math.sin on complex, imaginary part

Percentage Accurate: 53.4% → 99.6%

Time: 11.6s

Alternatives: 16

Speedup: 2.9×

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

• 48.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ t_1 := e^{im} \cdot -0.5\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\left(0.5 + t_1\right) \cdot \cos re\\ \mathbf{elif}\;t_0 \leq 0.001:\\ \;\;\;\;-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) - im \cdot \cos re\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \frac{0.5}{e^{im}} + t_1 \cdot \cos re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))) (t_1 (* (exp im) -0.5)))
   (if (<= t_0 (- INFINITY))
     (* (+ 0.5 t_1) (cos re))
     (if (<= t_0 0.001)
       (- (* -0.16666666666666666 (* (cos re) (pow im 3.0))) (* im (cos re)))
       (+ (* (cos re) (/ 0.5 (exp im))) (* t_1 (cos re)))))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double t_1 = exp(im) * -0.5;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (0.5 + t_1) * cos(re);
	} else if (t_0 <= 0.001) {
		tmp = (-0.16666666666666666 * (cos(re) * pow(im, 3.0))) - (im * cos(re));
	} else {
		tmp = (cos(re) * (0.5 / exp(im))) + (t_1 * cos(re));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(-im) - Math.exp(im);
	double t_1 = Math.exp(im) * -0.5;
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = (0.5 + t_1) * Math.cos(re);
	} else if (t_0 <= 0.001) {
		tmp = (-0.16666666666666666 * (Math.cos(re) * Math.pow(im, 3.0))) - (im * Math.cos(re));
	} else {
		tmp = (Math.cos(re) * (0.5 / Math.exp(im))) + (t_1 * Math.cos(re));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	t_1 = math.exp(im) * -0.5
	tmp = 0
	if t_0 <= -math.inf:
		tmp = (0.5 + t_1) * math.cos(re)
	elif t_0 <= 0.001:
		tmp = (-0.16666666666666666 * (math.cos(re) * math.pow(im, 3.0))) - (im * math.cos(re))
	else:
		tmp = (math.cos(re) * (0.5 / math.exp(im))) + (t_1 * math.cos(re))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	t_1 = Float64(exp(im) * -0.5)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(0.5 + t_1) * cos(re));
	elseif (t_0 <= 0.001)
		tmp = Float64(Float64(-0.16666666666666666 * Float64(cos(re) * (im ^ 3.0))) - Float64(im * cos(re)));
	else
		tmp = Float64(Float64(cos(re) * Float64(0.5 / exp(im))) + Float64(t_1 * cos(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	t_1 = exp(im) * -0.5;
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = (0.5 + t_1) * cos(re);
	elseif (t_0 <= 0.001)
		tmp = (-0.16666666666666666 * (cos(re) * (im ^ 3.0))) - (im * cos(re));
	else
		tmp = (cos(re) * (0.5 / exp(im))) + (t_1 * cos(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[im], $MachinePrecision] * -0.5), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(0.5 + t$95$1), $MachinePrecision] * N[Cos[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.001], N[(N[(-0.16666666666666666 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(im * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[re], $MachinePrecision] * N[(0.5 / N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < -inf.0

