math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 9.2s

Alternatives: 17

Speedup: 1.0×

• Unsound rule application detected in e-graph. Results may not be sound.

• 50.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp((0.0d0 - im)) + exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (Math.exp((0.0 - im)) + Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * (math.exp((0.0 - im)) + math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
end

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp((0.0d0 - im)) + exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (Math.exp((0.0 - im)) + Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * (math.exp((0.0 - im)) + math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
end

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

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(sin(re) * Float64(0.5 * Float64(exp(Float64(-im)) + exp(im))))
end

MATLAB

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

Wolfram

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

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

   2. associate-*l* 100.0%

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

   3. sub0-neg 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 96.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{if}\;im \leq -2.55 \cdot 10^{+77}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -0.011:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(re, e^{im}, \frac{re}{e^{im}}\right)\\ \mathbf{elif}\;im \leq 0.039:\\ \;\;\;\;\sin re \cdot \left(1 + im \cdot \left(0.5 \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(re \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (*
          (sin re)
          (+ 1.0 (* (* im im) (+ 0.5 (* (* im im) 0.041666666666666664)))))))
   (if (<= im -2.55e+77)
     t_0
     (if (<= im -0.011)
       (* 0.5 (fma re (exp im) (/ re (exp im))))
       (if (<= im 0.039)
         (* (sin re) (+ 1.0 (* im (* 0.5 im))))
         (if (<= im 2.6e+77) (* (+ (exp (- im)) (exp im)) (* re 0.5)) t_0))))))

C

double code(double re, double im) {
	double t_0 = sin(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	double tmp;
	if (im <= -2.55e+77) {
		tmp = t_0;
	} else if (im <= -0.011) {
		tmp = 0.5 * fma(re, exp(im), (re / exp(im)));
	} else if (im <= 0.039) {
		tmp = sin(re) * (1.0 + (im * (0.5 * im)));
	} else if (im <= 2.6e+77) {
		tmp = (exp(-im) + exp(im)) * (re * 0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(sin(re) * Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(Float64(im * im) * 0.041666666666666664)))))
	tmp = 0.0
	if (im <= -2.55e+77)
		tmp = t_0;
	elseif (im <= -0.011)
		tmp = Float64(0.5 * fma(re, exp(im), Float64(re / exp(im))));
	elseif (im <= 0.039)
		tmp = Float64(sin(re) * Float64(1.0 + Float64(im * Float64(0.5 * im))));
	elseif (im <= 2.6e+77)
		tmp = Float64(Float64(exp(Float64(-im)) + exp(im)) * Float64(re * 0.5));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Sin[re], $MachinePrecision] * N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -2.55e+77], t$95$0, If[LessEqual[im, -0.011], N[(0.5 * N[(re * N[Exp[im], $MachinePrecision] + N[(re / N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 0.039], N[(N[Sin[re], $MachinePrecision] * N[(1.0 + N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.6e+77], N[(N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(re * 0.5), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -2.54999999999999985e77 or 2.6000000000000002e77 < im

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

      1. *-rgt-identity 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*r* 100.0%

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

      6. distribute-rgt-out 100.0%

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

      7. distribute-lft-out 100.0%

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

      8. metadata-eval 100.0%

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

      9. pow-sqr 100.0%

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

      10. associate-*r* 100.0%

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

      11. distribute-rgt-out 100.0%

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

      12. unpow2 100.0%

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

      13. unpow2 100.0%

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

   6. Simplified 100.0%

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

   if -2.54999999999999985e77 < im < -0.010999999999999999

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*l* 99.9%

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

      3. sub0-neg 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in re around 0 71.4%

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

      1. distribute-lft-in 71.4%

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

      2. fma-def 71.4%

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

      3. exp-neg 71.4%

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

      4. associate-*r/ 71.4%

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

      5. *-rgt-identity 71.4%

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

   6. Simplified 71.4%

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

   if -0.010999999999999999 < im < 0.0389999999999999999

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

      3. distribute-rgt1-in 100.0%

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

      4. *-commutative 100.0%

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

      5. unpow2 100.0%

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

      6. associate-*l* 100.0%

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

   6. Simplified 100.0%

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

   if 0.0389999999999999999 < im < 2.6000000000000002e77

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 92.9%

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

      1. associate-*r* 92.9%

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

   6. Simplified 92.9%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.55 \cdot 10^{+77}:\\ \;\;\;\;\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{elif}\;im \leq -0.011:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(re, e^{im}, \frac{re}{e^{im}}\right)\\ \mathbf{elif}\;im \leq 0.039:\\ \;\;\;\;\sin re \cdot \left(1 + im \cdot \left(0.5 \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(re \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \]

Alternative 3: 96.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{-im} + e^{im}\right) \cdot \left(re \cdot 0.5\right)\\ t_1 := \sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{if}\;im \leq -2.55 \cdot 10^{+77}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.0125:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.022:\\ \;\;\;\;\sin re \cdot \left(1 + im \cdot \left(0.5 \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (+ (exp (- im)) (exp im)) (* re 0.5)))
        (t_1
         (*
          (sin re)
          (+ 1.0 (* (* im im) (+ 0.5 (* (* im im) 0.041666666666666664)))))))
   (if (<= im -2.55e+77)
     t_1
     (if (<= im -0.0125)
       t_0
       (if (<= im 0.022)
         (* (sin re) (+ 1.0 (* im (* 0.5 im))))
         (if (<= im 2.6e+77) t_0 t_1))))))

