math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 8.0s

Alternatives: 12

Speedup: 1.0×

• 48.8% 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 99.6%

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

   1. distribute-rgt-in 99.6%

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

   2. cancel-sign-sub 99.6%

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

   3. distribute-rgt-out-- 99.6%

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

   4. sub-neg 99.6%

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

   5. neg-sub0 99.6%

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

   6. remove-double-neg 99.6%

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

3. Simplified 99.6%

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

5. Final simplification 99.6%

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

Alternative 2: 84.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ \mathbf{if}\;im \leq 1.4 \cdot 10^{-5}:\\ \;\;\;\;t_0 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot {im}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))))
   (if (<= im 1.4e-5)
     (* t_0 (fma im im 2.0))
     (if (<= im 1.35e+154)
       (* (* 0.5 re) (+ (exp (- im)) (exp im)))
       (* t_0 (pow im 2.0))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sin(re);
	double tmp;
	if (im <= 1.4e-5) {
		tmp = t_0 * fma(im, im, 2.0);
	} else if (im <= 1.35e+154) {
		tmp = (0.5 * re) * (exp(-im) + exp(im));
	} else {
		tmp = t_0 * pow(im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * sin(re))
	tmp = 0.0
	if (im <= 1.4e-5)
		tmp = Float64(t_0 * fma(im, im, 2.0));
	elseif (im <= 1.35e+154)
		tmp = Float64(Float64(0.5 * re) * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(t_0 * (im ^ 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 1.4e-5], N[(t$95$0 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(N[(0.5 * re), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.39999999999999998e-5

   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 79.7%

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

   7. Simplified 79.7%

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

   if 1.39999999999999998e-5 < im < 1.35000000000000003e154

   1. Initial program 97.4%

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

      1. distribute-rgt-in 97.4%

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

      2. cancel-sign-sub 97.4%

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

      3. distribute-rgt-out-- 97.4%

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

      4. sub-neg 97.4%

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

      5. neg-sub0 97.4%

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

      6. remove-double-neg 97.4%

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

   3. Simplified 97.4%

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

   5. Taylor expanded in re around 0 62.2%

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

   7. Simplified 62.2%

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

   if 1.35000000000000003e154 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

   7. Simplified 100.0%

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

   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%

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

      3. associate-*r* 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.4 \cdot 10^{-5}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot {im}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.4 \cdot 10^{-5}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;im \leq 1.8 \cdot 10^{+51}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.001388888888888889 \cdot \left(\sin re \cdot {im}^{6}\right) + 12\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.4e-5)
   (* (* 0.5 (sin re)) (fma im im 2.0))
   (if (<= im 1.8e+51)
     (* (* 0.5 re) (+ (exp (- im)) (exp im)))
     (+ (* 0.001388888888888889 (* (sin re) (pow im 6.0))) 12.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.4e-5) {
		tmp = (0.5 * sin(re)) * fma(im, im, 2.0);
	} else if (im <= 1.8e+51) {
		tmp = (0.5 * re) * (exp(-im) + exp(im));
	} else {
		tmp = (0.001388888888888889 * (sin(re) * pow(im, 6.0))) + 12.0;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.4e-5)
		tmp = Float64(Float64(0.5 * sin(re)) * fma(im, im, 2.0));
	elseif (im <= 1.8e+51)
		tmp = Float64(Float64(0.5 * re) * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(Float64(0.001388888888888889 * Float64(sin(re) * (im ^ 6.0))) + 12.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.4e-5], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.8e+51], N[(N[(0.5 * re), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.001388888888888889 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 12.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.39999999999999998e-5

   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 79.7%

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

   7. Simplified 79.7%

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

   if 1.39999999999999998e-5 < im < 1.80000000000000005e51

   1. Initial program 91.1%

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

      1. distribute-rgt-in 91.3%

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

      2. cancel-sign-sub 91.3%

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

      3. distribute-rgt-out-- 91.1%

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

      4. sub-neg 91.1%

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

      5. neg-sub0 91.1%

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

      6. remove-double-neg 91.1%

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

   3. Simplified 91.1%

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

   5. Taylor expanded in re around 0 45.7%

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

   7. Simplified 45.7%

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

   if 1.80000000000000005e51 < 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 98.3%

      \[\leadsto \color{blue}{\sin re + \left(0.001388888888888889 \cdot \left({im}^{6} \cdot \sin re\right) + \left(0.041666666666666664 \cdot \left({im}^{4} \cdot \sin re\right) + 0.5 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]

