math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 7.3s

Alternatives: 13

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(im) + exp(Float64(-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. +-commutative 100.0%

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

   2. neg-sub0 100.0%

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

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: 84.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ \mathbf{if}\;im \leq 0.23:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;im \leq 1.34 \cdot 10^{+154}:\\ \;\;\;\;\left(e^{im} + e^{-im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot {im}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))))
   (if (<= im 0.23)
     (* t_0 (fma im im 2.0))
     (if (<= im 1.34e+154)
       (* (+ (exp im) (exp (- im))) (* 0.5 re))
       (* t_0 (pow im 2.0))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sin(re);
	double tmp;
	if (im <= 0.23) {
		tmp = t_0 * fma(im, im, 2.0);
	} else if (im <= 1.34e+154) {
		tmp = (exp(im) + exp(-im)) * (0.5 * re);
	} 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 <= 0.23)
		tmp = Float64(t_0 * fma(im, im, 2.0));
	elseif (im <= 1.34e+154)
		tmp = Float64(Float64(exp(im) + exp(Float64(-im))) * Float64(0.5 * re));
	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, 0.23], N[(t$95$0 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.34e+154], N[(N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.23000000000000001

   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. +-commutative 100.0%

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

      2. neg-sub0 100.0%

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

   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 85.6%

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

      1. +-commutative 43.0%

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

      2. unpow2 43.0%

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

      3. fma-define 43.0%

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

   7. Simplified 85.6%

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

   if 0.23000000000000001 < im < 1.34000000000000001e154

   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. +-commutative 100.0%

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

      2. neg-sub0 100.0%

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

   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 74.4%

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

      1. associate-*r* 74.4%

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

   7. Simplified 74.4%

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

   if 1.34000000000000001e154 < 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. +-commutative 100.0%

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

      2. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

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

      1. +-commutative 61.5%

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

      2. unpow2 61.5%

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

      3. fma-define 61.5%

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

   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 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.23:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;im \leq 1.34 \cdot 10^{+154}:\\ \;\;\;\;\left(e^{im} + e^{-im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot {im}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 64.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 60:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.8 \cdot 10^{+151}:\\ \;\;\;\;\left(re + 8\right) + {re}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot {im}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 60.0)
   (sin re)
   (if (<= im 1.8e+151)
     (+ (+ re 8.0) (* (pow re 3.0) -0.16666666666666666))
     (* (* 0.5 (sin re)) (pow im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 60.0) {
		tmp = sin(re);
	} else if (im <= 1.8e+151) {
		tmp = (re + 8.0) + (pow(re, 3.0) * -0.16666666666666666);
	} 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 <= 60.0d0) then
        tmp = sin(re)
    else if (im <= 1.8d+151) then
        tmp = (re + 8.0d0) + ((re ** 3.0d0) * (-0.16666666666666666d0))
    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 <= 60.0) {
		tmp = Math.sin(re);
	} else if (im <= 1.8e+151) {
		tmp = (re + 8.0) + (Math.pow(re, 3.0) * -0.16666666666666666);
	} else {
		tmp = (0.5 * Math.sin(re)) * Math.pow(im, 2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 60.0:
		tmp = math.sin(re)
	elif im <= 1.8e+151:
		tmp = (re + 8.0) + (math.pow(re, 3.0) * -0.16666666666666666)
	else:
		tmp = (0.5 * math.sin(re)) * math.pow(im, 2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 60.0)
		tmp = sin(re);
	elseif (im <= 1.8e+151)
		tmp = Float64(Float64(re + 8.0) + Float64((re ^ 3.0) * -0.16666666666666666));
	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 <= 60.0)
		tmp = sin(re);
	elseif (im <= 1.8e+151)
		tmp = (re + 8.0) + ((re ^ 3.0) * -0.16666666666666666);
	else
		tmp = (0.5 * sin(re)) * (im ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 60.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.8e+151], N[(N[(re + 8.0), $MachinePrecision] + N[(N[Power[re, 3.0], $MachinePrecision] * -0.16666666666666666), $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 < 60

   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. +-commutative 100.0%

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

      2. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 66.6%

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

   if 60 < im < 1.8e151

   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. +-commutative 100.0%

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

      2. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 48.8%

      \[\leadsto \color{blue}{\sin re + \left(0.041666666666666664 \cdot \left({im}^{4} \cdot \sin re\right) + 0.5 \cdot \left({im}^{2} \cdot \sin re\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 48.8%