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-lft-in 100.0%

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

      7. distribute-rgt-in 100.0%

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

      8. distribute-lft-neg-out 100.0%

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

      9. distribute-rgt-neg-in 100.0%

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

      10. metadata-eval 100.0%

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

      11. metadata-eval 100.0%

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

      12. fma-def 100.0%

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

      13. metadata-eval 100.0%

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

      14. exp-diff 100.0%

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

      15. associate-*l/ 100.0%

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

      16. exp-0 100.0%

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

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r/ 100.0%

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

      3. metadata-eval 100.0%

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

      4. *-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in im around 0 100.0%

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

   if -inf.0 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 1e-3

   1. Initial program 8.7%

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

      1. *-commutative 8.7%

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

      2. associate-*l* 8.7%

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

      3. sub-neg 8.7%

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

      4. +-commutative 8.7%

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

      5. distribute-lft-in 8.7%

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

      6. distribute-lft-in 8.7%

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

      7. distribute-rgt-in 8.7%

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

      8. distribute-lft-neg-out 8.7%

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

      9. distribute-rgt-neg-in 8.7%

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

      10. metadata-eval 8.7%

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

      11. metadata-eval 8.7%

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

      12. fma-def 8.7%

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

      13. metadata-eval 8.7%

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

      14. exp-diff 8.7%

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

      15. associate-*l/ 8.7%

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

      16. exp-0 8.7%

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

      17. metadata-eval 8.7%

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

   3. Simplified 8.7%

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

   4. Taylor expanded in im around 0 99.8%

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

      1. mul-1-neg 99.8%

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

      2. unsub-neg 99.8%

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

      3. *-commutative 99.8%

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

      4. *-commutative 99.8%

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

   6. Simplified 99.8%

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

   if 1e-3 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im))

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*l* 99.9%

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

      3. sub-neg 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-lft-in 99.9%

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

      6. distribute-lft-in 99.9%

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

      7. distribute-rgt-in 99.9%

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

      8. distribute-lft-neg-out 99.9%

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

      9. distribute-rgt-neg-in 99.9%

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

      10. metadata-eval 99.9%

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

      11. metadata-eval 99.9%

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

      12. fma-def 99.9%

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

      13. metadata-eval 99.9%

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

      14. exp-diff 99.9%

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

      15. associate-*l/ 99.9%

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

      16. exp-0 99.9%

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

      17. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in re around inf 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*r/ 99.9%

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

      3. metadata-eval 99.9%

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

      4. *-commutative 99.9%

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

   6. Simplified 99.9%

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

      1. *-commutative 99.9%

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

      2. distribute-rgt-in 100.0%

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

   8. Applied egg-rr 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -\infty:\\ \;\;\;\;\left(0.5 + e^{im} \cdot -0.5\right) \cdot \cos re\\ \mathbf{elif}\;e^{-im} - e^{im} \leq 0.001:\\ \;\;\;\;-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) - im \cdot \cos re\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \frac{0.5}{e^{im}} + \left(e^{im} \cdot -0.5\right) \cdot \cos re\\ \end{array} \]

Alternative 2: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ t_1 := e^{im} \cdot -0.5\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\left(0.5 + t_1\right) \cdot \cos re\\ \mathbf{elif}\;t_0 \leq 0.01:\\ \;\;\;\;\cos re \cdot \left(-0.008333333333333333 \cdot {im}^{5} + \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(t_1 + \frac{0.5}{e^{im}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))) (t_1 (* (exp im) -0.5)))
   (if (<= t_0 (- INFINITY))
     (* (+ 0.5 t_1) (cos re))
     (if (<= t_0 0.01)
       (*
        (cos re)
        (+
         (* -0.008333333333333333 (pow im 5.0))
         (- (* -0.16666666666666666 (pow im 3.0)) im)))
       (* (cos re) (+ t_1 (/ 0.5 (exp im))))))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double t_1 = exp(im) * -0.5;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (0.5 + t_1) * cos(re);
	} else if (t_0 <= 0.01) {
		tmp = cos(re) * ((-0.008333333333333333 * pow(im, 5.0)) + ((-0.16666666666666666 * pow(im, 3.0)) - im));
	} else {
		tmp = cos(re) * (t_1 + (0.5 / exp(im)));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(-im) - Math.exp(im);
	double t_1 = Math.exp(im) * -0.5;
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = (0.5 + t_1) * Math.cos(re);
	} else if (t_0 <= 0.01) {
		tmp = Math.cos(re) * ((-0.008333333333333333 * Math.pow(im, 5.0)) + ((-0.16666666666666666 * Math.pow(im, 3.0)) - im));
	} else {
		tmp = Math.cos(re) * (t_1 + (0.5 / Math.exp(im)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	t_1 = math.exp(im) * -0.5
	tmp = 0
	if t_0 <= -math.inf:
		tmp = (0.5 + t_1) * math.cos(re)
	elif t_0 <= 0.01:
		tmp = math.cos(re) * ((-0.008333333333333333 * math.pow(im, 5.0)) + ((-0.16666666666666666 * math.pow(im, 3.0)) - im))
	else:
		tmp = math.cos(re) * (t_1 + (0.5 / math.exp(im)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	t_1 = Float64(exp(im) * -0.5)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(0.5 + t_1) * cos(re));
	elseif (t_0 <= 0.01)
		tmp = Float64(cos(re) * Float64(Float64(-0.008333333333333333 * (im ^ 5.0)) + Float64(Float64(-0.16666666666666666 * (im ^ 3.0)) - im)));
	else
		tmp = Float64(cos(re) * Float64(t_1 + Float64(0.5 / exp(im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	t_1 = exp(im) * -0.5;
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = (0.5 + t_1) * cos(re);
	elseif (t_0 <= 0.01)
		tmp = cos(re) * ((-0.008333333333333333 * (im ^ 5.0)) + ((-0.16666666666666666 * (im ^ 3.0)) - im));
	else
		tmp = cos(re) * (t_1 + (0.5 / exp(im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[im], $MachinePrecision] * -0.5), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(0.5 + t$95$1), $MachinePrecision] * N[Cos[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.01], N[(N[Cos[re], $MachinePrecision] * N[(N[(-0.008333333333333333 * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.16666666666666666 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(t$95$1 + N[(0.5 / N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < -inf.0

