math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 10.7s

Alternatives: 17

Speedup: 1.0×

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

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. sub0-neg 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 92.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin re \cdot \left({im}^{4} \cdot 0.041666666666666664\right)\\ \mathbf{if}\;im \leq -1.16 \cdot 10^{+77}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -2400000000:\\ \;\;\;\;\log \left(\frac{-2}{e^{re}}\right)\\ \mathbf{elif}\;im \leq 6.8:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;re \cdot \left(0.5 + 0.5 \cdot e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (sin re) (* (pow im 4.0) 0.041666666666666664))))
   (if (<= im -1.16e+77)
     t_0
     (if (<= im -2400000000.0)
       (log (/ -2.0 (exp re)))
       (if (<= im 6.8)
         (* (* 0.5 (sin re)) (+ 2.0 (* im im)))
         (if (<= im 1.15e+77) (* re (+ 0.5 (* 0.5 (exp im)))) t_0))))))

C

double code(double re, double im) {
	double t_0 = sin(re) * (pow(im, 4.0) * 0.041666666666666664);
	double tmp;
	if (im <= -1.16e+77) {
		tmp = t_0;
	} else if (im <= -2400000000.0) {
		tmp = log((-2.0 / exp(re)));
	} else if (im <= 6.8) {
		tmp = (0.5 * sin(re)) * (2.0 + (im * im));
	} else if (im <= 1.15e+77) {
		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 = sin(re) * ((im ** 4.0d0) * 0.041666666666666664d0)
    if (im <= (-1.16d+77)) then
        tmp = t_0
    else if (im <= (-2400000000.0d0)) then
        tmp = log(((-2.0d0) / exp(re)))
    else if (im <= 6.8d0) then
        tmp = (0.5d0 * sin(re)) * (2.0d0 + (im * im))
    else if (im <= 1.15d+77) 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.sin(re) * (Math.pow(im, 4.0) * 0.041666666666666664);
	double tmp;
	if (im <= -1.16e+77) {
		tmp = t_0;
	} else if (im <= -2400000000.0) {
		tmp = Math.log((-2.0 / Math.exp(re)));
	} else if (im <= 6.8) {
		tmp = (0.5 * Math.sin(re)) * (2.0 + (im * im));
	} else if (im <= 1.15e+77) {
		tmp = re * (0.5 + (0.5 * Math.exp(im)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.sin(re) * (math.pow(im, 4.0) * 0.041666666666666664)
	tmp = 0
	if im <= -1.16e+77:
		tmp = t_0
	elif im <= -2400000000.0:
		tmp = math.log((-2.0 / math.exp(re)))
	elif im <= 6.8:
		tmp = (0.5 * math.sin(re)) * (2.0 + (im * im))
	elif im <= 1.15e+77:
		tmp = re * (0.5 + (0.5 * math.exp(im)))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(sin(re) * Float64((im ^ 4.0) * 0.041666666666666664))
	tmp = 0.0
	if (im <= -1.16e+77)
		tmp = t_0;
	elseif (im <= -2400000000.0)
		tmp = log(Float64(-2.0 / exp(re)));
	elseif (im <= 6.8)
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(2.0 + Float64(im * im)));
	elseif (im <= 1.15e+77)
		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 = sin(re) * ((im ^ 4.0) * 0.041666666666666664);
	tmp = 0.0;
	if (im <= -1.16e+77)
		tmp = t_0;
	elseif (im <= -2400000000.0)
		tmp = log((-2.0 / exp(re)));
	elseif (im <= 6.8)
		tmp = (0.5 * sin(re)) * (2.0 + (im * im));
	elseif (im <= 1.15e+77)
		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[Sin[re], $MachinePrecision] * N[(N[Power[im, 4.0], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.16e+77], t$95$0, If[LessEqual[im, -2400000000.0], N[Log[N[(-2.0 / N[Exp[re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[im, 6.8], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.15e+77], 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 < -1.1600000000000001e77 or 1.14999999999999997e77 < 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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

      1. unpow2 100.0%

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

      2. *-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in im around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

   9. Simplified 100.0%

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

   if -1.1600000000000001e77 < im < -2.4e9

   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 re around 0 53.3%

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

      1. distribute-lft-out 53.3%

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

      2. +-commutative 53.3%

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

      3. rec-exp 53.3%

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

   6. Simplified 53.3%

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

   8. Applied egg-rr 46.7%

      \[\leadsto \color{blue}{\log \left(\frac{-2}{e^{re}}\right)} \]

   if -2.4e9 < im < 6.79999999999999982

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 97.5%

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

      1. unpow2 97.5%

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

   6. Simplified 97.5%

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

   if 6.79999999999999982 < im < 1.14999999999999997e77

   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 71.4%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.16 \cdot 10^{+77}:\\ \;\;\;\;\sin re \cdot \left({im}^{4} \cdot 0.041666666666666664\right)\\ \mathbf{elif}\;im \leq -2400000000:\\ \;\;\;\;\log \left(\frac{-2}{e^{re}}\right)\\ \mathbf{elif}\;im \leq 6.8:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;re \cdot \left(0.5 + 0.5 \cdot e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{4} \cdot 0.041666666666666664\right)\\ \end{array} \]

Alternative 3: 97.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -3.2 \cdot 10^{+72}:\\ \;\;\;\;\sin re \cdot \left({im}^{4} \cdot 0.041666666666666664\right)\\ \mathbf{elif}\;im \leq -9.2 \cdot 10^{-10}:\\ \;\;\;\;0.5 \cdot \left(\frac{re}{e^{im}} + re \cdot e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 + 0.5 \cdot e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -3.2e+72)
   (* (sin re) (* (pow im 4.0) 0.041666666666666664))
   (if (<= im -9.2e-10)
     (* 0.5 (+ (/ re (exp im)) (* re (exp im))))
     (* (sin re) (+ 0.5 (* 0.5 (exp im)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -3.2e+72) {
		tmp = sin(re) * (pow(im, 4.0) * 0.041666666666666664);
	} else if (im <= -9.2e-10) {
		tmp = 0.5 * ((re / exp(im)) + (re * 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 <= (-3.2d+72)) then
        tmp = sin(re) * ((im ** 4.0d0) * 0.041666666666666664d0)
    else if (im <= (-9.2d-10)) then
        tmp = 0.5d0 * ((re / exp(im)) + (re * 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 <= -3.2e+72) {
		tmp = Math.sin(re) * (Math.pow(im, 4.0) * 0.041666666666666664);
	} else if (im <= -9.2e-10) {
		tmp = 0.5 * ((re / Math.exp(im)) + (re * 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 <= -3.2e+72:
		tmp = math.sin(re) * (math.pow(im, 4.0) * 0.041666666666666664)
	elif im <= -9.2e-10:
		tmp = 0.5 * ((re / math.exp(im)) + (re * 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 <= -3.2e+72)
		tmp = Float64(sin(re) * Float64((im ^ 4.0) * 0.041666666666666664));
	elseif (im <= -9.2e-10)
		tmp = Float64(0.5 * Float64(Float64(re / exp(im)) + Float64(re * exp(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 <= -3.2e+72)
		tmp = sin(re) * ((im ^ 4.0) * 0.041666666666666664);
	elseif (im <= -9.2e-10)
		tmp = 0.5 * ((re / exp(im)) + (re * exp(im)));
	else
		tmp = sin(re) * (0.5 + (0.5 * exp(im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -3.2e+72], N[(N[Sin[re], $MachinePrecision] * N[(N[Power[im, 4.0], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, -9.2e-10], N[(0.5 * N[(N[(re / N[Exp[im], $MachinePrecision]), $MachinePrecision] + N[(re * 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 < -3.2000000000000001e72

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 98.4%

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

      1. unpow2 98.4%

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

      2. *-commutative 98.4%

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

   6. Simplified 98.4%

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

   7. Taylor expanded in im around inf 98.4%

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

      1. *-commutative 98.4%

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

      2. associate-*l* 98.4%

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

   9. Simplified 98.4%

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

   if -3.2000000000000001e72 < im < -9.20000000000000028e-10

   1. Initial program 99.9%

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

      1. distribute-lft-in 99.8%

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

      2. +-commutative 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-*l* 99.8%

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

      5. *-commutative 99.8%

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

      6. *-commutative 99.8%

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

      7. associate-*r* 99.8%

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

      8. distribute-rgt-in 99.9%

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

      9. fma-def 99.9%

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

      10. exp-diff 99.8%

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

      11. associate-*l/ 99.8%

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

      12. exp-0 99.8%

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

      13. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in re around 0 58.1%

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

      1. +-commutative 58.1%

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

      2. distribute-lft-in 58.1%

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

      3. *-commutative 58.1%

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

      4. associate-*r* 58.1%

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

      5. *-commutative 58.1%

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

      6. associate-*r* 58.1%

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

      7. distribute-rgt-in 58.1%

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

      8. fma-def 58.1%

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

      9. associate-*r/ 58.1%

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

      10. *-rgt-identity 58.1%

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

   6. Simplified 58.1%

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

   7. Taylor expanded in im around inf 58.1%

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

   if -9.20000000000000028e-10 < 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 99.5%

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

      1. fma-udef 99.5%

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

   6. Applied egg-rr 99.5%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3.2 \cdot 10^{+72}:\\ \;\;\;\;\sin re \cdot \left({im}^{4} \cdot 0.041666666666666664\right)\\ \mathbf{elif}\;im \leq -9.2 \cdot 10^{-10}:\\ \;\;\;\;0.5 \cdot \left(\frac{re}{e^{im}} + re \cdot e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 + 0.5 \cdot e^{im}\right)\\ \end{array} \]