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

      2. associate-*r* 48.8%

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

      3. distribute-rgt-out 48.8%

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

   7. Simplified 48.8%

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

   9. Applied egg-rr 1.9%

      \[\leadsto \sin re + \color{blue}{8} \]

   10. Taylor expanded in re around 0 25.8%

       \[\leadsto \color{blue}{8 + \left(re + -0.16666666666666666 \cdot {re}^{3}\right)} \]
   11. Step-by-step derivation

       1. associate-+r+ 25.8%

          \[\leadsto \color{blue}{\left(8 + re\right) + -0.16666666666666666 \cdot {re}^{3}} \]

       2. +-commutative 25.8%

          \[\leadsto \color{blue}{\left(re + 8\right)} + -0.16666666666666666 \cdot {re}^{3} \]

       3. *-commutative 25.8%

          \[\leadsto \left(re + 8\right) + \color{blue}{{re}^{3} \cdot -0.16666666666666666} \]

   12. Simplified 25.8%

       \[\leadsto \color{blue}{\left(re + 8\right) + {re}^{3} \cdot -0.16666666666666666} \]

   if 1.8e151 < 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. +-commutative 100.0%

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

      2. neg-sub0 100.0%

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

   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 96.8%

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

      1. +-commutative 63.0%

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

      2. unpow2 63.0%

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

      3. fma-define 63.0%

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

   7. Simplified 96.8%

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

   8. Taylor expanded in im around inf 96.8%

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

      1. associate-*r* 96.8%

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

      2. *-commutative 96.8%

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

      3. associate-*r* 96.8%

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

   10. Simplified 96.8%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 60:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.8 \cdot 10^{+151}:\\ \;\;\;\;\left(re + 8\right) + {re}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot {im}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ \mathbf{if}\;im \leq 60:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;im \leq 2.15 \cdot 10^{+151}:\\ \;\;\;\;\left(re + 8\right) + {re}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot {im}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))))
   (if (<= im 60.0)
     (* t_0 (fma im im 2.0))
     (if (<= im 2.15e+151)
       (+ (+ re 8.0) (* (pow re 3.0) -0.16666666666666666))
       (* t_0 (pow im 2.0))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sin(re);
	double tmp;
	if (im <= 60.0) {
		tmp = t_0 * fma(im, im, 2.0);
	} else if (im <= 2.15e+151) {
		tmp = (re + 8.0) + (pow(re, 3.0) * -0.16666666666666666);
	} 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 <= 60.0)
		tmp = Float64(t_0 * fma(im, im, 2.0));
	elseif (im <= 2.15e+151)
		tmp = Float64(Float64(re + 8.0) + Float64((re ^ 3.0) * -0.16666666666666666));
	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, 60.0], N[(t$95$0 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.15e+151], N[(N[(re + 8.0), $MachinePrecision] + N[(N[Power[re, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < 60

   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. +-commutative 100.0%

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

      2. neg-sub0 100.0%

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

   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 84.9%

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

      1. +-commutative 42.7%

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

      2. unpow2 42.7%

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

      3. fma-define 42.7%

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

   7. Simplified 84.9%

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

   if 60 < im < 2.14999999999999991e151

   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. +-commutative 100.0%

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

      2. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 48.8%

      \[\leadsto \color{blue}{\sin re + \left(0.041666666666666664 \cdot \left({im}^{4} \cdot \sin re\right) + 0.5 \cdot \left({im}^{2} \cdot \sin re\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 48.8%

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

      2. associate-*r* 48.8%

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

      3. distribute-rgt-out 48.8%

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

   7. Simplified 48.8%

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

   9. Applied egg-rr 1.9%

      \[\leadsto \sin re + \color{blue}{8} \]

   10. Taylor expanded in re around 0 25.8%

       \[\leadsto \color{blue}{8 + \left(re + -0.16666666666666666 \cdot {re}^{3}\right)} \]
   11. Step-by-step derivation

       1. associate-+r+ 25.8%

          \[\leadsto \color{blue}{\left(8 + re\right) + -0.16666666666666666 \cdot {re}^{3}} \]

       2. +-commutative 25.8%

          \[\leadsto \color{blue}{\left(re + 8\right)} + -0.16666666666666666 \cdot {re}^{3} \]

       3. *-commutative 25.8%

          \[\leadsto \left(re + 8\right) + \color{blue}{{re}^{3} \cdot -0.16666666666666666} \]

   12. Simplified 25.8%

       \[\leadsto \color{blue}{\left(re + 8\right) + {re}^{3} \cdot -0.16666666666666666} \]

   if 2.14999999999999991e151 < 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. +-commutative 100.0%