   7. Simplified 98.3%

      \[\leadsto \color{blue}{\sin re \cdot \left(0.001388888888888889 \cdot {im}^{6} + 0.041666666666666664 \cdot {im}^{4}\right) + \left(0.5 \cdot {im}^{2} + 1\right) \cdot \sin re} \]

   9. Applied egg-rr 98.3%

      \[\leadsto \sin re \cdot \left(0.001388888888888889 \cdot {im}^{6} + 0.041666666666666664 \cdot {im}^{4}\right) + \color{blue}{12} \]

   10. Taylor expanded in im around inf 98.3%

       \[\leadsto \color{blue}{0.001388888888888889 \cdot \left({im}^{6} \cdot \sin re\right)} + 12 \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.4 \cdot 10^{-5}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;im \leq 1.8 \cdot 10^{+51}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.001388888888888889 \cdot \left(\sin re \cdot {im}^{6}\right) + 12\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 700:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{{\left(\frac{0.5}{re}\right)}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot {im}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 700.0)
   (sin re)
   (if (<= im 1.35e+154)
     (sqrt (pow (/ 0.5 re) 4.0))
     (* (* 0.5 (sin re)) (pow im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 700.0) {
		tmp = sin(re);
	} else if (im <= 1.35e+154) {
		tmp = sqrt(pow((0.5 / re), 4.0));
	} else {
		tmp = (0.5 * sin(re)) * 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 <= 700.0d0) then
        tmp = sin(re)
    else if (im <= 1.35d+154) then
        tmp = sqrt(((0.5d0 / re) ** 4.0d0))
    else
        tmp = (0.5d0 * sin(re)) * (im ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 700.0) {
		tmp = Math.sin(re);
	} else if (im <= 1.35e+154) {
		tmp = Math.sqrt(Math.pow((0.5 / re), 4.0));
	} else {
		tmp = (0.5 * Math.sin(re)) * Math.pow(im, 2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 700.0:
		tmp = math.sin(re)
	elif im <= 1.35e+154:
		tmp = math.sqrt(math.pow((0.5 / re), 4.0))
	else:
		tmp = (0.5 * math.sin(re)) * math.pow(im, 2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 700.0)
		tmp = sin(re);
	elseif (im <= 1.35e+154)
		tmp = sqrt((Float64(0.5 / re) ^ 4.0));
	else
		tmp = Float64(Float64(0.5 * sin(re)) * (im ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 700.0)
		tmp = sin(re);
	elseif (im <= 1.35e+154)
		tmp = sqrt(((0.5 / re) ^ 4.0));
	else
		tmp = (0.5 * sin(re)) * (im ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 700.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.35e+154], N[Sqrt[N[Power[N[(0.5 / re), $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 700

   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 62.8%

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

   if 700 < im < 1.35000000000000003e154

   1. Initial program 97.3%

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

      1. distribute-rgt-in 97.3%

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

      2. cancel-sign-sub 97.3%

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

      3. distribute-rgt-out-- 97.3%

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

      4. sub-neg 97.3%

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

      5. neg-sub0 97.3%

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

      6. remove-double-neg 97.3%

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

   3. Simplified 97.3%

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

   6. Applied egg-rr 16.3%

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

   7. Taylor expanded in re around 0 16.1%

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

      1. add-sqr-sqrt 16.1%

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

      2. sqrt-div 16.1%

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

      3. metadata-eval 16.1%

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

      4. unpow2 16.1%

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

      5. sqrt-prod 15.7%

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

      6. add-sqr-sqrt 30.2%

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

      7. sqrt-div 30.2%

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

      8. metadata-eval 30.2%

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

      9. unpow2 30.2%

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

      10. sqrt-prod 15.7%

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

      11. add-sqr-sqrt 16.1%

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

   9. Applied egg-rr 16.1%

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

       1. add-sqr-sqrt 16.1%

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

       2. sqrt-unprod 28.7%

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

       3. pow2 28.7%

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

       4. pow2 28.7%

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

       5. pow-prod-up 28.7%

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

       6. metadata-eval 28.7%

          \[\leadsto \sqrt{{\left(\frac{0.5}{re}\right)}^{\color{blue}{4}}} \]