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-lft-in 100.0%

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

      7. distribute-rgt-in 100.0%

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

      8. distribute-lft-neg-out 100.0%

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

      9. distribute-rgt-neg-in 100.0%

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

      10. metadata-eval 100.0%

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

      11. metadata-eval 100.0%

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

      12. fma-def 100.0%

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

      13. metadata-eval 100.0%

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

      14. exp-diff 100.0%

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

      15. associate-*l/ 100.0%

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

      16. exp-0 100.0%

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

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r/ 100.0%

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

      3. metadata-eval 100.0%

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

      4. *-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in im around 0 100.0%

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

   if -inf.0 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 0.0100000000000000002

   1. Initial program 9.3%

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

      1. *-commutative 9.3%

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

      2. associate-*l* 9.3%

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

      3. sub-neg 9.3%

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

      4. +-commutative 9.3%

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

      5. distribute-lft-in 9.3%

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

      6. distribute-lft-in 9.3%

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

      7. distribute-rgt-in 9.3%

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

      8. distribute-lft-neg-out 9.3%

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

      9. distribute-rgt-neg-in 9.3%

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

      10. metadata-eval 9.3%

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

      11. metadata-eval 9.3%

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

      12. fma-def 9.3%

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

      13. metadata-eval 9.3%

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

      14. exp-diff 9.3%

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

      15. associate-*l/ 9.3%

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

      16. exp-0 9.3%

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

      17. metadata-eval 9.3%

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

   3. Simplified 9.3%

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

   4. Taylor expanded in im around 0 99.8%

      \[\leadsto \cos re \cdot \color{blue}{\left(-0.008333333333333333 \cdot {im}^{5} + \left(-0.16666666666666666 \cdot {im}^{3} + -1 \cdot im\right)\right)} \]

   if 0.0100000000000000002 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im))

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-lft-in 100.0%

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

      7. distribute-rgt-in 100.0%

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

      8. distribute-lft-neg-out 100.0%

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

      9. distribute-rgt-neg-in 100.0%

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

      10. metadata-eval 100.0%

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

      11. metadata-eval 100.0%

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

      12. fma-def 100.0%

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

      13. metadata-eval 100.0%

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

      14. exp-diff 100.0%

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

      15. associate-*l/ 100.0%

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

      16. exp-0 100.0%

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

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r/ 100.0%

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

      3. metadata-eval 100.0%

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

      4. *-commutative 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -\infty:\\ \;\;\;\;\left(0.5 + e^{im} \cdot -0.5\right) \cdot \cos re\\ \mathbf{elif}\;e^{-im} - e^{im} \leq 0.01:\\ \;\;\;\;\cos re \cdot \left(-0.008333333333333333 \cdot {im}^{5} + \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(e^{im} \cdot -0.5 + \frac{0.5}{e^{im}}\right)\\ \end{array} \]