Alternative 4: 97.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -3.2 \cdot 10^{+72}:\\ \;\;\;\;\sin re \cdot \left({im}^{4} \cdot 0.041666666666666664\right)\\ \mathbf{elif}\;im \leq -9.2 \cdot 10^{-10}:\\ \;\;\;\;re \cdot \left(0.5 \cdot \left(e^{-im} + e^{im}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 + 0.5 \cdot e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -3.2e+72)
   (* (sin re) (* (pow im 4.0) 0.041666666666666664))
   (if (<= im -9.2e-10)
     (* re (* 0.5 (+ (exp (- im)) (exp im))))
     (* (sin re) (+ 0.5 (* 0.5 (exp im)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -3.2e+72) {
		tmp = sin(re) * (pow(im, 4.0) * 0.041666666666666664);
	} else if (im <= -9.2e-10) {
		tmp = re * (0.5 * (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 <= (-3.2d+72)) then
        tmp = sin(re) * ((im ** 4.0d0) * 0.041666666666666664d0)
    else if (im <= (-9.2d-10)) then
        tmp = re * (0.5d0 * (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 <= -3.2e+72) {
		tmp = Math.sin(re) * (Math.pow(im, 4.0) * 0.041666666666666664);
	} else if (im <= -9.2e-10) {
		tmp = re * (0.5 * (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 <= -3.2e+72:
		tmp = math.sin(re) * (math.pow(im, 4.0) * 0.041666666666666664)
	elif im <= -9.2e-10:
		tmp = re * (0.5 * (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 <= -3.2e+72)
		tmp = Float64(sin(re) * Float64((im ^ 4.0) * 0.041666666666666664));
	elseif (im <= -9.2e-10)
		tmp = Float64(re * Float64(0.5 * Float64(exp(Float64(-im)) + exp(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 <= -3.2e+72)
		tmp = sin(re) * ((im ^ 4.0) * 0.041666666666666664);
	elseif (im <= -9.2e-10)
		tmp = re * (0.5 * (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, -3.2e+72], N[(N[Sin[re], $MachinePrecision] * N[(N[Power[im, 4.0], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, -9.2e-10], N[(re * N[(0.5 * 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 < -3.2000000000000001e72

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 98.4%

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

      1. unpow2 98.4%

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

      2. *-commutative 98.4%

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

   6. Simplified 98.4%

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

   7. Taylor expanded in im around inf 98.4%

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

      1. *-commutative 98.4%

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

      2. associate-*l* 98.4%

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

   9. Simplified 98.4%

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

   if -3.2000000000000001e72 < im < -9.20000000000000028e-10

   1. Initial program 99.9%

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

      1. distribute-lft-in 99.8%

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

      2. +-commutative 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-*l* 99.8%

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

      5. *-commutative 99.8%

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

      6. *-commutative 99.8%

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

      7. associate-*r* 99.8%

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

      8. distribute-rgt-in 99.9%

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

      9. fma-def 99.9%

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

      10. exp-diff 99.8%

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

      11. associate-*l/ 99.8%

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

      12. exp-0 99.8%

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

      13. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in re around 0 58.1%

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

      1. distribute-lft-out 58.1%

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

      2. +-commutative 58.1%

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

      3. rec-exp 58.1%

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

   6. Simplified 58.1%

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

   if -9.20000000000000028e-10 < 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 99.5%

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

      1. fma-udef 99.5%

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

   6. Applied egg-rr 99.5%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3.2 \cdot 10^{+72}:\\ \;\;\;\;\sin re \cdot \left({im}^{4} \cdot 0.041666666666666664\right)\\ \mathbf{elif}\;im \leq -9.2 \cdot 10^{-10}:\\ \;\;\;\;re \cdot \left(0.5 \cdot \left(e^{-im} + e^{im}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 + 0.5 \cdot e^{im}\right)\\ \end{array} \]

Alternative 5: 93.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -7.2 \cdot 10^{+76}:\\ \;\;\;\;\sin re \cdot \left({im}^{4} \cdot 0.041666666666666664\right)\\ \mathbf{elif}\;im \leq -2400000000:\\ \;\;\;\;\log \left(\frac{-2}{e^{re}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 + 0.5 \cdot e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -7.2e+76)
   (* (sin re) (* (pow im 4.0) 0.041666666666666664))
   (if (<= im -2400000000.0)
     (log (/ -2.0 (exp re)))
     (* (sin re) (+ 0.5 (* 0.5 (exp im)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -7.2e+76) {
		tmp = sin(re) * (pow(im, 4.0) * 0.041666666666666664);
	} else if (im <= -2400000000.0) {
		tmp = log((-2.0 / exp(re)));
	} 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 <= (-7.2d+76)) then
        tmp = sin(re) * ((im ** 4.0d0) * 0.041666666666666664d0)
    else if (im <= (-2400000000.0d0)) then
        tmp = log(((-2.0d0) / exp(re)))
    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 <= -7.2e+76) {
		tmp = Math.sin(re) * (Math.pow(im, 4.0) * 0.041666666666666664);
	} else if (im <= -2400000000.0) {
		tmp = Math.log((-2.0 / Math.exp(re)));
	} else {
		tmp = Math.sin(re) * (0.5 + (0.5 * Math.exp(im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -7.2e+76:
		tmp = math.sin(re) * (math.pow(im, 4.0) * 0.041666666666666664)
	elif im <= -2400000000.0:
		tmp = math.log((-2.0 / math.exp(re)))
	else:
		tmp = math.sin(re) * (0.5 + (0.5 * math.exp(im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -7.2e+76)
		tmp = Float64(sin(re) * Float64((im ^ 4.0) * 0.041666666666666664));
	elseif (im <= -2400000000.0)
		tmp = log(Float64(-2.0 / exp(re)));
	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 <= -7.2e+76)
		tmp = sin(re) * ((im ^ 4.0) * 0.041666666666666664);
	elseif (im <= -2400000000.0)
		tmp = log((-2.0 / exp(re)));
	else
		tmp = sin(re) * (0.5 + (0.5 * exp(im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -7.2e+76], N[(N[Sin[re], $MachinePrecision] * N[(N[Power[im, 4.0], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, -2400000000.0], N[Log[N[(-2.0 / N[Exp[re], $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 < -7.2000000000000006e76

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

      1. unpow2 100.0%

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

      2. *-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in im around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

   9. Simplified 100.0%

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

   if -7.2000000000000006e76 < im < -2.4e9

   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 re around 0 53.3%

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

      1. distribute-lft-out 53.3%

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

      2. +-commutative 53.3%

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

      3. rec-exp 53.3%

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

   6. Simplified 53.3%

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

   8. Applied egg-rr 46.7%

      \[\leadsto \color{blue}{\log \left(\frac{-2}{e^{re}}\right)} \]

   if -2.4e9 < 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 97.5%

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

      1. fma-udef 97.5%

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

   6. Applied egg-rr 97.5%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -7.2 \cdot 10^{+76}:\\ \;\;\;\;\sin re \cdot \left({im}^{4} \cdot 0.041666666666666664\right)\\ \mathbf{elif}\;im \leq -2400000000:\\ \;\;\;\;\log \left(\frac{-2}{e^{re}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 + 0.5 \cdot e^{im}\right)\\ \end{array} \]

Alternative 6: 92.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + im \cdot im\\ t_1 := 0.5 \cdot \sin re\\ t_2 := t_1 \cdot \left(im \cdot im\right)\\ t_3 := t_1 \cdot \frac{t_0 \cdot t_0 - 1.1736111111111112}{t_0 - 1.0833333333333333}\\ \mathbf{if}\;im \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq -9 \cdot 10^{+76}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;im \leq -2400000000:\\ \;\;\;\;\log \left(\frac{-2}{e^{re}}\right)\\ \mathbf{elif}\;im \leq 7:\\ \;\;\;\;t_1 \cdot t_0\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;re \cdot \left(0.5 + 0.5 \cdot e^{im}\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ 2.0 (* im im)))
        (t_1 (* 0.5 (sin re)))
        (t_2 (* t_1 (* im im)))
        (t_3
         (*
          t_1
          (/ (- (* t_0 t_0) 1.1736111111111112) (- t_0 1.0833333333333333)))))
   (if (<= im -1.35e+154)
     t_2
     (if (<= im -9e+76)
       t_3
       (if (<= im -2400000000.0)
         (log (/ -2.0 (exp re)))
         (if (<= im 7.0)
           (* t_1 t_0)
           (if (<= im 1.15e+77)
             (* re (+ 0.5 (* 0.5 (exp im))))
             (if (<= im 1.35e+154) t_3 t_2))))))))