C

double code(double re, double im) {
	double t_0 = (exp(-im) + exp(im)) * (re * 0.5);
	double t_1 = sin(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	double tmp;
	if (im <= -2.55e+77) {
		tmp = t_1;
	} else if (im <= -0.0125) {
		tmp = t_0;
	} else if (im <= 0.022) {
		tmp = sin(re) * (1.0 + (im * (0.5 * im)));
	} else if (im <= 2.6e+77) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (exp(-im) + exp(im)) * (re * 0.5d0)
    t_1 = sin(re) * (1.0d0 + ((im * im) * (0.5d0 + ((im * im) * 0.041666666666666664d0))))
    if (im <= (-2.55d+77)) then
        tmp = t_1
    else if (im <= (-0.0125d0)) then
        tmp = t_0
    else if (im <= 0.022d0) then
        tmp = sin(re) * (1.0d0 + (im * (0.5d0 * im)))
    else if (im <= 2.6d+77) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (Math.exp(-im) + Math.exp(im)) * (re * 0.5);
	double t_1 = Math.sin(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	double tmp;
	if (im <= -2.55e+77) {
		tmp = t_1;
	} else if (im <= -0.0125) {
		tmp = t_0;
	} else if (im <= 0.022) {
		tmp = Math.sin(re) * (1.0 + (im * (0.5 * im)));
	} else if (im <= 2.6e+77) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (math.exp(-im) + math.exp(im)) * (re * 0.5)
	t_1 = math.sin(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))))
	tmp = 0
	if im <= -2.55e+77:
		tmp = t_1
	elif im <= -0.0125:
		tmp = t_0
	elif im <= 0.022:
		tmp = math.sin(re) * (1.0 + (im * (0.5 * im)))
	elif im <= 2.6e+77:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(exp(Float64(-im)) + exp(im)) * Float64(re * 0.5))
	t_1 = Float64(sin(re) * Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(Float64(im * im) * 0.041666666666666664)))))
	tmp = 0.0
	if (im <= -2.55e+77)
		tmp = t_1;
	elseif (im <= -0.0125)
		tmp = t_0;
	elseif (im <= 0.022)
		tmp = Float64(sin(re) * Float64(1.0 + Float64(im * Float64(0.5 * im))));
	elseif (im <= 2.6e+77)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (exp(-im) + exp(im)) * (re * 0.5);
	t_1 = sin(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	tmp = 0.0;
	if (im <= -2.55e+77)
		tmp = t_1;
	elseif (im <= -0.0125)
		tmp = t_0;
	elseif (im <= 0.022)
		tmp = sin(re) * (1.0 + (im * (0.5 * im)));
	elseif (im <= 2.6e+77)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[re], $MachinePrecision] * N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -2.55e+77], t$95$1, If[LessEqual[im, -0.0125], t$95$0, If[LessEqual[im, 0.022], N[(N[Sin[re], $MachinePrecision] * N[(1.0 + N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.6e+77], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -2.54999999999999985e77 or 2.6000000000000002e77 < im

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

      1. *-rgt-identity 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*r* 100.0%

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

      6. distribute-rgt-out 100.0%

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

      7. distribute-lft-out 100.0%

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

      8. metadata-eval 100.0%

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

      9. pow-sqr 100.0%

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

      10. associate-*r* 100.0%

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

      11. distribute-rgt-out 100.0%

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

      12. unpow2 100.0%

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

      13. unpow2 100.0%

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

   6. Simplified 100.0%

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

   if -2.54999999999999985e77 < im < -0.012500000000000001 or 0.021999999999999999 < im < 2.6000000000000002e77

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 78.5%

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

      1. associate-*r* 78.5%

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

   6. Simplified 78.5%

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

   if -0.012500000000000001 < im < 0.021999999999999999

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

      3. distribute-rgt1-in 100.0%

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

      4. *-commutative 100.0%

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

      5. unpow2 100.0%

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

      6. associate-*l* 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.55 \cdot 10^{+77}:\\ \;\;\;\;\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{elif}\;im \leq -0.0125:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(re \cdot 0.5\right)\\ \mathbf{elif}\;im \leq 0.022:\\ \;\;\;\;\sin re \cdot \left(1 + im \cdot \left(0.5 \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(re \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \]

