math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 9.2s

Alternatives: 14

Speedup: 1.0×

• 50.4% 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, 0.8× speedup

Math

\[\begin{array}{l} \\ \sin re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (sin re) (fma 0.5 (exp im) (/ 0.5 (exp im)))))

C

double code(double re, double im) {
	return sin(re) * fma(0.5, exp(im), (0.5 / exp(im)));
}

Julia

function code(re, im)
	return Float64(sin(re) * fma(0.5, exp(im), Float64(0.5 / exp(im))))
end

Wolfram

code[re_, im_] := N[(N[Sin[re], $MachinePrecision] * N[(0.5 * N[Exp[im], $MachinePrecision] + N[(0.5 / N[Exp[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. 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. +-commutative 100.0%

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

   3. associate-*r* 100.0%

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

   4. associate-*r* 100.0%

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

   5. distribute-rgt-out 100.0%

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

   6. cancel-sign-sub 100.0%

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

   7. distribute-lft-neg-in 100.0%

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

   8. neg-mul-1 100.0%

      \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 - \color{blue}{-1 \cdot \left(e^{0 - im} \cdot 0.5\right)}\right) \]

   9. *-commutative 100.0%

      \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 - \color{blue}{\left(e^{0 - im} \cdot 0.5\right) \cdot -1}\right) \]

   10. sub-neg 100.0%

       \[\leadsto \sin re \cdot \color{blue}{\left(e^{im} \cdot 0.5 + \left(-\left(e^{0 - im} \cdot 0.5\right) \cdot -1\right)\right)} \]

   11. *-commutative 100.0%

       \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 + \left(-\color{blue}{-1 \cdot \left(e^{0 - im} \cdot 0.5\right)}\right)\right) \]

   12. neg-mul-1 100.0%

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

   13. remove-double-neg 100.0%

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

   14. distribute-rgt-in 100.0%

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

   15. distribute-lft-in 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \sin re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right) \]

Alternative 2: 84.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.3:\\ \;\;\;\;\mathsf{fma}\left(im \cdot \left(\sin re \cdot 0.5\right), im, \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(0.5, e^{im}, 0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.3)
   (fma (* im (* (sin re) 0.5)) im (sin re))
   (* (sin re) (fma 0.5 (exp im) 0.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.3) {
		tmp = fma((im * (sin(re) * 0.5)), im, sin(re));
	} else {
		tmp = sin(re) * fma(0.5, exp(im), 0.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.3)
		tmp = fma(Float64(im * Float64(sin(re) * 0.5)), im, sin(re));
	else
		tmp = Float64(sin(re) * fma(0.5, exp(im), 0.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.3], N[(N[(im * N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * im + N[Sin[re], $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(0.5 * N[Exp[im], $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.30000000000000004

   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. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. cancel-sign-sub 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. neg-mul-1 100.0%

         \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 - \color{blue}{-1 \cdot \left(e^{0 - im} \cdot 0.5\right)}\right) \]

      9. *-commutative 100.0%

         \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 - \color{blue}{\left(e^{0 - im} \cdot 0.5\right) \cdot -1}\right) \]

      10. sub-neg 100.0%

          \[\leadsto \sin re \cdot \color{blue}{\left(e^{im} \cdot 0.5 + \left(-\left(e^{0 - im} \cdot 0.5\right) \cdot -1\right)\right)} \]

      11. *-commutative 100.0%

          \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 + \left(-\color{blue}{-1 \cdot \left(e^{0 - im} \cdot 0.5\right)}\right)\right) \]

      12. neg-mul-1 100.0%

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

      13. remove-double-neg 100.0%

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

      14. distribute-rgt-in 100.0%

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

      15. distribute-lft-in 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 87.6%

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

      1. +-commutative 87.6%

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

      2. associate-*r* 87.6%

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

      3. associate-*r* 87.6%

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

      4. distribute-rgt-out 87.6%

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

      5. fma-def 87.6%

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

   6. Simplified 87.6%

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

   7. Taylor expanded in im around 0 77.7%

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

      1. +-commutative 77.7%

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

      2. associate-*r* 77.7%

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

      3. unpow2 77.7%

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

      4. associate-*r* 74.7%

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

      5. fma-def 74.7%

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

   9. Applied egg-rr 74.7%

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

   if 1.30000000000000004 < 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. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. cancel-sign-sub 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. neg-mul-1 100.0%

         \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 - \color{blue}{-1 \cdot \left(e^{0 - im} \cdot 0.5\right)}\right) \]