   11. Applied egg-rr 28.7%

       \[\leadsto \color{blue}{\sqrt{{\left(\frac{0.5}{re}\right)}^{4}}} \]

   if 1.35000000000000003e154 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

   7. Simplified 100.0%

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

   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%

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

      3. associate-*r* 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 700:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{{\left(\frac{0.5}{re}\right)}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot {im}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ \mathbf{if}\;im \leq 850:\\ \;\;\;\;t_0 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{{\left(\frac{0.5}{re}\right)}^{4}}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot {im}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))))
   (if (<= im 850.0)
     (* t_0 (fma im im 2.0))
     (if (<= im 1.35e+154) (sqrt (pow (/ 0.5 re) 4.0)) (* t_0 (pow im 2.0))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sin(re);
	double tmp;
	if (im <= 850.0) {
		tmp = t_0 * fma(im, im, 2.0);
	} else if (im <= 1.35e+154) {
		tmp = sqrt(pow((0.5 / re), 4.0));
	} else {
		tmp = t_0 * pow(im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * sin(re))
	tmp = 0.0
	if (im <= 850.0)
		tmp = Float64(t_0 * fma(im, im, 2.0));
	elseif (im <= 1.35e+154)
		tmp = sqrt((Float64(0.5 / re) ^ 4.0));
	else
		tmp = Float64(t_0 * (im ^ 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 850.0], N[(t$95$0 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.35e+154], N[Sqrt[N[Power[N[(0.5 / re), $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision], N[(t$95$0 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < 850

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 79.6%

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

   7. Simplified 79.6%

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

   if 850 < im < 1.35000000000000003e154

   1. Initial program 97.3%

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

      1. distribute-rgt-in 97.3%

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

      2. cancel-sign-sub 97.3%

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

      3. distribute-rgt-out-- 97.3%

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

      4. sub-neg 97.3%

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

      5. neg-sub0 97.3%

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

      6. remove-double-neg 97.3%

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

   3. Simplified 97.3%

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

   6. Applied egg-rr 16.3%

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

   7. Taylor expanded in re around 0 16.1%

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

      1. add-sqr-sqrt 16.1%

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

      2. sqrt-div 16.1%

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

      3. metadata-eval 16.1%

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

      4. unpow2 16.1%

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

      5. sqrt-prod 15.7%

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

      6. add-sqr-sqrt 30.2%

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

      7. sqrt-div 30.2%

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

      8. metadata-eval 30.2%

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

      9. unpow2 30.2%

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

      10. sqrt-prod 15.7%

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

      11. add-sqr-sqrt 16.1%

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

   9. Applied egg-rr 16.1%

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

       1. add-sqr-sqrt 16.1%

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

       2. sqrt-unprod 28.7%

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

       3. pow2 28.7%

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

       4. pow2 28.7%

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

       5. pow-prod-up 28.7%

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

       6. metadata-eval 28.7%

          \[\leadsto \sqrt{{\left(\frac{0.5}{re}\right)}^{\color{blue}{4}}} \]

   11. Applied egg-rr 28.7%

       \[\leadsto \color{blue}{\sqrt{{\left(\frac{0.5}{re}\right)}^{4}}} \]