C

double code(double re, double im) {
	double t_0 = 2.0 + (im * im);
	double t_1 = 0.5 * sin(re);
	double t_2 = t_1 * (im * im);
	double t_3 = t_1 * (((t_0 * t_0) - 1.1736111111111112) / (t_0 - 1.0833333333333333));
	double tmp;
	if (im <= -1.35e+154) {
		tmp = t_2;
	} else if (im <= -9e+76) {
		tmp = t_3;
	} else if (im <= -2400000000.0) {
		tmp = log((-2.0 / exp(re)));
	} else if (im <= 7.0) {
		tmp = t_1 * t_0;
	} else if (im <= 1.15e+77) {
		tmp = re * (0.5 + (0.5 * exp(im)));
	} else if (im <= 1.35e+154) {
		tmp = t_3;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = 2.0d0 + (im * im)
    t_1 = 0.5d0 * sin(re)
    t_2 = t_1 * (im * im)
    t_3 = t_1 * (((t_0 * t_0) - 1.1736111111111112d0) / (t_0 - 1.0833333333333333d0))
    if (im <= (-1.35d+154)) then
        tmp = t_2
    else if (im <= (-9d+76)) then
        tmp = t_3
    else if (im <= (-2400000000.0d0)) then
        tmp = log(((-2.0d0) / exp(re)))
    else if (im <= 7.0d0) then
        tmp = t_1 * t_0
    else if (im <= 1.15d+77) then
        tmp = re * (0.5d0 + (0.5d0 * exp(im)))
    else if (im <= 1.35d+154) then
        tmp = t_3
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 2.0 + (im * im);
	double t_1 = 0.5 * Math.sin(re);
	double t_2 = t_1 * (im * im);
	double t_3 = t_1 * (((t_0 * t_0) - 1.1736111111111112) / (t_0 - 1.0833333333333333));
	double tmp;
	if (im <= -1.35e+154) {
		tmp = t_2;
	} else if (im <= -9e+76) {
		tmp = t_3;
	} else if (im <= -2400000000.0) {
		tmp = Math.log((-2.0 / Math.exp(re)));
	} else if (im <= 7.0) {
		tmp = t_1 * t_0;
	} else if (im <= 1.15e+77) {
		tmp = re * (0.5 + (0.5 * Math.exp(im)));
	} else if (im <= 1.35e+154) {
		tmp = t_3;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 2.0 + (im * im)
	t_1 = 0.5 * math.sin(re)
	t_2 = t_1 * (im * im)
	t_3 = t_1 * (((t_0 * t_0) - 1.1736111111111112) / (t_0 - 1.0833333333333333))
	tmp = 0
	if im <= -1.35e+154:
		tmp = t_2
	elif im <= -9e+76:
		tmp = t_3
	elif im <= -2400000000.0:
		tmp = math.log((-2.0 / math.exp(re)))
	elif im <= 7.0:
		tmp = t_1 * t_0
	elif im <= 1.15e+77:
		tmp = re * (0.5 + (0.5 * math.exp(im)))
	elif im <= 1.35e+154:
		tmp = t_3
	else:
		tmp = t_2
	return tmp

Julia

function code(re, im)
	t_0 = Float64(2.0 + Float64(im * im))
	t_1 = Float64(0.5 * sin(re))
	t_2 = Float64(t_1 * Float64(im * im))
	t_3 = Float64(t_1 * Float64(Float64(Float64(t_0 * t_0) - 1.1736111111111112) / Float64(t_0 - 1.0833333333333333)))
	tmp = 0.0
	if (im <= -1.35e+154)
		tmp = t_2;
	elseif (im <= -9e+76)
		tmp = t_3;
	elseif (im <= -2400000000.0)
		tmp = log(Float64(-2.0 / exp(re)));
	elseif (im <= 7.0)
		tmp = Float64(t_1 * t_0);
	elseif (im <= 1.15e+77)
		tmp = Float64(re * Float64(0.5 + Float64(0.5 * exp(im))));
	elseif (im <= 1.35e+154)
		tmp = t_3;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 2.0 + (im * im);
	t_1 = 0.5 * sin(re);
	t_2 = t_1 * (im * im);
	t_3 = t_1 * (((t_0 * t_0) - 1.1736111111111112) / (t_0 - 1.0833333333333333));
	tmp = 0.0;
	if (im <= -1.35e+154)
		tmp = t_2;
	elseif (im <= -9e+76)
		tmp = t_3;
	elseif (im <= -2400000000.0)
		tmp = log((-2.0 / exp(re)));
	elseif (im <= 7.0)
		tmp = t_1 * t_0;
	elseif (im <= 1.15e+77)
		tmp = re * (0.5 + (0.5 * exp(im)));
	elseif (im <= 1.35e+154)
		tmp = t_3;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(im * im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] - 1.1736111111111112), $MachinePrecision] / N[(t$95$0 - 1.0833333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.35e+154], t$95$2, If[LessEqual[im, -9e+76], t$95$3, If[LessEqual[im, -2400000000.0], N[Log[N[(-2.0 / N[Exp[re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[im, 7.0], N[(t$95$1 * t$95$0), $MachinePrecision], If[LessEqual[im, 1.15e+77], N[(re * N[(0.5 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.35e+154], t$95$3, t$95$2]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if im < -1.35000000000000003e154 or 1.35000000000000003e154 < im

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

      1. unpow2 100.0%

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

   6. Simplified 100.0%

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

   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. *-commutative 100.0%

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

      3. unpow2 100.0%

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

   9. Simplified 100.0%

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

   if -1.35000000000000003e154 < im < -8.9999999999999995e76 or 1.14999999999999997e77 < im < 1.35000000000000003e154

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

      1. unpow2 100.0%

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

      2. *-commutative 100.0%

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

   6. Simplified 100.0%

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

   8. Applied egg-rr 5.4%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(2 + \left(im \cdot im + \color{blue}{1.0833333333333333}\right)\right) \]
   9. Step-by-step derivation

      1. associate-+r+ 5.4%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(2 + im \cdot im\right) + 1.0833333333333333\right)} \]

      2. flip-+ 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\frac{\left(2 + im \cdot im\right) \cdot \left(2 + im \cdot im\right) - 1.0833333333333333 \cdot 1.0833333333333333}{\left(2 + im \cdot im\right) - 1.0833333333333333}} \]

      3. metadata-eval 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \frac{\left(2 + im \cdot im\right) \cdot \left(2 + im \cdot im\right) - \color{blue}{1.1736111111111112}}{\left(2 + im \cdot im\right) - 1.0833333333333333} \]

   10. Applied egg-rr 100.0%

       \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\frac{\left(2 + im \cdot im\right) \cdot \left(2 + im \cdot im\right) - 1.1736111111111112}{\left(2 + im \cdot im\right) - 1.0833333333333333}} \]

   if -8.9999999999999995e76 < im < -2.4e9

   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 re around 0 53.3%

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

      1. distribute-lft-out 53.3%

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

      2. +-commutative 53.3%

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

      3. rec-exp 53.3%

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

   6. Simplified 53.3%

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

   8. Applied egg-rr 46.7%

      \[\leadsto \color{blue}{\log \left(\frac{-2}{e^{re}}\right)} \]

   if -2.4e9 < im < 7

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 97.5%

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

      1. unpow2 97.5%

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

   6. Simplified 97.5%

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

   if 7 < im < 1.14999999999999997e77

   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 71.4%

      \[\leadsto \color{blue}{\left(0.5 + 0.5 \cdot e^{im}\right) \cdot re} \]
3. Recombined 5 regimes into one program.

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;im \leq -9 \cdot 10^{+76}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \frac{\left(2 + im \cdot im\right) \cdot \left(2 + im \cdot im\right) - 1.1736111111111112}{\left(2 + im \cdot im\right) - 1.0833333333333333}\\ \mathbf{elif}\;im \leq -2400000000:\\ \;\;\;\;\log \left(\frac{-2}{e^{re}}\right)\\ \mathbf{elif}\;im \leq 7:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;re \cdot \left(0.5 + 0.5 \cdot e^{im}\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \frac{\left(2 + im \cdot im\right) \cdot \left(2 + im \cdot im\right) - 1.1736111111111112}{\left(2 + im \cdot im\right) - 1.0833333333333333}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \end{array} \]

Alternative 7: 91.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + im \cdot im\\ t_1 := 0.5 \cdot \sin re\\ t_2 := t_1 \cdot \left(im \cdot im\right)\\ t_3 := t_1 \cdot \frac{t_0 \cdot t_0 - 1.1736111111111112}{t_0 - 1.0833333333333333}\\ \mathbf{if}\;im \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq -1.55:\\ \;\;\;\;t_3\\ \mathbf{elif}\;im \leq 6.8:\\ \;\;\;\;t_1 \cdot t_0\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;re \cdot \left(0.5 + 0.5 \cdot e^{im}\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ 2.0 (* im im)))
        (t_1 (* 0.5 (sin re)))
        (t_2 (* t_1 (* im im)))
        (t_3
         (*
          t_1
          (/ (- (* t_0 t_0) 1.1736111111111112) (- t_0 1.0833333333333333)))))
   (if (<= im -1.35e+154)
     t_2
     (if (<= im -1.55)
       t_3
       (if (<= im 6.8)
         (* t_1 t_0)
         (if (<= im 1.15e+77)
           (* re (+ 0.5 (* 0.5 (exp im))))
           (if (<= im 1.35e+154) t_3 t_2)))))))