      9. *-commutative 100.0%

         \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 - \color{blue}{\left(e^{0 - im} \cdot 0.5\right) \cdot -1}\right) \]

      10. sub-neg 100.0%

          \[\leadsto \sin re \cdot \color{blue}{\left(e^{im} \cdot 0.5 + \left(-\left(e^{0 - im} \cdot 0.5\right) \cdot -1\right)\right)} \]

      11. *-commutative 100.0%

          \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 + \left(-\color{blue}{-1 \cdot \left(e^{0 - im} \cdot 0.5\right)}\right)\right) \]

      12. neg-mul-1 100.0%

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

      13. remove-double-neg 100.0%

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

      14. distribute-rgt-in 100.0%

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

      15. distribute-lft-in 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 99.9%

      \[\leadsto \sin re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{0}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.3:\\ \;\;\;\;\mathsf{fma}\left(im \cdot \left(\sin re \cdot 0.5\right), im, \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(0.5, e^{im}, 0\right)\\ \end{array} \]

Alternative 3: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. distribute-lft-in 100.0%

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

   2. *-commutative 100.0%

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

   3. cancel-sign-sub 100.0%

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

   4. distribute-lft-neg-out 100.0%

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

   5. *-commutative 100.0%

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

   6. distribute-rgt-neg-out 100.0%

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

   7. neg-mul-1 100.0%

      \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

   8. associate-*r* 100.0%

      \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

   9. distribute-rgt-out-- 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

   10. sub-neg 100.0%

       \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

   11. neg-sub0 100.0%

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

   12. *-commutative 100.0%

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

   13. neg-mul-1 100.0%

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

   14. 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. Final simplification 100.0%

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

Alternative 4: 87.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.3:\\ \;\;\;\;\left(\sin re \cdot 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(0.5, e^{im}, 0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.3)
   (* (* (sin re) 0.5) (fma im im 2.0))
   (* (sin re) (fma 0.5 (exp im) 0.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.3) {
		tmp = (sin(re) * 0.5) * fma(im, im, 2.0);
	} else {
		tmp = sin(re) * fma(0.5, exp(im), 0.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.3)
		tmp = Float64(Float64(sin(re) * 0.5) * fma(im, im, 2.0));
	else
		tmp = Float64(sin(re) * fma(0.5, exp(im), 0.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.3], N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(0.5 * N[Exp[im], $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.30000000000000004

   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-lft-in 100.0%

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

      2. *-commutative 100.0%

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

      3. cancel-sign-sub 100.0%

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

      4. distribute-lft-neg-out 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-out 100.0%

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

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. 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. Taylor expanded in im around 0 77.7%

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

      1. +-commutative 77.7%

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

      2. unpow2 77.7%

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

      3. fma-def 77.7%

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

   6. Simplified 77.7%

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

   if 1.30000000000000004 < 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. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. cancel-sign-sub 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. neg-mul-1 100.0%

         \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 - \color{blue}{-1 \cdot \left(e^{0 - im} \cdot 0.5\right)}\right) \]

      9. *-commutative 100.0%

         \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 - \color{blue}{\left(e^{0 - im} \cdot 0.5\right) \cdot -1}\right) \]

      10. sub-neg 100.0%

          \[\leadsto \sin re \cdot \color{blue}{\left(e^{im} \cdot 0.5 + \left(-\left(e^{0 - im} \cdot 0.5\right) \cdot -1\right)\right)} \]

      11. *-commutative 100.0%

          \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 + \left(-\color{blue}{-1 \cdot \left(e^{0 - im} \cdot 0.5\right)}\right)\right) \]

      12. neg-mul-1 100.0%

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

      13. remove-double-neg 100.0%

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

      14. distribute-rgt-in 100.0%

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

      15. distribute-lft-in 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 99.9%

      \[\leadsto \sin re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{0}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.3:\\ \;\;\;\;\left(\sin re \cdot 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \mathsf{fma}\left(0.5, e^{im}, 0\right)\\ \end{array} \]

Alternative 5: 84.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1950 \lor \neg \left(im \leq 1.32 \cdot 10^{+154}\right):\\ \;\;\;\;\left(\sin re \cdot 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(e^{im} + e^{-im}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 1950.0) (not (<= im 1.32e+154)))
   (* (* (sin re) 0.5) (fma im im 2.0))
   (* 0.5 (* re (+ (exp im) (exp (- im)))))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 1950.0) || !(im <= 1.32e+154)) {
		tmp = (sin(re) * 0.5) * fma(im, im, 2.0);
	} else {
		tmp = 0.5 * (re * (exp(im) + exp(-im)));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 1950.0) || !(im <= 1.32e+154))
		tmp = Float64(Float64(sin(re) * 0.5) * fma(im, im, 2.0));
	else
		tmp = Float64(0.5 * Float64(re * Float64(exp(im) + exp(Float64(-im)))));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 1950.0], N[Not[LessEqual[im, 1.32e+154]], $MachinePrecision]], N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(re * N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1950 or 1.31999999999999998e154 < im