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

      2. neg-sub0 100.0%

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

   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 96.8%

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

      1. +-commutative 63.0%

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

      2. unpow2 63.0%

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

      3. fma-define 63.0%

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

   7. Simplified 96.8%

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

   8. Taylor expanded in im around inf 96.8%

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

      1. associate-*r* 96.8%

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

      2. *-commutative 96.8%

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

      3. associate-*r* 96.8%

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

   10. Simplified 96.8%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 60:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;im \leq 2.15 \cdot 10^{+151}:\\ \;\;\;\;\left(re + 8\right) + {re}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot {im}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 61.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 60:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.08 \cdot 10^{+129}:\\ \;\;\;\;\left(re + 8\right) + {re}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 60.0)
   (sin re)
   (if (<= im 1.08e+129)
     (+ (+ re 8.0) (* (pow re 3.0) -0.16666666666666666))
     (* (fma im im 2.0) (* 0.5 re)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 60.0) {
		tmp = sin(re);
	} else if (im <= 1.08e+129) {
		tmp = (re + 8.0) + (pow(re, 3.0) * -0.16666666666666666);
	} else {
		tmp = fma(im, im, 2.0) * (0.5 * re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 60.0)
		tmp = sin(re);
	elseif (im <= 1.08e+129)
		tmp = Float64(Float64(re + 8.0) + Float64((re ^ 3.0) * -0.16666666666666666));
	else
		tmp = Float64(fma(im, im, 2.0) * Float64(0.5 * re));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 60.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.08e+129], N[(N[(re + 8.0), $MachinePrecision] + N[(N[Power[re, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(N[(im * im + 2.0), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 60

   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. +-commutative 100.0%

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

      2. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 66.6%

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

   if 60 < im < 1.08e129

   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. +-commutative 100.0%

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

      2. neg-sub0 100.0%

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

   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 36.9%

      \[\leadsto \color{blue}{\sin re + \left(0.041666666666666664 \cdot \left({im}^{4} \cdot \sin re\right) + 0.5 \cdot \left({im}^{2} \cdot \sin re\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 36.9%

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

      2. associate-*r* 36.9%

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

      3. distribute-rgt-out 36.9%

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

   7. Simplified 36.9%

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

   9. Applied egg-rr 2.0%

      \[\leadsto \sin re + \color{blue}{8} \]

   10. Taylor expanded in re around 0 24.0%

       \[\leadsto \color{blue}{8 + \left(re + -0.16666666666666666 \cdot {re}^{3}\right)} \]
   11. Step-by-step derivation

       1. associate-+r+ 24.0%

          \[\leadsto \color{blue}{\left(8 + re\right) + -0.16666666666666666 \cdot {re}^{3}} \]

       2. +-commutative 24.0%

          \[\leadsto \color{blue}{\left(re + 8\right)} + -0.16666666666666666 \cdot {re}^{3} \]

       3. *-commutative 24.0%

          \[\leadsto \left(re + 8\right) + \color{blue}{{re}^{3} \cdot -0.16666666666666666} \]

   12. Simplified 24.0%

       \[\leadsto \color{blue}{\left(re + 8\right) + {re}^{3} \cdot -0.16666666666666666} \]

   if 1.08e129 < 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. +-commutative 100.0%

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

      2. neg-sub0 100.0%

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

   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 63.6%

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

      1. associate-*r* 63.6%

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

   7. Simplified 63.6%

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

   8. Taylor expanded in im around 0 60.7%

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

      1. +-commutative 60.7%

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

      2. unpow2 60.7%

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

      3. fma-define 60.7%

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

   10. Simplified 60.7%

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

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 60:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.08 \cdot 10^{+129}:\\ \;\;\;\;\left(re + 8\right) + {re}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.72 \cdot 10^{+22}:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.72e+22) (sin re) (* (fma im im 2.0) (* 0.5 re))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.72e+22) {
		tmp = sin(re);
	} else {
		tmp = fma(im, im, 2.0) * (0.5 * re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.72e+22)
		tmp = sin(re);
	else
		tmp = Float64(fma(im, im, 2.0) * Float64(0.5 * re));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.72e+22], N[Sin[re], $MachinePrecision], N[(N[(im * im + 2.0), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.72e22

   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. +-commutative 100.0%

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

      2. neg-sub0 100.0%

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

   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 64.4%

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

   if 1.72e22 < 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. +-commutative 100.0%