   if 1.35000000000000003e154 < im

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

   7. Simplified 100.0%

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

   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%

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

      3. associate-*r* 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 850:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{{\left(\frac{0.5}{re}\right)}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot {im}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 760:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.95 \cdot 10^{+164}:\\ \;\;\;\;\sqrt{\frac{0.0625}{{re}^{4}}}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot {im}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 760.0)
   (sin re)
   (if (<= im 1.95e+164)
     (sqrt (/ 0.0625 (pow re 4.0)))
     (* (* 0.5 re) (pow im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 760.0) {
		tmp = sin(re);
	} else if (im <= 1.95e+164) {
		tmp = sqrt((0.0625 / pow(re, 4.0)));
	} else {
		tmp = (0.5 * re) * 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 <= 760.0d0) then
        tmp = sin(re)
    else if (im <= 1.95d+164) then
        tmp = sqrt((0.0625d0 / (re ** 4.0d0)))
    else
        tmp = (0.5d0 * re) * (im ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 760.0) {
		tmp = Math.sin(re);
	} else if (im <= 1.95e+164) {
		tmp = Math.sqrt((0.0625 / Math.pow(re, 4.0)));
	} else {
		tmp = (0.5 * re) * Math.pow(im, 2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 760.0:
		tmp = math.sin(re)
	elif im <= 1.95e+164:
		tmp = math.sqrt((0.0625 / math.pow(re, 4.0)))
	else:
		tmp = (0.5 * re) * math.pow(im, 2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 760.0)
		tmp = sin(re);
	elseif (im <= 1.95e+164)
		tmp = sqrt(Float64(0.0625 / (re ^ 4.0)));
	else
		tmp = Float64(Float64(0.5 * re) * (im ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 760.0)
		tmp = sin(re);
	elseif (im <= 1.95e+164)
		tmp = sqrt((0.0625 / (re ^ 4.0)));
	else
		tmp = (0.5 * re) * (im ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 760.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.95e+164], N[Sqrt[N[(0.0625 / N[Power[re, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 760

   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 62.8%

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

   if 760 < im < 1.94999999999999993e164

   1. Initial program 97.4%

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

      1. distribute-rgt-in 97.4%

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

      2. cancel-sign-sub 97.4%

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

      3. distribute-rgt-out-- 97.4%

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

      4. sub-neg 97.4%

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

      5. neg-sub0 97.4%

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

      6. remove-double-neg 97.4%

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

   3. Simplified 97.4%

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

   6. Applied egg-rr 15.9%

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

   7. Taylor expanded in re around 0 15.7%

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

      1. add-sqr-sqrt 15.7%

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

      2. sqrt-unprod 28.0%

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

      3. frac-times 28.0%

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

      4. metadata-eval 28.0%

         \[\leadsto \sqrt{\frac{\color{blue}{0.0625}}{{re}^{2} \cdot {re}^{2}}} \]

      5. pow-prod-up 28.0%

         \[\leadsto \sqrt{\frac{0.0625}{\color{blue}{{re}^{\left(2 + 2\right)}}}} \]

      6. metadata-eval 28.0%

         \[\leadsto \sqrt{\frac{0.0625}{{re}^{\color{blue}{4}}}} \]

   9. Applied egg-rr 28.0%

      \[\leadsto \color{blue}{\sqrt{\frac{0.0625}{{re}^{4}}}} \]

   if 1.94999999999999993e164 < 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 79.2%

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

   7. Simplified 79.2%

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

   8. Taylor expanded in im around 0 79.2%

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

      1. +-commutative 79.2%

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

      2. unpow2 79.2%

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

      3. fma-def 79.2%

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

   10. Simplified 79.2%

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

   11. Taylor expanded in im around inf 79.2%

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

       1. associate-*r* 79.2%

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

       2. *-commutative 79.2%

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

       3. associate-*r* 79.2%

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

       4. *-commutative 79.2%

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

       5. *-commutative 79.2%

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

   13. Simplified 79.2%

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

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 760:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.95 \cdot 10^{+164}:\\ \;\;\;\;\sqrt{\frac{0.0625}{{re}^{4}}}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot {im}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 760:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.95 \cdot 10^{+164}:\\ \;\;\;\;\sqrt{{\left(\frac{0.5}{re}\right)}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot {im}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 760.0)
   (sin re)
   (if (<= im 1.95e+164)
     (sqrt (pow (/ 0.5 re) 4.0))
     (* (* 0.5 re) (pow im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 760.0) {
		tmp = sin(re);
	} else if (im <= 1.95e+164) {
		tmp = sqrt(pow((0.5 / re), 4.0));
	} else {
		tmp = (0.5 * re) * 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 <= 760.0d0) then
        tmp = sin(re)
    else if (im <= 1.95d+164) then
        tmp = sqrt(((0.5d0 / re) ** 4.0d0))
    else
        tmp = (0.5d0 * re) * (im ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 760.0) {
		tmp = Math.sin(re);
	} else if (im <= 1.95e+164) {
		tmp = Math.sqrt(Math.pow((0.5 / re), 4.0));
	} else {
		tmp = (0.5 * re) * Math.pow(im, 2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 760.0:
		tmp = math.sin(re)
	elif im <= 1.95e+164:
		tmp = math.sqrt(math.pow((0.5 / re), 4.0))
	else:
		tmp = (0.5 * re) * math.pow(im, 2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 760.0)
		tmp = sin(re);
	elseif (im <= 1.95e+164)
		tmp = sqrt((Float64(0.5 / re) ^ 4.0));
	else
		tmp = Float64(Float64(0.5 * re) * (im ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 760.0)
		tmp = sin(re);
	elseif (im <= 1.95e+164)
		tmp = sqrt(((0.5 / re) ^ 4.0));
	else
		tmp = (0.5 * re) * (im ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 760.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.95e+164], N[Sqrt[N[Power[N[(0.5 / re), $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 760