Alternative 3: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ t_1 := e^{im} \cdot -0.5\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\left(0.5 + t_1\right) \cdot \cos re\\ \mathbf{elif}\;t_0 \leq 0.001:\\ \;\;\;\;-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) - im \cdot \cos re\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(t_1 + \frac{0.5}{e^{im}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))) (t_1 (* (exp im) -0.5)))
   (if (<= t_0 (- INFINITY))
     (* (+ 0.5 t_1) (cos re))
     (if (<= t_0 0.001)
       (- (* -0.16666666666666666 (* (cos re) (pow im 3.0))) (* im (cos re)))
       (* (cos re) (+ t_1 (/ 0.5 (exp im))))))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double t_1 = exp(im) * -0.5;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (0.5 + t_1) * cos(re);
	} else if (t_0 <= 0.001) {
		tmp = (-0.16666666666666666 * (cos(re) * pow(im, 3.0))) - (im * cos(re));
	} else {
		tmp = cos(re) * (t_1 + (0.5 / exp(im)));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(-im) - Math.exp(im);
	double t_1 = Math.exp(im) * -0.5;
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = (0.5 + t_1) * Math.cos(re);
	} else if (t_0 <= 0.001) {
		tmp = (-0.16666666666666666 * (Math.cos(re) * Math.pow(im, 3.0))) - (im * Math.cos(re));
	} else {
		tmp = Math.cos(re) * (t_1 + (0.5 / Math.exp(im)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	t_1 = math.exp(im) * -0.5
	tmp = 0
	if t_0 <= -math.inf:
		tmp = (0.5 + t_1) * math.cos(re)
	elif t_0 <= 0.001:
		tmp = (-0.16666666666666666 * (math.cos(re) * math.pow(im, 3.0))) - (im * math.cos(re))
	else:
		tmp = math.cos(re) * (t_1 + (0.5 / math.exp(im)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	t_1 = Float64(exp(im) * -0.5)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(0.5 + t_1) * cos(re));
	elseif (t_0 <= 0.001)
		tmp = Float64(Float64(-0.16666666666666666 * Float64(cos(re) * (im ^ 3.0))) - Float64(im * cos(re)));
	else
		tmp = Float64(cos(re) * Float64(t_1 + Float64(0.5 / exp(im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	t_1 = exp(im) * -0.5;
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = (0.5 + t_1) * cos(re);
	elseif (t_0 <= 0.001)
		tmp = (-0.16666666666666666 * (cos(re) * (im ^ 3.0))) - (im * cos(re));
	else
		tmp = cos(re) * (t_1 + (0.5 / exp(im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[im], $MachinePrecision] * -0.5), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(0.5 + t$95$1), $MachinePrecision] * N[Cos[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.001], N[(N[(-0.16666666666666666 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(im * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(t$95$1 + N[(0.5 / N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < -inf.0

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-lft-in 100.0%

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

      7. distribute-rgt-in 100.0%

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

      8. distribute-lft-neg-out 100.0%

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

      9. distribute-rgt-neg-in 100.0%

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

      10. metadata-eval 100.0%

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

      11. metadata-eval 100.0%

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

      12. fma-def 100.0%

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

      13. metadata-eval 100.0%

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

      14. exp-diff 100.0%

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

      15. associate-*l/ 100.0%

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

      16. exp-0 100.0%

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

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r/ 100.0%

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

      3. metadata-eval 100.0%

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

      4. *-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in im around 0 100.0%

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

   if -inf.0 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 1e-3

   1. Initial program 8.7%

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

      1. *-commutative 8.7%

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

      2. associate-*l* 8.7%

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

      3. sub-neg 8.7%

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

      4. +-commutative 8.7%

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

      5. distribute-lft-in 8.7%

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

      6. distribute-lft-in 8.7%

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

      7. distribute-rgt-in 8.7%

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

      8. distribute-lft-neg-out 8.7%

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

      9. distribute-rgt-neg-in 8.7%

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

      10. metadata-eval 8.7%

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

      11. metadata-eval 8.7%

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

      12. fma-def 8.7%

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

      13. metadata-eval 8.7%

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

      14. exp-diff 8.7%

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

      15. associate-*l/ 8.7%

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

      16. exp-0 8.7%

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

      17. metadata-eval 8.7%

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

   3. Simplified 8.7%

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

   4. Taylor expanded in im around 0 99.8%

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

      1. mul-1-neg 99.8%

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

      2. unsub-neg 99.8%

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

      3. *-commutative 99.8%

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

      4. *-commutative 99.8%

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

   6. Simplified 99.8%

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

   if 1e-3 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im))