   1. Initial program 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. *-commutative 100.0%

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

      3. cancel-sign-sub 100.0%

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

      4. distribute-lft-neg-out 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-out 100.0%

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

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. 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. Taylor expanded in im around 0 80.9%

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

      1. +-commutative 80.9%

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

      2. unpow2 80.9%

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

      3. fma-def 80.9%

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

   6. Simplified 80.9%

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

   if 1950 < im < 1.31999999999999998e154

   1. Initial program 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. *-commutative 100.0%

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

      3. cancel-sign-sub 100.0%

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

      4. distribute-lft-neg-out 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-out 100.0%

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

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. 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. Taylor expanded in re around 0 67.7%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1950 \lor \neg \left(im \leq 1.32 \cdot 10^{+154}\right):\\ \;\;\;\;\left(\sin re \cdot 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(e^{im} + e^{-im}\right)\right)\\ \end{array} \]

Alternative 6: 77.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1100 \lor \neg \left(im \leq 2.02 \cdot 10^{+149}\right):\\ \;\;\;\;\left(\sin re \cdot 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {\left({re}^{4}\right)}^{0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 1100.0) (not (<= im 2.02e+149)))
   (* (* (sin re) 0.5) (fma im im 2.0))
   (* 0.5 (pow (pow re 4.0) 0.25))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 1100.0) || !(im <= 2.02e+149)) {
		tmp = (sin(re) * 0.5) * fma(im, im, 2.0);
	} else {
		tmp = 0.5 * pow(pow(re, 4.0), 0.25);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 1100.0) || !(im <= 2.02e+149))
		tmp = Float64(Float64(sin(re) * 0.5) * fma(im, im, 2.0));
	else
		tmp = Float64(0.5 * ((re ^ 4.0) ^ 0.25));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 1100.0], N[Not[LessEqual[im, 2.02e+149]], $MachinePrecision]], N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Power[N[Power[re, 4.0], $MachinePrecision], 0.25], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1100 or 2.02e149 < 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-lft-in 100.0%

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

      2. *-commutative 100.0%

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

      3. cancel-sign-sub 100.0%

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

      4. distribute-lft-neg-out 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-out 100.0%

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

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. 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. Taylor expanded in im around 0 81.2%

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

      1. +-commutative 81.2%

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

      2. unpow2 81.2%

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

      3. fma-def 81.2%

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

   6. Simplified 81.2%

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

   if 1100 < im < 2.02e149

   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-lft-in 100.0%

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

      2. *-commutative 100.0%

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

      3. cancel-sign-sub 100.0%

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

      4. distribute-lft-neg-out 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-out 100.0%

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

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. 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. Taylor expanded in re around 0 65.6%

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

   6. Applied egg-rr 16.8%

      \[\leadsto 0.5 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(re\right)\right)} \]
   7. Step-by-step derivation

      1. log1p-expm1-u 2.0%

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

      2. add-sqr-sqrt 1.1%

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

      3. sqrt-unprod 20.5%

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

      4. pow2 20.5%

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

   8. Applied egg-rr 20.5%

      \[\leadsto 0.5 \cdot \color{blue}{\sqrt{{re}^{2}}} \]
   9. Step-by-step derivation

      1. pow1/2 20.5%

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

      2. sqr-pow 20.5%

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

      3. pow-prod-down 32.2%

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

      4. pow-prod-up 32.2%

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

      5. metadata-eval 32.2%

         \[\leadsto 0.5 \cdot {\left({re}^{\color{blue}{4}}\right)}^{\left(\frac{0.5}{2}\right)} \]

      6. metadata-eval 32.2%

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

   10. Applied egg-rr 32.2%

       \[\leadsto 0.5 \cdot \color{blue}{{\left({re}^{4}\right)}^{0.25}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1100 \lor \neg \left(im \leq 2.02 \cdot 10^{+149}\right):\\ \;\;\;\;\left(\sin re \cdot 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {\left({re}^{4}\right)}^{0.25}\\ \end{array} \]