C

double code(double re, double im) {
	double t_0 = 2.0 + (im * im);
	double t_1 = 0.5 * sin(re);
	double t_2 = t_1 * (im * im);
	double t_3 = t_1 * (((t_0 * t_0) - 1.1736111111111112) / (t_0 - 1.0833333333333333));
	double tmp;
	if (im <= -1.35e+154) {
		tmp = t_2;
	} else if (im <= -1.55) {
		tmp = t_3;
	} else if (im <= 6.8) {
		tmp = t_1 * t_0;
	} else if (im <= 1.15e+77) {
		tmp = re * (0.5 + (0.5 * exp(im)));
	} else if (im <= 1.35e+154) {
		tmp = t_3;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = 2.0d0 + (im * im)
    t_1 = 0.5d0 * sin(re)
    t_2 = t_1 * (im * im)
    t_3 = t_1 * (((t_0 * t_0) - 1.1736111111111112d0) / (t_0 - 1.0833333333333333d0))
    if (im <= (-1.35d+154)) then
        tmp = t_2
    else if (im <= (-1.55d0)) then
        tmp = t_3
    else if (im <= 6.8d0) then
        tmp = t_1 * t_0
    else if (im <= 1.15d+77) then
        tmp = re * (0.5d0 + (0.5d0 * exp(im)))
    else if (im <= 1.35d+154) then
        tmp = t_3
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 2.0 + (im * im);
	double t_1 = 0.5 * Math.sin(re);
	double t_2 = t_1 * (im * im);
	double t_3 = t_1 * (((t_0 * t_0) - 1.1736111111111112) / (t_0 - 1.0833333333333333));
	double tmp;
	if (im <= -1.35e+154) {
		tmp = t_2;
	} else if (im <= -1.55) {
		tmp = t_3;
	} else if (im <= 6.8) {
		tmp = t_1 * t_0;
	} else if (im <= 1.15e+77) {
		tmp = re * (0.5 + (0.5 * Math.exp(im)));
	} else if (im <= 1.35e+154) {
		tmp = t_3;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 2.0 + (im * im)
	t_1 = 0.5 * math.sin(re)
	t_2 = t_1 * (im * im)
	t_3 = t_1 * (((t_0 * t_0) - 1.1736111111111112) / (t_0 - 1.0833333333333333))
	tmp = 0
	if im <= -1.35e+154:
		tmp = t_2
	elif im <= -1.55:
		tmp = t_3
	elif im <= 6.8:
		tmp = t_1 * t_0
	elif im <= 1.15e+77:
		tmp = re * (0.5 + (0.5 * math.exp(im)))
	elif im <= 1.35e+154:
		tmp = t_3
	else:
		tmp = t_2
	return tmp

Julia

function code(re, im)
	t_0 = Float64(2.0 + Float64(im * im))
	t_1 = Float64(0.5 * sin(re))
	t_2 = Float64(t_1 * Float64(im * im))
	t_3 = Float64(t_1 * Float64(Float64(Float64(t_0 * t_0) - 1.1736111111111112) / Float64(t_0 - 1.0833333333333333)))
	tmp = 0.0
	if (im <= -1.35e+154)
		tmp = t_2;
	elseif (im <= -1.55)
		tmp = t_3;
	elseif (im <= 6.8)
		tmp = Float64(t_1 * t_0);
	elseif (im <= 1.15e+77)
		tmp = Float64(re * Float64(0.5 + Float64(0.5 * exp(im))));
	elseif (im <= 1.35e+154)
		tmp = t_3;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 2.0 + (im * im);
	t_1 = 0.5 * sin(re);
	t_2 = t_1 * (im * im);
	t_3 = t_1 * (((t_0 * t_0) - 1.1736111111111112) / (t_0 - 1.0833333333333333));
	tmp = 0.0;
	if (im <= -1.35e+154)
		tmp = t_2;
	elseif (im <= -1.55)
		tmp = t_3;
	elseif (im <= 6.8)
		tmp = t_1 * t_0;
	elseif (im <= 1.15e+77)
		tmp = re * (0.5 + (0.5 * exp(im)));
	elseif (im <= 1.35e+154)
		tmp = t_3;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(im * im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] - 1.1736111111111112), $MachinePrecision] / N[(t$95$0 - 1.0833333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.35e+154], t$95$2, If[LessEqual[im, -1.55], t$95$3, If[LessEqual[im, 6.8], N[(t$95$1 * t$95$0), $MachinePrecision], If[LessEqual[im, 1.15e+77], N[(re * N[(0.5 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.35e+154], t$95$3, t$95$2]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -1.35000000000000003e154 or 1.35000000000000003e154 < im

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

      1. unpow2 100.0%

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

   6. Simplified 100.0%

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

   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. *-commutative 100.0%

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

      3. unpow2 100.0%

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

   9. Simplified 100.0%

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

   if -1.35000000000000003e154 < im < -1.55000000000000004 or 1.14999999999999997e77 < im < 1.35000000000000003e154

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 67.7%

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

      1. unpow2 67.7%

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

      2. *-commutative 67.7%

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

   6. Simplified 67.7%

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

   8. Applied egg-rr 4.9%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(2 + \left(im \cdot im + \color{blue}{1.0833333333333333}\right)\right) \]
   9. Step-by-step derivation

      1. associate-+r+ 4.9%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(2 + im \cdot im\right) + 1.0833333333333333\right)} \]

      2. flip-+ 67.4%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\frac{\left(2 + im \cdot im\right) \cdot \left(2 + im \cdot im\right) - 1.0833333333333333 \cdot 1.0833333333333333}{\left(2 + im \cdot im\right) - 1.0833333333333333}} \]

      3. metadata-eval 67.4%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \frac{\left(2 + im \cdot im\right) \cdot \left(2 + im \cdot im\right) - \color{blue}{1.1736111111111112}}{\left(2 + im \cdot im\right) - 1.0833333333333333} \]

   10. Applied egg-rr 67.4%

       \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\frac{\left(2 + im \cdot im\right) \cdot \left(2 + im \cdot im\right) - 1.1736111111111112}{\left(2 + im \cdot im\right) - 1.0833333333333333}} \]

   if -1.55000000000000004 < im < 6.79999999999999982

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 99.6%

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

      1. unpow2 99.6%

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

   6. Simplified 99.6%

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

   if 6.79999999999999982 < im < 1.14999999999999997e77

   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 71.4%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;im \leq -1.55:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \frac{\left(2 + im \cdot im\right) \cdot \left(2 + im \cdot im\right) - 1.1736111111111112}{\left(2 + im \cdot im\right) - 1.0833333333333333}\\ \mathbf{elif}\;im \leq 6.8:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;re \cdot \left(0.5 + 0.5 \cdot e^{im}\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \frac{\left(2 + im \cdot im\right) \cdot \left(2 + im \cdot im\right) - 1.1736111111111112}{\left(2 + im \cdot im\right) - 1.0833333333333333}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \end{array} \]

Alternative 8: 87.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ t_1 := t_0 \cdot \left(im \cdot im\right)\\ \mathbf{if}\;im \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -1.3 \cdot 10^{+95}:\\ \;\;\;\;\frac{\left(0.5 \cdot re\right) \cdot \left(4 - {im}^{4}\right)}{2 - im \cdot im}\\ \mathbf{elif}\;im \leq 5.6:\\ \;\;\;\;t_0 \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;re \cdot \left(0.5 + 0.5 \cdot e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))) (t_1 (* t_0 (* im im))))
   (if (<= im -1.35e+154)
     t_1
     (if (<= im -1.3e+95)
       (/ (* (* 0.5 re) (- 4.0 (pow im 4.0))) (- 2.0 (* im im)))
       (if (<= im 5.6)
         (* t_0 (+ 2.0 (* im im)))
         (if (<= im 1.35e+154) (* re (+ 0.5 (* 0.5 (exp im)))) t_1))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sin(re);
	double t_1 = t_0 * (im * im);
	double tmp;
	if (im <= -1.35e+154) {
		tmp = t_1;
	} else if (im <= -1.3e+95) {
		tmp = ((0.5 * re) * (4.0 - pow(im, 4.0))) / (2.0 - (im * im));
	} else if (im <= 5.6) {
		tmp = t_0 * (2.0 + (im * im));
	} else if (im <= 1.35e+154) {
		tmp = re * (0.5 + (0.5 * exp(im)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * sin(re)
    t_1 = t_0 * (im * im)
    if (im <= (-1.35d+154)) then
        tmp = t_1
    else if (im <= (-1.3d+95)) then
        tmp = ((0.5d0 * re) * (4.0d0 - (im ** 4.0d0))) / (2.0d0 - (im * im))
    else if (im <= 5.6d0) then
        tmp = t_0 * (2.0d0 + (im * im))
    else if (im <= 1.35d+154) then
        tmp = re * (0.5d0 + (0.5d0 * exp(im)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.sin(re);
	double t_1 = t_0 * (im * im);
	double tmp;
	if (im <= -1.35e+154) {
		tmp = t_1;
	} else if (im <= -1.3e+95) {
		tmp = ((0.5 * re) * (4.0 - Math.pow(im, 4.0))) / (2.0 - (im * im));
	} else if (im <= 5.6) {
		tmp = t_0 * (2.0 + (im * im));
	} else if (im <= 1.35e+154) {
		tmp = re * (0.5 + (0.5 * Math.exp(im)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sin(re)
	t_1 = t_0 * (im * im)
	tmp = 0
	if im <= -1.35e+154:
		tmp = t_1
	elif im <= -1.3e+95:
		tmp = ((0.5 * re) * (4.0 - math.pow(im, 4.0))) / (2.0 - (im * im))
	elif im <= 5.6:
		tmp = t_0 * (2.0 + (im * im))
	elif im <= 1.35e+154:
		tmp = re * (0.5 + (0.5 * math.exp(im)))
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sin(re))
	t_1 = Float64(t_0 * Float64(im * im))
	tmp = 0.0
	if (im <= -1.35e+154)
		tmp = t_1;
	elseif (im <= -1.3e+95)
		tmp = Float64(Float64(Float64(0.5 * re) * Float64(4.0 - (im ^ 4.0))) / Float64(2.0 - Float64(im * im)));
	elseif (im <= 5.6)
		tmp = Float64(t_0 * Float64(2.0 + Float64(im * im)));
	elseif (im <= 1.35e+154)
		tmp = Float64(re * Float64(0.5 + Float64(0.5 * exp(im))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * sin(re);
	t_1 = t_0 * (im * im);
	tmp = 0.0;
	if (im <= -1.35e+154)
		tmp = t_1;
	elseif (im <= -1.3e+95)
		tmp = ((0.5 * re) * (4.0 - (im ^ 4.0))) / (2.0 - (im * im));
	elseif (im <= 5.6)
		tmp = t_0 * (2.0 + (im * im));
	elseif (im <= 1.35e+154)
		tmp = re * (0.5 + (0.5 * exp(im)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(im * im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.35e+154], t$95$1, If[LessEqual[im, -1.3e+95], N[(N[(N[(0.5 * re), $MachinePrecision] * N[(4.0 - N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 - N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 5.6], N[(t$95$0 * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(re * N[(0.5 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -1.35000000000000003e154 or 1.35000000000000003e154 < im