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*l* 99.9%

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

      3. sub-neg 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-lft-in 99.9%

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

      6. distribute-lft-in 99.9%

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

      7. distribute-rgt-in 99.9%

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

      8. distribute-lft-neg-out 99.9%

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

      9. distribute-rgt-neg-in 99.9%

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

      10. metadata-eval 99.9%

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

      11. metadata-eval 99.9%

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

      12. fma-def 99.9%

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

      13. metadata-eval 99.9%

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

      14. exp-diff 99.9%

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

      15. associate-*l/ 99.9%

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

      16. exp-0 99.9%

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

      17. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in re around inf 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*r/ 99.9%

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

      3. metadata-eval 99.9%

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

      4. *-commutative 99.9%

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

   6. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -\infty:\\ \;\;\;\;\left(0.5 + e^{im} \cdot -0.5\right) \cdot \cos re\\ \mathbf{elif}\;e^{-im} - e^{im} \leq 0.001:\\ \;\;\;\;-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) - im \cdot \cos re\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(e^{im} \cdot -0.5 + \frac{0.5}{e^{im}}\right)\\ \end{array} \]

Alternative 4: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ t_1 := e^{im} \cdot -0.5\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\left(0.5 + t_1\right) \cdot \cos re\\ \mathbf{elif}\;t_0 \leq 0.001:\\ \;\;\;\;\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(t_1 + \frac{0.5}{e^{im}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))) (t_1 (* (exp im) -0.5)))
   (if (<= t_0 (- INFINITY))
     (* (+ 0.5 t_1) (cos re))
     (if (<= t_0 0.001)
       (* (cos re) (- (* -0.16666666666666666 (pow im 3.0)) im))
       (* (cos re) (+ t_1 (/ 0.5 (exp im))))))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double t_1 = exp(im) * -0.5;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (0.5 + t_1) * cos(re);
	} else if (t_0 <= 0.001) {
		tmp = cos(re) * ((-0.16666666666666666 * pow(im, 3.0)) - im);
	} else {
		tmp = cos(re) * (t_1 + (0.5 / exp(im)));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(-im) - Math.exp(im);
	double t_1 = Math.exp(im) * -0.5;
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = (0.5 + t_1) * Math.cos(re);
	} else if (t_0 <= 0.001) {
		tmp = Math.cos(re) * ((-0.16666666666666666 * Math.pow(im, 3.0)) - im);
	} else {
		tmp = Math.cos(re) * (t_1 + (0.5 / Math.exp(im)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	t_1 = math.exp(im) * -0.5
	tmp = 0
	if t_0 <= -math.inf:
		tmp = (0.5 + t_1) * math.cos(re)
	elif t_0 <= 0.001:
		tmp = math.cos(re) * ((-0.16666666666666666 * math.pow(im, 3.0)) - im)
	else:
		tmp = math.cos(re) * (t_1 + (0.5 / math.exp(im)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	t_1 = Float64(exp(im) * -0.5)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(0.5 + t_1) * cos(re));
	elseif (t_0 <= 0.001)
		tmp = Float64(cos(re) * Float64(Float64(-0.16666666666666666 * (im ^ 3.0)) - im));
	else
		tmp = Float64(cos(re) * Float64(t_1 + Float64(0.5 / exp(im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	t_1 = exp(im) * -0.5;
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = (0.5 + t_1) * cos(re);
	elseif (t_0 <= 0.001)
		tmp = cos(re) * ((-0.16666666666666666 * (im ^ 3.0)) - im);
	else
		tmp = cos(re) * (t_1 + (0.5 / exp(im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[im], $MachinePrecision] * -0.5), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(0.5 + t$95$1), $MachinePrecision] * N[Cos[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.001], N[(N[Cos[re], $MachinePrecision] * N[(N[(-0.16666666666666666 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(t$95$1 + N[(0.5 / N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < -inf.0