Alternative 7: 61.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 700:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 3.55 \cdot 10^{+92}:\\ \;\;\;\;0.5 \cdot {\left({re}^{3}\right)}^{0.3333333333333333}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(2 + {im}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 700.0)
   (sin re)
   (if (<= im 3.55e+92)
     (* 0.5 (pow (pow re 3.0) 0.3333333333333333))
     (* 0.5 (* re (+ 2.0 (pow im 2.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 700.0) {
		tmp = sin(re);
	} else if (im <= 3.55e+92) {
		tmp = 0.5 * pow(pow(re, 3.0), 0.3333333333333333);
	} else {
		tmp = 0.5 * (re * (2.0 + 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 <= 3.55d+92) then
        tmp = 0.5d0 * ((re ** 3.0d0) ** 0.3333333333333333d0)
    else
        tmp = 0.5d0 * (re * (2.0d0 + (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 <= 3.55e+92) {
		tmp = 0.5 * Math.pow(Math.pow(re, 3.0), 0.3333333333333333);
	} else {
		tmp = 0.5 * (re * (2.0 + Math.pow(im, 2.0)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 700.0:
		tmp = math.sin(re)
	elif im <= 3.55e+92:
		tmp = 0.5 * math.pow(math.pow(re, 3.0), 0.3333333333333333)
	else:
		tmp = 0.5 * (re * (2.0 + math.pow(im, 2.0)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 700.0)
		tmp = sin(re);
	elseif (im <= 3.55e+92)
		tmp = Float64(0.5 * ((re ^ 3.0) ^ 0.3333333333333333));
	else
		tmp = Float64(0.5 * Float64(re * Float64(2.0 + (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 <= 3.55e+92)
		tmp = 0.5 * ((re ^ 3.0) ^ 0.3333333333333333);
	else
		tmp = 0.5 * (re * (2.0 + (im ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 700.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 3.55e+92], N[(0.5 * N[Power[N[Power[re, 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(re * N[(2.0 + N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $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. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. cancel-sign-sub 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. neg-mul-1 100.0%

         \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 - \color{blue}{-1 \cdot \left(e^{0 - im} \cdot 0.5\right)}\right) \]

      9. *-commutative 100.0%

         \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 - \color{blue}{\left(e^{0 - im} \cdot 0.5\right) \cdot -1}\right) \]

      10. sub-neg 100.0%

          \[\leadsto \sin re \cdot \color{blue}{\left(e^{im} \cdot 0.5 + \left(-\left(e^{0 - im} \cdot 0.5\right) \cdot -1\right)\right)} \]

      11. *-commutative 100.0%

          \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 + \left(-\color{blue}{-1 \cdot \left(e^{0 - im} \cdot 0.5\right)}\right)\right) \]

      12. neg-mul-1 100.0%

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

      13. remove-double-neg 100.0%

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

      14. distribute-rgt-in 100.0%

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

      15. distribute-lft-in 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 63.0%

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

   if 700 < im < 3.55e92

   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-lft-in 100.0%

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

      2. *-commutative 100.0%

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

      3. cancel-sign-sub 100.0%

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

      4. distribute-lft-neg-out 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-out 100.0%

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

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. 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. Taylor expanded in re around 0 57.1%

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

   6. Applied egg-rr 15.4%

      \[\leadsto 0.5 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(re\right)\right)} \]
   7. Step-by-step derivation

      1. log1p-expm1-u 1.9%

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

      2. add-cbrt-cube 15.3%

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

      3. pow1/3 34.2%

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

      4. pow3 34.2%

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

   8. Applied egg-rr 34.2%

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

   if 3.55e92 < 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-lft-in 100.0%

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

      2. *-commutative 100.0%

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

      3. cancel-sign-sub 100.0%

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

      4. distribute-lft-neg-out 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-out 100.0%

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

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. 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. Taylor expanded in im around 0 78.8%

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

      1. +-commutative 78.8%

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

      2. unpow2 78.8%

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

      3. fma-def 78.8%

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

   6. Simplified 78.8%

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

   7. Taylor expanded in re around 0 57.9%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 700:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 3.55 \cdot 10^{+92}:\\ \;\;\;\;0.5 \cdot {\left({re}^{3}\right)}^{0.3333333333333333}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(2 + {im}^{2}\right)\right)\\ \end{array} \]

