math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 10.6s

Alternatives: 17

Speedup: 1.0×

• 50.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. distribute-lft-in 100.0%

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

   2. +-commutative 100.0%

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

   3. *-commutative 100.0%

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

   4. associate-*l* 100.0%

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

   5. *-commutative 100.0%

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

   6. *-commutative 100.0%

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

   7. associate-*r* 100.0%

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

   8. distribute-rgt-in 100.0%

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

   9. fma-def 100.0%

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

   10. exp-diff 100.0%

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

   11. associate-*l/ 100.0%

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

   12. exp-0 100.0%

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

   13. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 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(im) + exp(Float64(-im)))))
end

MATLAB

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

Wolfram

code[re_, im_] := N[(N[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 3: 93.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{4} \cdot \left(\sin re \cdot 0.041666666666666664\right)\\ \mathbf{if}\;im \leq -4.5 \cdot 10^{+72}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -460:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(re \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq 10:\\ \;\;\;\;\sin re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{+68}:\\ \;\;\;\;re \cdot \left(0.5 + 0.5 \cdot e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (pow im 4.0) (* (sin re) 0.041666666666666664))))
   (if (<= im -4.5e+72)
     t_0
     (if (<= im -460.0)
       (* 0.5 (log1p (expm1 (* re (* im im)))))
       (if (<= im 10.0)
         (* (sin re) (+ 1.0 (* 0.5 (* im im))))
         (if (<= im 4.2e+68) (* re (+ 0.5 (* 0.5 (exp im)))) t_0))))))

C

double code(double re, double im) {
	double t_0 = pow(im, 4.0) * (sin(re) * 0.041666666666666664);
	double tmp;
	if (im <= -4.5e+72) {
		tmp = t_0;
	} else if (im <= -460.0) {
		tmp = 0.5 * log1p(expm1((re * (im * im))));
	} else if (im <= 10.0) {
		tmp = sin(re) * (1.0 + (0.5 * (im * im)));
	} else if (im <= 4.2e+68) {
		tmp = re * (0.5 + (0.5 * exp(im)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = Math.pow(im, 4.0) * (Math.sin(re) * 0.041666666666666664);
	double tmp;
	if (im <= -4.5e+72) {
		tmp = t_0;
	} else if (im <= -460.0) {
		tmp = 0.5 * Math.log1p(Math.expm1((re * (im * im))));
	} else if (im <= 10.0) {
		tmp = Math.sin(re) * (1.0 + (0.5 * (im * im)));
	} else if (im <= 4.2e+68) {
		tmp = re * (0.5 + (0.5 * Math.exp(im)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.pow(im, 4.0) * (math.sin(re) * 0.041666666666666664)
	tmp = 0
	if im <= -4.5e+72:
		tmp = t_0
	elif im <= -460.0:
		tmp = 0.5 * math.log1p(math.expm1((re * (im * im))))
	elif im <= 10.0:
		tmp = math.sin(re) * (1.0 + (0.5 * (im * im)))
	elif im <= 4.2e+68:
		tmp = re * (0.5 + (0.5 * math.exp(im)))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64((im ^ 4.0) * Float64(sin(re) * 0.041666666666666664))
	tmp = 0.0
	if (im <= -4.5e+72)
		tmp = t_0;
	elseif (im <= -460.0)
		tmp = Float64(0.5 * log1p(expm1(Float64(re * Float64(im * im)))));
	elseif (im <= 10.0)
		tmp = Float64(sin(re) * Float64(1.0 + Float64(0.5 * Float64(im * im))));
	elseif (im <= 4.2e+68)
		tmp = Float64(re * Float64(0.5 + Float64(0.5 * exp(im))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Power[im, 4.0], $MachinePrecision] * N[(N[Sin[re], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -4.5e+72], t$95$0, If[LessEqual[im, -460.0], N[(0.5 * N[Log[1 + N[(Exp[N[(re * N[(im * im), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 10.0], N[(N[Sin[re], $MachinePrecision] * N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 4.2e+68], N[(re * N[(0.5 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -4.4999999999999998e72 or 4.20000000000000002e68 < 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 96.6%

      \[\leadsto \sin re \cdot \left(0.5 \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)}\right) \]
   5. Step-by-step derivation

      1. unpow2 96.6%

         \[\leadsto \sin re \cdot \left(0.5 \cdot \left(2 + \left(\color{blue}{im \cdot im} + 0.08333333333333333 \cdot {im}^{4}\right)\right)\right) \]

      2. *-commutative 96.6%

         \[\leadsto \sin re \cdot \left(0.5 \cdot \left(2 + \left(im \cdot im + \color{blue}{{im}^{4} \cdot 0.08333333333333333}\right)\right)\right) \]

   6. Simplified 96.6%

      \[\leadsto \sin re \cdot \left(0.5 \cdot \color{blue}{\left(2 + \left(im \cdot im + {im}^{4} \cdot 0.08333333333333333\right)\right)}\right) \]

   7. Taylor expanded in im around inf 96.6%

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

      1. *-commutative 96.6%

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

      2. *-commutative 96.6%

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

      3. associate-*l* 96.6%

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

   9. Simplified 96.6%

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

   if -4.4999999999999998e72 < im < -460

   1. Initial program 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. *-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*r* 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. fma-def 100.0%

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

      10. exp-diff 100.0%

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

      11. associate-*l/ 100.0%

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

      12. exp-0 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 3.2%

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

      1. unpow2 3.2%

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

   6. Simplified 3.2%

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

   7. Taylor expanded in im around inf 3.2%

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

      1. associate-*r* 3.2%

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

      2. *-commutative 3.2%

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

      3. unpow2 3.2%

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

   9. Simplified 3.2%

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

   10. Taylor expanded in re around 0 11.5%

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

       1. unpow2 11.5%

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

       2. *-commutative 11.5%

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

       3. associate-*l* 11.5%

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

   12. Simplified 11.5%

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

       1. log1p-expm1-u 59.4%

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

       2. associate-*r* 59.4%

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

       3. *-commutative 59.4%

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

   14. Applied egg-rr 59.4%

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

   if -460 < im < 10

   1. Initial program 100.0%

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

      1. distribute-lft-in 99.9%

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

      2. +-commutative 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-*l* 99.9%

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

      5. *-commutative 99.9%

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

      6. *-commutative 99.9%

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

      7. associate-*r* 99.9%

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

      8. distribute-rgt-in 100.0%

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

      9. fma-def 100.0%

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

      10. exp-diff 100.0%

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

      11. associate-*l/ 100.0%

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

      12. exp-0 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 99.1%

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

      1. unpow2 99.1%

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

   6. Simplified 99.1%

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

   if 10 < im < 4.20000000000000002e68

   1. Initial program 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. *-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*r* 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. fma-def 100.0%

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

      10. exp-diff 100.0%

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

      11. associate-*l/ 100.0%

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

      12. exp-0 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

   5. Taylor expanded in re around 0 90.0%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -4.5 \cdot 10^{+72}:\\ \;\;\;\;{im}^{4} \cdot \left(\sin re \cdot 0.041666666666666664\right)\\ \mathbf{elif}\;im \leq -460:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(re \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq 10:\\ \;\;\;\;\sin re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{+68}:\\ \;\;\;\;re \cdot \left(0.5 + 0.5 \cdot e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{4} \cdot \left(\sin re \cdot 0.041666666666666664\right)\\ \end{array} \]

Alternative 4: 91.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{4} \cdot \left(\sin re \cdot 0.041666666666666664\right)\\ \mathbf{if}\;im \leq -3.7:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 10:\\ \;\;\;\;\sin re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{+68}:\\ \;\;\;\;re \cdot \left(0.5 + 0.5 \cdot e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (pow im 4.0) (* (sin re) 0.041666666666666664))))
   (if (<= im -3.7)
     t_0
     (if (<= im 10.0)
       (* (sin re) (+ 1.0 (* 0.5 (* im im))))
       (if (<= im 4.2e+68) (* re (+ 0.5 (* 0.5 (exp im)))) t_0)))))