   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 62.8%

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

   if 760 < im < 1.94999999999999993e164

   1. Initial program 97.4%

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

      1. distribute-rgt-in 97.4%

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

      2. cancel-sign-sub 97.4%

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

      3. distribute-rgt-out-- 97.4%

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

      4. sub-neg 97.4%

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

      5. neg-sub0 97.4%

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

      6. remove-double-neg 97.4%

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

   3. Simplified 97.4%

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

   6. Applied egg-rr 15.9%

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

   7. Taylor expanded in re around 0 15.7%

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

      1. add-sqr-sqrt 15.7%

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

      2. sqrt-div 15.7%

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

      3. metadata-eval 15.7%

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

      4. unpow2 15.7%

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

      5. sqrt-prod 15.3%

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

      6. add-sqr-sqrt 29.4%

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

      7. sqrt-div 29.4%

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

      8. metadata-eval 29.4%

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

      9. unpow2 29.4%

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

      10. sqrt-prod 15.3%

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

      11. add-sqr-sqrt 15.7%

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

   9. Applied egg-rr 15.7%

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

       1. add-sqr-sqrt 15.7%

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

       2. sqrt-unprod 28.0%

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

       3. pow2 28.0%

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

       4. pow2 28.0%

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

       5. pow-prod-up 28.0%

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

       6. metadata-eval 28.0%

          \[\leadsto \sqrt{{\left(\frac{0.5}{re}\right)}^{\color{blue}{4}}} \]

   11. Applied egg-rr 28.0%

       \[\leadsto \color{blue}{\sqrt{{\left(\frac{0.5}{re}\right)}^{4}}} \]

   if 1.94999999999999993e164 < 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 79.2%

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

   7. Simplified 79.2%

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

   8. Taylor expanded in im around 0 79.2%

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

      1. +-commutative 79.2%

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

      2. unpow2 79.2%

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

      3. fma-def 79.2%

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

   10. Simplified 79.2%

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

   11. Taylor expanded in im around inf 79.2%

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

       1. associate-*r* 79.2%

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

       2. *-commutative 79.2%

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

       3. associate-*r* 79.2%

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

       4. *-commutative 79.2%

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

       5. *-commutative 79.2%

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

   13. Simplified 79.2%

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

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 760:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.95 \cdot 10^{+164}:\\ \;\;\;\;\sqrt{{\left(\frac{0.5}{re}\right)}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot {im}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 750:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 6.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{\frac{0.25}{re}}{re}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot {im}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 750.0)
   (sin re)
   (if (<= im 6.5e+153) (/ (/ 0.25 re) re) (* (* 0.5 re) (pow im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 750.0) {
		tmp = sin(re);
	} else if (im <= 6.5e+153) {
		tmp = (0.25 / re) / re;
	} else {
		tmp = (0.5 * re) * 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 <= 750.0d0) then
        tmp = sin(re)
    else if (im <= 6.5d+153) then
        tmp = (0.25d0 / re) / re
    else
        tmp = (0.5d0 * re) * (im ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 750.0) {
		tmp = Math.sin(re);
	} else if (im <= 6.5e+153) {
		tmp = (0.25 / re) / re;
	} else {
		tmp = (0.5 * re) * Math.pow(im, 2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 750.0:
		tmp = math.sin(re)
	elif im <= 6.5e+153:
		tmp = (0.25 / re) / re
	else:
		tmp = (0.5 * re) * math.pow(im, 2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 750.0)
		tmp = sin(re);
	elseif (im <= 6.5e+153)
		tmp = Float64(Float64(0.25 / re) / re);
	else
		tmp = Float64(Float64(0.5 * re) * (im ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 750.0)
		tmp = sin(re);
	elseif (im <= 6.5e+153)
		tmp = (0.25 / re) / re;
	else
		tmp = (0.5 * re) * (im ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 750.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 6.5e+153], N[(N[(0.25 / re), $MachinePrecision] / re), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[Power[im, 2.0], $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 62.8%