Alternative 8: 61.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 660:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 5.2 \cdot 10^{+89}:\\ \;\;\;\;0.5 \cdot {\left({re}^{4}\right)}^{0.25}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(2 + {im}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 660.0)
   (sin re)
   (if (<= im 5.2e+89)
     (* 0.5 (pow (pow re 4.0) 0.25))
     (* 0.5 (* re (+ 2.0 (pow im 2.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 660.0) {
		tmp = sin(re);
	} else if (im <= 5.2e+89) {
		tmp = 0.5 * pow(pow(re, 4.0), 0.25);
	} else {
		tmp = 0.5 * (re * (2.0 + 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 <= 660.0d0) then
        tmp = sin(re)
    else if (im <= 5.2d+89) then
        tmp = 0.5d0 * ((re ** 4.0d0) ** 0.25d0)
    else
        tmp = 0.5d0 * (re * (2.0d0 + (im ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 660.0) {
		tmp = Math.sin(re);
	} else if (im <= 5.2e+89) {
		tmp = 0.5 * Math.pow(Math.pow(re, 4.0), 0.25);
	} else {
		tmp = 0.5 * (re * (2.0 + Math.pow(im, 2.0)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 660.0:
		tmp = math.sin(re)
	elif im <= 5.2e+89:
		tmp = 0.5 * math.pow(math.pow(re, 4.0), 0.25)
	else:
		tmp = 0.5 * (re * (2.0 + math.pow(im, 2.0)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 660.0)
		tmp = sin(re);
	elseif (im <= 5.2e+89)
		tmp = Float64(0.5 * ((re ^ 4.0) ^ 0.25));
	else
		tmp = Float64(0.5 * Float64(re * Float64(2.0 + (im ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 660.0)
		tmp = sin(re);
	elseif (im <= 5.2e+89)
		tmp = 0.5 * ((re ^ 4.0) ^ 0.25);
	else
		tmp = 0.5 * (re * (2.0 + (im ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 660.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 5.2e+89], N[(0.5 * N[Power[N[Power[re, 4.0], $MachinePrecision], 0.25], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(re * N[(2.0 + N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 660

   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. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. cancel-sign-sub 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. neg-mul-1 100.0%

         \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 - \color{blue}{-1 \cdot \left(e^{0 - im} \cdot 0.5\right)}\right) \]

      9. *-commutative 100.0%

         \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 - \color{blue}{\left(e^{0 - im} \cdot 0.5\right) \cdot -1}\right) \]

      10. sub-neg 100.0%

          \[\leadsto \sin re \cdot \color{blue}{\left(e^{im} \cdot 0.5 + \left(-\left(e^{0 - im} \cdot 0.5\right) \cdot -1\right)\right)} \]

      11. *-commutative 100.0%

          \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 + \left(-\color{blue}{-1 \cdot \left(e^{0 - im} \cdot 0.5\right)}\right)\right) \]

      12. neg-mul-1 100.0%

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

      13. remove-double-neg 100.0%

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

      14. distribute-rgt-in 100.0%

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

      15. distribute-lft-in 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 63.0%

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

   if 660 < im < 5.2000000000000001e89

   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-lft-in 100.0%

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

      2. *-commutative 100.0%

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

      3. cancel-sign-sub 100.0%

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

      4. distribute-lft-neg-out 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-out 100.0%

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

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. 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. Taylor expanded in re around 0 60.0%

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

   6. Applied egg-rr 16.2%

      \[\leadsto 0.5 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(re\right)\right)} \]
   7. Step-by-step derivation

      1. log1p-expm1-u 2.0%

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

      2. add-sqr-sqrt 1.1%

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

      3. sqrt-unprod 26.6%

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

      4. pow2 26.6%

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

   8. Applied egg-rr 26.6%

      \[\leadsto 0.5 \cdot \color{blue}{\sqrt{{re}^{2}}} \]
   9. Step-by-step derivation

      1. pow1/2 26.6%

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

      2. sqr-pow 26.6%

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

      3. pow-prod-down 36.0%

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

      4. pow-prod-up 36.0%

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

      5. metadata-eval 36.0%

         \[\leadsto 0.5 \cdot {\left({re}^{\color{blue}{4}}\right)}^{\left(\frac{0.5}{2}\right)} \]

      6. metadata-eval 36.0%

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

   10. Applied egg-rr 36.0%

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

   if 5.2000000000000001e89 < 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-lft-in 100.0%

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

      2. *-commutative 100.0%

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

      3. cancel-sign-sub 100.0%

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

      4. distribute-lft-neg-out 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-out 100.0%

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

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. 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. Taylor expanded in im around 0 77.3%

