1.3500000000000001e121 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in re around 0 82.1%

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

   7. Simplified 82.1%

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

   8. Taylor expanded in im around 0 67.5%

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

   9. Taylor expanded in im around inf 67.5%

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

       1. *-commutative 67.5%

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

       2. associate-*r* 67.5%

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

       3. *-commutative 67.5%

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

       4. associate-*r* 67.5%

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

   11. Simplified 67.5%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 490:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+121}:\\ \;\;\;\;0.08333333333333333 + \frac{\frac{0.25}{re}}{re}\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 770:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;0.08333333333333333 + \frac{\frac{0.25}{re}}{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 770.0) (sin re) (+ 0.08333333333333333 (/ (/ 0.25 re) re))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 770.0) {
		tmp = sin(re);
	} else {
		tmp = 0.08333333333333333 + ((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 <= 770.0d0) then
        tmp = sin(re)
    else
        tmp = 0.08333333333333333d0 + ((0.25d0 / re) / re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 770.0) {
		tmp = Math.sin(re);
	} else {
		tmp = 0.08333333333333333 + ((0.25 / re) / re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 770.0:
		tmp = math.sin(re)
	else:
		tmp = 0.08333333333333333 + ((0.25 / re) / re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 770.0)
		tmp = sin(re);
	else
		tmp = Float64(0.08333333333333333 + Float64(Float64(0.25 / re) / re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 770.0)
		tmp = sin(re);
	else
		tmp = 0.08333333333333333 + ((0.25 / re) / re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 770.0], N[Sin[re], $MachinePrecision], N[(0.08333333333333333 + N[(N[(0.25 / re), $MachinePrecision] / re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 770

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 66.9%

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

   if 770 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 22.7%

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

   7. Taylor expanded in re around 0 22.7%

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

      1. associate-*r/ 22.7%

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

      2. metadata-eval 22.7%

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

   9. Simplified 22.7%

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

       1. add-sqr-sqrt 22.5%

          \[\leadsto \color{blue}{\sqrt{\frac{0.25}{{re}^{2}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}}} \]

       2. sqrt-div 22.5%

          \[\leadsto \color{blue}{\frac{\sqrt{0.25}}{\sqrt{{re}^{2}}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

       3. metadata-eval 22.5%

          \[\leadsto \frac{\color{blue}{0.5}}{\sqrt{{re}^{2}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

       4. unpow2 22.5%

          \[\leadsto \frac{0.5}{\sqrt{\color{blue}{re \cdot re}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

       5. sqrt-prod 22.2%

          \[\leadsto \frac{0.5}{\color{blue}{\sqrt{re} \cdot \sqrt{re}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

       6. add-sqr-sqrt 38.8%

          \[\leadsto \frac{0.5}{\color{blue}{re}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

       7. sqrt-div 38.8%

          \[\leadsto \frac{0.5}{re} \cdot \color{blue}{\frac{\sqrt{0.25}}{\sqrt{{re}^{2}}}} \]

       8. metadata-eval 38.8%

          \[\leadsto \frac{0.5}{re} \cdot \frac{\color{blue}{0.5}}{\sqrt{{re}^{2}}} \]

       9. unpow2 38.8%

          \[\leadsto \frac{0.5}{re} \cdot \frac{0.5}{\sqrt{\color{blue}{re \cdot re}}} \]

       10. sqrt-prod 22.2%

           \[\leadsto \frac{0.5}{re} \cdot \frac{0.5}{\color{blue}{\sqrt{re} \cdot \sqrt{re}}} \]

       11. add-sqr-sqrt 22.5%

           \[\leadsto \frac{0.5}{re} \cdot \frac{0.5}{\color{blue}{re}} \]

   11. Applied egg-rr 22.7%

       \[\leadsto 0.08333333333333333 + \color{blue}{\frac{0.5}{re} \cdot \frac{0.5}{re}} \]
   12. Step-by-step derivation

       1. associate-*l/ 22.5%

          \[\leadsto \color{blue}{\frac{0.5 \cdot \frac{0.5}{re}}{re}} \]

       2. associate-*r/ 22.5%

          \[\leadsto \frac{\color{blue}{\frac{0.5 \cdot 0.5}{re}}}{re} \]

