math.cos on complex, imaginary part

Percentage Accurate: 65.2% → 99.5%

Time: 9.2s

Alternatives: 15

Speedup: 2.7×

• 49.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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]

Initial Program: 65.2% 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]

Alternative 1: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 10^{-6}\right):\\ \;\;\;\;t_0 \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right) - im \cdot \sin re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 1e-6)))
     (* t_0 (* 0.5 (sin re)))
     (- (* -0.16666666666666666 (* (sin re) (pow im 3.0))) (* im (sin re))))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || !(t_0 <= 1e-6)) {
		tmp = t_0 * (0.5 * sin(re));
	} else {
		tmp = (-0.16666666666666666 * (sin(re) * pow(im, 3.0))) - (im * sin(re));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(-im) - Math.exp(im);
	double tmp;
	if ((t_0 <= -Double.POSITIVE_INFINITY) || !(t_0 <= 1e-6)) {
		tmp = t_0 * (0.5 * Math.sin(re));
	} else {
		tmp = (-0.16666666666666666 * (Math.sin(re) * Math.pow(im, 3.0))) - (im * Math.sin(re));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	tmp = 0
	if (t_0 <= -math.inf) or not (t_0 <= 1e-6):
		tmp = t_0 * (0.5 * math.sin(re))
	else:
		tmp = (-0.16666666666666666 * (math.sin(re) * math.pow(im, 3.0))) - (im * math.sin(re))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	tmp = 0.0
	if ((t_0 <= Float64(-Inf)) || !(t_0 <= 1e-6))
		tmp = Float64(t_0 * Float64(0.5 * sin(re)));
	else
		tmp = Float64(Float64(-0.16666666666666666 * Float64(sin(re) * (im ^ 3.0))) - Float64(im * sin(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	tmp = 0.0;
	if ((t_0 <= -Inf) || ~((t_0 <= 1e-6)))
		tmp = t_0 * (0.5 * sin(re));
	else
		tmp = (-0.16666666666666666 * (sin(re) * (im ^ 3.0))) - (im * sin(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, (-Infinity)], N[Not[LessEqual[t$95$0, 1e-6]], $MachinePrecision]], N[(t$95$0 * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.16666666666666666 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(im * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -inf.0 or 9.99999999999999955e-7 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 100.0%

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

   if -inf.0 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < 9.99999999999999955e-7

   1. Initial program 33.4%

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

      1. *-commutative 33.4%

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

      2. associate-*l* 33.4%

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

      3. sub-neg 33.4%

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

      4. +-commutative 33.4%

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

      5. distribute-rgt-in 33.4%

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

      6. distribute-lft-neg-out 33.4%

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

      7. distribute-rgt-neg-in 33.4%

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

      8. metadata-eval 33.4%

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

      9. metadata-eval 33.4%

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

      10. fma-def 33.4%

          \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(e^{im}, \frac{-1}{2}, e^{-im} \cdot 0.5\right)} \]

      11. metadata-eval 33.4%

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

      12. exp-neg 33.3%

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

      13. associate-*l/ 33.3%

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

      14. metadata-eval 33.3%

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

   3. Simplified 33.3%

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

   4. Taylor expanded in im around 0 99.8%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right) + -1 \cdot \left(\sin re \cdot im\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 99.8%

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

      2. unsub-neg 99.8%

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

      3. *-commutative 99.8%

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

      4. *-commutative 99.8%

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

   6. Simplified 99.8%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right) - im \cdot \sin re} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -\infty \lor \neg \left(e^{-im} - e^{im} \leq 10^{-6}\right):\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right) - im \cdot \sin re\\ \end{array} \]

Alternative 2: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 10^{-6}\right):\\ \;\;\;\;t_0 \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 1e-6)))
     (* t_0 (* 0.5 (sin re)))
     (* (sin re) (- (* -0.16666666666666666 (pow im 3.0)) im)))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || !(t_0 <= 1e-6)) {
		tmp = t_0 * (0.5 * sin(re));
	} else {
		tmp = sin(re) * ((-0.16666666666666666 * pow(im, 3.0)) - im);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(-im) - Math.exp(im);
	double tmp;
	if ((t_0 <= -Double.POSITIVE_INFINITY) || !(t_0 <= 1e-6)) {
		tmp = t_0 * (0.5 * Math.sin(re));
	} else {
		tmp = Math.sin(re) * ((-0.16666666666666666 * Math.pow(im, 3.0)) - im);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	tmp = 0
	if (t_0 <= -math.inf) or not (t_0 <= 1e-6):
		tmp = t_0 * (0.5 * math.sin(re))
	else:
		tmp = math.sin(re) * ((-0.16666666666666666 * math.pow(im, 3.0)) - im)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	tmp = 0.0
	if ((t_0 <= Float64(-Inf)) || !(t_0 <= 1e-6))
		tmp = Float64(t_0 * Float64(0.5 * sin(re)));
	else
		tmp = Float64(sin(re) * Float64(Float64(-0.16666666666666666 * (im ^ 3.0)) - im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	tmp = 0.0;
	if ((t_0 <= -Inf) || ~((t_0 <= 1e-6)))
		tmp = t_0 * (0.5 * sin(re));
	else
		tmp = sin(re) * ((-0.16666666666666666 * (im ^ 3.0)) - im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, (-Infinity)], N[Not[LessEqual[t$95$0, 1e-6]], $MachinePrecision]], N[(t$95$0 * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(N[(-0.16666666666666666 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -inf.0 or 9.99999999999999955e-7 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 100.0%

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

   if -inf.0 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < 9.99999999999999955e-7

   1. Initial program 33.4%

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

      1. *-commutative 33.4%

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

      2. associate-*l* 33.4%

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

      3. sub-neg 33.4%

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

      4. +-commutative 33.4%

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

      5. distribute-rgt-in 33.4%

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

      6. distribute-lft-neg-out 33.4%

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

      7. distribute-rgt-neg-in 33.4%

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

      8. metadata-eval 33.4%

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

      9. metadata-eval 33.4%

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

      10. fma-def 33.4%

          \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(e^{im}, \frac{-1}{2}, e^{-im} \cdot 0.5\right)} \]

      11. metadata-eval 33.4%

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

      12. exp-neg 33.3%

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

      13. associate-*l/ 33.3%

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

      14. metadata-eval 33.3%

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

   3. Simplified 33.3%

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

   4. Taylor expanded in im around 0 99.8%

      \[\leadsto \sin re \cdot \color{blue}{\left(-0.16666666666666666 \cdot {im}^{3} + -1 \cdot im\right)} \]
   5. Step-by-step derivation

      1. neg-mul-1 99.8%

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

      2. unsub-neg 99.8%

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

   6. Simplified 99.8%

      \[\leadsto \sin re \cdot \color{blue}{\left(-0.16666666666666666 \cdot {im}^{3} - im\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -\infty \lor \neg \left(e^{-im} - e^{im} \leq 10^{-6}\right):\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \end{array} \]

Alternative 3: 94.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + -0.08333333333333333 \cdot \left(re \cdot re\right)\right)\right)\\ t_1 := -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\\ \mathbf{if}\;im \leq -8 \cdot 10^{+91}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.34:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.032:\\ \;\;\;\;\sin re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \mathbf{elif}\;im \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (*
          re
          (*
           (- (exp (- im)) (exp im))
           (+ 0.5 (* -0.08333333333333333 (* re re))))))
        (t_1 (* -0.16666666666666666 (* (sin re) (pow im 3.0)))))
   (if (<= im -8e+91)
     t_1
     (if (<= im -0.34)
       t_0
       (if (<= im 0.032)
         (* (sin re) (- (* -0.16666666666666666 (pow im 3.0)) im))
         (if (<= im 5.6e+102) t_0 t_1))))))