C

double code(double re, double im) {
	double t_0 = pow(im, 4.0) * (sin(re) * 0.041666666666666664);
	double tmp;
	if (im <= -3.7) {
		tmp = t_0;
	} else if (im <= 10.0) {
		tmp = sin(re) * (1.0 + (0.5 * (im * im)));
	} else if (im <= 4.2e+68) {
		tmp = re * (0.5 + (0.5 * exp(im)));
	} 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) :: tmp
    t_0 = (im ** 4.0d0) * (sin(re) * 0.041666666666666664d0)
    if (im <= (-3.7d0)) then
        tmp = t_0
    else if (im <= 10.0d0) then
        tmp = sin(re) * (1.0d0 + (0.5d0 * (im * im)))
    else if (im <= 4.2d+68) then
        tmp = re * (0.5d0 + (0.5d0 * exp(im)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.pow(im, 4.0) * (Math.sin(re) * 0.041666666666666664);
	double tmp;
	if (im <= -3.7) {
		tmp = t_0;
	} else if (im <= 10.0) {
		tmp = Math.sin(re) * (1.0 + (0.5 * (im * im)));
	} else if (im <= 4.2e+68) {
		tmp = re * (0.5 + (0.5 * Math.exp(im)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.pow(im, 4.0) * (math.sin(re) * 0.041666666666666664)
	tmp = 0
	if im <= -3.7:
		tmp = t_0
	elif im <= 10.0:
		tmp = math.sin(re) * (1.0 + (0.5 * (im * im)))
	elif im <= 4.2e+68:
		tmp = re * (0.5 + (0.5 * math.exp(im)))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64((im ^ 4.0) * Float64(sin(re) * 0.041666666666666664))
	tmp = 0.0
	if (im <= -3.7)
		tmp = t_0;
	elseif (im <= 10.0)
		tmp = Float64(sin(re) * Float64(1.0 + Float64(0.5 * Float64(im * im))));
	elseif (im <= 4.2e+68)
		tmp = Float64(re * Float64(0.5 + Float64(0.5 * exp(im))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im ^ 4.0) * (sin(re) * 0.041666666666666664);
	tmp = 0.0;
	if (im <= -3.7)
		tmp = t_0;
	elseif (im <= 10.0)
		tmp = sin(re) * (1.0 + (0.5 * (im * im)));
	elseif (im <= 4.2e+68)
		tmp = re * (0.5 + (0.5 * exp(im)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Power[im, 4.0], $MachinePrecision] * N[(N[Sin[re], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -3.7], t$95$0, If[LessEqual[im, 10.0], N[(N[Sin[re], $MachinePrecision] * N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 4.2e+68], N[(re * N[(0.5 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -3.7000000000000002 or 4.20000000000000002e68 < 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 87.5%

      \[\leadsto \sin re \cdot \left(0.5 \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)}\right) \]
   5. Step-by-step derivation

      1. unpow2 87.5%

         \[\leadsto \sin re \cdot \left(0.5 \cdot \left(2 + \left(\color{blue}{im \cdot im} + 0.08333333333333333 \cdot {im}^{4}\right)\right)\right) \]

      2. *-commutative 87.5%

         \[\leadsto \sin re \cdot \left(0.5 \cdot \left(2 + \left(im \cdot im + \color{blue}{{im}^{4} \cdot 0.08333333333333333}\right)\right)\right) \]

   6. Simplified 87.5%

      \[\leadsto \sin re \cdot \left(0.5 \cdot \color{blue}{\left(2 + \left(im \cdot im + {im}^{4} \cdot 0.08333333333333333\right)\right)}\right) \]

   7. Taylor expanded in im around inf 87.5%

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

      1. *-commutative 87.5%

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

      2. *-commutative 87.5%

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

      3. associate-*l* 87.5%

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

   9. Simplified 87.5%

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

   if -3.7000000000000002 < im < 10

   1. Initial program 100.0%

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

      1. distribute-lft-in 99.9%

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

      2. +-commutative 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-*l* 99.9%

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

      5. *-commutative 99.9%

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

      6. *-commutative 99.9%

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

      7. associate-*r* 99.9%

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

      8. distribute-rgt-in 100.0%

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

      9. fma-def 100.0%

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

      10. exp-diff 100.0%

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

      11. associate-*l/ 100.0%

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

      12. exp-0 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 99.1%

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

      1. unpow2 99.1%

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

   6. Simplified 99.1%

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

   if 10 < im < 4.20000000000000002e68

   1. Initial program 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. *-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*r* 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. fma-def 100.0%

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

      10. exp-diff 100.0%

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

      11. associate-*l/ 100.0%

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

      12. exp-0 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

   5. Taylor expanded in re around 0 90.0%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3.7:\\ \;\;\;\;{im}^{4} \cdot \left(\sin re \cdot 0.041666666666666664\right)\\ \mathbf{elif}\;im \leq 10:\\ \;\;\;\;\sin re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{+68}:\\ \;\;\;\;re \cdot \left(0.5 + 0.5 \cdot e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{4} \cdot \left(\sin re \cdot 0.041666666666666664\right)\\ \end{array} \]

Alternative 5: 97.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -4.5 \cdot 10^{+72}:\\ \;\;\;\;{im}^{4} \cdot \left(\sin re \cdot 0.041666666666666664\right)\\ \mathbf{elif}\;im \leq -6.9 \cdot 10^{-6}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(e^{im} + e^{-im}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 + 0.5 \cdot e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -4.5e+72)
   (* (pow im 4.0) (* (sin re) 0.041666666666666664))
   (if (<= im -6.9e-6)
     (* 0.5 (* re (+ (exp im) (exp (- im)))))
     (* (sin re) (+ 0.5 (* 0.5 (exp im)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -4.5e+72) {
		tmp = pow(im, 4.0) * (sin(re) * 0.041666666666666664);
	} else if (im <= -6.9e-6) {
		tmp = 0.5 * (re * (exp(im) + exp(-im)));
	} else {
		tmp = sin(re) * (0.5 + (0.5 * exp(im)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= (-4.5d+72)) then
        tmp = (im ** 4.0d0) * (sin(re) * 0.041666666666666664d0)
    else if (im <= (-6.9d-6)) then
        tmp = 0.5d0 * (re * (exp(im) + exp(-im)))
    else
        tmp = sin(re) * (0.5d0 + (0.5d0 * exp(im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -4.5e+72) {
		tmp = Math.pow(im, 4.0) * (Math.sin(re) * 0.041666666666666664);
	} else if (im <= -6.9e-6) {
		tmp = 0.5 * (re * (Math.exp(im) + Math.exp(-im)));
	} else {
		tmp = Math.sin(re) * (0.5 + (0.5 * Math.exp(im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -4.5e+72:
		tmp = math.pow(im, 4.0) * (math.sin(re) * 0.041666666666666664)
	elif im <= -6.9e-6:
		tmp = 0.5 * (re * (math.exp(im) + math.exp(-im)))
	else:
		tmp = math.sin(re) * (0.5 + (0.5 * math.exp(im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -4.5e+72)
		tmp = Float64((im ^ 4.0) * Float64(sin(re) * 0.041666666666666664));
	elseif (im <= -6.9e-6)
		tmp = Float64(0.5 * Float64(re * Float64(exp(im) + exp(Float64(-im)))));
	else
		tmp = Float64(sin(re) * Float64(0.5 + Float64(0.5 * exp(im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -4.5e+72)
		tmp = (im ^ 4.0) * (sin(re) * 0.041666666666666664);
	elseif (im <= -6.9e-6)
		tmp = 0.5 * (re * (exp(im) + exp(-im)));
	else
		tmp = sin(re) * (0.5 + (0.5 * exp(im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -4.5e+72], N[(N[Power[im, 4.0], $MachinePrecision] * N[(N[Sin[re], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, -6.9e-6], N[(0.5 * N[(re * N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(0.5 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < -4.4999999999999998e72

   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 98.3%

      \[\leadsto \sin re \cdot \left(0.5 \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)}\right) \]
   5. Step-by-step derivation

      1. unpow2 98.3%

         \[\leadsto \sin re \cdot \left(0.5 \cdot \left(2 + \left(\color{blue}{im \cdot im} + 0.08333333333333333 \cdot {im}^{4}\right)\right)\right) \]

      2. *-commutative 98.3%

         \[\leadsto \sin re \cdot \left(0.5 \cdot \left(2 + \left(im \cdot im + \color{blue}{{im}^{4} \cdot 0.08333333333333333}\right)\right)\right) \]

   6. Simplified 98.3%

      \[\leadsto \sin re \cdot \left(0.5 \cdot \color{blue}{\left(2 + \left(im \cdot im + {im}^{4} \cdot 0.08333333333333333\right)\right)}\right) \]

   7. Taylor expanded in im around inf 98.3%

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

      1. *-commutative 98.3%

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

      2. *-commutative 98.3%

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

      3. associate-*l* 98.3%

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

   9. Simplified 98.3%

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

   if -4.4999999999999998e72 < im < -6.9e-6

   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 0 100.0%

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

   if -6.9e-6 < im

   1. Initial program 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. *-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*r* 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. fma-def 100.0%