Alternative 4: 85.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(\sin re \cdot \left(im \cdot im\right)\right)\\ t_1 := re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{if}\;im \leq -2.35 \cdot 10^{+156}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -0.00075:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 370:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 3.6 \cdot 10^{+137}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (* (sin re) (* im im))))
        (t_1
         (*
          re
          (+ 1.0 (* (* im im) (+ 0.5 (* (* im im) 0.041666666666666664)))))))
   (if (<= im -2.35e+156)
     t_0
     (if (<= im -0.00075)
       t_1
       (if (<= im 370.0) (sin re) (if (<= im 3.6e+137) t_1 t_0))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (sin(re) * (im * im));
	double t_1 = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	double tmp;
	if (im <= -2.35e+156) {
		tmp = t_0;
	} else if (im <= -0.00075) {
		tmp = t_1;
	} else if (im <= 370.0) {
		tmp = sin(re);
	} else if (im <= 3.6e+137) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * (sin(re) * (im * im))
    t_1 = re * (1.0d0 + ((im * im) * (0.5d0 + ((im * im) * 0.041666666666666664d0))))
    if (im <= (-2.35d+156)) then
        tmp = t_0
    else if (im <= (-0.00075d0)) then
        tmp = t_1
    else if (im <= 370.0d0) then
        tmp = sin(re)
    else if (im <= 3.6d+137) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * (Math.sin(re) * (im * im));
	double t_1 = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	double tmp;
	if (im <= -2.35e+156) {
		tmp = t_0;
	} else if (im <= -0.00075) {
		tmp = t_1;
	} else if (im <= 370.0) {
		tmp = Math.sin(re);
	} else if (im <= 3.6e+137) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * (math.sin(re) * (im * im))
	t_1 = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))))
	tmp = 0
	if im <= -2.35e+156:
		tmp = t_0
	elif im <= -0.00075:
		tmp = t_1
	elif im <= 370.0:
		tmp = math.sin(re)
	elif im <= 3.6e+137:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(sin(re) * Float64(im * im)))
	t_1 = Float64(re * Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(Float64(im * im) * 0.041666666666666664)))))
	tmp = 0.0
	if (im <= -2.35e+156)
		tmp = t_0;
	elseif (im <= -0.00075)
		tmp = t_1;
	elseif (im <= 370.0)
		tmp = sin(re);
	elseif (im <= 3.6e+137)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (sin(re) * (im * im));
	t_1 = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	tmp = 0.0;
	if (im <= -2.35e+156)
		tmp = t_0;
	elseif (im <= -0.00075)
		tmp = t_1;
	elseif (im <= 370.0)
		tmp = sin(re);
	elseif (im <= 3.6e+137)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(N[Sin[re], $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(re * N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -2.35e+156], t$95$0, If[LessEqual[im, -0.00075], t$95$1, If[LessEqual[im, 370.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 3.6e+137], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -2.35e156 or 3.6e137 < im

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 98.5%

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

      1. *-commutative 98.5%

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

      2. associate-*r* 98.5%

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

      3. distribute-rgt1-in 98.5%

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

      4. *-commutative 98.5%

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

      5. unpow2 98.5%

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

      6. associate-*l* 98.5%

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

   6. Simplified 98.5%

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

   7. Taylor expanded in im around inf 98.5%

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

      1. *-commutative 98.5%

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

      2. unpow2 98.5%

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

   9. Simplified 98.5%

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

   if -2.35e156 < im < -7.5000000000000002e-4 or 370 < im < 3.6e137

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 40.5%

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

      1. *-rgt-identity 40.5%

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

      2. *-commutative 40.5%

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

      3. associate-*r* 40.5%

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

      4. *-commutative 40.5%

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

      5. associate-*r* 40.5%

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

      6. distribute-rgt-out 40.5%

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

      7. distribute-lft-out 40.5%

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

      8. metadata-eval 40.5%

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

      9. pow-sqr 40.5%

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

      10. associate-*r* 40.5%

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

      11. distribute-rgt-out 40.5%

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

      12. unpow2 40.5%

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

      13. unpow2 40.5%

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

   6. Simplified 40.5%

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

   7. Taylor expanded in re around 0 39.2%

      \[\leadsto \color{blue}{\left(1 + \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right) \cdot {im}^{2}\right) \cdot re} \]
   8. Step-by-step derivation

      1. *-commutative 39.2%

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

      2. unpow2 39.2%

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

      3. unpow2 39.2%

         \[\leadsto \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \cdot re \]

   9. Simplified 39.2%

      \[\leadsto \color{blue}{\left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right) \cdot re} \]

   if -7.5000000000000002e-4 < im < 370

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 99.8%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.35 \cdot 10^{+156}:\\ \;\;\;\;0.5 \cdot \left(\sin re \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq -0.00075:\\ \;\;\;\;re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{elif}\;im \leq 370:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 3.6 \cdot 10^{+137}:\\ \;\;\;\;re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sin re \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 5: 85.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(\sin re \cdot \left(im \cdot im\right)\right)\\ t_1 := re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{if}\;im \leq -2.35 \cdot 10^{+156}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -0.014:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 950:\\ \;\;\;\;\sin re \cdot \left(1 + im \cdot \left(0.5 \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 3.6 \cdot 10^{+137}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (* (sin re) (* im im))))
        (t_1
         (*
          re
          (+ 1.0 (* (* im im) (+ 0.5 (* (* im im) 0.041666666666666664)))))))
   (if (<= im -2.35e+156)
     t_0
     (if (<= im -0.014)
       t_1
       (if (<= im 950.0)
         (* (sin re) (+ 1.0 (* im (* 0.5 im))))
         (if (<= im 3.6e+137) t_1 t_0))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (sin(re) * (im * im));
	double t_1 = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	double tmp;
	if (im <= -2.35e+156) {
		tmp = t_0;
	} else if (im <= -0.014) {
		tmp = t_1;
	} else if (im <= 950.0) {
		tmp = sin(re) * (1.0 + (im * (0.5 * im)));
	} else if (im <= 3.6e+137) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * (sin(re) * (im * im))
    t_1 = re * (1.0d0 + ((im * im) * (0.5d0 + ((im * im) * 0.041666666666666664d0))))
    if (im <= (-2.35d+156)) then
        tmp = t_0
    else if (im <= (-0.014d0)) then
        tmp = t_1
    else if (im <= 950.0d0) then
        tmp = sin(re) * (1.0d0 + (im * (0.5d0 * im)))
    else if (im <= 3.6d+137) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * (Math.sin(re) * (im * im));
	double t_1 = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	double tmp;
	if (im <= -2.35e+156) {
		tmp = t_0;
	} else if (im <= -0.014) {
		tmp = t_1;
	} else if (im <= 950.0) {
		tmp = Math.sin(re) * (1.0 + (im * (0.5 * im)));
	} else if (im <= 3.6e+137) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * (math.sin(re) * (im * im))
	t_1 = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))))
	tmp = 0
	if im <= -2.35e+156:
		tmp = t_0
	elif im <= -0.014:
		tmp = t_1
	elif im <= 950.0:
		tmp = math.sin(re) * (1.0 + (im * (0.5 * im)))
	elif im <= 3.6e+137:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(sin(re) * Float64(im * im)))
	t_1 = Float64(re * Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(Float64(im * im) * 0.041666666666666664)))))
	tmp = 0.0
	if (im <= -2.35e+156)
		tmp = t_0;
	elseif (im <= -0.014)
		tmp = t_1;
	elseif (im <= 950.0)
		tmp = Float64(sin(re) * Float64(1.0 + Float64(im * Float64(0.5 * im))));
	elseif (im <= 3.6e+137)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (sin(re) * (im * im));
	t_1 = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	tmp = 0.0;
	if (im <= -2.35e+156)
		tmp = t_0;
	elseif (im <= -0.014)
		tmp = t_1;
	elseif (im <= 950.0)
		tmp = sin(re) * (1.0 + (im * (0.5 * im)));
	elseif (im <= 3.6e+137)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(N[Sin[re], $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(re * N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -2.35e+156], t$95$0, If[LessEqual[im, -0.014], t$95$1, If[LessEqual[im, 950.0], N[(N[Sin[re], $MachinePrecision] * N[(1.0 + N[(im * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 3.6e+137], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -2.35e156 or 3.6e137 < im