C

double code(double re, double im) {
	double t_0 = re * ((exp(-im) - exp(im)) * (0.5 + (-0.08333333333333333 * (re * re))));
	double t_1 = -0.16666666666666666 * (sin(re) * pow(im, 3.0));
	double tmp;
	if (im <= -8e+91) {
		tmp = t_1;
	} else if (im <= -0.34) {
		tmp = t_0;
	} else if (im <= 0.032) {
		tmp = sin(re) * ((-0.16666666666666666 * pow(im, 3.0)) - im);
	} else if (im <= 5.6e+102) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = re * ((exp(-im) - exp(im)) * (0.5d0 + ((-0.08333333333333333d0) * (re * re))))
    t_1 = (-0.16666666666666666d0) * (sin(re) * (im ** 3.0d0))
    if (im <= (-8d+91)) then
        tmp = t_1
    else if (im <= (-0.34d0)) then
        tmp = t_0
    else if (im <= 0.032d0) then
        tmp = sin(re) * (((-0.16666666666666666d0) * (im ** 3.0d0)) - im)
    else if (im <= 5.6d+102) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re * ((Math.exp(-im) - Math.exp(im)) * (0.5 + (-0.08333333333333333 * (re * re))));
	double t_1 = -0.16666666666666666 * (Math.sin(re) * Math.pow(im, 3.0));
	double tmp;
	if (im <= -8e+91) {
		tmp = t_1;
	} else if (im <= -0.34) {
		tmp = t_0;
	} else if (im <= 0.032) {
		tmp = Math.sin(re) * ((-0.16666666666666666 * Math.pow(im, 3.0)) - im);
	} else if (im <= 5.6e+102) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * ((math.exp(-im) - math.exp(im)) * (0.5 + (-0.08333333333333333 * (re * re))))
	t_1 = -0.16666666666666666 * (math.sin(re) * math.pow(im, 3.0))
	tmp = 0
	if im <= -8e+91:
		tmp = t_1
	elif im <= -0.34:
		tmp = t_0
	elif im <= 0.032:
		tmp = math.sin(re) * ((-0.16666666666666666 * math.pow(im, 3.0)) - im)
	elif im <= 5.6e+102:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(Float64(exp(Float64(-im)) - exp(im)) * Float64(0.5 + Float64(-0.08333333333333333 * Float64(re * re)))))
	t_1 = Float64(-0.16666666666666666 * Float64(sin(re) * (im ^ 3.0)))
	tmp = 0.0
	if (im <= -8e+91)
		tmp = t_1;
	elseif (im <= -0.34)
		tmp = t_0;
	elseif (im <= 0.032)
		tmp = Float64(sin(re) * Float64(Float64(-0.16666666666666666 * (im ^ 3.0)) - im));
	elseif (im <= 5.6e+102)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * ((exp(-im) - exp(im)) * (0.5 + (-0.08333333333333333 * (re * re))));
	t_1 = -0.16666666666666666 * (sin(re) * (im ^ 3.0));
	tmp = 0.0;
	if (im <= -8e+91)
		tmp = t_1;
	elseif (im <= -0.34)
		tmp = t_0;
	elseif (im <= 0.032)
		tmp = sin(re) * ((-0.16666666666666666 * (im ^ 3.0)) - im);
	elseif (im <= 5.6e+102)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(-0.08333333333333333 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-0.16666666666666666 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -8e+91], t$95$1, If[LessEqual[im, -0.34], t$95$0, If[LessEqual[im, 0.032], N[(N[Sin[re], $MachinePrecision] * N[(N[(-0.16666666666666666 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 5.6e+102], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -8.00000000000000064e91 or 5.60000000000000037e102 < im

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-rgt-in 100.0%

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

      6. distribute-lft-neg-out 100.0%

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

      7. distribute-rgt-neg-in 100.0%

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

      8. metadata-eval 100.0%

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

      9. metadata-eval 100.0%

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

      10. fma-def 100.0%

          \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(e^{im}, \frac{-1}{2}, e^{-im} \cdot 0.5\right)} \]

      11. metadata-eval 100.0%

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

      12. exp-neg 100.0%

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

      13. associate-*l/ 100.0%

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

      14. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 98.9%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right) + -1 \cdot \left(\sin re \cdot im\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 98.9%

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

      2. unsub-neg 98.9%

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

      3. *-commutative 98.9%

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

      4. *-commutative 98.9%

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

   6. Simplified 98.9%

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

   7. Taylor expanded in im around 0 98.9%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right) + -1 \cdot \left(\sin re \cdot im\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg 98.9%

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

      2. sub-neg 98.9%

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

      3. *-commutative 98.9%

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

      4. associate-*l* 98.9%

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

      5. *-commutative 98.9%

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

      6. distribute-lft-out-- 98.9%

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

      7. *-commutative 98.9%

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

   9. Simplified 98.9%

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

   10. Taylor expanded in im around inf 98.9%

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

       1. *-commutative 98.9%

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

   12. Simplified 98.9%

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

   if -8.00000000000000064e91 < im < -0.340000000000000024 or 0.032000000000000001 < im < 5.60000000000000037e102

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 0.0%

      \[\leadsto \color{blue}{-0.08333333333333333 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot {re}^{3}\right) + 0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)} \]
   3. Step-by-step derivation

      1. associate-*r* 0.0%

         \[\leadsto \color{blue}{\left(-0.08333333333333333 \cdot \left(e^{-im} - e^{im}\right)\right) \cdot {re}^{3}} + 0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right) \]

      2. unpow3 0.0%

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

      3. associate-*r* 0.0%

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

      4. associate-*r* 0.0%

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

      5. distribute-rgt-out 0.0%

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

      6. *-commutative 0.0%

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

      7. associate-*l* 0.0%

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

      8. *-commutative 0.0%

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

      9. distribute-lft-out 81.6%

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

   4. Simplified 81.6%

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

   if -0.340000000000000024 < im < 0.032000000000000001

   1. Initial program 33.4%

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

      1. *-commutative 33.4%

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

      2. associate-*l* 33.4%

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

      3. sub-neg 33.4%

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

      4. +-commutative 33.4%

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

      5. distribute-rgt-in 33.4%

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

      6. distribute-lft-neg-out 33.4%

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

      7. distribute-rgt-neg-in 33.4%

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

      8. metadata-eval 33.4%

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

      9. metadata-eval 33.4%

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

      10. fma-def 33.4%

          \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(e^{im}, \frac{-1}{2}, e^{-im} \cdot 0.5\right)} \]

      11. metadata-eval 33.4%

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

      12. exp-neg 33.3%

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

      13. associate-*l/ 33.3%

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

      14. metadata-eval 33.3%

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

   3. Simplified 33.3%

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

   4. Taylor expanded in im around 0 99.8%

      \[\leadsto \sin re \cdot \color{blue}{\left(-0.16666666666666666 \cdot {im}^{3} + -1 \cdot im\right)} \]
   5. Step-by-step derivation

      1. neg-mul-1 99.8%

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

      2. unsub-neg 99.8%

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

   6. Simplified 99.8%

      \[\leadsto \sin re \cdot \color{blue}{\left(-0.16666666666666666 \cdot {im}^{3} - im\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -8 \cdot 10^{+91}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\\ \mathbf{elif}\;im \leq -0.34:\\ \;\;\;\;re \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + -0.08333333333333333 \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{elif}\;im \leq 0.032:\\ \;\;\;\;\sin re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \mathbf{elif}\;im \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;re \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + -0.08333333333333333 \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\\ \end{array} \]

Alternative 4: 94.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)\\ t_1 := -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\\ \mathbf{if}\;im \leq -7.5 \cdot 10^{+109}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -170000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 14000:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 9.5 \cdot 10^{+101}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (* (- (exp (- im)) (exp im)) re)))
        (t_1 (* -0.16666666666666666 (* (sin re) (pow im 3.0)))))
   (if (<= im -7.5e+109)
     t_1
     (if (<= im -170000000.0)
       t_0
       (if (<= im 14000.0)
         (* im (- (sin re)))
         (if (<= im 9.5e+101) t_0 t_1))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * ((exp(-im) - exp(im)) * re);
	double t_1 = -0.16666666666666666 * (sin(re) * pow(im, 3.0));
	double tmp;
	if (im <= -7.5e+109) {
		tmp = t_1;
	} else if (im <= -170000000.0) {
		tmp = t_0;
	} else if (im <= 14000.0) {
		tmp = im * -sin(re);
	} else if (im <= 9.5e+101) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * ((exp(-im) - exp(im)) * re)
    t_1 = (-0.16666666666666666d0) * (sin(re) * (im ** 3.0d0))
    if (im <= (-7.5d+109)) then
        tmp = t_1
    else if (im <= (-170000000.0d0)) then
        tmp = t_0
    else if (im <= 14000.0d0) then
        tmp = im * -sin(re)
    else if (im <= 9.5d+101) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * ((Math.exp(-im) - Math.exp(im)) * re);
	double t_1 = -0.16666666666666666 * (Math.sin(re) * Math.pow(im, 3.0));
	double tmp;
	if (im <= -7.5e+109) {
		tmp = t_1;
	} else if (im <= -170000000.0) {
		tmp = t_0;
	} else if (im <= 14000.0) {
		tmp = im * -Math.sin(re);
	} else if (im <= 9.5e+101) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * ((math.exp(-im) - math.exp(im)) * re)
	t_1 = -0.16666666666666666 * (math.sin(re) * math.pow(im, 3.0))
	tmp = 0
	if im <= -7.5e+109:
		tmp = t_1
	elif im <= -170000000.0:
		tmp = t_0
	elif im <= 14000.0:
		tmp = im * -math.sin(re)
	elif im <= 9.5e+101:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(Float64(exp(Float64(-im)) - exp(im)) * re))
	t_1 = Float64(-0.16666666666666666 * Float64(sin(re) * (im ^ 3.0)))
	tmp = 0.0
	if (im <= -7.5e+109)
		tmp = t_1;
	elseif (im <= -170000000.0)
		tmp = t_0;
	elseif (im <= 14000.0)
		tmp = Float64(im * Float64(-sin(re)));
	elseif (im <= 9.5e+101)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * ((exp(-im) - exp(im)) * re);
	t_1 = -0.16666666666666666 * (sin(re) * (im ^ 3.0));
	tmp = 0.0;
	if (im <= -7.5e+109)
		tmp = t_1;
	elseif (im <= -170000000.0)
		tmp = t_0;
	elseif (im <= 14000.0)
		tmp = im * -sin(re);
	elseif (im <= 9.5e+101)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-0.16666666666666666 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -7.5e+109], t$95$1, If[LessEqual[im, -170000000.0], t$95$0, If[LessEqual[im, 14000.0], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im, 9.5e+101], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -7.50000000000000018e109 or 9.49999999999999947e101 < im