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

      1. +-commutative 77.3%

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

      2. unpow2 77.3%

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

      3. fma-def 77.3%

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

   6. Simplified 77.3%

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

   7. Taylor expanded in re around 0 56.7%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 660:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 5.2 \cdot 10^{+89}:\\ \;\;\;\;0.5 \cdot {\left({re}^{4}\right)}^{0.25}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(2 + {im}^{2}\right)\right)\\ \end{array} \]

Alternative 9: 62.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1950:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 9.6 \cdot 10^{+114}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(2 + {im}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1950.0)
   (sin re)
   (if (<= im 9.6e+114)
     (* 0.5 (log1p (expm1 re)))
     (* 0.5 (* re (+ 2.0 (pow im 2.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1950.0) {
		tmp = sin(re);
	} else if (im <= 9.6e+114) {
		tmp = 0.5 * log1p(expm1(re));
	} else {
		tmp = 0.5 * (re * (2.0 + pow(im, 2.0)));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1950.0) {
		tmp = Math.sin(re);
	} else if (im <= 9.6e+114) {
		tmp = 0.5 * Math.log1p(Math.expm1(re));
	} else {
		tmp = 0.5 * (re * (2.0 + Math.pow(im, 2.0)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1950.0:
		tmp = math.sin(re)
	elif im <= 9.6e+114:
		tmp = 0.5 * math.log1p(math.expm1(re))
	else:
		tmp = 0.5 * (re * (2.0 + math.pow(im, 2.0)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1950.0)
		tmp = sin(re);
	elseif (im <= 9.6e+114)
		tmp = Float64(0.5 * log1p(expm1(re)));
	else
		tmp = Float64(0.5 * Float64(re * Float64(2.0 + (im ^ 2.0))));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1950.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 9.6e+114], N[(0.5 * N[Log[1 + N[(Exp[re] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(re * N[(2.0 + N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1950

   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. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. cancel-sign-sub 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. neg-mul-1 100.0%

         \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 - \color{blue}{-1 \cdot \left(e^{0 - im} \cdot 0.5\right)}\right) \]

      9. *-commutative 100.0%

         \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 - \color{blue}{\left(e^{0 - im} \cdot 0.5\right) \cdot -1}\right) \]

      10. sub-neg 100.0%

          \[\leadsto \sin re \cdot \color{blue}{\left(e^{im} \cdot 0.5 + \left(-\left(e^{0 - im} \cdot 0.5\right) \cdot -1\right)\right)} \]

      11. *-commutative 100.0%

          \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 + \left(-\color{blue}{-1 \cdot \left(e^{0 - im} \cdot 0.5\right)}\right)\right) \]

      12. neg-mul-1 100.0%

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

      13. remove-double-neg 100.0%

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

      14. distribute-rgt-in 100.0%

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

      15. distribute-lft-in 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 62.7%

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

   if 1950 < im < 9.6e114

   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-lft-in 100.0%

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

      2. *-commutative 100.0%

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

      3. cancel-sign-sub 100.0%

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

      4. distribute-lft-neg-out 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-out 100.0%

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

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. 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. 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. Applied egg-rr 16.7%

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

   if 9.6e114 < 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-lft-in 100.0%

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

      2. *-commutative 100.0%

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

      3. cancel-sign-sub 100.0%

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

      4. distribute-lft-neg-out 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-out 100.0%

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

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. 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. Taylor expanded in im around 0 89.2%

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

      1. +-commutative 89.2%

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

      2. unpow2 89.2%

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

      3. fma-def 89.2%

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

   6. Simplified 89.2%

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

   7. Taylor expanded in re around 0 65.3%

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

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1950:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 9.6 \cdot 10^{+114}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(2 + {im}^{2}\right)\right)\\ \end{array} \]

Alternative 10: 60.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 360000000:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 4.4 \cdot 10^{+136}:\\ \;\;\;\;-0.5 \cdot {re}^{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(2 + {im}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 360000000.0)
   (sin re)
   (if (<= im 4.4e+136)
     (* -0.5 (pow re 2.0))
     (* 0.5 (* re (+ 2.0 (pow im 2.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 360000000.0) {
		tmp = sin(re);
	} else if (im <= 4.4e+136) {
		tmp = -0.5 * pow(re, 2.0);
	} else {
		tmp = 0.5 * (re * (2.0 + 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 <= 360000000.0d0) then
        tmp = sin(re)
    else if (im <= 4.4d+136) then
        tmp = (-0.5d0) * (re ** 2.0d0)
    else
        tmp = 0.5d0 * (re * (2.0d0 + (im ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 360000000.0) {
		tmp = Math.sin(re);
	} else if (im <= 4.4e+136) {
		tmp = -0.5 * Math.pow(re, 2.0);
	} else {
		tmp = 0.5 * (re * (2.0 + Math.pow(im, 2.0)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 360000000.0:
		tmp = math.sin(re)
	elif im <= 4.4e+136:
		tmp = -0.5 * math.pow(re, 2.0)
	else:
		tmp = 0.5 * (re * (2.0 + math.pow(im, 2.0)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 360000000.0)
		tmp = sin(re);
	elseif (im <= 4.4e+136)
		tmp = Float64(-0.5 * (re ^ 2.0));
	else
		tmp = Float64(0.5 * Float64(re * Float64(2.0 + (im ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 360000000.0)
		tmp = sin(re);
	elseif (im <= 4.4e+136)
		tmp = -0.5 * (re ^ 2.0);
	else
		tmp = 0.5 * (re * (2.0 + (im ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 360000000.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 4.4e+136], N[(-0.5 * N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(re * N[(2.0 + N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 3.6e8