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

      10. exp-diff 100.0%

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

      11. associate-*l/ 100.0%

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

      12. exp-0 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 98.7%

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

   5. Taylor expanded in re around inf 98.7%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -4.5 \cdot 10^{+72}:\\ \;\;\;\;{im}^{4} \cdot \left(\sin re \cdot 0.041666666666666664\right)\\ \mathbf{elif}\;im \leq -6.9 \cdot 10^{-6}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(e^{im} + e^{-im}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 + 0.5 \cdot e^{im}\right)\\ \end{array} \]

Alternative 6: 94.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -4.5 \cdot 10^{+72}:\\ \;\;\;\;{im}^{4} \cdot \left(\sin re \cdot 0.041666666666666664\right)\\ \mathbf{elif}\;im \leq -510:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(re \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 + 0.5 \cdot e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -4.5e+72)
   (* (pow im 4.0) (* (sin re) 0.041666666666666664))
   (if (<= im -510.0)
     (* 0.5 (log1p (expm1 (* re (* im im)))))
     (* (sin re) (+ 0.5 (* 0.5 (exp im)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -4.5e+72) {
		tmp = pow(im, 4.0) * (sin(re) * 0.041666666666666664);
	} else if (im <= -510.0) {
		tmp = 0.5 * log1p(expm1((re * (im * im))));
	} else {
		tmp = sin(re) * (0.5 + (0.5 * exp(im)));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -4.5e+72) {
		tmp = Math.pow(im, 4.0) * (Math.sin(re) * 0.041666666666666664);
	} else if (im <= -510.0) {
		tmp = 0.5 * Math.log1p(Math.expm1((re * (im * im))));
	} else {
		tmp = Math.sin(re) * (0.5 + (0.5 * Math.exp(im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -4.5e+72:
		tmp = math.pow(im, 4.0) * (math.sin(re) * 0.041666666666666664)
	elif im <= -510.0:
		tmp = 0.5 * math.log1p(math.expm1((re * (im * im))))
	else:
		tmp = math.sin(re) * (0.5 + (0.5 * math.exp(im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -4.5e+72)
		tmp = Float64((im ^ 4.0) * Float64(sin(re) * 0.041666666666666664));
	elseif (im <= -510.0)
		tmp = Float64(0.5 * log1p(expm1(Float64(re * Float64(im * im)))));
	else
		tmp = Float64(sin(re) * Float64(0.5 + Float64(0.5 * exp(im))));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, -4.5e+72], N[(N[Power[im, 4.0], $MachinePrecision] * N[(N[Sin[re], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, -510.0], N[(0.5 * N[Log[1 + N[(Exp[N[(re * N[(im * im), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(0.5 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < -4.4999999999999998e72

   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 98.3%

      \[\leadsto \sin re \cdot \left(0.5 \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)}\right) \]
   5. Step-by-step derivation

      1. unpow2 98.3%

         \[\leadsto \sin re \cdot \left(0.5 \cdot \left(2 + \left(\color{blue}{im \cdot im} + 0.08333333333333333 \cdot {im}^{4}\right)\right)\right) \]

      2. *-commutative 98.3%

         \[\leadsto \sin re \cdot \left(0.5 \cdot \left(2 + \left(im \cdot im + \color{blue}{{im}^{4} \cdot 0.08333333333333333}\right)\right)\right) \]

   6. Simplified 98.3%

      \[\leadsto \sin re \cdot \left(0.5 \cdot \color{blue}{\left(2 + \left(im \cdot im + {im}^{4} \cdot 0.08333333333333333\right)\right)}\right) \]

   7. Taylor expanded in im around inf 98.3%

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

      1. *-commutative 98.3%

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

      2. *-commutative 98.3%

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

      3. associate-*l* 98.3%

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

   9. Simplified 98.3%

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

   if -4.4999999999999998e72 < im < -510

   1. Initial program 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. *-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*r* 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. fma-def 100.0%

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

      10. exp-diff 100.0%

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

      11. associate-*l/ 100.0%

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

      12. exp-0 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 3.2%

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

      1. unpow2 3.2%

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

   6. Simplified 3.2%

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

   7. Taylor expanded in im around inf 3.2%

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

      1. associate-*r* 3.2%

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

      2. *-commutative 3.2%

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

      3. unpow2 3.2%

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

   9. Simplified 3.2%

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

   10. Taylor expanded in re around 0 11.5%

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

       1. unpow2 11.5%

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

       2. *-commutative 11.5%

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

       3. associate-*l* 11.5%

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

   12. Simplified 11.5%

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

       1. log1p-expm1-u 59.4%

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

       2. associate-*r* 59.4%

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

       3. *-commutative 59.4%

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

   14. Applied egg-rr 59.4%

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

   if -510 < im

   1. Initial program 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. *-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*r* 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. fma-def 100.0%

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

      10. exp-diff 100.0%

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

      11. associate-*l/ 100.0%

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

      12. exp-0 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 98.7%

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

   5. Taylor expanded in re around inf 98.7%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -4.5 \cdot 10^{+72}:\\ \;\;\;\;{im}^{4} \cdot \left(\sin re \cdot 0.041666666666666664\right)\\ \mathbf{elif}\;im \leq -510:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(re \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 + 0.5 \cdot e^{im}\right)\\ \end{array} \]

Alternative 7: 89.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(im \cdot im\right) \cdot \left(\sin re \cdot 0.5\right)\\ \mathbf{if}\;im \leq -3 \cdot 10^{+196}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -200:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(2 + \left(im \cdot im + {im}^{4} \cdot 0.08333333333333333\right)\right)\right)\\ \mathbf{elif}\;im \leq 10:\\ \;\;\;\;\sin re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+150}:\\ \;\;\;\;re \cdot \left(0.5 + 0.5 \cdot e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* im im) (* (sin re) 0.5))))
   (if (<= im -3e+196)
     t_0
     (if (<= im -200.0)
       (*
        0.5
        (* re (+ 2.0 (+ (* im im) (* (pow im 4.0) 0.08333333333333333)))))
       (if (<= im 10.0)
         (* (sin re) (+ 1.0 (* 0.5 (* im im))))
         (if (<= im 1.35e+150) (* re (+ 0.5 (* 0.5 (exp im)))) t_0))))))

C

double code(double re, double im) {
	double t_0 = (im * im) * (sin(re) * 0.5);
	double tmp;
	if (im <= -3e+196) {
		tmp = t_0;
	} else if (im <= -200.0) {
		tmp = 0.5 * (re * (2.0 + ((im * im) + (pow(im, 4.0) * 0.08333333333333333))));
	} else if (im <= 10.0) {
		tmp = sin(re) * (1.0 + (0.5 * (im * im)));
	} else if (im <= 1.35e+150) {
		tmp = re * (0.5 + (0.5 * exp(im)));
	} 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) :: tmp
    t_0 = (im * im) * (sin(re) * 0.5d0)
    if (im <= (-3d+196)) then
        tmp = t_0
    else if (im <= (-200.0d0)) then
        tmp = 0.5d0 * (re * (2.0d0 + ((im * im) + ((im ** 4.0d0) * 0.08333333333333333d0))))
    else if (im <= 10.0d0) then
        tmp = sin(re) * (1.0d0 + (0.5d0 * (im * im)))
    else if (im <= 1.35d+150) then
        tmp = re * (0.5d0 + (0.5d0 * exp(im)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (im * im) * (Math.sin(re) * 0.5);
	double tmp;
	if (im <= -3e+196) {
		tmp = t_0;
	} else if (im <= -200.0) {
		tmp = 0.5 * (re * (2.0 + ((im * im) + (Math.pow(im, 4.0) * 0.08333333333333333))));
	} else if (im <= 10.0) {
		tmp = Math.sin(re) * (1.0 + (0.5 * (im * im)));
	} else if (im <= 1.35e+150) {
		tmp = re * (0.5 + (0.5 * Math.exp(im)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (im * im) * (math.sin(re) * 0.5)
	tmp = 0
	if im <= -3e+196:
		tmp = t_0
	elif im <= -200.0:
		tmp = 0.5 * (re * (2.0 + ((im * im) + (math.pow(im, 4.0) * 0.08333333333333333))))
	elif im <= 10.0:
		tmp = math.sin(re) * (1.0 + (0.5 * (im * im)))
	elif im <= 1.35e+150:
		tmp = re * (0.5 + (0.5 * math.exp(im)))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(im * im) * Float64(sin(re) * 0.5))
	tmp = 0.0
	if (im <= -3e+196)
		tmp = t_0;
	elseif (im <= -200.0)
		tmp = Float64(0.5 * Float64(re * Float64(2.0 + Float64(Float64(im * im) + Float64((im ^ 4.0) * 0.08333333333333333)))));
	elseif (im <= 10.0)
		tmp = Float64(sin(re) * Float64(1.0 + Float64(0.5 * Float64(im * im))));
	elseif (im <= 1.35e+150)
		tmp = Float64(re * Float64(0.5 + Float64(0.5 * exp(im))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im * im) * (sin(re) * 0.5);
	tmp = 0.0;
	if (im <= -3e+196)
		tmp = t_0;
	elseif (im <= -200.0)
		tmp = 0.5 * (re * (2.0 + ((im * im) + ((im ^ 4.0) * 0.08333333333333333))));
	elseif (im <= 10.0)
		tmp = sin(re) * (1.0 + (0.5 * (im * im)));
	elseif (im <= 1.35e+150)
		tmp = re * (0.5 + (0.5 * exp(im)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(im * im), $MachinePrecision] * N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -3e+196], t$95$0, If[LessEqual[im, -200.0], N[(0.5 * N[(re * N[(2.0 + N[(N[(im * im), $MachinePrecision] + N[(N[Power[im, 4.0], $MachinePrecision] * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 10.0], N[(N[Sin[re], $MachinePrecision] * N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.35e+150], N[(re * N[(0.5 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -2.9999999999999999e196 or 1.35000000000000004e150 < im