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-lft-in 100.0%

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

      7. distribute-rgt-in 100.0%

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

      8. distribute-lft-neg-out 100.0%

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

      9. distribute-rgt-neg-in 100.0%

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

      10. metadata-eval 100.0%

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

      11. metadata-eval 100.0%

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

      12. fma-def 100.0%

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

      13. metadata-eval 100.0%

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

      14. exp-diff 100.0%

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

      15. associate-*l/ 100.0%

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

      16. exp-0 100.0%

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

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r/ 100.0%

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

      3. metadata-eval 100.0%

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

      4. *-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in im around 0 100.0%

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

   if -inf.0 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 1e-3

   1. Initial program 8.7%

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

      1. *-commutative 8.7%

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

      2. associate-*l* 8.7%

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

      3. sub-neg 8.7%

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

      4. +-commutative 8.7%

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

      5. distribute-lft-in 8.7%

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

      6. distribute-lft-in 8.7%

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

      7. distribute-rgt-in 8.7%

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

      8. distribute-lft-neg-out 8.7%

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

      9. distribute-rgt-neg-in 8.7%

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

      10. metadata-eval 8.7%

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

      11. metadata-eval 8.7%

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

      12. fma-def 8.7%

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

      13. metadata-eval 8.7%

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

      14. exp-diff 8.7%

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

      15. associate-*l/ 8.7%

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

      16. exp-0 8.7%

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

      17. metadata-eval 8.7%

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

   3. Simplified 8.7%

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

   4. Taylor expanded in im around 0 99.8%

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

      1. neg-mul-1 99.8%

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

      2. unsub-neg 99.8%

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

   6. Simplified 99.8%

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

   if 1e-3 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im))

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*l* 99.9%

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

      3. sub-neg 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-lft-in 99.9%

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

      6. distribute-lft-in 99.9%

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

      7. distribute-rgt-in 99.9%

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

      8. distribute-lft-neg-out 99.9%

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

      9. distribute-rgt-neg-in 99.9%

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

      10. metadata-eval 99.9%

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

      11. metadata-eval 99.9%

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

      12. fma-def 99.9%

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

      13. metadata-eval 99.9%

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

      14. exp-diff 99.9%

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

      15. associate-*l/ 99.9%

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

      16. exp-0 99.9%

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

      17. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in re around inf 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*r/ 99.9%

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

      3. metadata-eval 99.9%

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

      4. *-commutative 99.9%

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

   6. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -\infty:\\ \;\;\;\;\left(0.5 + e^{im} \cdot -0.5\right) \cdot \cos re\\ \mathbf{elif}\;e^{-im} - e^{im} \leq 0.001:\\ \;\;\;\;\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(e^{im} \cdot -0.5 + \frac{0.5}{e^{im}}\right)\\ \end{array} \]

Alternative 5: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\left(0.5 + e^{im} \cdot -0.5\right) \cdot \cos re\\ \mathbf{elif}\;t_0 \leq 0.001:\\ \;\;\;\;\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (<= t_0 (- INFINITY))
     (* (+ 0.5 (* (exp im) -0.5)) (cos re))
     (if (<= t_0 0.001)
       (* (cos re) (- (* -0.16666666666666666 (pow im 3.0)) im))
       (* (* 0.5 (cos re)) t_0)))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (0.5 + (exp(im) * -0.5)) * cos(re);
	} else if (t_0 <= 0.001) {
		tmp = cos(re) * ((-0.16666666666666666 * pow(im, 3.0)) - im);
	} else {
		tmp = (0.5 * cos(re)) * t_0;
	}
	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) {
		tmp = (0.5 + (Math.exp(im) * -0.5)) * Math.cos(re);
	} else if (t_0 <= 0.001) {
		tmp = Math.cos(re) * ((-0.16666666666666666 * Math.pow(im, 3.0)) - im);
	} else {
		tmp = (0.5 * Math.cos(re)) * t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	tmp = 0
	if t_0 <= -math.inf:
		tmp = (0.5 + (math.exp(im) * -0.5)) * math.cos(re)
	elif t_0 <= 0.001:
		tmp = math.cos(re) * ((-0.16666666666666666 * math.pow(im, 3.0)) - im)
	else:
		tmp = (0.5 * math.cos(re)) * t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(0.5 + Float64(exp(im) * -0.5)) * cos(re));
	elseif (t_0 <= 0.001)
		tmp = Float64(cos(re) * Float64(Float64(-0.16666666666666666 * (im ^ 3.0)) - im));
	else
		tmp = Float64(Float64(0.5 * cos(re)) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = (0.5 + (exp(im) * -0.5)) * cos(re);
	elseif (t_0 <= 0.001)
		tmp = cos(re) * ((-0.16666666666666666 * (im ^ 3.0)) - im);
	else
		tmp = (0.5 * cos(re)) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(0.5 + N[(N[Exp[im], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] * N[Cos[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.001], N[(N[Cos[re], $MachinePrecision] * N[(N[(-0.16666666666666666 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < -inf.0