   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. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. cancel-sign-sub 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. neg-mul-1 100.0%

         \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 - \color{blue}{-1 \cdot \left(e^{0 - im} \cdot 0.5\right)}\right) \]

      9. *-commutative 100.0%

         \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 - \color{blue}{\left(e^{0 - im} \cdot 0.5\right) \cdot -1}\right) \]

      10. sub-neg 100.0%

          \[\leadsto \sin re \cdot \color{blue}{\left(e^{im} \cdot 0.5 + \left(-\left(e^{0 - im} \cdot 0.5\right) \cdot -1\right)\right)} \]

      11. *-commutative 100.0%

          \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 + \left(-\color{blue}{-1 \cdot \left(e^{0 - im} \cdot 0.5\right)}\right)\right) \]

      12. neg-mul-1 100.0%

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

      13. remove-double-neg 100.0%

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

      14. distribute-rgt-in 100.0%

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

      15. distribute-lft-in 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 61.2%

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

   if 3.6e8 < im < 4.3999999999999999e136

   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-lft-in 100.0%

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

      2. *-commutative 100.0%

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

      3. cancel-sign-sub 100.0%

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

      4. distribute-lft-neg-out 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-out 100.0%

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

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. 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. Taylor expanded in im around 0 3.7%

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

      1. +-commutative 3.7%

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

      2. unpow2 3.7%

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

      3. fma-def 3.7%

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

   6. Simplified 3.7%

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

   8. Applied egg-rr 1.7%

      \[\leadsto \color{blue}{-1 - -1 \cdot \cos re} \]

   9. Taylor expanded in re around 0 10.8%

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

   if 4.3999999999999999e136 < 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-lft-in 100.0%

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

      2. *-commutative 100.0%

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

      3. cancel-sign-sub 100.0%

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

      4. distribute-lft-neg-out 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-out 100.0%

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

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. 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. Taylor expanded in im around 0 89.2%

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

      1. +-commutative 89.2%

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

      2. unpow2 89.2%

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

      3. fma-def 89.2%

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

   6. Simplified 89.2%

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

   7. Taylor expanded in re around 0 65.3%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 360000000:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 4.4 \cdot 10^{+136}:\\ \;\;\;\;-0.5 \cdot {re}^{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(2 + {im}^{2}\right)\right)\\ \end{array} \]

Alternative 11: 52.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 380000000:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot {re}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 380000000.0) (sin re) (* -0.5 (pow re 2.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 380000000.0) {
		tmp = sin(re);
	} else {
		tmp = -0.5 * 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 <= 380000000.0d0) then
        tmp = sin(re)
    else
        tmp = (-0.5d0) * (re ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 380000000.0) {
		tmp = Math.sin(re);
	} else {
		tmp = -0.5 * Math.pow(re, 2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 380000000.0:
		tmp = math.sin(re)
	else:
		tmp = -0.5 * math.pow(re, 2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 380000000.0)
		tmp = sin(re);
	else
		tmp = Float64(-0.5 * (re ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 380000000.0)
		tmp = sin(re);
	else
		tmp = -0.5 * (re ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 380000000.0], N[Sin[re], $MachinePrecision], N[(-0.5 * N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 3.8e8

   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. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. cancel-sign-sub 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. neg-mul-1 100.0%

         \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 - \color{blue}{-1 \cdot \left(e^{0 - im} \cdot 0.5\right)}\right) \]

      9. *-commutative 100.0%

         \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 - \color{blue}{\left(e^{0 - im} \cdot 0.5\right) \cdot -1}\right) \]