   1. Initial program 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. *-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*r* 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. fma-def 100.0%

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

      10. exp-diff 100.0%

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

      11. associate-*l/ 100.0%

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

      12. exp-0 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 98.6%

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

      1. unpow2 98.6%

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

   6. Simplified 98.6%

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

   7. Taylor expanded in im around inf 98.6%

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

      1. associate-*r* 98.6%

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

      2. *-commutative 98.6%

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

      3. unpow2 98.6%

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

   9. Simplified 98.6%

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

   if -2.9999999999999999e196 < im < -200

   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 84.6%

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

   5. Taylor expanded in im around 0 60.0%

      \[\leadsto 0.5 \cdot \left(re \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)}\right) \]
   6. Step-by-step derivation

      1. unpow2 60.0%

         \[\leadsto 0.5 \cdot \left(re \cdot \left(2 + \left(\color{blue}{im \cdot im} + 0.08333333333333333 \cdot {im}^{4}\right)\right)\right) \]

      2. *-commutative 60.0%

         \[\leadsto 0.5 \cdot \left(re \cdot \left(2 + \left(im \cdot im + \color{blue}{{im}^{4} \cdot 0.08333333333333333}\right)\right)\right) \]

   7. Simplified 60.0%

      \[\leadsto 0.5 \cdot \left(re \cdot \color{blue}{\left(2 + \left(im \cdot im + {im}^{4} \cdot 0.08333333333333333\right)\right)}\right) \]

   if -200 < im < 10

   1. Initial program 100.0%

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

      1. distribute-lft-in 99.9%

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

      2. +-commutative 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-*l* 99.9%

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

      5. *-commutative 99.9%

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

      6. *-commutative 99.9%

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

      7. associate-*r* 99.9%

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

      8. distribute-rgt-in 100.0%

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

      9. fma-def 100.0%

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

      10. exp-diff 100.0%

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

      11. associate-*l/ 100.0%

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

      12. exp-0 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 99.1%

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

      1. unpow2 99.1%

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

   6. Simplified 99.1%

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

   if 10 < im < 1.35000000000000004e150

   1. Initial program 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. *-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*r* 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. fma-def 100.0%

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

      10. exp-diff 100.0%

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

      11. associate-*l/ 100.0%

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

      12. exp-0 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

   5. Taylor expanded in re around 0 72.4%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3 \cdot 10^{+196}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(\sin re \cdot 0.5\right)\\ \mathbf{elif}\;im \leq -200:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(2 + \left(im \cdot im + {im}^{4} \cdot 0.08333333333333333\right)\right)\right)\\ \mathbf{elif}\;im \leq 10:\\ \;\;\;\;\sin re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+150}:\\ \;\;\;\;re \cdot \left(0.5 + 0.5 \cdot e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(\sin re \cdot 0.5\right)\\ \end{array} \]

Alternative 8: 72.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(0.5 \cdot \left(\sin re \cdot im\right)\right)\\ t_1 := re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{if}\;im \leq -2.35 \cdot 10^{+198}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -0.00042:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 3 \cdot 10^{+28}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2.03 \cdot 10^{+210} \lor \neg \left(im \leq 2.45 \cdot 10^{+227}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (* 0.5 (* (sin re) im))))
        (t_1 (* re (+ 1.0 (* 0.5 (* im im))))))
   (if (<= im -2.35e+198)
     t_0
     (if (<= im -0.00042)
       t_1
       (if (<= im 3e+28)
         (sin re)
         (if (or (<= im 2.03e+210) (not (<= im 2.45e+227))) t_1 t_0))))))

C

double code(double re, double im) {
	double t_0 = im * (0.5 * (sin(re) * im));
	double t_1 = re * (1.0 + (0.5 * (im * im)));
	double tmp;
	if (im <= -2.35e+198) {
		tmp = t_0;
	} else if (im <= -0.00042) {
		tmp = t_1;
	} else if (im <= 3e+28) {
		tmp = sin(re);
	} else if ((im <= 2.03e+210) || !(im <= 2.45e+227)) {
		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 = im * (0.5d0 * (sin(re) * im))
    t_1 = re * (1.0d0 + (0.5d0 * (im * im)))
    if (im <= (-2.35d+198)) then
        tmp = t_0
    else if (im <= (-0.00042d0)) then
        tmp = t_1
    else if (im <= 3d+28) then
        tmp = sin(re)
    else if ((im <= 2.03d+210) .or. (.not. (im <= 2.45d+227))) 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 = im * (0.5 * (Math.sin(re) * im));
	double t_1 = re * (1.0 + (0.5 * (im * im)));
	double tmp;
	if (im <= -2.35e+198) {
		tmp = t_0;
	} else if (im <= -0.00042) {
		tmp = t_1;
	} else if (im <= 3e+28) {
		tmp = Math.sin(re);
	} else if ((im <= 2.03e+210) || !(im <= 2.45e+227)) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (0.5 * (math.sin(re) * im))
	t_1 = re * (1.0 + (0.5 * (im * im)))
	tmp = 0
	if im <= -2.35e+198:
		tmp = t_0
	elif im <= -0.00042:
		tmp = t_1
	elif im <= 3e+28:
		tmp = math.sin(re)
	elif (im <= 2.03e+210) or not (im <= 2.45e+227):
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(0.5 * Float64(sin(re) * im)))
	t_1 = Float64(re * Float64(1.0 + Float64(0.5 * Float64(im * im))))
	tmp = 0.0
	if (im <= -2.35e+198)
		tmp = t_0;
	elseif (im <= -0.00042)
		tmp = t_1;
	elseif (im <= 3e+28)
		tmp = sin(re);
	elseif ((im <= 2.03e+210) || !(im <= 2.45e+227))
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (0.5 * (sin(re) * im));
	t_1 = re * (1.0 + (0.5 * (im * im)));
	tmp = 0.0;
	if (im <= -2.35e+198)
		tmp = t_0;
	elseif (im <= -0.00042)
		tmp = t_1;
	elseif (im <= 3e+28)
		tmp = sin(re);
	elseif ((im <= 2.03e+210) || ~((im <= 2.45e+227)))
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(0.5 * N[(N[Sin[re], $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(re * N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -2.35e+198], t$95$0, If[LessEqual[im, -0.00042], t$95$1, If[LessEqual[im, 3e+28], N[Sin[re], $MachinePrecision], If[Or[LessEqual[im, 2.03e+210], N[Not[LessEqual[im, 2.45e+227]], $MachinePrecision]], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -2.3500000000000001e198 or 2.0299999999999999e210 < im < 2.45000000000000002e227

   1. Initial program 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. *-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*r* 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. fma-def 100.0%

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

      10. exp-diff 100.0%

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

      11. associate-*l/ 100.0%

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

      12. exp-0 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

      1. unpow2 100.0%

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

   6. Simplified 100.0%

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

   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. associate-*r* 100.0%

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

      2. unpow2 100.0%

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

      3. associate-*r* 88.7%

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

      4. associate-*r* 88.7%

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

      5. *-commutative 88.7%

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

      6. *-commutative 88.7%

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

   9. Simplified 88.7%

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

   if -2.3500000000000001e198 < im < -4.2000000000000002e-4 or 3.0000000000000001e28 < im < 2.0299999999999999e210 or 2.45000000000000002e227 < im

   1. Initial program 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. *-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*r* 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. fma-def 100.0%