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-rgt-in 100.0%

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

      6. distribute-lft-neg-out 100.0%

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

      7. distribute-rgt-neg-in 100.0%

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

      8. metadata-eval 100.0%

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

      9. metadata-eval 100.0%

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

      10. fma-def 100.0%

          \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(e^{im}, \frac{-1}{2}, e^{-im} \cdot 0.5\right)} \]

      11. metadata-eval 100.0%

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

      12. exp-neg 100.0%

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

      13. associate-*l/ 100.0%

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

      14. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 99.0%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right) + -1 \cdot \left(\sin re \cdot im\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 99.0%

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

      2. unsub-neg 99.0%

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

      3. *-commutative 99.0%

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

      4. *-commutative 99.0%

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

   6. Simplified 99.0%

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

   7. Taylor expanded in im around 0 99.0%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right) + -1 \cdot \left(\sin re \cdot im\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg 99.0%

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

      2. sub-neg 99.0%

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

      3. *-commutative 99.0%

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

      4. associate-*l* 99.0%

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

      5. *-commutative 99.0%

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

      6. distribute-lft-out-- 99.0%

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

      7. *-commutative 99.0%

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

   9. Simplified 99.0%

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

   10. Taylor expanded in im around inf 99.0%

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

       1. *-commutative 99.0%

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

   12. Simplified 99.0%

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

   if -7.50000000000000018e109 < im < -1.7e8 or 14000 < im < 9.49999999999999947e101

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 75.7%

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

   if -1.7e8 < im < 14000

   1. Initial program 34.4%

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

      1. *-commutative 34.4%

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

      2. associate-*l* 34.4%

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

      3. sub-neg 34.4%

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

      4. +-commutative 34.4%

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

      5. distribute-rgt-in 34.4%

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

      6. distribute-lft-neg-out 34.4%

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

      7. distribute-rgt-neg-in 34.4%

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

      8. metadata-eval 34.4%

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

      9. metadata-eval 34.4%

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

      10. fma-def 34.4%

          \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(e^{im}, \frac{-1}{2}, e^{-im} \cdot 0.5\right)} \]

      11. metadata-eval 34.4%

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

      12. exp-neg 34.3%

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

      13. associate-*l/ 34.3%

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

      14. metadata-eval 34.3%

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

   3. Simplified 34.3%

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

   4. Taylor expanded in im around 0 98.3%

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

      1. neg-mul-1 98.3%

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

   6. Simplified 98.3%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -7.5 \cdot 10^{+109}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\\ \mathbf{elif}\;im \leq -170000000:\\ \;\;\;\;0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)\\ \mathbf{elif}\;im \leq 14000:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 9.5 \cdot 10^{+101}:\\ \;\;\;\;0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\\ \end{array} \]

Alternative 5: 95.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)\\ t_1 := -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\\ \mathbf{if}\;im \leq -7.5 \cdot 10^{+109}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -170000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 14000:\\ \;\;\;\;\sin re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \mathbf{elif}\;im \leq 9.5 \cdot 10^{+101}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (* (- (exp (- im)) (exp im)) re)))
        (t_1 (* -0.16666666666666666 (* (sin re) (pow im 3.0)))))
   (if (<= im -7.5e+109)
     t_1
     (if (<= im -170000000.0)
       t_0
       (if (<= im 14000.0)
         (* (sin re) (- (* -0.16666666666666666 (pow im 3.0)) im))
         (if (<= im 9.5e+101) t_0 t_1))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * ((exp(-im) - exp(im)) * re);
	double t_1 = -0.16666666666666666 * (sin(re) * pow(im, 3.0));
	double tmp;
	if (im <= -7.5e+109) {
		tmp = t_1;
	} else if (im <= -170000000.0) {
		tmp = t_0;
	} else if (im <= 14000.0) {
		tmp = sin(re) * ((-0.16666666666666666 * pow(im, 3.0)) - im);
	} else if (im <= 9.5e+101) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * ((exp(-im) - exp(im)) * re)
    t_1 = (-0.16666666666666666d0) * (sin(re) * (im ** 3.0d0))
    if (im <= (-7.5d+109)) then
        tmp = t_1
    else if (im <= (-170000000.0d0)) then
        tmp = t_0
    else if (im <= 14000.0d0) then
        tmp = sin(re) * (((-0.16666666666666666d0) * (im ** 3.0d0)) - im)
    else if (im <= 9.5d+101) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * ((Math.exp(-im) - Math.exp(im)) * re);
	double t_1 = -0.16666666666666666 * (Math.sin(re) * Math.pow(im, 3.0));
	double tmp;
	if (im <= -7.5e+109) {
		tmp = t_1;
	} else if (im <= -170000000.0) {
		tmp = t_0;
	} else if (im <= 14000.0) {
		tmp = Math.sin(re) * ((-0.16666666666666666 * Math.pow(im, 3.0)) - im);
	} else if (im <= 9.5e+101) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * ((math.exp(-im) - math.exp(im)) * re)
	t_1 = -0.16666666666666666 * (math.sin(re) * math.pow(im, 3.0))
	tmp = 0
	if im <= -7.5e+109:
		tmp = t_1
	elif im <= -170000000.0:
		tmp = t_0
	elif im <= 14000.0:
		tmp = math.sin(re) * ((-0.16666666666666666 * math.pow(im, 3.0)) - im)
	elif im <= 9.5e+101:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(Float64(exp(Float64(-im)) - exp(im)) * re))
	t_1 = Float64(-0.16666666666666666 * Float64(sin(re) * (im ^ 3.0)))
	tmp = 0.0
	if (im <= -7.5e+109)
		tmp = t_1;
	elseif (im <= -170000000.0)
		tmp = t_0;
	elseif (im <= 14000.0)
		tmp = Float64(sin(re) * Float64(Float64(-0.16666666666666666 * (im ^ 3.0)) - im));
	elseif (im <= 9.5e+101)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * ((exp(-im) - exp(im)) * re);
	t_1 = -0.16666666666666666 * (sin(re) * (im ^ 3.0));
	tmp = 0.0;
	if (im <= -7.5e+109)
		tmp = t_1;
	elseif (im <= -170000000.0)
		tmp = t_0;
	elseif (im <= 14000.0)
		tmp = sin(re) * ((-0.16666666666666666 * (im ^ 3.0)) - im);
	elseif (im <= 9.5e+101)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-0.16666666666666666 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -7.5e+109], t$95$1, If[LessEqual[im, -170000000.0], t$95$0, If[LessEqual[im, 14000.0], N[(N[Sin[re], $MachinePrecision] * N[(N[(-0.16666666666666666 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 9.5e+101], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -7.50000000000000018e109 or 9.49999999999999947e101 < im

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-rgt-in 100.0%

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

      6. distribute-lft-neg-out 100.0%

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

      7. distribute-rgt-neg-in 100.0%

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

      8. metadata-eval 100.0%

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

      9. metadata-eval 100.0%

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

      10. fma-def 100.0%

          \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(e^{im}, \frac{-1}{2}, e^{-im} \cdot 0.5\right)} \]

      11. metadata-eval 100.0%

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

      12. exp-neg 100.0%

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

      13. associate-*l/ 100.0%

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

      14. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 99.0%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right) + -1 \cdot \left(\sin re \cdot im\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 99.0%