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 98.5%

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

      1. *-commutative 98.5%

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

      2. associate-*r* 98.5%

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

      3. distribute-rgt1-in 98.5%

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

      4. *-commutative 98.5%

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

      5. unpow2 98.5%

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

      6. associate-*l* 98.5%

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

   6. Simplified 98.5%

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

   7. Taylor expanded in im around inf 98.5%

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

      1. *-commutative 98.5%

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

      2. unpow2 98.5%

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

   9. Simplified 98.5%

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

   if -2.35e156 < im < -0.0140000000000000003 or 950 < im < 3.6e137

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 40.5%

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

      1. *-rgt-identity 40.5%

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

      2. *-commutative 40.5%

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

      3. associate-*r* 40.5%

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

      4. *-commutative 40.5%

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

      5. associate-*r* 40.5%

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

      6. distribute-rgt-out 40.5%

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

      7. distribute-lft-out 40.5%

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

      8. metadata-eval 40.5%

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

      9. pow-sqr 40.5%

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

      10. associate-*r* 40.5%

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

      11. distribute-rgt-out 40.5%

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

      12. unpow2 40.5%

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

      13. unpow2 40.5%

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

   6. Simplified 40.5%

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

   7. Taylor expanded in re around 0 39.2%

      \[\leadsto \color{blue}{\left(1 + \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right) \cdot {im}^{2}\right) \cdot re} \]
   8. Step-by-step derivation

      1. *-commutative 39.2%

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

      2. unpow2 39.2%

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

      3. unpow2 39.2%

         \[\leadsto \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \cdot re \]

   9. Simplified 39.2%

      \[\leadsto \color{blue}{\left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right) \cdot re} \]

   if -0.0140000000000000003 < im < 950

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

      3. distribute-rgt1-in 100.0%

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

      4. *-commutative 100.0%

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

      5. unpow2 100.0%

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

      6. associate-*l* 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.35 \cdot 10^{+156}:\\ \;\;\;\;0.5 \cdot \left(\sin re \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq -0.014:\\ \;\;\;\;re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{elif}\;im \leq 950:\\ \;\;\;\;\sin re \cdot \left(1 + im \cdot \left(0.5 \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 3.6 \cdot 10^{+137}:\\ \;\;\;\;re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sin re \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 6: 88.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (sin re) (+ 1.0 (* (* im im) (+ 0.5 (* (* im im) 0.041666666666666664))))))

C

double code(double re, double im) {
	return sin(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = sin(re) * (1.0d0 + ((im * im) * (0.5d0 + ((im * im) * 0.041666666666666664d0))))
end function

Java

public static double code(double re, double im) {
	return Math.sin(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
}

Python

def code(re, im):
	return math.sin(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))))

Julia

function code(re, im)
	return Float64(sin(re) * Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(Float64(im * im) * 0.041666666666666664)))))
end

MATLAB

function tmp = code(re, im)
	tmp = sin(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
end

Wolfram

code[re_, im_] := N[(N[Sin[re], $MachinePrecision] * N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. *-commutative 100.0%

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

   2. associate-*l* 100.0%

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

   3. sub0-neg 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in im around 0 84.6%

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

   1. *-rgt-identity 84.6%

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

   2. *-commutative 84.6%

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

   3. associate-*r* 84.6%

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

   4. *-commutative 84.6%

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

   5. associate-*r* 84.6%

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

   6. distribute-rgt-out 84.6%

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

   7. distribute-lft-out 84.6%

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

   8. metadata-eval 84.6%

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

   9. pow-sqr 84.6%

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

   10. associate-*r* 84.6%

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

   11. distribute-rgt-out 84.6%

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

   12. unpow2 84.6%

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

   13. unpow2 84.6%

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

6. Simplified 84.6%

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

7. Final simplification 84.6%

   \[\leadsto \sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right) \]