       3. metadata-eval 22.5%

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

   13. Applied egg-rr 22.7%

       \[\leadsto 0.08333333333333333 + \color{blue}{\frac{\frac{0.25}{re}}{re}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 770:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;0.08333333333333333 + \frac{\frac{0.25}{re}}{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 29.0% accurate, 25.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.29:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;0.08333333333333333 + \frac{\frac{0.25}{re}}{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.29) re (+ 0.08333333333333333 (/ (/ 0.25 re) re))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.29) {
		tmp = re;
	} else {
		tmp = 0.08333333333333333 + ((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 <= 0.29d0) then
        tmp = re
    else
        tmp = 0.08333333333333333d0 + ((0.25d0 / re) / re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.29) {
		tmp = re;
	} else {
		tmp = 0.08333333333333333 + ((0.25 / re) / re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.29:
		tmp = re
	else:
		tmp = 0.08333333333333333 + ((0.25 / re) / re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.29)
		tmp = re;
	else
		tmp = Float64(0.08333333333333333 + Float64(Float64(0.25 / re) / re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.29)
		tmp = re;
	else
		tmp = 0.08333333333333333 + ((0.25 / re) / re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.29], re, N[(0.08333333333333333 + N[(N[(0.25 / re), $MachinePrecision] / re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 0.28999999999999998

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in re around 0 64.3%

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

   7. Simplified 64.3%

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

   8. Taylor expanded in im around 0 38.2%

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

   if 0.28999999999999998 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 21.9%

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

   7. Taylor expanded in re around 0 21.9%

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

      1. associate-*r/ 21.9%

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

      2. metadata-eval 21.9%

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

   9. Simplified 21.9%

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

       1. add-sqr-sqrt 21.8%

          \[\leadsto \color{blue}{\sqrt{\frac{0.25}{{re}^{2}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}}} \]

       2. sqrt-div 21.8%

          \[\leadsto \color{blue}{\frac{\sqrt{0.25}}{\sqrt{{re}^{2}}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

       3. metadata-eval 21.8%

          \[\leadsto \frac{\color{blue}{0.5}}{\sqrt{{re}^{2}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

       4. unpow2 21.8%

          \[\leadsto \frac{0.5}{\sqrt{\color{blue}{re \cdot re}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

       5. sqrt-prod 21.5%

          \[\leadsto \frac{0.5}{\color{blue}{\sqrt{re} \cdot \sqrt{re}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

       6. add-sqr-sqrt 37.6%

          \[\leadsto \frac{0.5}{\color{blue}{re}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

       7. sqrt-div 37.6%

          \[\leadsto \frac{0.5}{re} \cdot \color{blue}{\frac{\sqrt{0.25}}{\sqrt{{re}^{2}}}} \]

       8. metadata-eval 37.6%

          \[\leadsto \frac{0.5}{re} \cdot \frac{\color{blue}{0.5}}{\sqrt{{re}^{2}}} \]

       9. unpow2 37.6%

          \[\leadsto \frac{0.5}{re} \cdot \frac{0.5}{\sqrt{\color{blue}{re \cdot re}}} \]

       10. sqrt-prod 21.5%

           \[\leadsto \frac{0.5}{re} \cdot \frac{0.5}{\color{blue}{\sqrt{re} \cdot \sqrt{re}}} \]

       11. add-sqr-sqrt 21.8%

           \[\leadsto \frac{0.5}{re} \cdot \frac{0.5}{\color{blue}{re}} \]

   11. Applied egg-rr 21.9%

       \[\leadsto 0.08333333333333333 + \color{blue}{\frac{0.5}{re} \cdot \frac{0.5}{re}} \]
   12. Step-by-step derivation

       1. associate-*l/ 21.8%

          \[\leadsto \color{blue}{\frac{0.5 \cdot \frac{0.5}{re}}{re}} \]

       2. associate-*r/ 21.8%

          \[\leadsto \frac{\color{blue}{\frac{0.5 \cdot 0.5}{re}}}{re} \]

       3. metadata-eval 21.8%

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

   13. Applied egg-rr 21.9%

       \[\leadsto 0.08333333333333333 + \color{blue}{\frac{\frac{0.25}{re}}{re}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.29:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;0.08333333333333333 + \frac{\frac{0.25}{re}}{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 29.0% accurate, 30.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1080:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25}{re}}{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 1080.0) re (/ (/ 0.25 re) re)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1080.0) {
		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 <= 1080.0d0) 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 <= 1080.0) {
		tmp = re;
	} else {
		tmp = (0.25 / re) / re;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1080.0:
		tmp = re
	else:
		tmp = (0.25 / re) / re
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1080.0)
		tmp = re;
	else
		tmp = Float64(Float64(0.25 / re) / re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1080.0)
		tmp = re;
	else
		tmp = (0.25 / re) / re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1080.0], re, N[(N[(0.25 / re), $MachinePrecision] / re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1080