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

      2. unsub-neg 99.0%

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

      3. *-commutative 99.0%

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

      4. *-commutative 99.0%

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

   6. Simplified 99.0%

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

   7. Taylor expanded in im around 0 99.0%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right) + -1 \cdot \left(\sin re \cdot im\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg 99.0%

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

      2. sub-neg 99.0%

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

      3. *-commutative 99.0%

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

      4. associate-*l* 99.0%

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

      5. *-commutative 99.0%

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

      6. distribute-lft-out-- 99.0%

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

      7. *-commutative 99.0%

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

   9. Simplified 99.0%

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

   10. Taylor expanded in im around inf 99.0%

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

       1. *-commutative 99.0%

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

   12. Simplified 99.0%

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

   if -7.50000000000000018e109 < im < -1.7e8 or 14000 < im < 9.49999999999999947e101

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 75.7%

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

   if -1.7e8 < im < 14000

   1. Initial program 34.4%

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

      1. *-commutative 34.4%

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

      2. associate-*l* 34.4%

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

      3. sub-neg 34.4%

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

      4. +-commutative 34.4%

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

      5. distribute-rgt-in 34.4%

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

      6. distribute-lft-neg-out 34.4%

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

      7. distribute-rgt-neg-in 34.4%

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

      8. metadata-eval 34.4%

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

      9. metadata-eval 34.4%

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

      10. fma-def 34.4%

          \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(e^{im}, \frac{-1}{2}, e^{-im} \cdot 0.5\right)} \]

      11. metadata-eval 34.4%

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

      12. exp-neg 34.3%

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

      13. associate-*l/ 34.3%

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

      14. metadata-eval 34.3%

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

   3. Simplified 34.3%

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

   4. Taylor expanded in im around 0 98.4%

      \[\leadsto \sin re \cdot \color{blue}{\left(-0.16666666666666666 \cdot {im}^{3} + -1 \cdot im\right)} \]
   5. Step-by-step derivation

      1. neg-mul-1 98.4%

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

      2. unsub-neg 98.4%

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

   6. Simplified 98.4%

      \[\leadsto \sin re \cdot \color{blue}{\left(-0.16666666666666666 \cdot {im}^{3} - im\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -7.5 \cdot 10^{+109}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\\ \mathbf{elif}\;im \leq -170000000:\\ \;\;\;\;0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)\\ \mathbf{elif}\;im \leq 14000:\\ \;\;\;\;\sin re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \mathbf{elif}\;im \leq 9.5 \cdot 10^{+101}:\\ \;\;\;\;0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\\ \end{array} \]

Alternative 6: 83.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -2.5 \lor \neg \left(im \leq 2.5\right):\\ \;\;\;\;-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -2.5) (not (<= im 2.5)))
   (* -0.16666666666666666 (* (sin re) (pow im 3.0)))
   (* im (- (sin re)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -2.5) || !(im <= 2.5)) {
		tmp = -0.16666666666666666 * (sin(re) * pow(im, 3.0));
	} else {
		tmp = im * -sin(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-2.5d0)) .or. (.not. (im <= 2.5d0))) then
        tmp = (-0.16666666666666666d0) * (sin(re) * (im ** 3.0d0))
    else
        tmp = im * -sin(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -2.5) || !(im <= 2.5)) {
		tmp = -0.16666666666666666 * (Math.sin(re) * Math.pow(im, 3.0));
	} else {
		tmp = im * -Math.sin(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -2.5) or not (im <= 2.5):
		tmp = -0.16666666666666666 * (math.sin(re) * math.pow(im, 3.0))
	else:
		tmp = im * -math.sin(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -2.5) || !(im <= 2.5))
		tmp = Float64(-0.16666666666666666 * Float64(sin(re) * (im ^ 3.0)));
	else
		tmp = Float64(im * Float64(-sin(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -2.5) || ~((im <= 2.5)))
		tmp = -0.16666666666666666 * (sin(re) * (im ^ 3.0));
	else
		tmp = im * -sin(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -2.5], N[Not[LessEqual[im, 2.5]], $MachinePrecision]], N[(-0.16666666666666666 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -2.5 or 2.5 < im

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-rgt-in 100.0%

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

      6. distribute-lft-neg-out 100.0%

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

      7. distribute-rgt-neg-in 100.0%

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

      8. metadata-eval 100.0%

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

      9. metadata-eval 100.0%

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

      10. fma-def 100.0%

          \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(e^{im}, \frac{-1}{2}, e^{-im} \cdot 0.5\right)} \]

      11. metadata-eval 100.0%

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

      12. exp-neg 100.0%

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

      13. associate-*l/ 100.0%

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

      14. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 69.7%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right) + -1 \cdot \left(\sin re \cdot im\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 69.7%

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

      2. unsub-neg 69.7%

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

      3. *-commutative 69.7%

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

      4. *-commutative 69.7%

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

   6. Simplified 69.7%

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

   7. Taylor expanded in im around 0 69.7%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right) + -1 \cdot \left(\sin re \cdot im\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg 69.7%

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

      2. sub-neg 69.7%

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

      3. *-commutative 69.7%

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

      4. associate-*l* 69.7%

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

      5. *-commutative 69.7%

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

      6. distribute-lft-out-- 69.7%

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

      7. *-commutative 69.7%

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

   9. Simplified 69.7%

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

   10. Taylor expanded in im around inf 69.7%

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

       1. *-commutative 69.7%

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

   12. Simplified 69.7%

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

   if -2.5 < im < 2.5

   1. Initial program 33.4%

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

      1. *-commutative 33.4%

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

      2. associate-*l* 33.4%

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

      3. sub-neg 33.4%

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

      4. +-commutative 33.4%

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

      5. distribute-rgt-in 33.4%

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

      6. distribute-lft-neg-out 33.4%

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

      7. distribute-rgt-neg-in 33.4%

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

      8. metadata-eval 33.4%

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

      9. metadata-eval 33.4%

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

      10. fma-def 33.4%

          \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(e^{im}, \frac{-1}{2}, e^{-im} \cdot 0.5\right)} \]

      11. metadata-eval 33.4%

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

      12. exp-neg 33.3%

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

      13. associate-*l/ 33.3%

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

      14. metadata-eval 33.3%

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

   3. Simplified 33.3%

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

   4. Taylor expanded in im around 0 99.7%

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

      1. neg-mul-1 99.7%

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

   6. Simplified 99.7%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.5 \lor \neg \left(im \leq 2.5\right):\\ \;\;\;\;-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \]

Alternative 7: 76.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \mathbf{if}\;im \leq -2.8 \cdot 10^{+50}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+23}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 1.55 \cdot 10^{+272} \lor \neg \left(im \leq 2.05 \cdot 10^{+288}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-1270932914164.5 - im \cdot \left(re + -0.16666666666666666 \cdot {re}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (- (* -0.16666666666666666 (pow im 3.0)) im))))
   (if (<= im -2.8e+50)
     t_0
     (if (<= im 2e+23)
       (* im (- (sin re)))
       (if (or (<= im 1.55e+272) (not (<= im 2.05e+288)))
         t_0
         (-
          -1270932914164.5
          (* im (+ re (* -0.16666666666666666 (pow re 3.0))))))))))