Alternative 7: 80.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -2.3 \cdot 10^{+243}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(\sin re \cdot im\right)\right)\\ \mathbf{elif}\;im \leq -0.00014 \lor \neg \left(im \leq 400\right):\\ \;\;\;\;re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -2.3e+243)
   (* 0.5 (* im (* (sin re) im)))
   (if (or (<= im -0.00014) (not (<= im 400.0)))
     (* re (+ 1.0 (* (* im im) (+ 0.5 (* (* im im) 0.041666666666666664)))))
     (sin re))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -2.3e+243) {
		tmp = 0.5 * (im * (sin(re) * im));
	} else if ((im <= -0.00014) || !(im <= 400.0)) {
		tmp = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	} else {
		tmp = sin(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= (-2.3d+243)) then
        tmp = 0.5d0 * (im * (sin(re) * im))
    else if ((im <= (-0.00014d0)) .or. (.not. (im <= 400.0d0))) then
        tmp = re * (1.0d0 + ((im * im) * (0.5d0 + ((im * im) * 0.041666666666666664d0))))
    else
        tmp = sin(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -2.3e+243) {
		tmp = 0.5 * (im * (Math.sin(re) * im));
	} else if ((im <= -0.00014) || !(im <= 400.0)) {
		tmp = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	} else {
		tmp = Math.sin(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -2.3e+243:
		tmp = 0.5 * (im * (math.sin(re) * im))
	elif (im <= -0.00014) or not (im <= 400.0):
		tmp = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))))
	else:
		tmp = math.sin(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -2.3e+243)
		tmp = Float64(0.5 * Float64(im * Float64(sin(re) * im)));
	elseif ((im <= -0.00014) || !(im <= 400.0))
		tmp = Float64(re * Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(Float64(im * im) * 0.041666666666666664)))));
	else
		tmp = sin(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -2.3e+243)
		tmp = 0.5 * (im * (sin(re) * im));
	elseif ((im <= -0.00014) || ~((im <= 400.0)))
		tmp = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	else
		tmp = sin(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -2.3e+243], N[(0.5 * N[(im * N[(N[Sin[re], $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[im, -0.00014], N[Not[LessEqual[im, 400.0]], $MachinePrecision]], N[(re * N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < -2.29999999999999987e243

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

      3. distribute-rgt1-in 100.0%

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

      4. *-commutative 100.0%

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

      5. unpow2 100.0%

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

      6. associate-*l* 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in im around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. unpow2 100.0%

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

      3. associate-*l* 100.0%

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

   9. Simplified 100.0%

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

   if -2.29999999999999987e243 < im < -1.3999999999999999e-4 or 400 < im

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 65.5%

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

      1. *-rgt-identity 65.5%

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

      2. *-commutative 65.5%

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

      3. associate-*r* 65.5%

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

      4. *-commutative 65.5%

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

      5. associate-*r* 65.5%

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

      6. distribute-rgt-out 65.5%

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

      7. distribute-lft-out 65.5%

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

      8. metadata-eval 65.5%

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

      9. pow-sqr 65.5%

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

      10. associate-*r* 65.5%

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

      11. distribute-rgt-out 65.5%

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

      12. unpow2 65.5%

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

      13. unpow2 65.5%

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

   6. Simplified 65.5%

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

   7. Taylor expanded in re around 0 54.2%

      \[\leadsto \color{blue}{\left(1 + \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right) \cdot {im}^{2}\right) \cdot re} \]
   8. Step-by-step derivation

      1. *-commutative 54.2%

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

      2. unpow2 54.2%

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

      3. unpow2 54.2%

         \[\leadsto \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \cdot re \]

   9. Simplified 54.2%

      \[\leadsto \color{blue}{\left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right) \cdot re} \]

   if -1.3999999999999999e-4 < im < 400

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 99.8%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.3 \cdot 10^{+243}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(\sin re \cdot im\right)\right)\\ \mathbf{elif}\;im \leq -0.00014 \lor \neg \left(im \leq 400\right):\\ \;\;\;\;re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re\\ \end{array} \]

Alternative 8: 80.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -0.00037 \lor \neg \left(im \leq 250\right):\\ \;\;\;\;re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -0.00037) (not (<= im 250.0)))
   (* re (+ 1.0 (* (* im im) (+ 0.5 (* (* im im) 0.041666666666666664)))))
   (sin re)))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -0.00037) || !(im <= 250.0)) {
		tmp = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	} else {
		tmp = sin(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-0.00037d0)) .or. (.not. (im <= 250.0d0))) then
        tmp = re * (1.0d0 + ((im * im) * (0.5d0 + ((im * im) * 0.041666666666666664d0))))
    else
        tmp = sin(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -0.00037) || !(im <= 250.0)) {
		tmp = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	} else {
		tmp = Math.sin(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -0.00037) or not (im <= 250.0):
		tmp = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))))
	else:
		tmp = math.sin(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -0.00037) || !(im <= 250.0))
		tmp = Float64(re * Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(Float64(im * im) * 0.041666666666666664)))));
	else
		tmp = sin(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -0.00037) || ~((im <= 250.0)))
		tmp = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	else
		tmp = sin(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -0.00037], N[Not[LessEqual[im, 250.0]], $MachinePrecision]], N[(re * N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -3.6999999999999999e-4 or 250 < im