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

   if 750 < im < 6.49999999999999972e153

   1. Initial program 97.3%

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

      1. distribute-rgt-in 97.3%

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

      2. cancel-sign-sub 97.3%

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

      3. distribute-rgt-out-- 97.3%

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

      4. sub-neg 97.3%

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

      5. neg-sub0 97.3%

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

      6. remove-double-neg 97.3%

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

   3. Simplified 97.3%

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

   6. Applied egg-rr 16.3%

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

   7. Taylor expanded in re around 0 16.1%

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

      1. add-sqr-sqrt 16.1%

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

      2. sqrt-div 16.1%

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

      3. metadata-eval 16.1%

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

      4. unpow2 16.1%

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

      5. sqrt-prod 15.7%

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

      6. add-sqr-sqrt 30.2%

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

      7. sqrt-div 30.2%

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

      8. metadata-eval 30.2%

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

      9. unpow2 30.2%

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

      10. sqrt-prod 15.7%

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

      11. add-sqr-sqrt 16.1%

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

   9. Applied egg-rr 16.1%

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

       1. associate-*l/ 16.1%

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

       2. associate-*r/ 16.1%

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

       3. metadata-eval 16.1%

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

   11. Applied egg-rr 16.1%

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

   if 6.49999999999999972e153 < 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 76.0%

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

   7. Simplified 76.0%

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

   8. Taylor expanded in im around 0 76.0%

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

      1. +-commutative 76.0%

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

      2. unpow2 76.0%

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

      3. fma-def 76.0%

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

   10. Simplified 76.0%

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

   11. Taylor expanded in im around inf 76.0%

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

       1. associate-*r* 76.0%

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

       2. *-commutative 76.0%

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

       3. associate-*r* 76.0%

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

       4. *-commutative 76.0%

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

       5. *-commutative 76.0%

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

   13. Simplified 76.0%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 750:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 6.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{\frac{0.25}{re}}{re}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot {im}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 54.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 680:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot {re}^{-2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 680.0) (sin re) (* 0.25 (pow re -2.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 680.0) {
		tmp = sin(re);
	} else {
		tmp = 0.25 * pow(re, -2.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 680.0d0) then
        tmp = sin(re)
    else
        tmp = 0.25d0 * (re ** (-2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 680.0) {
		tmp = Math.sin(re);
	} else {
		tmp = 0.25 * Math.pow(re, -2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 680.0:
		tmp = math.sin(re)
	else:
		tmp = 0.25 * math.pow(re, -2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 680.0)
		tmp = sin(re);
	else
		tmp = Float64(0.25 * (re ^ -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 680.0)
		tmp = sin(re);
	else
		tmp = 0.25 * (re ^ -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 680.0], N[Sin[re], $MachinePrecision], N[(0.25 * N[Power[re, -2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 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 62.8%

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

   if 680 < im

   1. Initial program 98.4%

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

      1. distribute-rgt-in 98.4%

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

      2. cancel-sign-sub 98.4%

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

      3. distribute-rgt-out-- 98.4%

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

      4. sub-neg 98.4%

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

      5. neg-sub0 98.4%

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

      6. remove-double-neg 98.4%

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

   3. Simplified 98.4%

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

   6. Applied egg-rr 14.0%

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

   7. Taylor expanded in re around 0 13.9%

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

      1. clear-num 13.9%

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

      2. associate-/r/ 15.2%

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

      3. pow-flip 15.2%

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

      4. metadata-eval 15.2%

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

   9. Applied egg-rr 15.2%

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

4. Final simplification 51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 680:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot {re}^{-2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 54.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 550:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25}{re}}{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 550.0) (sin re) (/ (/ 0.25 re) re)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 550.0) {
		tmp = sin(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 <= 550.0d0) then
        tmp = sin(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 <= 550.0) {
		tmp = Math.sin(re);
	} else {
		tmp = (0.25 / re) / re;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 550.0:
		tmp = math.sin(re)
	else:
		tmp = (0.25 / re) / re
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 550.0)
		tmp = sin(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 <= 550.0)
		tmp = sin(re);
	else
		tmp = (0.25 / re) / re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 550.0], N[Sin[re], $MachinePrecision], N[(N[(0.25 / re), $MachinePrecision] / re), $MachinePrecision]]