C

double code(double re, double im) {
	double t_0 = re * ((-0.16666666666666666 * pow(im, 3.0)) - im);
	double tmp;
	if (im <= -2.8e+50) {
		tmp = t_0;
	} else if (im <= 2e+23) {
		tmp = im * -sin(re);
	} else if ((im <= 1.55e+272) || !(im <= 2.05e+288)) {
		tmp = t_0;
	} else {
		tmp = -1270932914164.5 - (im * (re + (-0.16666666666666666 * pow(re, 3.0))));
	}
	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 = re * (((-0.16666666666666666d0) * (im ** 3.0d0)) - im)
    if (im <= (-2.8d+50)) then
        tmp = t_0
    else if (im <= 2d+23) then
        tmp = im * -sin(re)
    else if ((im <= 1.55d+272) .or. (.not. (im <= 2.05d+288))) then
        tmp = t_0
    else
        tmp = (-1270932914164.5d0) - (im * (re + ((-0.16666666666666666d0) * (re ** 3.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re * ((-0.16666666666666666 * Math.pow(im, 3.0)) - im);
	double tmp;
	if (im <= -2.8e+50) {
		tmp = t_0;
	} else if (im <= 2e+23) {
		tmp = im * -Math.sin(re);
	} else if ((im <= 1.55e+272) || !(im <= 2.05e+288)) {
		tmp = t_0;
	} else {
		tmp = -1270932914164.5 - (im * (re + (-0.16666666666666666 * Math.pow(re, 3.0))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * ((-0.16666666666666666 * math.pow(im, 3.0)) - im)
	tmp = 0
	if im <= -2.8e+50:
		tmp = t_0
	elif im <= 2e+23:
		tmp = im * -math.sin(re)
	elif (im <= 1.55e+272) or not (im <= 2.05e+288):
		tmp = t_0
	else:
		tmp = -1270932914164.5 - (im * (re + (-0.16666666666666666 * math.pow(re, 3.0))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(Float64(-0.16666666666666666 * (im ^ 3.0)) - im))
	tmp = 0.0
	if (im <= -2.8e+50)
		tmp = t_0;
	elseif (im <= 2e+23)
		tmp = Float64(im * Float64(-sin(re)));
	elseif ((im <= 1.55e+272) || !(im <= 2.05e+288))
		tmp = t_0;
	else
		tmp = Float64(-1270932914164.5 - Float64(im * Float64(re + Float64(-0.16666666666666666 * (re ^ 3.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * ((-0.16666666666666666 * (im ^ 3.0)) - im);
	tmp = 0.0;
	if (im <= -2.8e+50)
		tmp = t_0;
	elseif (im <= 2e+23)
		tmp = im * -sin(re);
	elseif ((im <= 1.55e+272) || ~((im <= 2.05e+288)))
		tmp = t_0;
	else
		tmp = -1270932914164.5 - (im * (re + (-0.16666666666666666 * (re ^ 3.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(N[(-0.16666666666666666 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -2.8e+50], t$95$0, If[LessEqual[im, 2e+23], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], If[Or[LessEqual[im, 1.55e+272], N[Not[LessEqual[im, 2.05e+288]], $MachinePrecision]], t$95$0, N[(-1270932914164.5 - N[(im * N[(re + N[(-0.16666666666666666 * N[Power[re, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -2.7999999999999998e50 or 1.9999999999999998e23 < im < 1.54999999999999986e272 or 2.0499999999999999e288 < im

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-rgt-in 100.0%

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

      6. distribute-lft-neg-out 100.0%

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

      7. distribute-rgt-neg-in 100.0%

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

      8. metadata-eval 100.0%

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

      9. metadata-eval 100.0%

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

      10. fma-def 100.0%

          \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(e^{im}, \frac{-1}{2}, e^{-im} \cdot 0.5\right)} \]

      11. metadata-eval 100.0%

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

      12. exp-neg 100.0%

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

      13. associate-*l/ 100.0%

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

      14. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 77.9%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right) + -1 \cdot \left(\sin re \cdot im\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 77.9%

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

      2. unsub-neg 77.9%

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

      3. *-commutative 77.9%

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

      4. *-commutative 77.9%

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

   6. Simplified 77.9%

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

   7. Taylor expanded in re around 0 63.7%

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

      1. *-commutative 63.7%

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

   9. Simplified 63.7%

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

   if -2.7999999999999998e50 < im < 1.9999999999999998e23

   1. Initial program 40.1%

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

      1. *-commutative 40.1%

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

      2. associate-*l* 40.1%

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

      3. sub-neg 40.1%

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

      4. +-commutative 40.1%

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

      5. distribute-rgt-in 40.1%

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

      6. distribute-lft-neg-out 40.1%

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

      7. distribute-rgt-neg-in 40.1%

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

      8. metadata-eval 40.1%

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

      9. metadata-eval 40.1%

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

      10. fma-def 40.1%

          \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(e^{im}, \frac{-1}{2}, e^{-im} \cdot 0.5\right)} \]

      11. metadata-eval 40.1%

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

      12. exp-neg 40.0%

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

      13. associate-*l/ 40.0%

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

      14. metadata-eval 40.0%

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

   3. Simplified 40.0%

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

   4. Taylor expanded in im around 0 89.9%

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

      1. neg-mul-1 89.9%

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

   6. Simplified 89.9%

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

   if 1.54999999999999986e272 < im < 2.0499999999999999e288

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-rgt-in 100.0%

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

      6. distribute-lft-neg-out 100.0%

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

      7. distribute-rgt-neg-in 100.0%

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

      8. metadata-eval 100.0%

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

      9. metadata-eval 100.0%

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

      10. fma-def 100.0%

          \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(e^{im}, \frac{-1}{2}, e^{-im} \cdot 0.5\right)} \]

      11. metadata-eval 100.0%

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

      12. exp-neg 100.0%

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

      13. associate-*l/ 100.0%

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

      14. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right) + -1 \cdot \left(\sin re \cdot im\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. *-commutative 100.0%

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

      4. *-commutative 100.0%

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

   6. Simplified 100.0%

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

   8. Applied egg-rr 8.3%

      \[\leadsto -0.16666666666666666 \cdot \color{blue}{7625597484987} - im \cdot \sin re \]

   9. Taylor expanded in re around 0 81.9%

      \[\leadsto -0.16666666666666666 \cdot 7625597484987 - im \cdot \color{blue}{\left(re + -0.16666666666666666 \cdot {re}^{3}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.8 \cdot 10^{+50}:\\ \;\;\;\;re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+23}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 1.55 \cdot 10^{+272} \lor \neg \left(im \leq 2.05 \cdot 10^{+288}\right):\\ \;\;\;\;re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \mathbf{else}:\\ \;\;\;\;-1270932914164.5 - im \cdot \left(re + -0.16666666666666666 \cdot {re}^{3}\right)\\ \end{array} \]

Alternative 8: 76.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -3.1 \cdot 10^{-7}:\\ \;\;\;\;0.5 \cdot \left(-0.3333333333333333 \cdot \left(re \cdot {im}^{3}\right) + -2 \cdot \left(im \cdot re\right)\right)\\ \mathbf{elif}\;im \leq 5.6 \cdot 10^{+24}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 1.55 \cdot 10^{+272} \lor \neg \left(im \leq 1.22 \cdot 10^{+288}\right):\\ \;\;\;\;re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \mathbf{else}:\\ \;\;\;\;-1270932914164.5 - im \cdot \left(re + -0.16666666666666666 \cdot {re}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -3.1e-7)
   (* 0.5 (+ (* -0.3333333333333333 (* re (pow im 3.0))) (* -2.0 (* im re))))
   (if (<= im 5.6e+24)
     (* im (- (sin re)))
     (if (or (<= im 1.55e+272) (not (<= im 1.22e+288)))
       (* re (- (* -0.16666666666666666 (pow im 3.0)) im))
       (-
        -1270932914164.5
        (* im (+ re (* -0.16666666666666666 (pow re 3.0)))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -3.1e-7) {
		tmp = 0.5 * ((-0.3333333333333333 * (re * pow(im, 3.0))) + (-2.0 * (im * re)));
	} else if (im <= 5.6e+24) {
		tmp = im * -sin(re);
	} else if ((im <= 1.55e+272) || !(im <= 1.22e+288)) {
		tmp = re * ((-0.16666666666666666 * pow(im, 3.0)) - im);
	} else {
		tmp = -1270932914164.5 - (im * (re + (-0.16666666666666666 * pow(re, 3.0))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= (-3.1d-7)) then
        tmp = 0.5d0 * (((-0.3333333333333333d0) * (re * (im ** 3.0d0))) + ((-2.0d0) * (im * re)))
    else if (im <= 5.6d+24) then
        tmp = im * -sin(re)
    else if ((im <= 1.55d+272) .or. (.not. (im <= 1.22d+288))) then
        tmp = re * (((-0.16666666666666666d0) * (im ** 3.0d0)) - im)
    else
        tmp = (-1270932914164.5d0) - (im * (re + ((-0.16666666666666666d0) * (re ** 3.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -3.1e-7) {
		tmp = 0.5 * ((-0.3333333333333333 * (re * Math.pow(im, 3.0))) + (-2.0 * (im * re)));
	} else if (im <= 5.6e+24) {
		tmp = im * -Math.sin(re);
	} else if ((im <= 1.55e+272) || !(im <= 1.22e+288)) {
		tmp = re * ((-0.16666666666666666 * Math.pow(im, 3.0)) - im);
	} else {
		tmp = -1270932914164.5 - (im * (re + (-0.16666666666666666 * Math.pow(re, 3.0))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -3.1e-7:
		tmp = 0.5 * ((-0.3333333333333333 * (re * math.pow(im, 3.0))) + (-2.0 * (im * re)))
	elif im <= 5.6e+24:
		tmp = im * -math.sin(re)
	elif (im <= 1.55e+272) or not (im <= 1.22e+288):
		tmp = re * ((-0.16666666666666666 * math.pow(im, 3.0)) - im)
	else:
		tmp = -1270932914164.5 - (im * (re + (-0.16666666666666666 * math.pow(re, 3.0))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -3.1e-7)
		tmp = Float64(0.5 * Float64(Float64(-0.3333333333333333 * Float64(re * (im ^ 3.0))) + Float64(-2.0 * Float64(im * re))));
	elseif (im <= 5.6e+24)
		tmp = Float64(im * Float64(-sin(re)));
	elseif ((im <= 1.55e+272) || !(im <= 1.22e+288))
		tmp = Float64(re * Float64(Float64(-0.16666666666666666 * (im ^ 3.0)) - im));
	else
		tmp = Float64(-1270932914164.5 - Float64(im * Float64(re + Float64(-0.16666666666666666 * (re ^ 3.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -3.1e-7)
		tmp = 0.5 * ((-0.3333333333333333 * (re * (im ^ 3.0))) + (-2.0 * (im * re)));
	elseif (im <= 5.6e+24)
		tmp = im * -sin(re);
	elseif ((im <= 1.55e+272) || ~((im <= 1.22e+288)))
		tmp = re * ((-0.16666666666666666 * (im ^ 3.0)) - im);
	else
		tmp = -1270932914164.5 - (im * (re + (-0.16666666666666666 * (re ^ 3.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -3.1e-7], N[(0.5 * N[(N[(-0.3333333333333333 * N[(re * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(im * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 5.6e+24], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], If[Or[LessEqual[im, 1.55e+272], N[Not[LessEqual[im, 1.22e+288]], $MachinePrecision]], N[(re * N[(N[(-0.16666666666666666 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], N[(-1270932914164.5 - N[(im * N[(re + N[(-0.16666666666666666 * N[Power[re, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < -3.1e-7