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

      10. exp-diff 100.0%

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

      11. associate-*l/ 100.0%

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

      12. exp-0 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 39.4%

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

      1. unpow2 39.4%

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

   6. Simplified 39.4%

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

   7. Taylor expanded in re around 0 49.7%

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

      1. *-commutative 49.7%

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

      2. *-commutative 49.7%

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

      3. unpow2 49.7%

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

   9. Simplified 49.7%

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

   if -4.2000000000000002e-4 < im < 3.0000000000000001e28

   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 97.1%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.35 \cdot 10^{+198}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(\sin re \cdot im\right)\right)\\ \mathbf{elif}\;im \leq -0.00042:\\ \;\;\;\;re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 3 \cdot 10^{+28}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2.03 \cdot 10^{+210} \lor \neg \left(im \leq 2.45 \cdot 10^{+227}\right):\\ \;\;\;\;re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(\sin re \cdot im\right)\right)\\ \end{array} \]

Alternative 9: 77.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(im \cdot im\right) \cdot \left(\sin re \cdot 0.5\right)\\ \mathbf{if}\;im \leq -2.6 \cdot 10^{+196}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -1760:\\ \;\;\;\;re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 3.8 \cdot 10^{+30}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+150}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(re \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* im im) (* (sin re) 0.5))))
   (if (<= im -2.6e+196)
     t_0
     (if (<= im -1760.0)
       (* re (+ 1.0 (* 0.5 (* im im))))
       (if (<= im 3.8e+30)
         (sin re)
         (if (<= im 1.35e+150) (* 0.5 (* im (* re im))) t_0))))))

C

double code(double re, double im) {
	double t_0 = (im * im) * (sin(re) * 0.5);
	double tmp;
	if (im <= -2.6e+196) {
		tmp = t_0;
	} else if (im <= -1760.0) {
		tmp = re * (1.0 + (0.5 * (im * im)));
	} else if (im <= 3.8e+30) {
		tmp = sin(re);
	} else if (im <= 1.35e+150) {
		tmp = 0.5 * (im * (re * im));
	} 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) :: tmp
    t_0 = (im * im) * (sin(re) * 0.5d0)
    if (im <= (-2.6d+196)) then
        tmp = t_0
    else if (im <= (-1760.0d0)) then
        tmp = re * (1.0d0 + (0.5d0 * (im * im)))
    else if (im <= 3.8d+30) then
        tmp = sin(re)
    else if (im <= 1.35d+150) then
        tmp = 0.5d0 * (im * (re * im))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (im * im) * (Math.sin(re) * 0.5);
	double tmp;
	if (im <= -2.6e+196) {
		tmp = t_0;
	} else if (im <= -1760.0) {
		tmp = re * (1.0 + (0.5 * (im * im)));
	} else if (im <= 3.8e+30) {
		tmp = Math.sin(re);
	} else if (im <= 1.35e+150) {
		tmp = 0.5 * (im * (re * im));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (im * im) * (math.sin(re) * 0.5)
	tmp = 0
	if im <= -2.6e+196:
		tmp = t_0
	elif im <= -1760.0:
		tmp = re * (1.0 + (0.5 * (im * im)))
	elif im <= 3.8e+30:
		tmp = math.sin(re)
	elif im <= 1.35e+150:
		tmp = 0.5 * (im * (re * im))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(im * im) * Float64(sin(re) * 0.5))
	tmp = 0.0
	if (im <= -2.6e+196)
		tmp = t_0;
	elseif (im <= -1760.0)
		tmp = Float64(re * Float64(1.0 + Float64(0.5 * Float64(im * im))));
	elseif (im <= 3.8e+30)
		tmp = sin(re);
	elseif (im <= 1.35e+150)
		tmp = Float64(0.5 * Float64(im * Float64(re * im)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im * im) * (sin(re) * 0.5);
	tmp = 0.0;
	if (im <= -2.6e+196)
		tmp = t_0;
	elseif (im <= -1760.0)
		tmp = re * (1.0 + (0.5 * (im * im)));
	elseif (im <= 3.8e+30)
		tmp = sin(re);
	elseif (im <= 1.35e+150)
		tmp = 0.5 * (im * (re * im));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(im * im), $MachinePrecision] * N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -2.6e+196], t$95$0, If[LessEqual[im, -1760.0], N[(re * N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 3.8e+30], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.35e+150], N[(0.5 * N[(im * N[(re * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -2.60000000000000012e196 or 1.35000000000000004e150 < im

   1. Initial program 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. *-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*r* 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. fma-def 100.0%

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

      10. exp-diff 100.0%

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

      11. associate-*l/ 100.0%

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

      12. exp-0 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 98.6%

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

      1. unpow2 98.6%

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

   6. Simplified 98.6%

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

   7. Taylor expanded in im around inf 98.6%

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

      1. associate-*r* 98.6%

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

      2. *-commutative 98.6%

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

      3. unpow2 98.6%

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

   9. Simplified 98.6%

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

   if -2.60000000000000012e196 < im < -1760

   1. Initial program 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. *-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*r* 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. fma-def 100.0%

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

      10. exp-diff 100.0%

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

      11. associate-*l/ 100.0%

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

      12. exp-0 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 17.3%

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

      1. unpow2 17.3%

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

   6. Simplified 17.3%

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

   7. Taylor expanded in re around 0 35.3%

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

      1. *-commutative 35.3%

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

      2. *-commutative 35.3%

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

      3. unpow2 35.3%

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

   9. Simplified 35.3%

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

   if -1760 < im < 3.8000000000000001e30

   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 97.1%

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

   if 3.8000000000000001e30 < im < 1.35000000000000004e150

   1. Initial program 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. *-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*r* 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. fma-def 100.0%

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

      10. exp-diff 100.0%

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

      11. associate-*l/ 100.0%

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

      12. exp-0 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 5.0%

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

      1. unpow2 5.0%

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

   6. Simplified 5.0%

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

   7. Taylor expanded in im around inf 5.0%

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

      1. associate-*r* 5.0%

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

      2. *-commutative 5.0%

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

      3. unpow2 5.0%

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

   9. Simplified 5.0%

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

   10. Taylor expanded in re around 0 27.8%

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

       1. unpow2 27.8%

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

       2. *-commutative 27.8%

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

       3. associate-*l* 27.8%

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

   12. Simplified 27.8%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.6 \cdot 10^{+196}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(\sin re \cdot 0.5\right)\\ \mathbf{elif}\;im \leq -1760:\\ \;\;\;\;re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 3.8 \cdot 10^{+30}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+150}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(re \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(\sin re \cdot 0.5\right)\\ \end{array} \]

Alternative 10: 84.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(im \cdot im\right) \cdot \left(\sin re \cdot 0.5\right)\\ \mathbf{if}\;im \leq -2.6 \cdot 10^{+196}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -650:\\ \;\;\;\;re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 10:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+150}:\\ \;\;\;\;re \cdot \left(0.5 + 0.5 \cdot e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* im im) (* (sin re) 0.5))))
   (if (<= im -2.6e+196)
     t_0
     (if (<= im -650.0)
       (* re (+ 1.0 (* 0.5 (* im im))))
       (if (<= im 10.0)
         (sin re)
         (if (<= im 1.35e+150) (* re (+ 0.5 (* 0.5 (exp im)))) t_0))))))

C

double code(double re, double im) {
	double t_0 = (im * im) * (sin(re) * 0.5);
	double tmp;
	if (im <= -2.6e+196) {
		tmp = t_0;
	} else if (im <= -650.0) {
		tmp = re * (1.0 + (0.5 * (im * im)));
	} else if (im <= 10.0) {
		tmp = sin(re);
	} else if (im <= 1.35e+150) {
		tmp = re * (0.5 + (0.5 * exp(im)));
	} 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) :: tmp
    t_0 = (im * im) * (sin(re) * 0.5d0)
    if (im <= (-2.6d+196)) then
        tmp = t_0
    else if (im <= (-650.0d0)) then
        tmp = re * (1.0d0 + (0.5d0 * (im * im)))
    else if (im <= 10.0d0) then
        tmp = sin(re)
    else if (im <= 1.35d+150) then
        tmp = re * (0.5d0 + (0.5d0 * exp(im)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (im * im) * (Math.sin(re) * 0.5);
	double tmp;
	if (im <= -2.6e+196) {
		tmp = t_0;
	} else if (im <= -650.0) {
		tmp = re * (1.0 + (0.5 * (im * im)));
	} else if (im <= 10.0) {
		tmp = Math.sin(re);
	} else if (im <= 1.35e+150) {
		tmp = re * (0.5 + (0.5 * Math.exp(im)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (im * im) * (math.sin(re) * 0.5)
	tmp = 0
	if im <= -2.6e+196:
		tmp = t_0
	elif im <= -650.0:
		tmp = re * (1.0 + (0.5 * (im * im)))
	elif im <= 10.0:
		tmp = math.sin(re)
	elif im <= 1.35e+150:
		tmp = re * (0.5 + (0.5 * math.exp(im)))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(im * im) * Float64(sin(re) * 0.5))
	tmp = 0.0
	if (im <= -2.6e+196)
		tmp = t_0;
	elseif (im <= -650.0)
		tmp = Float64(re * Float64(1.0 + Float64(0.5 * Float64(im * im))));
	elseif (im <= 10.0)
		tmp = sin(re);
	elseif (im <= 1.35e+150)
		tmp = Float64(re * Float64(0.5 + Float64(0.5 * exp(im))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im * im) * (sin(re) * 0.5);
	tmp = 0.0;
	if (im <= -2.6e+196)
		tmp = t_0;
	elseif (im <= -650.0)
		tmp = re * (1.0 + (0.5 * (im * im)));
	elseif (im <= 10.0)
		tmp = sin(re);
	elseif (im <= 1.35e+150)
		tmp = re * (0.5 + (0.5 * exp(im)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(im * im), $MachinePrecision] * N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -2.6e+196], t$95$0, If[LessEqual[im, -650.0], N[(re * N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 10.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.35e+150], N[(re * N[(0.5 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -2.60000000000000012e196 or 1.35000000000000004e150 < im