Derivation

1. Split input into 2 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. 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 62.8%

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

   if 550 < im

   1. Initial program 98.4%

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

      1. distribute-rgt-in 98.4%

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

      2. cancel-sign-sub 98.4%

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

      3. distribute-rgt-out-- 98.4%

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

      4. sub-neg 98.4%

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

      5. neg-sub0 98.4%

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

      6. remove-double-neg 98.4%

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

   3. Simplified 98.4%

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

   6. Applied egg-rr 14.0%

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

   7. Taylor expanded in re around 0 13.9%

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

      1. add-sqr-sqrt 13.9%

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

      2. sqrt-div 13.9%

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

      3. metadata-eval 13.9%

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

      4. unpow2 13.9%

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

      5. sqrt-prod 13.5%

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

      6. add-sqr-sqrt 30.8%

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

      7. sqrt-div 30.8%

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

      8. metadata-eval 30.8%

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

      9. unpow2 30.8%

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

      10. sqrt-prod 13.5%

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

      11. add-sqr-sqrt 13.9%

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

   9. Applied egg-rr 13.9%

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

       1. associate-*l/ 13.9%

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

       2. associate-*r/ 13.9%

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

       3. metadata-eval 13.9%

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

   11. Applied egg-rr 13.9%

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

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 550:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25}{re}}{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 29.6% accurate, 30.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 700:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25}{re}}{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 700.0) re (/ (/ 0.25 re) re)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 700.0) {
		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 <= 700.0d0) 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 <= 700.0) {
		tmp = re;
	} else {
		tmp = (0.25 / re) / re;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 700.0:
		tmp = re
	else:
		tmp = (0.25 / re) / re
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 700.0)
		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 <= 700.0)
		tmp = re;
	else
		tmp = (0.25 / re) / re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 700.0], re, N[(N[(0.25 / re), $MachinePrecision] / re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 700

   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 58.8%

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

   7. Simplified 58.8%

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

   8. Taylor expanded in im around 0 29.8%

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

   if 700 < im

   1. Initial program 98.4%

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

      1. distribute-rgt-in 98.4%

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

      2. cancel-sign-sub 98.4%

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

      3. distribute-rgt-out-- 98.4%

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

      4. sub-neg 98.4%

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

      5. neg-sub0 98.4%

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

      6. remove-double-neg 98.4%

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

   3. Simplified 98.4%

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

   6. Applied egg-rr 14.0%

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

   7. Taylor expanded in re around 0 13.9%

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

      1. add-sqr-sqrt 13.9%

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

      2. sqrt-div 13.9%

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

      3. metadata-eval 13.9%

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

      4. unpow2 13.9%

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

      5. sqrt-prod 13.5%

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

      6. add-sqr-sqrt 30.8%

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

      7. sqrt-div 30.8%

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

      8. metadata-eval 30.8%

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

      9. unpow2 30.8%

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

      10. sqrt-prod 13.5%

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

      11. add-sqr-sqrt 13.9%

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

   9. Applied egg-rr 13.9%

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

       1. associate-*l/ 13.9%

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

       2. associate-*r/ 13.9%

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

       3. metadata-eval 13.9%

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

   11. Applied egg-rr 13.9%

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

4. Final simplification 26.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 700:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25}{re}}{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 26.7% 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 99.6%

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

   1. distribute-rgt-in 99.6%

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

   2. cancel-sign-sub 99.6%

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

   3. distribute-rgt-out-- 99.6%

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

   4. sub-neg 99.6%

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

   5. neg-sub0 99.6%

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

   6. remove-double-neg 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in re around 0 60.8%

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

7. Simplified 60.8%

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

8. Taylor expanded in im around 0 23.2%

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

9. Final simplification 23.2%

   \[\leadsto re \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024021 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))