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

      1. unpow2 100.0%

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

   6. Simplified 100.0%

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

   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. *-commutative 100.0%

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

      3. unpow2 100.0%

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

   9. Simplified 100.0%

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

   if -1.35000000000000003e154 < im < -1.29999999999999995e95

   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 re around 0 64.3%

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

      1. distribute-lft-out 64.3%

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

      2. +-commutative 64.3%

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

      3. rec-exp 64.3%

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

   6. Simplified 64.3%

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

   7. Taylor expanded in im around 0 10.2%

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

      1. unpow2 10.2%

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

   9. Simplified 10.2%

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

       1. associate-*r* 10.2%

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

       2. flip-+ 64.3%

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

       3. associate-*r/ 64.3%

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

       4. metadata-eval 64.3%

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

       5. pow2 64.3%

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

       6. pow2 64.3%

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

       7. pow-prod-up 64.3%

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

       8. metadata-eval 64.3%

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

   11. Applied egg-rr 64.3%

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

   if -1.29999999999999995e95 < im < 5.5999999999999996

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 86.6%

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

      1. unpow2 86.6%

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

   6. Simplified 86.6%

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

   if 5.5999999999999996 < im < 1.35000000000000003e154

   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 69.7%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;im \leq -1.3 \cdot 10^{+95}:\\ \;\;\;\;\frac{\left(0.5 \cdot re\right) \cdot \left(4 - {im}^{4}\right)}{2 - im \cdot im}\\ \mathbf{elif}\;im \leq 5.6:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;re \cdot \left(0.5 + 0.5 \cdot e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \end{array} \]

Alternative 9: 78.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ t_1 := 8874444426961748000 + \left(\frac{26623333280885244000}{re \cdot re} + \left(re \cdot re\right) \cdot 1774888885392349700\right)\\ \mathbf{if}\;im \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -2300000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 3 \cdot 10^{+56}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2.3 \cdot 10^{+139}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (* im im)))
        (t_1
         (+
          8874444426961748000.0
          (+
           (/ 26623333280885244000.0 (* re re))
           (* (* re re) 1774888885392349700.0)))))
   (if (<= im -1.35e+154)
     t_0
     (if (<= im -2300000000000.0)
       t_1
       (if (<= im 3e+56) (sin re) (if (<= im 2.3e+139) t_1 t_0))))))

C

double code(double re, double im) {
	double t_0 = (0.5 * sin(re)) * (im * im);
	double t_1 = 8874444426961748000.0 + ((26623333280885244000.0 / (re * re)) + ((re * re) * 1774888885392349700.0));
	double tmp;
	if (im <= -1.35e+154) {
		tmp = t_0;
	} else if (im <= -2300000000000.0) {
		tmp = t_1;
	} else if (im <= 3e+56) {
		tmp = sin(re);
	} else if (im <= 2.3e+139) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (0.5d0 * sin(re)) * (im * im)
    t_1 = 8874444426961748000.0d0 + ((26623333280885244000.0d0 / (re * re)) + ((re * re) * 1774888885392349700.0d0))
    if (im <= (-1.35d+154)) then
        tmp = t_0
    else if (im <= (-2300000000000.0d0)) then
        tmp = t_1
    else if (im <= 3d+56) then
        tmp = sin(re)
    else if (im <= 2.3d+139) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (0.5 * Math.sin(re)) * (im * im);
	double t_1 = 8874444426961748000.0 + ((26623333280885244000.0 / (re * re)) + ((re * re) * 1774888885392349700.0));
	double tmp;
	if (im <= -1.35e+154) {
		tmp = t_0;
	} else if (im <= -2300000000000.0) {
		tmp = t_1;
	} else if (im <= 3e+56) {
		tmp = Math.sin(re);
	} else if (im <= 2.3e+139) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (0.5 * math.sin(re)) * (im * im)
	t_1 = 8874444426961748000.0 + ((26623333280885244000.0 / (re * re)) + ((re * re) * 1774888885392349700.0))
	tmp = 0
	if im <= -1.35e+154:
		tmp = t_0
	elif im <= -2300000000000.0:
		tmp = t_1
	elif im <= 3e+56:
		tmp = math.sin(re)
	elif im <= 2.3e+139:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(im * im))
	t_1 = Float64(8874444426961748000.0 + Float64(Float64(26623333280885244000.0 / Float64(re * re)) + Float64(Float64(re * re) * 1774888885392349700.0)))
	tmp = 0.0
	if (im <= -1.35e+154)
		tmp = t_0;
	elseif (im <= -2300000000000.0)
		tmp = t_1;
	elseif (im <= 3e+56)
		tmp = sin(re);
	elseif (im <= 2.3e+139)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (0.5 * sin(re)) * (im * im);
	t_1 = 8874444426961748000.0 + ((26623333280885244000.0 / (re * re)) + ((re * re) * 1774888885392349700.0));
	tmp = 0.0;
	if (im <= -1.35e+154)
		tmp = t_0;
	elseif (im <= -2300000000000.0)
		tmp = t_1;
	elseif (im <= 3e+56)
		tmp = sin(re);
	elseif (im <= 2.3e+139)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(8874444426961748000.0 + N[(N[(26623333280885244000.0 / N[(re * re), $MachinePrecision]), $MachinePrecision] + N[(N[(re * re), $MachinePrecision] * 1774888885392349700.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.35e+154], t$95$0, If[LessEqual[im, -2300000000000.0], t$95$1, If[LessEqual[im, 3e+56], N[Sin[re], $MachinePrecision], If[LessEqual[im, 2.3e+139], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.35000000000000003e154 or 2.3e139 < 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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 97.2%

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

      1. unpow2 97.2%

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

   6. Simplified 97.2%

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

   7. Taylor expanded in im around inf 97.2%

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

      1. associate-*r* 97.2%

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

      2. *-commutative 97.2%

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

      3. unpow2 97.2%

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

   9. Simplified 97.2%

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

   if -1.35000000000000003e154 < im < -2.3e12 or 3.00000000000000006e56 < im < 2.3e139

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 66.3%

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

      1. unpow2 66.3%

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

      2. *-commutative 66.3%

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

   6. Simplified 66.3%

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

   8. Applied egg-rr 16.9%

      \[\leadsto \color{blue}{{\left(\sin re \cdot 1.9380669946781485 \cdot 10^{-10}\right)}^{-2}} \]

   9. Taylor expanded in re around 0 24.6%

      \[\leadsto \color{blue}{8874444426961748000 + \left(26623333280885244000 \cdot \frac{1}{{re}^{2}} + 1774888885392349700 \cdot {re}^{2}\right)} \]
   10. Step-by-step derivation

       1. associate-*r/ 24.6%

          \[\leadsto 8874444426961748000 + \left(\color{blue}{\frac{26623333280885244000 \cdot 1}{{re}^{2}}} + 1774888885392349700 \cdot {re}^{2}\right) \]

       2. metadata-eval 24.6%

          \[\leadsto 8874444426961748000 + \left(\frac{\color{blue}{26623333280885244000}}{{re}^{2}} + 1774888885392349700 \cdot {re}^{2}\right) \]

       3. unpow2 24.6%

          \[\leadsto 8874444426961748000 + \left(\frac{26623333280885244000}{\color{blue}{re \cdot re}} + 1774888885392349700 \cdot {re}^{2}\right) \]

       4. *-commutative 24.6%

          \[\leadsto 8874444426961748000 + \left(\frac{26623333280885244000}{re \cdot re} + \color{blue}{{re}^{2} \cdot 1774888885392349700}\right) \]

       5. unpow2 24.6%

          \[\leadsto 8874444426961748000 + \left(\frac{26623333280885244000}{re \cdot re} + \color{blue}{\left(re \cdot re\right)} \cdot 1774888885392349700\right) \]

   11. Simplified 24.6%

       \[\leadsto \color{blue}{8874444426961748000 + \left(\frac{26623333280885244000}{re \cdot re} + \left(re \cdot re\right) \cdot 1774888885392349700\right)} \]

   if -2.3e12 < im < 3.00000000000000006e56

   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 90.0%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;im \leq -2300000000000:\\ \;\;\;\;8874444426961748000 + \left(\frac{26623333280885244000}{re \cdot re} + \left(re \cdot re\right) \cdot 1774888885392349700\right)\\ \mathbf{elif}\;im \leq 3 \cdot 10^{+56}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2.3 \cdot 10^{+139}:\\ \;\;\;\;8874444426961748000 + \left(\frac{26623333280885244000}{re \cdot re} + \left(re \cdot re\right) \cdot 1774888885392349700\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \end{array} \]