   1. Initial program 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. *-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*r* 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. fma-def 100.0%

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

      10. exp-diff 100.0%

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

      11. associate-*l/ 100.0%

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

      12. exp-0 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 98.6%

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

      1. unpow2 98.6%

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

   6. Simplified 98.6%

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

   7. Taylor expanded in im around inf 98.6%

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

      1. associate-*r* 98.6%

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

      2. *-commutative 98.6%

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

      3. unpow2 98.6%

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

   9. Simplified 98.6%

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

   if -2.60000000000000012e196 < im < -650

   1. Initial program 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. *-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*r* 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. fma-def 100.0%

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

      10. exp-diff 100.0%

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

      11. associate-*l/ 100.0%

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

      12. exp-0 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 17.3%

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

      1. unpow2 17.3%

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

   6. Simplified 17.3%

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

   7. Taylor expanded in re around 0 35.3%

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

      1. *-commutative 35.3%

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

      2. *-commutative 35.3%

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

      3. unpow2 35.3%

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

   9. Simplified 35.3%

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

   if -650 < im < 10

   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 98.6%

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

   if 10 < im < 1.35000000000000004e150

   1. Initial program 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. *-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*r* 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. fma-def 100.0%

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

      10. exp-diff 100.0%

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

      11. associate-*l/ 100.0%

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

      12. exp-0 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

   5. Taylor expanded in re around 0 72.4%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.6 \cdot 10^{+196}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(\sin re \cdot 0.5\right)\\ \mathbf{elif}\;im \leq -650:\\ \;\;\;\;re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 10:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+150}:\\ \;\;\;\;re \cdot \left(0.5 + 0.5 \cdot e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(\sin re \cdot 0.5\right)\\ \end{array} \]

Alternative 11: 88.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(im \cdot im\right) \cdot \left(\sin re \cdot 0.5\right)\\ \mathbf{if}\;im \leq -2.6 \cdot 10^{+196}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -300:\\ \;\;\;\;0.5 \cdot \left({im}^{4} \cdot \left(re \cdot 0.08333333333333333\right)\right)\\ \mathbf{elif}\;im \leq 10:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+150}:\\ \;\;\;\;re \cdot \left(0.5 + 0.5 \cdot e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* im im) (* (sin re) 0.5))))
   (if (<= im -2.6e+196)
     t_0
     (if (<= im -300.0)
       (* 0.5 (* (pow im 4.0) (* re 0.08333333333333333)))
       (if (<= im 10.0)
         (sin re)
         (if (<= im 1.35e+150) (* re (+ 0.5 (* 0.5 (exp im)))) t_0))))))

C

double code(double re, double im) {
	double t_0 = (im * im) * (sin(re) * 0.5);
	double tmp;
	if (im <= -2.6e+196) {
		tmp = t_0;
	} else if (im <= -300.0) {
		tmp = 0.5 * (pow(im, 4.0) * (re * 0.08333333333333333));
	} else if (im <= 10.0) {
		tmp = sin(re);
	} else if (im <= 1.35e+150) {
		tmp = re * (0.5 + (0.5 * exp(im)));
	} 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) :: tmp
    t_0 = (im * im) * (sin(re) * 0.5d0)
    if (im <= (-2.6d+196)) then
        tmp = t_0
    else if (im <= (-300.0d0)) then
        tmp = 0.5d0 * ((im ** 4.0d0) * (re * 0.08333333333333333d0))
    else if (im <= 10.0d0) then
        tmp = sin(re)
    else if (im <= 1.35d+150) then
        tmp = re * (0.5d0 + (0.5d0 * exp(im)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (im * im) * (Math.sin(re) * 0.5);
	double tmp;
	if (im <= -2.6e+196) {
		tmp = t_0;
	} else if (im <= -300.0) {
		tmp = 0.5 * (Math.pow(im, 4.0) * (re * 0.08333333333333333));
	} else if (im <= 10.0) {
		tmp = Math.sin(re);
	} else if (im <= 1.35e+150) {
		tmp = re * (0.5 + (0.5 * Math.exp(im)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (im * im) * (math.sin(re) * 0.5)
	tmp = 0
	if im <= -2.6e+196:
		tmp = t_0
	elif im <= -300.0:
		tmp = 0.5 * (math.pow(im, 4.0) * (re * 0.08333333333333333))
	elif im <= 10.0:
		tmp = math.sin(re)
	elif im <= 1.35e+150:
		tmp = re * (0.5 + (0.5 * math.exp(im)))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(im * im) * Float64(sin(re) * 0.5))
	tmp = 0.0
	if (im <= -2.6e+196)
		tmp = t_0;
	elseif (im <= -300.0)
		tmp = Float64(0.5 * Float64((im ^ 4.0) * Float64(re * 0.08333333333333333)));
	elseif (im <= 10.0)
		tmp = sin(re);
	elseif (im <= 1.35e+150)
		tmp = Float64(re * Float64(0.5 + Float64(0.5 * exp(im))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im * im) * (sin(re) * 0.5);
	tmp = 0.0;
	if (im <= -2.6e+196)
		tmp = t_0;
	elseif (im <= -300.0)
		tmp = 0.5 * ((im ^ 4.0) * (re * 0.08333333333333333));
	elseif (im <= 10.0)
		tmp = sin(re);
	elseif (im <= 1.35e+150)
		tmp = re * (0.5 + (0.5 * exp(im)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(im * im), $MachinePrecision] * N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -2.6e+196], t$95$0, If[LessEqual[im, -300.0], N[(0.5 * N[(N[Power[im, 4.0], $MachinePrecision] * N[(re * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 10.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.35e+150], N[(re * N[(0.5 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -2.60000000000000012e196 or 1.35000000000000004e150 < im

   1. Initial program 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. *-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*r* 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. fma-def 100.0%

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

      10. exp-diff 100.0%

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

      11. associate-*l/ 100.0%

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

      12. exp-0 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 98.6%

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

      1. unpow2 98.6%

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

   6. Simplified 98.6%

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

   7. Taylor expanded in im around inf 98.6%

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

      1. associate-*r* 98.6%

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

      2. *-commutative 98.6%

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

      3. unpow2 98.6%

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

   9. Simplified 98.6%

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

   if -2.60000000000000012e196 < im < -300

   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 84.6%

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

   5. Taylor expanded in im around 0 60.0%

      \[\leadsto 0.5 \cdot \left(re \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)}\right) \]
   6. Step-by-step derivation

      1. unpow2 60.0%

         \[\leadsto 0.5 \cdot \left(re \cdot \left(2 + \left(\color{blue}{im \cdot im} + 0.08333333333333333 \cdot {im}^{4}\right)\right)\right) \]

      2. *-commutative 60.0%

         \[\leadsto 0.5 \cdot \left(re \cdot \left(2 + \left(im \cdot im + \color{blue}{{im}^{4} \cdot 0.08333333333333333}\right)\right)\right) \]

   7. Simplified 60.0%

      \[\leadsto 0.5 \cdot \left(re \cdot \color{blue}{\left(2 + \left(im \cdot im + {im}^{4} \cdot 0.08333333333333333\right)\right)}\right) \]

   8. Taylor expanded in im around inf 60.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(0.08333333333333333 \cdot \left(re \cdot {im}^{4}\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 60.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(0.08333333333333333 \cdot re\right) \cdot {im}^{4}\right)} \]