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in re around 0 63.7%

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

   7. Simplified 63.7%

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

   8. Taylor expanded in im around 0 37.9%

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

   if 1080 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 22.7%

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

   7. Taylor expanded in re around 0 22.5%

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

      1. add-sqr-sqrt 22.5%

         \[\leadsto \color{blue}{\sqrt{\frac{0.25}{{re}^{2}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}}} \]

      2. sqrt-div 22.5%

         \[\leadsto \color{blue}{\frac{\sqrt{0.25}}{\sqrt{{re}^{2}}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

      3. metadata-eval 22.5%

         \[\leadsto \frac{\color{blue}{0.5}}{\sqrt{{re}^{2}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

      4. unpow2 22.5%

         \[\leadsto \frac{0.5}{\sqrt{\color{blue}{re \cdot re}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

      5. sqrt-prod 22.2%

         \[\leadsto \frac{0.5}{\color{blue}{\sqrt{re} \cdot \sqrt{re}}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

      6. add-sqr-sqrt 38.8%

         \[\leadsto \frac{0.5}{\color{blue}{re}} \cdot \sqrt{\frac{0.25}{{re}^{2}}} \]

      7. sqrt-div 38.8%

         \[\leadsto \frac{0.5}{re} \cdot \color{blue}{\frac{\sqrt{0.25}}{\sqrt{{re}^{2}}}} \]

      8. metadata-eval 38.8%

         \[\leadsto \frac{0.5}{re} \cdot \frac{\color{blue}{0.5}}{\sqrt{{re}^{2}}} \]

      9. unpow2 38.8%

         \[\leadsto \frac{0.5}{re} \cdot \frac{0.5}{\sqrt{\color{blue}{re \cdot re}}} \]

      10. sqrt-prod 22.2%

          \[\leadsto \frac{0.5}{re} \cdot \frac{0.5}{\color{blue}{\sqrt{re} \cdot \sqrt{re}}} \]

      11. add-sqr-sqrt 22.5%

          \[\leadsto \frac{0.5}{re} \cdot \frac{0.5}{\color{blue}{re}} \]

   9. Applied egg-rr 22.5%

      \[\leadsto \color{blue}{\frac{0.5}{re} \cdot \frac{0.5}{re}} \]
   10. Step-by-step derivation

       1. associate-*l/ 22.5%

          \[\leadsto \color{blue}{\frac{0.5 \cdot \frac{0.5}{re}}{re}} \]

       2. associate-*r/ 22.5%

          \[\leadsto \frac{\color{blue}{\frac{0.5 \cdot 0.5}{re}}}{re} \]

       3. metadata-eval 22.5%

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

   11. Applied egg-rr 22.5%

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

4. Final simplification 34.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1080:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25}{re}}{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 4.1% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ -4 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 -4.0)

C

double code(double re, double im) {
	return -4.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = -4.0d0
end function

Java

public static double code(double re, double im) {
	return -4.0;
}

Python

def code(re, im):
	return -4.0

Julia

function code(re, im)
	return -4.0
end

MATLAB

function tmp = code(re, im)
	tmp = -4.0;
end

Wolfram

code[re_, im_] := -4.0

Derivation

1. Initial program 100.0%

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

   1. distribute-rgt-in 100.0%

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

   2. cancel-sign-sub 100.0%

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

   3. distribute-rgt-out-- 100.0%

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

   4. sub-neg 100.0%

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

   5. neg-sub0 100.0%

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

   6. remove-double-neg 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in im around 0 77.9%

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

7. Simplified 77.9%

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

9. Applied egg-rr 4.2%

   \[\leadsto \color{blue}{-4} \]

10. Final simplification 4.2%

    \[\leadsto -4 \]
11. Add Preprocessing

Alternative 11: 4.7% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 -1.0)

C

double code(double re, double im) {
	return -1.0;
}

Fortran

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

Java

public static double code(double re, double im) {
	return -1.0;
}

Python

def code(re, im):
	return -1.0

Julia

function code(re, im)
	return -1.0
end

MATLAB

function tmp = code(re, im)
	tmp = -1.0;
end

Wolfram

code[re_, im_] := -1.0

Derivation

1. Initial program 100.0%

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

   1. distribute-rgt-in 100.0%

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

   2. cancel-sign-sub 100.0%

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

   3. distribute-rgt-out-- 100.0%

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

   4. sub-neg 100.0%

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

   5. neg-sub0 100.0%

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

   6. remove-double-neg 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in im around 0 77.9%

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

7. Simplified 77.9%

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

9. Applied egg-rr 4.9%

   \[\leadsto \color{blue}{-1} \]

10. Final simplification 4.9%

    \[\leadsto -1 \]
11. Add Preprocessing

Alternative 12: 4.2% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ 0.0625 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.0625)