Alternative 10: 86.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \mathbf{if}\;im \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -2300000000000:\\ \;\;\;\;8874444426961748000 + \left(\frac{26623333280885244000}{re \cdot re} + \left(re \cdot re\right) \cdot 1774888885392349700\right)\\ \mathbf{elif}\;im \leq 3.8:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;re \cdot \left(0.5 + 0.5 \cdot e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (* im im))))
   (if (<= im -1.35e+154)
     t_0
     (if (<= im -2300000000000.0)
       (+
        8874444426961748000.0
        (+
         (/ 26623333280885244000.0 (* re re))
         (* (* re re) 1774888885392349700.0)))
       (if (<= im 3.8)
         (sin re)
         (if (<= im 1.35e+154) (* re (+ 0.5 (* 0.5 (exp im)))) t_0))))))

C

double code(double re, double im) {
	double t_0 = (0.5 * sin(re)) * (im * im);
	double tmp;
	if (im <= -1.35e+154) {
		tmp = t_0;
	} else if (im <= -2300000000000.0) {
		tmp = 8874444426961748000.0 + ((26623333280885244000.0 / (re * re)) + ((re * re) * 1774888885392349700.0));
	} else if (im <= 3.8) {
		tmp = sin(re);
	} else if (im <= 1.35e+154) {
		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 = (0.5d0 * sin(re)) * (im * im)
    if (im <= (-1.35d+154)) then
        tmp = t_0
    else if (im <= (-2300000000000.0d0)) then
        tmp = 8874444426961748000.0d0 + ((26623333280885244000.0d0 / (re * re)) + ((re * re) * 1774888885392349700.0d0))
    else if (im <= 3.8d0) then
        tmp = sin(re)
    else if (im <= 1.35d+154) 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 = (0.5 * Math.sin(re)) * (im * im);
	double tmp;
	if (im <= -1.35e+154) {
		tmp = t_0;
	} else if (im <= -2300000000000.0) {
		tmp = 8874444426961748000.0 + ((26623333280885244000.0 / (re * re)) + ((re * re) * 1774888885392349700.0));
	} else if (im <= 3.8) {
		tmp = Math.sin(re);
	} else if (im <= 1.35e+154) {
		tmp = re * (0.5 + (0.5 * Math.exp(im)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (0.5 * math.sin(re)) * (im * im)
	tmp = 0
	if im <= -1.35e+154:
		tmp = t_0
	elif im <= -2300000000000.0:
		tmp = 8874444426961748000.0 + ((26623333280885244000.0 / (re * re)) + ((re * re) * 1774888885392349700.0))
	elif im <= 3.8:
		tmp = math.sin(re)
	elif im <= 1.35e+154:
		tmp = re * (0.5 + (0.5 * math.exp(im)))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(im * im))
	tmp = 0.0
	if (im <= -1.35e+154)
		tmp = t_0;
	elseif (im <= -2300000000000.0)
		tmp = Float64(8874444426961748000.0 + Float64(Float64(26623333280885244000.0 / Float64(re * re)) + Float64(Float64(re * re) * 1774888885392349700.0)));
	elseif (im <= 3.8)
		tmp = sin(re);
	elseif (im <= 1.35e+154)
		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 = (0.5 * sin(re)) * (im * im);
	tmp = 0.0;
	if (im <= -1.35e+154)
		tmp = t_0;
	elseif (im <= -2300000000000.0)
		tmp = 8874444426961748000.0 + ((26623333280885244000.0 / (re * re)) + ((re * re) * 1774888885392349700.0));
	elseif (im <= 3.8)
		tmp = sin(re);
	elseif (im <= 1.35e+154)
		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[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.35e+154], t$95$0, If[LessEqual[im, -2300000000000.0], N[(8874444426961748000.0 + N[(N[(26623333280885244000.0 / N[(re * re), $MachinePrecision]), $MachinePrecision] + N[(N[(re * re), $MachinePrecision] * 1774888885392349700.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 3.8], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.35e+154], 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 < -1.35000000000000003e154 or 1.35000000000000003e154 < im

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

      1. unpow2 100.0%

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

   6. Simplified 100.0%

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

   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. *-commutative 100.0%

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

      3. unpow2 100.0%

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

   9. Simplified 100.0%

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

   if -1.35000000000000003e154 < im < -2.3e12

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 55.3%

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

      1. unpow2 55.3%

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

      2. *-commutative 55.3%

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

   6. Simplified 55.3%

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

   8. Applied egg-rr 14.4%

      \[\leadsto \color{blue}{{\left(\sin re \cdot 1.9380669946781485 \cdot 10^{-10}\right)}^{-2}} \]

   9. Taylor expanded in re around 0 21.2%

      \[\leadsto \color{blue}{8874444426961748000 + \left(26623333280885244000 \cdot \frac{1}{{re}^{2}} + 1774888885392349700 \cdot {re}^{2}\right)} \]
   10. Step-by-step derivation

       1. associate-*r/ 21.2%

          \[\leadsto 8874444426961748000 + \left(\color{blue}{\frac{26623333280885244000 \cdot 1}{{re}^{2}}} + 1774888885392349700 \cdot {re}^{2}\right) \]

       2. metadata-eval 21.2%

          \[\leadsto 8874444426961748000 + \left(\frac{\color{blue}{26623333280885244000}}{{re}^{2}} + 1774888885392349700 \cdot {re}^{2}\right) \]

       3. unpow2 21.2%

          \[\leadsto 8874444426961748000 + \left(\frac{26623333280885244000}{\color{blue}{re \cdot re}} + 1774888885392349700 \cdot {re}^{2}\right) \]

       4. *-commutative 21.2%

          \[\leadsto 8874444426961748000 + \left(\frac{26623333280885244000}{re \cdot re} + \color{blue}{{re}^{2} \cdot 1774888885392349700}\right) \]

       5. unpow2 21.2%

          \[\leadsto 8874444426961748000 + \left(\frac{26623333280885244000}{re \cdot re} + \color{blue}{\left(re \cdot re\right)} \cdot 1774888885392349700\right) \]

   11. Simplified 21.2%

       \[\leadsto \color{blue}{8874444426961748000 + \left(\frac{26623333280885244000}{re \cdot re} + \left(re \cdot re\right) \cdot 1774888885392349700\right)} \]

   if -2.3e12 < im < 3.7999999999999998

   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 96.6%

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

   if 3.7999999999999998 < im < 1.35000000000000003e154

   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 69.7%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;im \leq -2300000000000:\\ \;\;\;\;8874444426961748000 + \left(\frac{26623333280885244000}{re \cdot re} + \left(re \cdot re\right) \cdot 1774888885392349700\right)\\ \mathbf{elif}\;im \leq 3.8:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;re \cdot \left(0.5 + 0.5 \cdot e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \end{array} \]

Alternative 11: 86.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ t_1 := t_0 \cdot \left(im \cdot im\right)\\ \mathbf{if}\;im \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -2300000000000:\\ \;\;\;\;8874444426961748000 + \left(\frac{26623333280885244000}{re \cdot re} + \left(re \cdot re\right) \cdot 1774888885392349700\right)\\ \mathbf{elif}\;im \leq 7.4:\\ \;\;\;\;t_0 \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;re \cdot \left(0.5 + 0.5 \cdot e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))) (t_1 (* t_0 (* im im))))
   (if (<= im -1.35e+154)
     t_1
     (if (<= im -2300000000000.0)
       (+
        8874444426961748000.0
        (+
         (/ 26623333280885244000.0 (* re re))
         (* (* re re) 1774888885392349700.0)))
       (if (<= im 7.4)
         (* t_0 (+ 2.0 (* im im)))
         (if (<= im 1.35e+154) (* re (+ 0.5 (* 0.5 (exp im)))) t_1))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sin(re);
	double t_1 = t_0 * (im * im);
	double tmp;
	if (im <= -1.35e+154) {
		tmp = t_1;
	} else if (im <= -2300000000000.0) {
		tmp = 8874444426961748000.0 + ((26623333280885244000.0 / (re * re)) + ((re * re) * 1774888885392349700.0));
	} else if (im <= 7.4) {
		tmp = t_0 * (2.0 + (im * im));
	} else if (im <= 1.35e+154) {
		tmp = re * (0.5 + (0.5 * exp(im)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * sin(re)
    t_1 = t_0 * (im * im)
    if (im <= (-1.35d+154)) then
        tmp = t_1
    else if (im <= (-2300000000000.0d0)) then
        tmp = 8874444426961748000.0d0 + ((26623333280885244000.0d0 / (re * re)) + ((re * re) * 1774888885392349700.0d0))
    else if (im <= 7.4d0) then
        tmp = t_0 * (2.0d0 + (im * im))
    else if (im <= 1.35d+154) then
        tmp = re * (0.5d0 + (0.5d0 * exp(im)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.sin(re);
	double t_1 = t_0 * (im * im);
	double tmp;
	if (im <= -1.35e+154) {
		tmp = t_1;
	} else if (im <= -2300000000000.0) {
		tmp = 8874444426961748000.0 + ((26623333280885244000.0 / (re * re)) + ((re * re) * 1774888885392349700.0));
	} else if (im <= 7.4) {
		tmp = t_0 * (2.0 + (im * im));
	} else if (im <= 1.35e+154) {
		tmp = re * (0.5 + (0.5 * Math.exp(im)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sin(re)
	t_1 = t_0 * (im * im)
	tmp = 0
	if im <= -1.35e+154:
		tmp = t_1
	elif im <= -2300000000000.0:
		tmp = 8874444426961748000.0 + ((26623333280885244000.0 / (re * re)) + ((re * re) * 1774888885392349700.0))
	elif im <= 7.4:
		tmp = t_0 * (2.0 + (im * im))
	elif im <= 1.35e+154:
		tmp = re * (0.5 + (0.5 * math.exp(im)))
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sin(re))
	t_1 = Float64(t_0 * Float64(im * im))
	tmp = 0.0
	if (im <= -1.35e+154)
		tmp = t_1;
	elseif (im <= -2300000000000.0)
		tmp = Float64(8874444426961748000.0 + Float64(Float64(26623333280885244000.0 / Float64(re * re)) + Float64(Float64(re * re) * 1774888885392349700.0)));
	elseif (im <= 7.4)
		tmp = Float64(t_0 * Float64(2.0 + Float64(im * im)));
	elseif (im <= 1.35e+154)
		tmp = Float64(re * Float64(0.5 + Float64(0.5 * exp(im))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * sin(re);
	t_1 = t_0 * (im * im);
	tmp = 0.0;
	if (im <= -1.35e+154)
		tmp = t_1;
	elseif (im <= -2300000000000.0)
		tmp = 8874444426961748000.0 + ((26623333280885244000.0 / (re * re)) + ((re * re) * 1774888885392349700.0));
	elseif (im <= 7.4)
		tmp = t_0 * (2.0 + (im * im));
	elseif (im <= 1.35e+154)
		tmp = re * (0.5 + (0.5 * exp(im)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(im * im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.35e+154], t$95$1, If[LessEqual[im, -2300000000000.0], N[(8874444426961748000.0 + N[(N[(26623333280885244000.0 / N[(re * re), $MachinePrecision]), $MachinePrecision] + N[(N[(re * re), $MachinePrecision] * 1774888885392349700.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 7.4], N[(t$95$0 * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(re * N[(0.5 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -1.35000000000000003e154 or 1.35000000000000003e154 < im