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-lft-in 100.0%

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

      7. distribute-rgt-in 100.0%

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

      8. distribute-lft-neg-out 100.0%

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

      9. distribute-rgt-neg-in 100.0%

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

      10. metadata-eval 100.0%

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

      11. metadata-eval 100.0%

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

      12. fma-def 100.0%

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

      13. metadata-eval 100.0%

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

      14. exp-diff 100.0%

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

      15. associate-*l/ 100.0%

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

      16. exp-0 100.0%

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

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r/ 100.0%

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

      3. metadata-eval 100.0%

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

      4. *-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in im around 0 100.0%

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

   if -inf.0 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 1e-3

   1. Initial program 8.7%

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

      1. *-commutative 8.7%

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

      2. associate-*l* 8.7%

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

      3. sub-neg 8.7%

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

      4. +-commutative 8.7%

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

      5. distribute-lft-in 8.7%

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

      6. distribute-lft-in 8.7%

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

      7. distribute-rgt-in 8.7%

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

      8. distribute-lft-neg-out 8.7%

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

      9. distribute-rgt-neg-in 8.7%

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

      10. metadata-eval 8.7%

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

      11. metadata-eval 8.7%

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

      12. fma-def 8.7%

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

      13. metadata-eval 8.7%

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

      14. exp-diff 8.7%

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

      15. associate-*l/ 8.7%

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

      16. exp-0 8.7%

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

      17. metadata-eval 8.7%

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

   3. Simplified 8.7%

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

   4. Taylor expanded in im around 0 99.8%

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

      1. neg-mul-1 99.8%

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

      2. unsub-neg 99.8%

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

   6. Simplified 99.8%

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

   if 1e-3 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im))

   1. Initial program 99.9%

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

      1. sub0-neg 99.9%

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

   3. Simplified 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -\infty:\\ \;\;\;\;\left(0.5 + e^{im} \cdot -0.5\right) \cdot \cos re\\ \mathbf{elif}\;e^{-im} - e^{im} \leq 0.001:\\ \;\;\;\;\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \]