   1. Initial program 99.5%

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

   2. Taylor expanded in re around 0 72.9%

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

   3. Taylor expanded in im around 0 58.9%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.3333333333333333 \cdot \left(re \cdot {im}^{3}\right) + -2 \cdot \left(re \cdot im\right)\right)} \]

   if -3.1e-7 < im < 5.6000000000000003e24

   1. Initial program 35.1%

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

      1. *-commutative 35.1%

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

      2. associate-*l* 35.1%

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

      3. sub-neg 35.1%

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

      4. +-commutative 35.1%

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

      5. distribute-rgt-in 35.1%

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

      6. distribute-lft-neg-out 35.1%

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

      7. distribute-rgt-neg-in 35.1%

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

      8. metadata-eval 35.1%

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

      9. metadata-eval 35.1%

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

      10. fma-def 35.1%

          \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(e^{im}, \frac{-1}{2}, e^{-im} \cdot 0.5\right)} \]

      11. metadata-eval 35.1%

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

      12. exp-neg 35.0%

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

      13. associate-*l/ 35.0%

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

      14. metadata-eval 35.0%

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

   3. Simplified 35.0%

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

   4. Taylor expanded in im around 0 96.9%

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

      1. neg-mul-1 96.9%

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

   6. Simplified 96.9%

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

   if 5.6000000000000003e24 < im < 1.54999999999999986e272 or 1.22000000000000011e288 < im

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-rgt-in 100.0%

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

      6. distribute-lft-neg-out 100.0%

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

      7. distribute-rgt-neg-in 100.0%

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

      8. metadata-eval 100.0%

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

      9. metadata-eval 100.0%

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

      10. fma-def 100.0%

          \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(e^{im}, \frac{-1}{2}, e^{-im} \cdot 0.5\right)} \]

      11. metadata-eval 100.0%

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

      12. exp-neg 100.0%

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

      13. associate-*l/ 100.0%

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

      14. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 72.3%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right) + -1 \cdot \left(\sin re \cdot im\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 72.3%

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

      2. unsub-neg 72.3%

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

      3. *-commutative 72.3%

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

      4. *-commutative 72.3%

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

   6. Simplified 72.3%

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

   7. Taylor expanded in re around 0 57.2%

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

      1. *-commutative 57.2%

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

   9. Simplified 57.2%

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

   if 1.54999999999999986e272 < im < 1.22000000000000011e288

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-rgt-in 100.0%

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

      6. distribute-lft-neg-out 100.0%

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

      7. distribute-rgt-neg-in 100.0%

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

      8. metadata-eval 100.0%

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

      9. metadata-eval 100.0%

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

      10. fma-def 100.0%

          \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(e^{im}, \frac{-1}{2}, e^{-im} \cdot 0.5\right)} \]

      11. metadata-eval 100.0%

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

      12. exp-neg 100.0%

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

      13. associate-*l/ 100.0%

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

      14. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right) + -1 \cdot \left(\sin re \cdot im\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. *-commutative 100.0%

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

      4. *-commutative 100.0%

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

   6. Simplified 100.0%

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

   8. Applied egg-rr 8.3%

      \[\leadsto -0.16666666666666666 \cdot \color{blue}{7625597484987} - im \cdot \sin re \]

   9. Taylor expanded in re around 0 81.9%

      \[\leadsto -0.16666666666666666 \cdot 7625597484987 - im \cdot \color{blue}{\left(re + -0.16666666666666666 \cdot {re}^{3}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3.1 \cdot 10^{-7}:\\ \;\;\;\;0.5 \cdot \left(-0.3333333333333333 \cdot \left(re \cdot {im}^{3}\right) + -2 \cdot \left(im \cdot re\right)\right)\\ \mathbf{elif}\;im \leq 5.6 \cdot 10^{+24}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 1.55 \cdot 10^{+272} \lor \neg \left(im \leq 1.22 \cdot 10^{+288}\right):\\ \;\;\;\;re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \mathbf{else}:\\ \;\;\;\;-1270932914164.5 - im \cdot \left(re + -0.16666666666666666 \cdot {re}^{3}\right)\\ \end{array} \]

Alternative 9: 77.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -2.8 \cdot 10^{+50} \lor \neg \left(im \leq 1.5 \cdot 10^{+25}\right):\\ \;\;\;\;re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -2.8e+50) (not (<= im 1.5e+25)))
   (* re (- (* -0.16666666666666666 (pow im 3.0)) im))
   (* im (- (sin re)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -2.8e+50) || !(im <= 1.5e+25)) {
		tmp = re * ((-0.16666666666666666 * pow(im, 3.0)) - im);
	} else {
		tmp = im * -sin(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-2.8d+50)) .or. (.not. (im <= 1.5d+25))) then
        tmp = re * (((-0.16666666666666666d0) * (im ** 3.0d0)) - im)
    else
        tmp = im * -sin(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -2.8e+50) || !(im <= 1.5e+25)) {
		tmp = re * ((-0.16666666666666666 * Math.pow(im, 3.0)) - im);
	} else {
		tmp = im * -Math.sin(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -2.8e+50) or not (im <= 1.5e+25):
		tmp = re * ((-0.16666666666666666 * math.pow(im, 3.0)) - im)
	else:
		tmp = im * -math.sin(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -2.8e+50) || !(im <= 1.5e+25))
		tmp = Float64(re * Float64(Float64(-0.16666666666666666 * (im ^ 3.0)) - im));
	else
		tmp = Float64(im * Float64(-sin(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -2.8e+50) || ~((im <= 1.5e+25)))
		tmp = re * ((-0.16666666666666666 * (im ^ 3.0)) - im);
	else
		tmp = im * -sin(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -2.8e+50], N[Not[LessEqual[im, 1.5e+25]], $MachinePrecision]], N[(re * N[(N[(-0.16666666666666666 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -2.7999999999999998e50 or 1.50000000000000003e25 < im

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-rgt-in 100.0%

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

      6. distribute-lft-neg-out 100.0%

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

      7. distribute-rgt-neg-in 100.0%

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

      8. metadata-eval 100.0%

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

      9. metadata-eval 100.0%

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

      10. fma-def 100.0%

          \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(e^{im}, \frac{-1}{2}, e^{-im} \cdot 0.5\right)} \]

      11. metadata-eval 100.0%

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

      12. exp-neg 100.0%

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

      13. associate-*l/ 100.0%

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

      14. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 78.9%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right) + -1 \cdot \left(\sin re \cdot im\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 78.9%