      10. sub-neg 100.0%

          \[\leadsto \sin re \cdot \color{blue}{\left(e^{im} \cdot 0.5 + \left(-\left(e^{0 - im} \cdot 0.5\right) \cdot -1\right)\right)} \]

      11. *-commutative 100.0%

          \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 + \left(-\color{blue}{-1 \cdot \left(e^{0 - im} \cdot 0.5\right)}\right)\right) \]

      12. neg-mul-1 100.0%

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

      13. remove-double-neg 100.0%

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

      14. distribute-rgt-in 100.0%

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

      15. distribute-lft-in 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 61.2%

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

   if 3.8e8 < 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-lft-in 100.0%

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

      2. *-commutative 100.0%

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

      3. cancel-sign-sub 100.0%

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

      4. distribute-lft-neg-out 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-out 100.0%

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

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. 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. Taylor expanded in im around 0 61.2%

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

      1. +-commutative 61.2%

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

      2. unpow2 61.2%

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

      3. fma-def 61.2%

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

   6. Simplified 61.2%

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

   8. Applied egg-rr 1.7%

      \[\leadsto \color{blue}{-1 - -1 \cdot \cos re} \]

   9. Taylor expanded in re around 0 15.2%

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

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 380000000:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot {re}^{2}\\ \end{array} \]

Alternative 12: 50.2% accurate, 3.1× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 (sin re))

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return math.sin(re)

Julia

function code(re, im)
	return sin(re)
end

MATLAB

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

Wolfram

code[re_, im_] := N[Sin[re], $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. distribute-rgt-in 100.0%

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

   2. +-commutative 100.0%

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

   3. associate-*r* 100.0%

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

   4. associate-*r* 100.0%

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

   5. distribute-rgt-out 100.0%

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

   6. cancel-sign-sub 100.0%

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

   7. distribute-lft-neg-in 100.0%

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

   8. neg-mul-1 100.0%

      \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 - \color{blue}{-1 \cdot \left(e^{0 - im} \cdot 0.5\right)}\right) \]

   9. *-commutative 100.0%

      \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 - \color{blue}{\left(e^{0 - im} \cdot 0.5\right) \cdot -1}\right) \]

   10. sub-neg 100.0%

       \[\leadsto \sin re \cdot \color{blue}{\left(e^{im} \cdot 0.5 + \left(-\left(e^{0 - im} \cdot 0.5\right) \cdot -1\right)\right)} \]

   11. *-commutative 100.0%

       \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 + \left(-\color{blue}{-1 \cdot \left(e^{0 - im} \cdot 0.5\right)}\right)\right) \]

   12. neg-mul-1 100.0%

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

   13. remove-double-neg 100.0%

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

   14. distribute-rgt-in 100.0%

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

   15. distribute-lft-in 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in im around 0 46.6%

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

5. Final simplification 46.6%

   \[\leadsto \sin re \]

Alternative 13: 26.0% accurate, 61.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. distribute-lft-in 100.0%

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

   2. *-commutative 100.0%

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

   3. cancel-sign-sub 100.0%

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

   4. distribute-lft-neg-out 100.0%

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

   5. *-commutative 100.0%

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

   6. distribute-rgt-neg-out 100.0%

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

   7. neg-mul-1 100.0%

      \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

   8. associate-*r* 100.0%

      \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

   9. distribute-rgt-out-- 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

   10. sub-neg 100.0%

       \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

   11. neg-sub0 100.0%

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

   12. *-commutative 100.0%

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

   13. neg-mul-1 100.0%

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

   14. 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. Taylor expanded in re around 0 62.9%

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

6. Applied egg-rr 27.0%

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

7. Final simplification 27.0%

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

Alternative 14: 6.7% accurate, 103.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. distribute-lft-in 100.0%

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

   2. *-commutative 100.0%

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

   3. cancel-sign-sub 100.0%

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

   4. distribute-lft-neg-out 100.0%

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

   5. *-commutative 100.0%

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

   6. distribute-rgt-neg-out 100.0%

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

   7. neg-mul-1 100.0%

      \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

   8. associate-*r* 100.0%

      \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

   9. distribute-rgt-out-- 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

   10. sub-neg 100.0%

       \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

   11. neg-sub0 100.0%

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

   12. *-commutative 100.0%

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

   13. neg-mul-1 100.0%

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

   14. 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. Taylor expanded in re around 0 62.9%

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

6. Applied egg-rr 17.6%

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

7. Taylor expanded in re around 0 6.9%

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

8. Final simplification 6.9%

   \[\leadsto re \cdot 0.5 \]

Reproduce

herbie shell --seed 2023320 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))