C

double code(double re, double im) {
	return 0.0625;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.0625d0
end function

Java

public static double code(double re, double im) {
	return 0.0625;
}

Python

def code(re, im):
	return 0.0625

Julia

function code(re, im)
	return 0.0625
end

MATLAB

function tmp = code(re, im)
	tmp = 0.0625;
end

Wolfram

code[re_, im_] := 0.0625

Derivation

1. Initial program 100.0%

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

   1. distribute-rgt-in 100.0%

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

   2. cancel-sign-sub 100.0%

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

   3. distribute-rgt-out-- 100.0%

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

   4. sub-neg 100.0%

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

   5. neg-sub0 100.0%

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

   6. remove-double-neg 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in im around 0 77.9%

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

7. Simplified 77.9%

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

9. Applied egg-rr 4.2%

   \[\leadsto \color{blue}{0.0625} \]

10. Final simplification 4.2%

    \[\leadsto 0.0625 \]
11. Add Preprocessing

Alternative 13: 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. distribute-rgt-in 100.0%

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

   2. cancel-sign-sub 100.0%

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

   3. distribute-rgt-out-- 100.0%

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

   4. sub-neg 100.0%

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

   5. neg-sub0 100.0%

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

   6. remove-double-neg 100.0%

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

3. Simplified 100.0%

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

6. Applied egg-rr 12.1%

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

7. Taylor expanded in re around 0 12.1%

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

   1. associate-*r/ 12.1%

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

   2. metadata-eval 12.1%

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

9. Simplified 12.1%

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

10. Taylor expanded in re around inf 4.2%

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

11. Final simplification 4.2%

    \[\leadsto 0.08333333333333333 \]
12. Add Preprocessing

Alternative 14: 4.4% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ 0.25 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.25)

C

double code(double re, double im) {
	return 0.25;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.25d0
end function

Java

public static double code(double re, double im) {
	return 0.25;
}

Python

def code(re, im):
	return 0.25

Julia

function code(re, im)
	return 0.25
end

MATLAB

function tmp = code(re, im)
	tmp = 0.25;
end

Wolfram

code[re_, im_] := 0.25

Derivation

1. Initial program 100.0%

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

   1. distribute-rgt-in 100.0%

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

   2. cancel-sign-sub 100.0%

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

   3. distribute-rgt-out-- 100.0%

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

   4. sub-neg 100.0%

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

   5. neg-sub0 100.0%

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

   6. remove-double-neg 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in im around 0 77.9%

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

7. Simplified 77.9%

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

9. Applied egg-rr 4.3%

   \[\leadsto \color{blue}{0.25} \]

10. Final simplification 4.3%

    \[\leadsto 0.25 \]
11. Add Preprocessing

Alternative 15: 4.6% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.5)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return 0.5

Julia

function code(re, im)
	return 0.5
end

MATLAB

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

Wolfram

code[re_, im_] := 0.5

Derivation

1. Initial program 100.0%

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

   1. distribute-rgt-in 100.0%

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

   2. cancel-sign-sub 100.0%

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

   3. distribute-rgt-out-- 100.0%

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

   4. sub-neg 100.0%

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

   5. neg-sub0 100.0%

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

   6. remove-double-neg 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in im around 0 77.9%

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

7. Simplified 77.9%

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

9. Applied egg-rr 4.4%

   \[\leadsto \color{blue}{0.5} \]

10. Final simplification 4.4%

    \[\leadsto 0.5 \]
11. Add Preprocessing

Alternative 16: 4.8% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return 1.0

Julia

function code(re, im)
	return 1.0
end

MATLAB

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

Wolfram

code[re_, im_] := 1.0

Derivation

1. Initial program 100.0%

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

   1. distribute-rgt-in 100.0%

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

   2. cancel-sign-sub 100.0%

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

   3. distribute-rgt-out-- 100.0%

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

   4. sub-neg 100.0%

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

   5. neg-sub0 100.0%

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

   6. remove-double-neg 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in im around 0 77.9%

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

7. Simplified 77.9%

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

9. Applied egg-rr 4.5%

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

10. Final simplification 4.5%

    \[\leadsto 1 \]
11. Add Preprocessing

Alternative 17: 26.3% 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. distribute-rgt-in 100.0%

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

   2. cancel-sign-sub 100.0%

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

   3. distribute-rgt-out-- 100.0%

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

   4. sub-neg 100.0%

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

   5. neg-sub0 100.0%

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

   6. remove-double-neg 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in re around 0 68.6%

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

7. Simplified 68.6%

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

8. Taylor expanded in im around 0 30.1%

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

9. Final simplification 30.1%

   \[\leadsto re \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024040 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))