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

      2. neg-sub0 100.0%

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

   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 69.2%

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

      1. associate-*r* 69.2%

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

   7. Simplified 69.2%

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

   8. Taylor expanded in im around 0 41.5%

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

      1. +-commutative 41.5%

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

      2. unpow2 41.5%

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

      3. fma-define 41.5%

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

   10. Simplified 41.5%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.72 \cdot 10^{+22}:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 60.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4.3 \cdot 10^{+21}:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot {im}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 4.3e+21) (sin re) (* 0.5 (* re (pow im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 4.3e+21) {
		tmp = sin(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 <= 4.3d+21) then
        tmp = sin(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 <= 4.3e+21) {
		tmp = Math.sin(re);
	} else {
		tmp = 0.5 * (re * Math.pow(im, 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 4.3e+21:
		tmp = math.sin(re)
	else:
		tmp = 0.5 * (re * math.pow(im, 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 4.3e+21)
		tmp = sin(re);
	else
		tmp = Float64(0.5 * Float64(re * (im ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 4.3e+21)
		tmp = sin(re);
	else
		tmp = 0.5 * (re * (im ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 4.3e+21], N[Sin[re], $MachinePrecision], N[(0.5 * N[(re * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 4.3e21

   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. +-commutative 100.0%

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

      2. neg-sub0 100.0%

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

   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 64.4%

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

   if 4.3e21 < 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. +-commutative 100.0%

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

      2. neg-sub0 100.0%

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

   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 69.2%

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

      1. associate-*r* 69.2%

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

   7. Simplified 69.2%

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

   8. Taylor expanded in im around 0 41.5%

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

   9. Taylor expanded in im around inf 41.5%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.3 \cdot 10^{+21}:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot {im}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.3 \cdot 10^{+22}:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;re + 0.5 \cdot \left(im \cdot \left(re \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.3e+22) (sin re) (+ re (* 0.5 (* im (* re im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.3e+22) {
		tmp = sin(re);
	} else {
		tmp = re + (0.5 * (im * (re * im)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 2.3d+22) then
        tmp = sin(re)
    else
        tmp = re + (0.5d0 * (im * (re * im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.3e+22) {
		tmp = Math.sin(re);
	} else {
		tmp = re + (0.5 * (im * (re * im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.3e+22:
		tmp = math.sin(re)
	else:
		tmp = re + (0.5 * (im * (re * im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.3e+22)
		tmp = sin(re);
	else
		tmp = Float64(re + Float64(0.5 * Float64(im * Float64(re * im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.3e+22)
		tmp = sin(re);
	else
		tmp = re + (0.5 * (im * (re * im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.3e+22], N[Sin[re], $MachinePrecision], N[(re + N[(0.5 * N[(im * N[(re * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.3000000000000002e22

   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. +-commutative 100.0%

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

      2. neg-sub0 100.0%

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

   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 64.4%

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

   if 2.3000000000000002e22 < 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. +-commutative 100.0%

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

      2. neg-sub0 100.0%

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

   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 69.2%

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

      1. associate-*r* 69.2%

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

   7. Simplified 69.2%

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

   8. Taylor expanded in im around 0 41.5%

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

      1. add0 41.5%

         \[\leadsto re + 0.5 \cdot \color{blue}{\left({im}^{2} \cdot re + 0\right)} \]

      2. fma-define 41.5%

         \[\leadsto re + 0.5 \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, re, 0\right)} \]

      3. metadata-eval 41.5%

         \[\leadsto re + 0.5 \cdot \mathsf{fma}\left({im}^{2}, re, \color{blue}{-0}\right) \]

      4. fma-neg 41.5%

         \[\leadsto re + 0.5 \cdot \color{blue}{\left({im}^{2} \cdot re - 0\right)} \]

      5. *-commutative 41.5%

         \[\leadsto re + 0.5 \cdot \left(\color{blue}{re \cdot {im}^{2}} - 0\right) \]

      6. unpow2 41.5%

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

      7. associate-*r* 32.5%

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

      8. fma-neg 32.5%

         \[\leadsto re + 0.5 \cdot \color{blue}{\mathsf{fma}\left(re \cdot im, im, -0\right)} \]

      9. metadata-eval 32.5%

         \[\leadsto re + 0.5 \cdot \mathsf{fma}\left(re \cdot im, im, \color{blue}{0}\right) \]

   10. Applied egg-rr 32.5%

       \[\leadsto re + 0.5 \cdot \color{blue}{\mathsf{fma}\left(re \cdot im, im, 0\right)} \]
   11. Step-by-step derivation