      2. *-commutative 60.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left({im}^{4} \cdot \left(0.08333333333333333 \cdot re\right)\right)} \]

   10. Simplified 60.0%

       \[\leadsto 0.5 \cdot \color{blue}{\left({im}^{4} \cdot \left(0.08333333333333333 \cdot re\right)\right)} \]

   if -300 < im < 10

   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 98.6%

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

   if 10 < im < 1.35000000000000004e150

   1. Initial program 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. *-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*r* 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. fma-def 100.0%

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

      10. exp-diff 100.0%

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

      11. associate-*l/ 100.0%

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

      12. exp-0 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

   5. Taylor expanded in re around 0 72.4%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.6 \cdot 10^{+196}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(\sin re \cdot 0.5\right)\\ \mathbf{elif}\;im \leq -300:\\ \;\;\;\;0.5 \cdot \left({im}^{4} \cdot \left(re \cdot 0.08333333333333333\right)\right)\\ \mathbf{elif}\;im \leq 10:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+150}:\\ \;\;\;\;re \cdot \left(0.5 + 0.5 \cdot e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(\sin re \cdot 0.5\right)\\ \end{array} \]

Alternative 12: 89.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(im \cdot im\right) \cdot \left(\sin re \cdot 0.5\right)\\ \mathbf{if}\;im \leq -2 \cdot 10^{+198}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -420:\\ \;\;\;\;0.5 \cdot \left({im}^{4} \cdot \left(re \cdot 0.08333333333333333\right)\right)\\ \mathbf{elif}\;im \leq 10:\\ \;\;\;\;\sin re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+150}:\\ \;\;\;\;re \cdot \left(0.5 + 0.5 \cdot e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* im im) (* (sin re) 0.5))))
   (if (<= im -2e+198)
     t_0
     (if (<= im -420.0)
       (* 0.5 (* (pow im 4.0) (* re 0.08333333333333333)))
       (if (<= im 10.0)
         (* (sin re) (+ 1.0 (* 0.5 (* im im))))
         (if (<= im 1.35e+150) (* re (+ 0.5 (* 0.5 (exp im)))) t_0))))))

C

double code(double re, double im) {
	double t_0 = (im * im) * (sin(re) * 0.5);
	double tmp;
	if (im <= -2e+198) {
		tmp = t_0;
	} else if (im <= -420.0) {
		tmp = 0.5 * (pow(im, 4.0) * (re * 0.08333333333333333));
	} else if (im <= 10.0) {
		tmp = sin(re) * (1.0 + (0.5 * (im * im)));
	} else if (im <= 1.35e+150) {
		tmp = re * (0.5 + (0.5 * exp(im)));
	} 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) :: tmp
    t_0 = (im * im) * (sin(re) * 0.5d0)
    if (im <= (-2d+198)) then
        tmp = t_0
    else if (im <= (-420.0d0)) then
        tmp = 0.5d0 * ((im ** 4.0d0) * (re * 0.08333333333333333d0))
    else if (im <= 10.0d0) then
        tmp = sin(re) * (1.0d0 + (0.5d0 * (im * im)))
    else if (im <= 1.35d+150) then
        tmp = re * (0.5d0 + (0.5d0 * exp(im)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (im * im) * (Math.sin(re) * 0.5);
	double tmp;
	if (im <= -2e+198) {
		tmp = t_0;
	} else if (im <= -420.0) {
		tmp = 0.5 * (Math.pow(im, 4.0) * (re * 0.08333333333333333));
	} else if (im <= 10.0) {
		tmp = Math.sin(re) * (1.0 + (0.5 * (im * im)));
	} else if (im <= 1.35e+150) {
		tmp = re * (0.5 + (0.5 * Math.exp(im)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (im * im) * (math.sin(re) * 0.5)
	tmp = 0
	if im <= -2e+198:
		tmp = t_0
	elif im <= -420.0:
		tmp = 0.5 * (math.pow(im, 4.0) * (re * 0.08333333333333333))
	elif im <= 10.0:
		tmp = math.sin(re) * (1.0 + (0.5 * (im * im)))
	elif im <= 1.35e+150:
		tmp = re * (0.5 + (0.5 * math.exp(im)))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(im * im) * Float64(sin(re) * 0.5))
	tmp = 0.0
	if (im <= -2e+198)
		tmp = t_0;
	elseif (im <= -420.0)
		tmp = Float64(0.5 * Float64((im ^ 4.0) * Float64(re * 0.08333333333333333)));
	elseif (im <= 10.0)
		tmp = Float64(sin(re) * Float64(1.0 + Float64(0.5 * Float64(im * im))));
	elseif (im <= 1.35e+150)
		tmp = Float64(re * Float64(0.5 + Float64(0.5 * exp(im))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im * im) * (sin(re) * 0.5);
	tmp = 0.0;
	if (im <= -2e+198)
		tmp = t_0;
	elseif (im <= -420.0)
		tmp = 0.5 * ((im ^ 4.0) * (re * 0.08333333333333333));
	elseif (im <= 10.0)
		tmp = sin(re) * (1.0 + (0.5 * (im * im)));
	elseif (im <= 1.35e+150)
		tmp = re * (0.5 + (0.5 * exp(im)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(im * im), $MachinePrecision] * N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -2e+198], t$95$0, If[LessEqual[im, -420.0], N[(0.5 * N[(N[Power[im, 4.0], $MachinePrecision] * N[(re * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 10.0], N[(N[Sin[re], $MachinePrecision] * N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.35e+150], N[(re * N[(0.5 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -2.00000000000000004e198 or 1.35000000000000004e150 < im

   1. Initial program 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. *-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*r* 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. fma-def 100.0%

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

      10. exp-diff 100.0%

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

      11. associate-*l/ 100.0%

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

      12. exp-0 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 98.6%

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

      1. unpow2 98.6%

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

   6. Simplified 98.6%

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

   7. Taylor expanded in im around inf 98.6%

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

      1. associate-*r* 98.6%

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

      2. *-commutative 98.6%

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

      3. unpow2 98.6%

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

   9. Simplified 98.6%

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

   if -2.00000000000000004e198 < im < -420

   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 84.6%

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

   5. Taylor expanded in im around 0 60.0%

      \[\leadsto 0.5 \cdot \left(re \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)}\right) \]
   6. Step-by-step derivation

      1. unpow2 60.0%

         \[\leadsto 0.5 \cdot \left(re \cdot \left(2 + \left(\color{blue}{im \cdot im} + 0.08333333333333333 \cdot {im}^{4}\right)\right)\right) \]

      2. *-commutative 60.0%

         \[\leadsto 0.5 \cdot \left(re \cdot \left(2 + \left(im \cdot im + \color{blue}{{im}^{4} \cdot 0.08333333333333333}\right)\right)\right) \]

   7. Simplified 60.0%

      \[\leadsto 0.5 \cdot \left(re \cdot \color{blue}{\left(2 + \left(im \cdot im + {im}^{4} \cdot 0.08333333333333333\right)\right)}\right) \]

   8. Taylor expanded in im around inf 60.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(0.08333333333333333 \cdot \left(re \cdot {im}^{4}\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 60.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(0.08333333333333333 \cdot re\right) \cdot {im}^{4}\right)} \]

      2. *-commutative 60.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left({im}^{4} \cdot \left(0.08333333333333333 \cdot re\right)\right)} \]

   10. Simplified 60.0%

       \[\leadsto 0.5 \cdot \color{blue}{\left({im}^{4} \cdot \left(0.08333333333333333 \cdot re\right)\right)} \]

   if -420 < im < 10

   1. Initial program 100.0%

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

      1. distribute-lft-in 99.9%

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

      2. +-commutative 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-*l* 99.9%

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

      5. *-commutative 99.9%

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

      6. *-commutative 99.9%

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

      7. associate-*r* 99.9%

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

      8. distribute-rgt-in 100.0%

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

      9. fma-def 100.0%

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

      10. exp-diff 100.0%

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

      11. associate-*l/ 100.0%

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

      12. exp-0 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 99.1%

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

      1. unpow2 99.1%

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

   6. Simplified 99.1%

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

   if 10 < im < 1.35000000000000004e150

   1. Initial program 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. *-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*r* 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. fma-def 100.0%