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

      2. unsub-neg 78.9%

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

      3. *-commutative 78.9%

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

      4. *-commutative 78.9%

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

   6. Simplified 78.9%

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

   7. Taylor expanded in re around 0 60.7%

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

      1. *-commutative 60.7%

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

   9. Simplified 60.7%

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

   if -2.7999999999999998e50 < im < 1.50000000000000003e25

   1. Initial program 40.1%

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

      1. *-commutative 40.1%

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

      2. associate-*l* 40.1%

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

      3. sub-neg 40.1%

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

      4. +-commutative 40.1%

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

      5. distribute-rgt-in 40.1%

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

      6. distribute-lft-neg-out 40.1%

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

      7. distribute-rgt-neg-in 40.1%

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

      8. metadata-eval 40.1%

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

      9. metadata-eval 40.1%

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

      10. fma-def 40.1%

          \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(e^{im}, \frac{-1}{2}, e^{-im} \cdot 0.5\right)} \]

      11. metadata-eval 40.1%

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

      12. exp-neg 40.0%

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

      13. associate-*l/ 40.0%

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

      14. metadata-eval 40.0%

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

   3. Simplified 40.0%

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

   4. Taylor expanded in im around 0 89.9%

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

      1. neg-mul-1 89.9%

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

   6. Simplified 89.9%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.8 \cdot 10^{+50} \lor \neg \left(im \leq 1.5 \cdot 10^{+25}\right):\\ \;\;\;\;re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \]

Alternative 10: 58.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot {re}^{3}\\ \mathbf{if}\;im \leq -1.95 \cdot 10^{+34}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 680:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 5.5 \cdot 10^{+288}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-2 \cdot \left(im \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (pow re 3.0))))
   (if (<= im -1.95e+34)
     t_0
     (if (<= im 680.0)
       (* im (- (sin re)))
       (if (<= im 5.5e+288) t_0 (* 0.5 (* -2.0 (* im re))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * pow(re, 3.0);
	double tmp;
	if (im <= -1.95e+34) {
		tmp = t_0;
	} else if (im <= 680.0) {
		tmp = im * -sin(re);
	} else if (im <= 5.5e+288) {
		tmp = t_0;
	} else {
		tmp = 0.5 * (-2.0 * (im * re));
	}
	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 * (re ** 3.0d0)
    if (im <= (-1.95d+34)) then
        tmp = t_0
    else if (im <= 680.0d0) then
        tmp = im * -sin(re)
    else if (im <= 5.5d+288) then
        tmp = t_0
    else
        tmp = 0.5d0 * ((-2.0d0) * (im * re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.pow(re, 3.0);
	double tmp;
	if (im <= -1.95e+34) {
		tmp = t_0;
	} else if (im <= 680.0) {
		tmp = im * -Math.sin(re);
	} else if (im <= 5.5e+288) {
		tmp = t_0;
	} else {
		tmp = 0.5 * (-2.0 * (im * re));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.pow(re, 3.0)
	tmp = 0
	if im <= -1.95e+34:
		tmp = t_0
	elif im <= 680.0:
		tmp = im * -math.sin(re)
	elif im <= 5.5e+288:
		tmp = t_0
	else:
		tmp = 0.5 * (-2.0 * (im * re))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * (re ^ 3.0))
	tmp = 0.0
	if (im <= -1.95e+34)
		tmp = t_0;
	elseif (im <= 680.0)
		tmp = Float64(im * Float64(-sin(re)));
	elseif (im <= 5.5e+288)
		tmp = t_0;
	else
		tmp = Float64(0.5 * Float64(-2.0 * Float64(im * re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (re ^ 3.0);
	tmp = 0.0;
	if (im <= -1.95e+34)
		tmp = t_0;
	elseif (im <= 680.0)
		tmp = im * -sin(re);
	elseif (im <= 5.5e+288)
		tmp = t_0;
	else
		tmp = 0.5 * (-2.0 * (im * re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Power[re, 3.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.95e+34], t$95$0, If[LessEqual[im, 680.0], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im, 5.5e+288], t$95$0, N[(0.5 * N[(-2.0 * N[(im * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.9500000000000001e34 or 680 < im < 5.5e288

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-rgt-in 100.0%

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

      6. distribute-lft-neg-out 100.0%

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

      7. distribute-rgt-neg-in 100.0%

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

      8. metadata-eval 100.0%

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

      9. metadata-eval 100.0%

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

      10. fma-def 100.0%

          \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(e^{im}, \frac{-1}{2}, e^{-im} \cdot 0.5\right)} \]

      11. metadata-eval 100.0%

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

      12. exp-neg 100.0%

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

      13. associate-*l/ 100.0%

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

      14. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 1.8%

      \[\leadsto \sin re \cdot \color{blue}{-3} \]

   6. Taylor expanded in re around 0 21.1%

      \[\leadsto \color{blue}{-3 \cdot re + 0.5 \cdot {re}^{3}} \]

   7. Taylor expanded in re around inf 21.0%

      \[\leadsto \color{blue}{0.5 \cdot {re}^{3}} \]
   8. Step-by-step derivation

      1. *-commutative 21.0%

         \[\leadsto \color{blue}{{re}^{3} \cdot 0.5} \]

   9. Simplified 21.0%

      \[\leadsto \color{blue}{{re}^{3} \cdot 0.5} \]

   if -1.9500000000000001e34 < im < 680

   1. Initial program 35.8%

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

      1. *-commutative 35.8%

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

      2. associate-*l* 35.8%

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

      3. sub-neg 35.8%

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

      4. +-commutative 35.8%

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

      5. distribute-rgt-in 35.8%

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

      6. distribute-lft-neg-out 35.8%

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

      7. distribute-rgt-neg-in 35.8%

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

      8. metadata-eval 35.8%

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

      9. metadata-eval 35.8%

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

      10. fma-def 35.8%

          \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(e^{im}, \frac{-1}{2}, e^{-im} \cdot 0.5\right)} \]

      11. metadata-eval 35.8%

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

      12. exp-neg 35.7%

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

      13. associate-*l/ 35.7%

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

      14. metadata-eval 35.7%

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

   3. Simplified 35.7%

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

   4. Taylor expanded in im around 0 96.2%

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

      1. neg-mul-1 96.2%

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

   6. Simplified 96.2%

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

   if 5.5e288 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 100.0%

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

   3. Taylor expanded in im around 0 69.3%

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

      1. *-commutative 69.3%

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

   5. Simplified 69.3%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.95 \cdot 10^{+34}:\\ \;\;\;\;0.5 \cdot {re}^{3}\\ \mathbf{elif}\;im \leq 680:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 5.5 \cdot 10^{+288}:\\ \;\;\;\;0.5 \cdot {re}^{3}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-2 \cdot \left(im \cdot re\right)\right)\\ \end{array} \]

Alternative 11: 56.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -6.5 \cdot 10^{+51} \lor \neg \left(im \leq 4.6 \cdot 10^{+222}\right):\\ \;\;\;\;0.5 \cdot \left(-2 \cdot \left(im \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -6.5e+51) (not (<= im 4.6e+222)))
   (* 0.5 (* -2.0 (* im re)))
   (* im (- (sin re)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -6.5e+51) || !(im <= 4.6e+222)) {
		tmp = 0.5 * (-2.0 * (im * re));
	} else {
		tmp = im * -sin(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-6.5d+51)) .or. (.not. (im <= 4.6d+222))) then
        tmp = 0.5d0 * ((-2.0d0) * (im * re))
    else
        tmp = im * -sin(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -6.5e+51) || !(im <= 4.6e+222)) {
		tmp = 0.5 * (-2.0 * (im * re));
	} else {
		tmp = im * -Math.sin(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -6.5e+51) or not (im <= 4.6e+222):
		tmp = 0.5 * (-2.0 * (im * re))
	else:
		tmp = im * -math.sin(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -6.5e+51) || !(im <= 4.6e+222))
		tmp = Float64(0.5 * Float64(-2.0 * Float64(im * re)));
	else
		tmp = Float64(im * Float64(-sin(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -6.5e+51) || ~((im <= 4.6e+222)))
		tmp = 0.5 * (-2.0 * (im * re));
	else
		tmp = im * -sin(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -6.5e+51], N[Not[LessEqual[im, 4.6e+222]], $MachinePrecision]], N[(0.5 * N[(-2.0 * N[(im * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -6.5e51 or 4.60000000000000021e222 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 74.6%

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

   3. Taylor expanded in im around 0 23.4%

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

      1. *-commutative 23.4%

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

   5. Simplified 23.4%

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

   if -6.5e51 < im < 4.60000000000000021e222

   1. Initial program 53.1%

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

      1. *-commutative 53.1%

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

      2. associate-*l* 53.1%

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

      3. sub-neg 53.1%

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

      4. +-commutative 53.1%

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

      5. distribute-rgt-in 53.1%

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

      6. distribute-lft-neg-out 53.1%

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

      7. distribute-rgt-neg-in 53.1%

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

      8. metadata-eval 53.1%

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

      9. metadata-eval 53.1%

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

      10. fma-def 53.1%

          \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(e^{im}, \frac{-1}{2}, e^{-im} \cdot 0.5\right)} \]