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

      1. unpow2 100.0%

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

   6. Simplified 100.0%

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

   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. *-commutative 100.0%

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

      3. unpow2 100.0%

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

   9. Simplified 100.0%

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

   if -1.35000000000000003e154 < im < -2.3e12

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 55.3%

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

      1. unpow2 55.3%

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

      2. *-commutative 55.3%

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

   6. Simplified 55.3%

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

   8. Applied egg-rr 14.4%

      \[\leadsto \color{blue}{{\left(\sin re \cdot 1.9380669946781485 \cdot 10^{-10}\right)}^{-2}} \]

   9. Taylor expanded in re around 0 21.2%

      \[\leadsto \color{blue}{8874444426961748000 + \left(26623333280885244000 \cdot \frac{1}{{re}^{2}} + 1774888885392349700 \cdot {re}^{2}\right)} \]
   10. Step-by-step derivation

       1. associate-*r/ 21.2%

          \[\leadsto 8874444426961748000 + \left(\color{blue}{\frac{26623333280885244000 \cdot 1}{{re}^{2}}} + 1774888885392349700 \cdot {re}^{2}\right) \]

       2. metadata-eval 21.2%

          \[\leadsto 8874444426961748000 + \left(\frac{\color{blue}{26623333280885244000}}{{re}^{2}} + 1774888885392349700 \cdot {re}^{2}\right) \]

       3. unpow2 21.2%

          \[\leadsto 8874444426961748000 + \left(\frac{26623333280885244000}{\color{blue}{re \cdot re}} + 1774888885392349700 \cdot {re}^{2}\right) \]

       4. *-commutative 21.2%

          \[\leadsto 8874444426961748000 + \left(\frac{26623333280885244000}{re \cdot re} + \color{blue}{{re}^{2} \cdot 1774888885392349700}\right) \]

       5. unpow2 21.2%

          \[\leadsto 8874444426961748000 + \left(\frac{26623333280885244000}{re \cdot re} + \color{blue}{\left(re \cdot re\right)} \cdot 1774888885392349700\right) \]

   11. Simplified 21.2%

       \[\leadsto \color{blue}{8874444426961748000 + \left(\frac{26623333280885244000}{re \cdot re} + \left(re \cdot re\right) \cdot 1774888885392349700\right)} \]

   if -2.3e12 < im < 7.4000000000000004

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 96.8%

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

      1. unpow2 96.8%

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

   6. Simplified 96.8%

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

   if 7.4000000000000004 < im < 1.35000000000000003e154

   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 69.7%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;im \leq -2300000000000:\\ \;\;\;\;8874444426961748000 + \left(\frac{26623333280885244000}{re \cdot re} + \left(re \cdot re\right) \cdot 1774888885392349700\right)\\ \mathbf{elif}\;im \leq 7.4:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;re \cdot \left(0.5 + 0.5 \cdot e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \end{array} \]

Alternative 12: 72.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(re \cdot \left(im \cdot im\right)\right)\\ t_1 := 8874444426961748000 + \left(\frac{26623333280885244000}{re \cdot re} + \left(re \cdot re\right) \cdot 1774888885392349700\right)\\ \mathbf{if}\;im \leq -8.2 \cdot 10^{+152}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -2300000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 3 \cdot 10^{+56}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2.3 \cdot 10^{+138}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (* re (* im im))))
        (t_1
         (+
          8874444426961748000.0
          (+
           (/ 26623333280885244000.0 (* re re))
           (* (* re re) 1774888885392349700.0)))))
   (if (<= im -8.2e+152)
     t_0
     (if (<= im -2300000000000.0)
       t_1
       (if (<= im 3e+56) (sin re) (if (<= im 2.3e+138) t_1 t_0))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (re * (im * im));
	double t_1 = 8874444426961748000.0 + ((26623333280885244000.0 / (re * re)) + ((re * re) * 1774888885392349700.0));
	double tmp;
	if (im <= -8.2e+152) {
		tmp = t_0;
	} else if (im <= -2300000000000.0) {
		tmp = t_1;
	} else if (im <= 3e+56) {
		tmp = sin(re);
	} else if (im <= 2.3e+138) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * (re * (im * im))
    t_1 = 8874444426961748000.0d0 + ((26623333280885244000.0d0 / (re * re)) + ((re * re) * 1774888885392349700.0d0))
    if (im <= (-8.2d+152)) then
        tmp = t_0
    else if (im <= (-2300000000000.0d0)) then
        tmp = t_1
    else if (im <= 3d+56) then
        tmp = sin(re)
    else if (im <= 2.3d+138) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * (re * (im * im));
	double t_1 = 8874444426961748000.0 + ((26623333280885244000.0 / (re * re)) + ((re * re) * 1774888885392349700.0));
	double tmp;
	if (im <= -8.2e+152) {
		tmp = t_0;
	} else if (im <= -2300000000000.0) {
		tmp = t_1;
	} else if (im <= 3e+56) {
		tmp = Math.sin(re);
	} else if (im <= 2.3e+138) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * (re * (im * im))
	t_1 = 8874444426961748000.0 + ((26623333280885244000.0 / (re * re)) + ((re * re) * 1774888885392349700.0))
	tmp = 0
	if im <= -8.2e+152:
		tmp = t_0
	elif im <= -2300000000000.0:
		tmp = t_1
	elif im <= 3e+56:
		tmp = math.sin(re)
	elif im <= 2.3e+138:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(re * Float64(im * im)))
	t_1 = Float64(8874444426961748000.0 + Float64(Float64(26623333280885244000.0 / Float64(re * re)) + Float64(Float64(re * re) * 1774888885392349700.0)))
	tmp = 0.0
	if (im <= -8.2e+152)
		tmp = t_0;
	elseif (im <= -2300000000000.0)
		tmp = t_1;
	elseif (im <= 3e+56)
		tmp = sin(re);
	elseif (im <= 2.3e+138)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (re * (im * im));
	t_1 = 8874444426961748000.0 + ((26623333280885244000.0 / (re * re)) + ((re * re) * 1774888885392349700.0));
	tmp = 0.0;
	if (im <= -8.2e+152)
		tmp = t_0;
	elseif (im <= -2300000000000.0)
		tmp = t_1;
	elseif (im <= 3e+56)
		tmp = sin(re);
	elseif (im <= 2.3e+138)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(re * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(8874444426961748000.0 + N[(N[(26623333280885244000.0 / N[(re * re), $MachinePrecision]), $MachinePrecision] + N[(N[(re * re), $MachinePrecision] * 1774888885392349700.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -8.2e+152], t$95$0, If[LessEqual[im, -2300000000000.0], t$95$1, If[LessEqual[im, 3e+56], N[Sin[re], $MachinePrecision], If[LessEqual[im, 2.3e+138], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -8.1999999999999996e152 or 2.30000000000000008e138 < 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 re around 0 75.0%

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

      1. distribute-lft-out 75.0%

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

      2. +-commutative 75.0%

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

      3. rec-exp 75.0%

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

   6. Simplified 75.0%

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

   7. Taylor expanded in im around 0 75.0%

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

      1. unpow2 75.0%

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

   9. Simplified 75.0%

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

   10. Taylor expanded in im around inf 75.0%

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

       1. unpow2 75.0%

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

   12. Simplified 75.0%

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

   if -8.1999999999999996e152 < im < -2.3e12 or 3.00000000000000006e56 < im < 2.30000000000000008e138

   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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 66.3%

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

      1. unpow2 66.3%

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

      2. *-commutative 66.3%

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

   6. Simplified 66.3%

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

   8. Applied egg-rr 16.9%

      \[\leadsto \color{blue}{{\left(\sin re \cdot 1.9380669946781485 \cdot 10^{-10}\right)}^{-2}} \]