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

      10. exp-diff 100.0%

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

      11. associate-*l/ 100.0%

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

      12. exp-0 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

   5. Taylor expanded in re around 0 72.4%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2 \cdot 10^{+198}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(\sin re \cdot 0.5\right)\\ \mathbf{elif}\;im \leq -420:\\ \;\;\;\;0.5 \cdot \left({im}^{4} \cdot \left(re \cdot 0.08333333333333333\right)\right)\\ \mathbf{elif}\;im \leq 10:\\ \;\;\;\;\sin re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+150}:\\ \;\;\;\;re \cdot \left(0.5 + 0.5 \cdot e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(\sin re \cdot 0.5\right)\\ \end{array} \]

Alternative 13: 72.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -0.000102 \lor \neg \left(im \leq 2.8 \cdot 10^{+30}\right):\\ \;\;\;\;re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -0.000102) (not (<= im 2.8e+30)))
   (* re (+ 1.0 (* 0.5 (* im im))))
   (sin re)))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -0.000102) || !(im <= 2.8e+30)) {
		tmp = re * (1.0 + (0.5 * (im * im)));
	} 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 <= (-0.000102d0)) .or. (.not. (im <= 2.8d+30))) then
        tmp = re * (1.0d0 + (0.5d0 * (im * im)))
    else
        tmp = sin(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -0.000102) || !(im <= 2.8e+30)) {
		tmp = re * (1.0 + (0.5 * (im * im)));
	} else {
		tmp = Math.sin(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -0.000102) or not (im <= 2.8e+30):
		tmp = re * (1.0 + (0.5 * (im * im)))
	else:
		tmp = math.sin(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -0.000102) || !(im <= 2.8e+30))
		tmp = Float64(re * Float64(1.0 + Float64(0.5 * Float64(im * im))));
	else
		tmp = sin(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -0.000102) || ~((im <= 2.8e+30)))
		tmp = re * (1.0 + (0.5 * (im * im)));
	else
		tmp = sin(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -0.000102], N[Not[LessEqual[im, 2.8e+30]], $MachinePrecision]], N[(re * N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -1.01999999999999999e-4 or 2.79999999999999983e30 < im

   1. Initial program 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. *-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*r* 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. fma-def 100.0%

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

      10. exp-diff 100.0%

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

      11. associate-*l/ 100.0%

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

      12. exp-0 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 54.4%

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

      1. unpow2 54.4%

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

   6. Simplified 54.4%

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

   7. Taylor expanded in re around 0 52.1%

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

      1. *-commutative 52.1%

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

      2. *-commutative 52.1%

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

      3. unpow2 52.1%

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

   9. Simplified 52.1%

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

   if -1.01999999999999999e-4 < im < 2.79999999999999983e30

   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 97.1%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -0.000102 \lor \neg \left(im \leq 2.8 \cdot 10^{+30}\right):\\ \;\;\;\;re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re\\ \end{array} \]

Alternative 14: 43.0% accurate, 27.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1.42 \lor \neg \left(im \leq 0.0085\right):\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(re \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -1.42) (not (<= im 0.0085))) (* 0.5 (* im (* re im))) re))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -1.42) || !(im <= 0.0085)) {
		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 <= 0.0085d0))) 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 <= 0.0085)) {
		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 <= 0.0085):
		tmp = 0.5 * (im * (re * im))
	else:
		tmp = re
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -1.42) || !(im <= 0.0085))
		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 <= 0.0085)))
		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, 0.0085]], $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 0.0085000000000000006 < im

   1. Initial program 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. *-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*r* 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. fma-def 100.0%

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

      10. exp-diff 100.0%

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

      11. associate-*l/ 100.0%

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

      12. exp-0 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 53.5%

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

      1. unpow2 53.5%

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

   6. Simplified 53.5%

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

   7. Taylor expanded in im around inf 53.1%

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

      1. associate-*r* 53.1%

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

      2. *-commutative 53.1%

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

      3. unpow2 53.1%

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

   9. Simplified 53.1%

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

   10. Taylor expanded in re around 0 50.7%

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

       1. unpow2 50.7%

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

       2. *-commutative 50.7%

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

       3. associate-*l* 40.9%

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

   12. Simplified 40.9%

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

   if -1.4199999999999999 < im < 0.0085000000000000006

   1. Initial program 100.0%

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

      1. distribute-lft-in 99.9%

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

      2. +-commutative 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-*l* 99.9%

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

      5. *-commutative 99.9%

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

      6. *-commutative 99.9%

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

      7. associate-*r* 99.9%

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

      8. distribute-rgt-in 100.0%

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

      9. fma-def 100.0%

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

      10. exp-diff 100.0%

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

      11. associate-*l/ 100.0%

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

      12. exp-0 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

      1. unpow2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around 0 55.6%

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

      1. *-commutative 55.6%

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

      2. *-commutative 55.6%

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

      3. unpow2 55.6%

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

   9. Simplified 55.6%

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

   10. Taylor expanded in im around 0 55.3%

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

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.42 \lor \neg \left(im \leq 0.0085\right):\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(re \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array} \]

Alternative 15: 48.7% accurate, 27.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1.42 \lor \neg \left(im \leq 0.0085\right):\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -1.42) (not (<= im 0.0085))) (* 0.5 (* re (* im im))) re))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -1.42) || !(im <= 0.0085)) {
		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 <= 0.0085d0))) 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 <= 0.0085)) {
		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 <= 0.0085):
		tmp = 0.5 * (re * (im * im))
	else:
		tmp = re
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -1.42) || !(im <= 0.0085))
		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 <= 0.0085)))
		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, 0.0085]], $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 0.0085000000000000006 < im

   1. Initial program 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. *-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*r* 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. fma-def 100.0%

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

      10. exp-diff 100.0%

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

      11. associate-*l/ 100.0%

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

      12. exp-0 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 53.5%

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

      1. unpow2 53.5%

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

   6. Simplified 53.5%

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

   7. Taylor expanded in im around inf 53.1%

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

      1. associate-*r* 53.1%

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

      2. *-commutative 53.1%

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

      3. unpow2 53.1%

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

   9. Simplified 53.1%

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

   10. Taylor expanded in re around 0 50.7%

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

       1. unpow2 50.7%

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

   12. Simplified 50.7%

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

   if -1.4199999999999999 < im < 0.0085000000000000006

   1. Initial program 100.0%

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

      1. distribute-lft-in 99.9%

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

      2. +-commutative 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-*l* 99.9%

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

      5. *-commutative 99.9%

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

      6. *-commutative 99.9%

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

      7. associate-*r* 99.9%

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

      8. distribute-rgt-in 100.0%

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

      9. fma-def 100.0%

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

      10. exp-diff 100.0%

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

      11. associate-*l/ 100.0%

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

      12. exp-0 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

      1. unpow2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around 0 55.6%

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

      1. *-commutative 55.6%

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

      2. *-commutative 55.6%

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

      3. unpow2 55.6%

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

   9. Simplified 55.6%

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

   10. Taylor expanded in im around 0 55.3%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.42 \lor \neg \left(im \leq 0.0085\right):\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array} \]

Alternative 16: 48.9% 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. distribute-lft-in 100.0%

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

   2. +-commutative 100.0%

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

   3. *-commutative 100.0%

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

   4. associate-*l* 100.0%

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

   5. *-commutative 100.0%

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

   6. *-commutative 100.0%

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

   7. associate-*r* 100.0%

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

   8. distribute-rgt-in 100.0%

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

   9. fma-def 100.0%

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

   10. exp-diff 100.0%

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

   11. associate-*l/ 100.0%

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

   12. exp-0 100.0%

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

   13. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in im around 0 75.8%

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

   1. unpow2 75.8%

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

6. Simplified 75.8%

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

7. Taylor expanded in re around 0 53.0%

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

   1. *-commutative 53.0%

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

   2. *-commutative 53.0%

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

   3. unpow2 53.0%

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

9. Simplified 53.0%

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

10. Final simplification 53.0%

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

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. distribute-lft-in 100.0%

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

   2. +-commutative 100.0%

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

   3. *-commutative 100.0%

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

   4. associate-*l* 100.0%

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

   5. *-commutative 100.0%

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

   6. *-commutative 100.0%

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

   7. associate-*r* 100.0%

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

   8. distribute-rgt-in 100.0%

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

   9. fma-def 100.0%

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

   10. exp-diff 100.0%

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

   11. associate-*l/ 100.0%

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

   12. exp-0 100.0%

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

   13. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in im around 0 75.8%

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

   1. unpow2 75.8%

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

6. Simplified 75.8%

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

7. Taylor expanded in re around 0 53.0%

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

   1. *-commutative 53.0%

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

   2. *-commutative 53.0%

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

   3. unpow2 53.0%

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

9. Simplified 53.0%

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

10. Taylor expanded in im around 0 28.0%

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

11. Final simplification 28.0%

    \[\leadsto re \]

Reproduce

herbie shell --seed 2023181 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))