      11. metadata-eval 53.1%

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

      12. exp-neg 53.1%

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

      13. associate-*l/ 53.1%

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

      14. metadata-eval 53.1%

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

   3. Simplified 53.1%

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

   4. Taylor expanded in im around 0 71.2%

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

      1. neg-mul-1 71.2%

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

   6. Simplified 71.2%

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

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -6.5 \cdot 10^{+51} \lor \neg \left(im \leq 4.6 \cdot 10^{+222}\right):\\ \;\;\;\;0.5 \cdot \left(-2 \cdot \left(im \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \]

Alternative 12: 33.5% accurate, 44.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(0.5 * N[(-2.0 * N[(im * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.4%

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

2. Taylor expanded in re around 0 51.0%

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

3. Taylor expanded in im around 0 35.3%

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

   1. *-commutative 35.3%

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

5. Simplified 35.3%

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

6. Final simplification 35.3%

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

Alternative 13: 3.2% accurate, 102.7× speedup

Math

\[\begin{array}{l} \\ re \cdot -3 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* re -3.0))

C

double code(double re, double im) {
	return re * -3.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = re * (-3.0d0)
end function

Java

public static double code(double re, double im) {
	return re * -3.0;
}

Python

def code(re, im):
	return re * -3.0

Julia

function code(re, im)
	return Float64(re * -3.0)
end

MATLAB

function tmp = code(re, im)
	tmp = re * -3.0;
end

Wolfram

code[re_, im_] := N[(re * -3.0), $MachinePrecision]

Derivation

1. Initial program 65.4%

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

   1. *-commutative 65.4%

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

   2. associate-*l* 65.4%

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

   3. sub-neg 65.4%

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

   4. +-commutative 65.4%

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

   5. distribute-rgt-in 65.4%

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

   6. distribute-lft-neg-out 65.4%

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

   7. distribute-rgt-neg-in 65.4%

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

   8. metadata-eval 65.4%

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

   9. metadata-eval 65.4%

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

   10. fma-def 65.4%

       \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(e^{im}, \frac{-1}{2}, e^{-im} \cdot 0.5\right)} \]

   11. metadata-eval 65.4%

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

   12. exp-neg 65.3%

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

   13. associate-*l/ 65.3%

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

   14. metadata-eval 65.3%

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

3. Simplified 65.3%

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

5. Applied egg-rr 3.3%

   \[\leadsto \sin re \cdot \color{blue}{-3} \]

6. Taylor expanded in re around 0 3.1%

   \[\leadsto \color{blue}{-3 \cdot re} \]
7. Step-by-step derivation

   1. *-commutative 3.1%

      \[\leadsto \color{blue}{re \cdot -3} \]

8. Simplified 3.1%

   \[\leadsto \color{blue}{re \cdot -3} \]

9. Final simplification 3.1%

   \[\leadsto re \cdot -3 \]

Alternative 14: 3.2% accurate, 102.7× speedup

Math

\[\begin{array}{l} \\ re \cdot -0.001953125 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* re -0.001953125))

C

double code(double re, double im) {
	return re * -0.001953125;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = re * (-0.001953125d0)
end function

Java

public static double code(double re, double im) {
	return re * -0.001953125;
}

Python

def code(re, im):
	return re * -0.001953125

Julia

function code(re, im)
	return Float64(re * -0.001953125)
end

MATLAB

function tmp = code(re, im)
	tmp = re * -0.001953125;
end

Wolfram

code[re_, im_] := N[(re * -0.001953125), $MachinePrecision]

Derivation

1. Initial program 65.4%

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

   1. *-commutative 65.4%

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

   2. associate-*l* 65.4%

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

   3. sub-neg 65.4%

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

   4. +-commutative 65.4%

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

   5. distribute-rgt-in 65.4%

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

   6. distribute-lft-neg-out 65.4%

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

   7. distribute-rgt-neg-in 65.4%

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

   8. metadata-eval 65.4%

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

   9. metadata-eval 65.4%

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

   10. fma-def 65.4%

       \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(e^{im}, \frac{-1}{2}, e^{-im} \cdot 0.5\right)} \]

   11. metadata-eval 65.4%

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

   12. exp-neg 65.3%

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

   13. associate-*l/ 65.3%

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

   14. metadata-eval 65.3%

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

3. Simplified 65.3%

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

5. Applied egg-rr 3.4%

   \[\leadsto \sin re \cdot \color{blue}{-0.001953125} \]

6. Taylor expanded in re around 0 3.2%

   \[\leadsto \color{blue}{re} \cdot -0.001953125 \]

7. Final simplification 3.2%

   \[\leadsto re \cdot -0.001953125 \]

Alternative 15: 14.8% accurate, 102.7× speedup

Math

\[\begin{array}{l} \\ re \cdot 0 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* re 0.0))

C

double code(double re, double im) {
	return re * 0.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = re * 0.0d0
end function

Java

public static double code(double re, double im) {
	return re * 0.0;
}

Python

def code(re, im):
	return re * 0.0

Julia

function code(re, im)
	return Float64(re * 0.0)
end

MATLAB

function tmp = code(re, im)
	tmp = re * 0.0;
end

Wolfram

code[re_, im_] := N[(re * 0.0), $MachinePrecision]

Derivation

1. Initial program 65.4%

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

   1. *-commutative 65.4%

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

   2. associate-*l* 65.4%

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

   3. sub-neg 65.4%

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

   4. +-commutative 65.4%

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

   5. distribute-rgt-in 65.4%

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

   6. distribute-lft-neg-out 65.4%

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

   7. distribute-rgt-neg-in 65.4%

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

   8. metadata-eval 65.4%

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

   9. metadata-eval 65.4%

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

   10. fma-def 65.4%

       \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(e^{im}, \frac{-1}{2}, e^{-im} \cdot 0.5\right)} \]

   11. metadata-eval 65.4%

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

   12. exp-neg 65.3%

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

   13. associate-*l/ 65.3%

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

   14. metadata-eval 65.3%

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

3. Simplified 65.3%

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

5. Applied egg-rr 16.7%

   \[\leadsto \sin re \cdot \color{blue}{0} \]

6. Taylor expanded in re around 0 16.7%

   \[\leadsto \color{blue}{re} \cdot 0 \]

7. Final simplification 16.7%

   \[\leadsto re \cdot 0 \]

Developer target: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|im\right| < 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(0.16666666666666666 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (< (fabs im) 1.0)
   (-
    (*
     (sin re)
     (+
      (+ im (* (* (* 0.16666666666666666 im) im) im))
      (* (* (* (* (* 0.008333333333333333 im) im) im) im) im))))
   (* (* 0.5 (sin re)) (- (exp (- im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if (fabs(im) < 1.0) {
		tmp = -(sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * sin(re)) * (exp(-im) - 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 (abs(im) < 1.0d0) then
        tmp = -(sin(re) * ((im + (((0.16666666666666666d0 * im) * im) * im)) + (((((0.008333333333333333d0 * im) * im) * im) * im) * im)))
    else
        tmp = (0.5d0 * sin(re)) * (exp(-im) - exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.abs(im) < 1.0) {
		tmp = -(Math.sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * Math.sin(re)) * (Math.exp(-im) - Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.fabs(im) < 1.0:
		tmp = -(math.sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)))
	else:
		tmp = (0.5 * math.sin(re)) * (math.exp(-im) - math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (abs(im) < 1.0)
		tmp = Float64(-Float64(sin(re) * Float64(Float64(im + Float64(Float64(Float64(0.16666666666666666 * im) * im) * im)) + Float64(Float64(Float64(Float64(Float64(0.008333333333333333 * im) * im) * im) * im) * im))));
	else
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (abs(im) < 1.0)
		tmp = -(sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	else
		tmp = (0.5 * sin(re)) * (exp(-im) - exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Less[N[Abs[im], $MachinePrecision], 1.0], (-N[(N[Sin[re], $MachinePrecision] * N[(N[(im + N[(N[(N[(0.16666666666666666 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[(0.008333333333333333 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2023188 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"
  :precision binary64

  :herbie-target
  (if (< (fabs im) 1.0) (- (* (sin re) (+ (+ im (* (* (* 0.16666666666666666 im) im) im)) (* (* (* (* (* 0.008333333333333333 im) im) im) im) im)))) (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))

  (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))