   9. Taylor expanded in re around 0 24.6%

      \[\leadsto \color{blue}{8874444426961748000 + \left(26623333280885244000 \cdot \frac{1}{{re}^{2}} + 1774888885392349700 \cdot {re}^{2}\right)} \]
   10. Step-by-step derivation

       1. associate-*r/ 24.6%

          \[\leadsto 8874444426961748000 + \left(\color{blue}{\frac{26623333280885244000 \cdot 1}{{re}^{2}}} + 1774888885392349700 \cdot {re}^{2}\right) \]

       2. metadata-eval 24.6%

          \[\leadsto 8874444426961748000 + \left(\frac{\color{blue}{26623333280885244000}}{{re}^{2}} + 1774888885392349700 \cdot {re}^{2}\right) \]

       3. unpow2 24.6%

          \[\leadsto 8874444426961748000 + \left(\frac{26623333280885244000}{\color{blue}{re \cdot re}} + 1774888885392349700 \cdot {re}^{2}\right) \]

       4. *-commutative 24.6%

          \[\leadsto 8874444426961748000 + \left(\frac{26623333280885244000}{re \cdot re} + \color{blue}{{re}^{2} \cdot 1774888885392349700}\right) \]

       5. unpow2 24.6%

          \[\leadsto 8874444426961748000 + \left(\frac{26623333280885244000}{re \cdot re} + \color{blue}{\left(re \cdot re\right)} \cdot 1774888885392349700\right) \]

   11. Simplified 24.6%

       \[\leadsto \color{blue}{8874444426961748000 + \left(\frac{26623333280885244000}{re \cdot re} + \left(re \cdot re\right) \cdot 1774888885392349700\right)} \]

   if -2.3e12 < im < 3.00000000000000006e56

   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 90.0%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -8.2 \cdot 10^{+152}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq -2300000000000:\\ \;\;\;\;8874444426961748000 + \left(\frac{26623333280885244000}{re \cdot re} + \left(re \cdot re\right) \cdot 1774888885392349700\right)\\ \mathbf{elif}\;im \leq 3 \cdot 10^{+56}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2.3 \cdot 10^{+138}:\\ \;\;\;\;8874444426961748000 + \left(\frac{26623333280885244000}{re \cdot re} + \left(re \cdot re\right) \cdot 1774888885392349700\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 13: 46.6% accurate, 27.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -19.5 \lor \neg \left(im \leq 1.45\right):\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -19.5) (not (<= im 1.45))) (* 0.5 (* re (* im im))) re))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -19.5) || !(im <= 1.45)) {
		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 <= (-19.5d0)) .or. (.not. (im <= 1.45d0))) 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 <= -19.5) || !(im <= 1.45)) {
		tmp = 0.5 * (re * (im * im));
	} else {
		tmp = re;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -19.5) or not (im <= 1.45):
		tmp = 0.5 * (re * (im * im))
	else:
		tmp = re
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -19.5) || !(im <= 1.45))
		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 <= -19.5) || ~((im <= 1.45)))
		tmp = 0.5 * (re * (im * im));
	else
		tmp = re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -19.5], N[Not[LessEqual[im, 1.45]], $MachinePrecision]], N[(0.5 * N[(re * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], re]

Derivation

1. Split input into 2 regimes

2. if im < -19.5 or 1.44999999999999996 < 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 re around 0 68.7%

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

      1. distribute-lft-out 68.7%

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

      2. +-commutative 68.7%

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

      3. rec-exp 68.7%

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

   6. Simplified 68.7%

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

   7. Taylor expanded in im around 0 40.3%

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

      1. unpow2 40.3%

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

   9. Simplified 40.3%

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

   10. Taylor expanded in im around inf 40.3%

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

       1. unpow2 40.3%

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

   12. Simplified 40.3%

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

   if -19.5 < im < 1.44999999999999996

   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 re around 0 46.7%

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

      1. distribute-lft-out 46.7%

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

      2. +-commutative 46.7%

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

      3. rec-exp 46.7%

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

   6. Simplified 46.7%

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

   7. Taylor expanded in im around 0 46.7%

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

4. Final simplification 43.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -19.5 \lor \neg \left(im \leq 1.45\right):\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array} \]

Alternative 14: 31.6% accurate, 33.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1.5 \cdot 10^{+37} \lor \neg \left(im \leq 4.7 \cdot 10^{+47}\right):\\ \;\;\;\;\frac{26623333280885244000}{re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -1.5e+37) (not (<= im 4.7e+47)))
   (/ 26623333280885244000.0 (* re re))
   re))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -1.5e+37) || !(im <= 4.7e+47)) {
		tmp = 26623333280885244000.0 / (re * re);
	} else {
		tmp = re;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-1.5d+37)) .or. (.not. (im <= 4.7d+47))) then
        tmp = 26623333280885244000.0d0 / (re * re)
    else
        tmp = re
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -1.5e+37) || !(im <= 4.7e+47)) {
		tmp = 26623333280885244000.0 / (re * re);
	} else {
		tmp = re;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -1.5e+37) or not (im <= 4.7e+47):
		tmp = 26623333280885244000.0 / (re * re)
	else:
		tmp = re
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -1.5e+37) || !(im <= 4.7e+47))
		tmp = Float64(26623333280885244000.0 / Float64(re * re));
	else
		tmp = re;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -1.5e+37) || ~((im <= 4.7e+47)))
		tmp = 26623333280885244000.0 / (re * re);
	else
		tmp = re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -1.5e+37], N[Not[LessEqual[im, 4.7e+47]], $MachinePrecision]], N[(26623333280885244000.0 / N[(re * re), $MachinePrecision]), $MachinePrecision], re]

Derivation

1. Split input into 2 regimes

2. if im < -1.50000000000000011e37 or 4.69999999999999964e47 < 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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 88.0%

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

      1. unpow2 88.0%

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

      2. *-commutative 88.0%

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

   6. Simplified 88.0%

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

   8. Applied egg-rr 14.2%

      \[\leadsto \color{blue}{{\left(\sin re \cdot 1.9380669946781485 \cdot 10^{-10}\right)}^{-2}} \]

   9. Taylor expanded in re around 0 14.1%

      \[\leadsto \color{blue}{\frac{26623333280885244000}{{re}^{2}}} \]
   10. Step-by-step derivation

       1. unpow2 14.1%

          \[\leadsto \frac{26623333280885244000}{\color{blue}{re \cdot re}} \]

   11. Simplified 14.1%

       \[\leadsto \color{blue}{\frac{26623333280885244000}{re \cdot re}} \]

   if -1.50000000000000011e37 < im < 4.69999999999999964e47

   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 re around 0 48.1%

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

      1. distribute-lft-out 48.1%

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

      2. +-commutative 48.1%

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

      3. rec-exp 48.1%

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

   6. Simplified 48.1%

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

   7. Taylor expanded in im around 0 41.5%

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

4. Final simplification 29.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.5 \cdot 10^{+37} \lor \neg \left(im \leq 4.7 \cdot 10^{+47}\right):\\ \;\;\;\;\frac{26623333280885244000}{re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array} \]

Alternative 15: 46.8% accurate, 34.3× speedup

Math

\[\begin{array}{l} \\ re \cdot \left(0.5 \cdot \left(2 + im \cdot im\right)\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* re (* 0.5 (+ 2.0 (* im im)))))

C

double code(double re, double im) {
	return re * (0.5 * (2.0 + (im * im)));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = re * (0.5d0 * (2.0d0 + (im * im)))
end function

Java

public static double code(double re, double im) {
	return re * (0.5 * (2.0 + (im * im)));
}

Python

def code(re, im):
	return re * (0.5 * (2.0 + (im * im)))

Julia

function code(re, im)
	return Float64(re * Float64(0.5 * Float64(2.0 + Float64(im * im))))
end

MATLAB

function tmp = code(re, im)
	tmp = re * (0.5 * (2.0 + (im * im)));
end

Wolfram

code[re_, im_] := N[(re * N[(0.5 * N[(2.0 + 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 re around 0 57.7%

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

   1. distribute-lft-out 57.7%

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

   2. +-commutative 57.7%

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

   3. rec-exp 57.7%

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

6. Simplified 57.7%

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

7. Taylor expanded in im around 0 43.5%

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

   1. unpow2 43.5%

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

9. Simplified 43.5%

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

10. Final simplification 43.5%

    \[\leadsto re \cdot \left(0.5 \cdot \left(2 + im \cdot im\right)\right) \]

Alternative 16: 4.7% accurate, 309.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return 1.0

Julia

function code(re, im)
	return 1.0
end

MATLAB

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

Wolfram

code[re_, im_] := 1.0

Derivation

1. Initial program 100.0%

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

   1. sub0-neg 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in im around 0 88.0%

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

   1. unpow2 88.0%

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

   2. *-commutative 88.0%

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

6. Simplified 88.0%

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

8. Applied egg-rr 4.4%

   \[\leadsto \color{blue}{\frac{\sin re - \sin re \cdot 1.9380669946781485 \cdot 10^{-10}}{\sin re - \sin re \cdot 1.9380669946781485 \cdot 10^{-10}}} \]
9. Step-by-step derivation

   1. *-inverses 4.4%

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

10. Simplified 4.4%

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

11. Final simplification 4.4%

    \[\leadsto 1 \]

Alternative 17: 26.3% 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 re around 0 57.7%

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

   1. distribute-lft-out 57.7%

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

   2. +-commutative 57.7%

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

   3. rec-exp 57.7%

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

6. Simplified 57.7%

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

7. Taylor expanded in im around 0 24.5%

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

8. Final simplification 24.5%

   \[\leadsto re \]

Reproduce

herbie shell --seed 2023178 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))