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 69.5%

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

      1. *-rgt-identity 69.5%

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

      2. *-commutative 69.5%

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

      3. associate-*r* 69.5%

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

      4. *-commutative 69.5%

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

      5. associate-*r* 69.5%

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

      6. distribute-rgt-out 69.5%

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

      7. distribute-lft-out 69.5%

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

      8. metadata-eval 69.5%

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

      9. pow-sqr 69.5%

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

      10. associate-*r* 69.5%

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

      11. distribute-rgt-out 69.5%

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

      12. unpow2 69.5%

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

      13. unpow2 69.5%

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

   6. Simplified 69.5%

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

   7. Taylor expanded in re around 0 55.7%

      \[\leadsto \color{blue}{\left(1 + \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right) \cdot {im}^{2}\right) \cdot re} \]
   8. Step-by-step derivation

      1. *-commutative 55.7%

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

      2. unpow2 55.7%

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

      3. unpow2 55.7%

         \[\leadsto \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \cdot re \]

   9. Simplified 55.7%

      \[\leadsto \color{blue}{\left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right) \cdot re} \]

   if -3.6999999999999999e-4 < im < 250

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 99.8%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -0.00037 \lor \neg \left(im \leq 250\right):\\ \;\;\;\;re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re\\ \end{array} \]

Alternative 9: 56.6% accurate, 20.6× speedup

Math

\[\begin{array}{l} \\ re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* re (+ 1.0 (* (* im im) (+ 0.5 (* (* im im) 0.041666666666666664))))))

C

double code(double re, double im) {
	return re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = re * (1.0d0 + ((im * im) * (0.5d0 + ((im * im) * 0.041666666666666664d0))))
end function

Java

public static double code(double re, double im) {
	return re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
}

Python

def code(re, im):
	return re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))))

Julia

function code(re, im)
	return Float64(re * Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(Float64(im * im) * 0.041666666666666664)))))
end

MATLAB

function tmp = code(re, im)
	tmp = re * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
end

Wolfram

code[re_, im_] := N[(re * N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. *-commutative 100.0%

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

   2. associate-*l* 100.0%

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

   3. sub0-neg 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in im around 0 84.6%

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

   1. *-rgt-identity 84.6%

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

   2. *-commutative 84.6%

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

   3. associate-*r* 84.6%

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

   4. *-commutative 84.6%

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

   5. associate-*r* 84.6%

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

   6. distribute-rgt-out 84.6%

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

   7. distribute-lft-out 84.6%

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

   8. metadata-eval 84.6%

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

   9. pow-sqr 84.6%

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

   10. associate-*r* 84.6%

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

   11. distribute-rgt-out 84.6%

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

   12. unpow2 84.6%

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

   13. unpow2 84.6%

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

6. Simplified 84.6%

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

7. Taylor expanded in re around 0 53.4%

   \[\leadsto \color{blue}{\left(1 + \left(0.5 + 0.041666666666666664 \cdot {im}^{2}\right) \cdot {im}^{2}\right) \cdot re} \]
8. Step-by-step derivation

   1. *-commutative 53.4%

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

   2. unpow2 53.4%

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

   3. unpow2 53.4%

      \[\leadsto \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \color{blue}{\left(im \cdot im\right)}\right)\right) \cdot re \]

9. Simplified 53.4%

   \[\leadsto \color{blue}{\left(1 + \left(im \cdot im\right) \cdot \left(0.5 + 0.041666666666666664 \cdot \left(im \cdot im\right)\right)\right) \cdot re} \]

10. Final simplification 53.4%

    \[\leadsto re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right) \]

Alternative 10: 41.1% accurate, 27.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1.42 \lor \neg \left(im \leq 1.28 \cdot 10^{-33}\right):\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(re \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -1.42) (not (<= im 1.28e-33))) (* 0.5 (* im (* re im))) re))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -1.42) || !(im <= 1.28e-33)) {
		tmp = 0.5 * (im * (re * im));
	} else {
		tmp = re;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-1.42d0)) .or. (.not. (im <= 1.28d-33))) then
        tmp = 0.5d0 * (im * (re * im))
    else
        tmp = re
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -1.42) || !(im <= 1.28e-33)) {
		tmp = 0.5 * (im * (re * im));
	} else {
		tmp = re;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -1.42) or not (im <= 1.28e-33):
		tmp = 0.5 * (im * (re * im))
	else:
		tmp = re
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -1.42) || !(im <= 1.28e-33))
		tmp = Float64(0.5 * Float64(im * Float64(re * im)));
	else
		tmp = re;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -1.42) || ~((im <= 1.28e-33)))
		tmp = 0.5 * (im * (re * im));
	else
		tmp = re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -1.42], N[Not[LessEqual[im, 1.28e-33]], $MachinePrecision]], N[(0.5 * N[(im * N[(re * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], re]

Derivation

1. Split input into 2 regimes

2. if im < -1.4199999999999999 or 1.28000000000000001e-33 < im

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 52.7%

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

      1. *-commutative 52.7%

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

      2. associate-*r* 52.7%

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

      3. distribute-rgt1-in 52.7%

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

      4. *-commutative 52.7%

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

      5. unpow2 52.7%

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

      6. associate-*l* 52.7%

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

   6. Simplified 52.7%

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

   7. Taylor expanded in im around inf 49.9%

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

      1. *-commutative 49.9%

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

      2. unpow2 49.9%

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

      3. associate-*l* 36.4%

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

   9. Simplified 36.4%

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

   10. Taylor expanded in re around 0 29.4%

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

   if -1.4199999999999999 < im < 1.28000000000000001e-33

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 53.1%

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

      1. associate-*r* 53.1%

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

   6. Simplified 53.1%

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

   7. Taylor expanded in im around 0 52.6%

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

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.42 \lor \neg \left(im \leq 1.28 \cdot 10^{-33}\right):\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(re \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array} \]