       1. fma-undefine 32.5%

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

       2. add0 32.5%

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

       3. *-commutative 32.5%

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

   12. Applied egg-rr 32.5%

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

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.3 \cdot 10^{+22}:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;re + 0.5 \cdot \left(im \cdot \left(re \cdot im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 42.1% accurate, 34.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. +-commutative 100.0%

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

   2. neg-sub0 100.0%

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

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 54.4%

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

   1. associate-*r* 54.4%

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

7. Simplified 54.4%

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

8. Taylor expanded in im around 0 41.3%

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

   1. add0 41.3%

      \[\leadsto re + 0.5 \cdot \color{blue}{\left({im}^{2} \cdot re + 0\right)} \]

   2. fma-define 41.3%

      \[\leadsto re + 0.5 \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, re, 0\right)} \]

   3. metadata-eval 41.3%

      \[\leadsto re + 0.5 \cdot \mathsf{fma}\left({im}^{2}, re, \color{blue}{-0}\right) \]

   4. fma-neg 41.3%

      \[\leadsto re + 0.5 \cdot \color{blue}{\left({im}^{2} \cdot re - 0\right)} \]

   5. *-commutative 41.3%

      \[\leadsto re + 0.5 \cdot \left(\color{blue}{re \cdot {im}^{2}} - 0\right) \]

   6. unpow2 41.3%

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

   7. associate-*r* 36.5%

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

   8. fma-neg 36.5%

      \[\leadsto re + 0.5 \cdot \color{blue}{\mathsf{fma}\left(re \cdot im, im, -0\right)} \]

   9. metadata-eval 36.5%

      \[\leadsto re + 0.5 \cdot \mathsf{fma}\left(re \cdot im, im, \color{blue}{0}\right) \]

10. Applied egg-rr 36.5%

    \[\leadsto re + 0.5 \cdot \color{blue}{\mathsf{fma}\left(re \cdot im, im, 0\right)} \]
11. Step-by-step derivation

    1. fma-undefine 36.5%

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

    2. add0 36.5%

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

    3. *-commutative 36.5%

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

12. Applied egg-rr 36.5%

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

13. Final simplification 36.5%

    \[\leadsto re + 0.5 \cdot \left(im \cdot \left(re \cdot im\right)\right) \]
14. Add Preprocessing

Alternative 10: 27.2% accurate, 51.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 380000000000:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= re 380000000000.0) re 0.5))

C

double code(double re, double im) {
	double tmp;
	if (re <= 380000000000.0) {
		tmp = re;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 380000000000.0d0) then
        tmp = re
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 380000000000.0) {
		tmp = re;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 380000000000.0:
		tmp = re
	else:
		tmp = 0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 380000000000.0)
		tmp = re;
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 380000000000.0)
		tmp = re;
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 380000000000.0], re, 0.5]

Derivation

1. Split input into 2 regimes

2. if re < 3.8e11

   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. +-commutative 100.0%

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

      2. neg-sub0 100.0%

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

   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 66.6%

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

      1. associate-*r* 66.6%

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

   7. Simplified 66.6%

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

   8. Taylor expanded in im around 0 30.8%

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

   if 3.8e11 < re

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. neg-sub0 100.0%

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

   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 6.9%

      \[\leadsto \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 24.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 380000000000:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. 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. +-commutative 100.0%

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

   2. neg-sub0 100.0%

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

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 87.3%

   \[\leadsto \color{blue}{\sin re + \left(0.041666666666666664 \cdot \left({im}^{4} \cdot \sin re\right) + 0.5 \cdot \left({im}^{2} \cdot \sin re\right)\right)} \]
6. Step-by-step derivation

   1. associate-*r* 87.3%

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

   2. associate-*r* 87.3%

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

   3. distribute-rgt-out 87.3%

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

7. Simplified 87.3%

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

9. Applied egg-rr 5.0%

   \[\leadsto \sin re + \color{blue}{-1} \]

10. Taylor expanded in re around 0 5.6%

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

11. Final simplification 5.6%

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

Alternative 12: 4.3% 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. +-commutative 100.0%

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

   2. neg-sub0 100.0%

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

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 4.5%

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

7. Final simplification 4.5%

   \[\leadsto 0.25 \]
8. Add Preprocessing

Alternative 13: 4.5% 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. +-commutative 100.0%

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

   2. neg-sub0 100.0%

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

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 4.6%

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

7. Final simplification 4.6%

   \[\leadsto 0.5 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024034 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))