Alternative 11: 47.2% accurate, 27.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1.42 \lor \neg \left(im \leq 1.28 \cdot 10^{-33}\right):\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -1.42) (not (<= im 1.28e-33))) (* 0.5 (* re (* im im))) re))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -1.42) || !(im <= 1.28e-33)) {
		tmp = 0.5 * (re * (im * im));
	} else {
		tmp = re;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-1.42d0)) .or. (.not. (im <= 1.28d-33))) then
        tmp = 0.5d0 * (re * (im * im))
    else
        tmp = re
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -1.42) || !(im <= 1.28e-33)) {
		tmp = 0.5 * (re * (im * im));
	} else {
		tmp = re;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -1.42) or not (im <= 1.28e-33):
		tmp = 0.5 * (re * (im * im))
	else:
		tmp = re
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -1.42) || !(im <= 1.28e-33))
		tmp = Float64(0.5 * Float64(re * Float64(im * im)));
	else
		tmp = re;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -1.42) || ~((im <= 1.28e-33)))
		tmp = 0.5 * (re * (im * im));
	else
		tmp = re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -1.42], N[Not[LessEqual[im, 1.28e-33]], $MachinePrecision]], N[(0.5 * N[(re * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], re]

Derivation

1. Split input into 2 regimes

2. if im < -1.4199999999999999 or 1.28000000000000001e-33 < im

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 52.7%

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

      1. *-commutative 52.7%

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

      2. associate-*r* 52.7%

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

      3. distribute-rgt1-in 52.7%

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

      4. *-commutative 52.7%

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

      5. unpow2 52.7%

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

      6. associate-*l* 52.7%

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

   6. Simplified 52.7%

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

   7. Taylor expanded in im around inf 49.9%

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

      1. *-commutative 49.9%

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

      2. unpow2 49.9%

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

      3. associate-*l* 36.4%

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

   9. Simplified 36.4%

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

   10. Taylor expanded in re around 0 42.9%

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

       1. *-commutative 42.9%

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

       2. unpow2 42.9%

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

   12. Simplified 42.9%

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

   if -1.4199999999999999 < im < 1.28000000000000001e-33

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 53.1%

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

      1. associate-*r* 53.1%

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

   6. Simplified 53.1%

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

   7. Taylor expanded in im around 0 52.6%

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

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.42 \lor \neg \left(im \leq 1.28 \cdot 10^{-33}\right):\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array} \]

Alternative 12: 30.7% accurate, 33.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1800000000000 \lor \neg \left(im \leq 1.28 \cdot 10^{-33}\right):\\ \;\;\;\;\frac{0.25}{re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -1800000000000.0) (not (<= im 1.28e-33)))
   (/ 0.25 (* re re))
   re))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -1800000000000.0) || !(im <= 1.28e-33)) {
		tmp = 0.25 / (re * re);
	} else {
		tmp = re;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-1800000000000.0d0)) .or. (.not. (im <= 1.28d-33))) then
        tmp = 0.25d0 / (re * re)
    else
        tmp = re
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -1800000000000.0) || !(im <= 1.28e-33)) {
		tmp = 0.25 / (re * re);
	} else {
		tmp = re;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -1800000000000.0) or not (im <= 1.28e-33):
		tmp = 0.25 / (re * re)
	else:
		tmp = re
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -1800000000000.0) || !(im <= 1.28e-33))
		tmp = Float64(0.25 / Float64(re * re));
	else
		tmp = re;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -1800000000000.0) || ~((im <= 1.28e-33)))
		tmp = 0.25 / (re * re);
	else
		tmp = re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -1800000000000.0], N[Not[LessEqual[im, 1.28e-33]], $MachinePrecision]], N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision], re]

Derivation

1. Split input into 2 regimes

2. if im < -1.8e12 or 1.28000000000000001e-33 < im

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 13.1%

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

   6. Taylor expanded in re around 0 12.7%

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

      1. unpow2 12.7%

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

   8. Simplified 12.7%

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

   if -1.8e12 < im < 1.28000000000000001e-33

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 54.8%

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

      1. associate-*r* 54.8%

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

   6. Simplified 54.8%

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

   7. Taylor expanded in im around 0 49.9%

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

4. Final simplification 31.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1800000000000 \lor \neg \left(im \leq 1.28 \cdot 10^{-33}\right):\\ \;\;\;\;\frac{0.25}{re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array} \]

Alternative 13: 30.7% accurate, 33.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1800000000000:\\ \;\;\;\;\frac{0.25}{re \cdot re}\\ \mathbf{elif}\;im \leq 1.28 \cdot 10^{-33}:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25}{re}}{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -1800000000000.0)
   (/ 0.25 (* re re))
   (if (<= im 1.28e-33) re (/ (/ 0.25 re) re))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -1800000000000.0) {
		tmp = 0.25 / (re * re);
	} else if (im <= 1.28e-33) {
		tmp = re;
	} else {
		tmp = (0.25 / re) / re;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= (-1800000000000.0d0)) then
        tmp = 0.25d0 / (re * re)
    else if (im <= 1.28d-33) then
        tmp = re
    else
        tmp = (0.25d0 / re) / re
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -1800000000000.0) {
		tmp = 0.25 / (re * re);
	} else if (im <= 1.28e-33) {
		tmp = re;
	} else {
		tmp = (0.25 / re) / re;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -1800000000000.0:
		tmp = 0.25 / (re * re)
	elif im <= 1.28e-33:
		tmp = re
	else:
		tmp = (0.25 / re) / re
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -1800000000000.0)
		tmp = Float64(0.25 / Float64(re * re));
	elseif (im <= 1.28e-33)
		tmp = re;
	else
		tmp = Float64(Float64(0.25 / re) / re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -1800000000000.0)
		tmp = 0.25 / (re * re);
	elseif (im <= 1.28e-33)
		tmp = re;
	else
		tmp = (0.25 / re) / re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -1800000000000.0], N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.28e-33], re, N[(N[(0.25 / re), $MachinePrecision] / re), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.8e12

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 9.0%

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

   6. Taylor expanded in re around 0 9.0%

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

      1. unpow2 9.0%

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

   8. Simplified 9.0%

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

   if -1.8e12 < im < 1.28000000000000001e-33

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 54.8%

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

      1. associate-*r* 54.8%

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

   6. Simplified 54.8%

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

   7. Taylor expanded in im around 0 49.9%

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

   if 1.28000000000000001e-33 < im

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 18.0%

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

   6. Taylor expanded in re around 0 17.2%

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

      1. unpow2 17.2%

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

   8. Simplified 17.2%

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

   9. Taylor expanded in re around 0 17.2%

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

       1. unpow2 17.2%

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

       2. associate-/r* 17.2%

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

   11. Simplified 17.2%

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

4. Final simplification 31.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1800000000000:\\ \;\;\;\;\frac{0.25}{re \cdot re}\\ \mathbf{elif}\;im \leq 1.28 \cdot 10^{-33}:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25}{re}}{re}\\ \end{array} \]

Alternative 14: 48.7% accurate, 34.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(re * N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. *-commutative 100.0%

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

   2. associate-*l* 100.0%

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

   3. sub0-neg 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in im around 0 75.5%

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

   1. *-commutative 75.5%

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

   2. associate-*r* 75.5%

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

   3. distribute-rgt1-in 75.5%

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

   4. *-commutative 75.5%

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

   5. unpow2 75.5%

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

   6. associate-*l* 75.5%

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

6. Simplified 75.5%

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

7. Taylor expanded in re around 0 47.7%

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

   1. *-commutative 47.7%

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

   2. unpow2 47.7%

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

9. Simplified 47.7%

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

10. Final simplification 47.7%

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

Alternative 15: 3.7% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ -512 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 -512.0)

C

double code(double re, double im) {
	return -512.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = -512.0d0
end function

Java

public static double code(double re, double im) {
	return -512.0;
}

Python

def code(re, im):
	return -512.0

Julia

function code(re, im)
	return -512.0
end

MATLAB

function tmp = code(re, im)
	tmp = -512.0;
end

Wolfram

code[re_, im_] := -512.0

Derivation

1. Initial program 100.0%

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

   1. *-commutative 100.0%

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

   2. associate-*l* 100.0%

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

   3. sub0-neg 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in im around 0 84.6%

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

   1. *-rgt-identity 84.6%

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

   2. *-commutative 84.6%

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

   3. associate-*r* 84.6%

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

   4. *-commutative 84.6%

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

   5. associate-*r* 84.6%

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

   6. distribute-rgt-out 84.6%

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

   7. distribute-lft-out 84.6%

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

   8. metadata-eval 84.6%

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

   9. pow-sqr 84.6%

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

   10. associate-*r* 84.6%

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

   11. distribute-rgt-out 84.6%

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

   12. unpow2 84.6%

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

   13. unpow2 84.6%

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

6. Simplified 84.6%

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

8. Applied egg-rr 4.0%

   \[\leadsto \color{blue}{-512} \]

9. Final simplification 4.0%

   \[\leadsto -512 \]

Alternative 16: 4.6% accurate, 309.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return -1.0

Julia

function code(re, im)
	return -1.0
end

MATLAB

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

Wolfram

code[re_, im_] := -1.0

Derivation

1. Initial program 100.0%

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

   1. *-commutative 100.0%

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

   2. associate-*l* 100.0%

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

   3. sub0-neg 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in im around 0 84.6%

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

   1. *-rgt-identity 84.6%

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

   2. *-commutative 84.6%

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

   3. associate-*r* 84.6%

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

   4. *-commutative 84.6%

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

   5. associate-*r* 84.6%

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

   6. distribute-rgt-out 84.6%

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

   7. distribute-lft-out 84.6%

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

   8. metadata-eval 84.6%

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

   9. pow-sqr 84.6%

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

   10. associate-*r* 84.6%

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

   11. distribute-rgt-out 84.6%

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

   12. unpow2 84.6%

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

   13. unpow2 84.6%

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

6. Simplified 84.6%

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

8. Applied egg-rr 5.1%

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

9. Final simplification 5.1%

   \[\leadsto -1 \]

Alternative 17: 26.6% accurate, 309.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 re)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return re

Julia

function code(re, im)
	return re
end

MATLAB

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

Wolfram

code[re_, im_] := re

Derivation

1. Initial program 100.0%

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

   1. *-commutative 100.0%

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

   2. associate-*l* 100.0%

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

   3. sub0-neg 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in re around 0 62.1%

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

   1. associate-*r* 62.1%

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

6. Simplified 62.1%

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

7. Taylor expanded in im around 0 26.8%

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

8. Final simplification 26.8%

   \[\leadsto re \]

Reproduce

herbie shell --seed 2023171 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))