Alternative 6: 96.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(e^{-im} - e^{im}\right)\\ t_1 := \cos re \cdot \left(-0.008333333333333333 \cdot {im}^{5}\right)\\ \mathbf{if}\;im \leq -3.4 \cdot 10^{+53}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.0038:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 165000:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (- (exp (- im)) (exp im))))
        (t_1 (* (cos re) (* -0.008333333333333333 (pow im 5.0)))))
   (if (<= im -3.4e+53)
     t_1
     (if (<= im -0.0038)
       t_0
       (if (<= im 165000.0)
         (* im (- (cos re)))
         (if (<= im 4.5e+61) t_0 t_1))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (exp(-im) - exp(im));
	double t_1 = cos(re) * (-0.008333333333333333 * pow(im, 5.0));
	double tmp;
	if (im <= -3.4e+53) {
		tmp = t_1;
	} else if (im <= -0.0038) {
		tmp = t_0;
	} else if (im <= 165000.0) {
		tmp = im * -cos(re);
	} else if (im <= 4.5e+61) {
		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))
    t_1 = cos(re) * ((-0.008333333333333333d0) * (im ** 5.0d0))
    if (im <= (-3.4d+53)) then
        tmp = t_1
    else if (im <= (-0.0038d0)) then
        tmp = t_0
    else if (im <= 165000.0d0) then
        tmp = im * -cos(re)
    else if (im <= 4.5d+61) 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));
	double t_1 = Math.cos(re) * (-0.008333333333333333 * Math.pow(im, 5.0));
	double tmp;
	if (im <= -3.4e+53) {
		tmp = t_1;
	} else if (im <= -0.0038) {
		tmp = t_0;
	} else if (im <= 165000.0) {
		tmp = im * -Math.cos(re);
	} else if (im <= 4.5e+61) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * (math.exp(-im) - math.exp(im))
	t_1 = math.cos(re) * (-0.008333333333333333 * math.pow(im, 5.0))
	tmp = 0
	if im <= -3.4e+53:
		tmp = t_1
	elif im <= -0.0038:
		tmp = t_0
	elif im <= 165000.0:
		tmp = im * -math.cos(re)
	elif im <= 4.5e+61:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(exp(Float64(-im)) - exp(im)))
	t_1 = Float64(cos(re) * Float64(-0.008333333333333333 * (im ^ 5.0)))
	tmp = 0.0
	if (im <= -3.4e+53)
		tmp = t_1;
	elseif (im <= -0.0038)
		tmp = t_0;
	elseif (im <= 165000.0)
		tmp = Float64(im * Float64(-cos(re)));
	elseif (im <= 4.5e+61)
		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));
	t_1 = cos(re) * (-0.008333333333333333 * (im ^ 5.0));
	tmp = 0.0;
	if (im <= -3.4e+53)
		tmp = t_1;
	elseif (im <= -0.0038)
		tmp = t_0;
	elseif (im <= 165000.0)
		tmp = im * -cos(re);
	elseif (im <= 4.5e+61)
		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[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[re], $MachinePrecision] * N[(-0.008333333333333333 * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -3.4e+53], t$95$1, If[LessEqual[im, -0.0038], t$95$0, If[LessEqual[im, 165000.0], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im, 4.5e+61], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -3.39999999999999998e53 or 4.5e61 < im

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-lft-in 100.0%

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

      7. distribute-rgt-in 100.0%

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

      8. distribute-lft-neg-out 100.0%

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

      9. distribute-rgt-neg-in 100.0%

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

      10. metadata-eval 100.0%

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

      11. metadata-eval 100.0%

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

      12. fma-def 100.0%

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

      13. metadata-eval 100.0%

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

      14. exp-diff 100.0%

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

      15. associate-*l/ 100.0%

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

      16. exp-0 100.0%

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

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r/ 100.0%

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

      3. metadata-eval 100.0%

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

      4. *-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in im around 0 99.1%

      \[\leadsto \color{blue}{\left(-0.008333333333333333 \cdot {im}^{5} + \left(-0.16666666666666666 \cdot {im}^{3} + -1 \cdot im\right)\right)} \cdot \cos re \]

   8. Taylor expanded in im around inf 99.1%

      \[\leadsto \color{blue}{\left(-0.008333333333333333 \cdot {im}^{5}\right)} \cdot \cos re \]

   if -3.39999999999999998e53 < im < -0.00379999999999999999 or 165000 < im < 4.5e61

   1. Initial program 99.9%

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

      1. sub0-neg 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in re around 0 75.8%

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

   if -0.00379999999999999999 < im < 165000

   1. Initial program 10.7%

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

      1. *-commutative 10.7%

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

      2. associate-*l* 10.7%

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

      3. sub-neg 10.7%

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

      4. +-commutative 10.7%

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

      5. distribute-lft-in 10.7%

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

      6. distribute-lft-in 10.7%

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

      7. distribute-rgt-in 10.7%

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

      8. distribute-lft-neg-out 10.7%

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

      9. distribute-rgt-neg-in 10.7%

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

      10. metadata-eval 10.7%

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

      11. metadata-eval 10.7%

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

      12. fma-def 10.7%

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

      13. metadata-eval 10.7%

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

      14. exp-diff 10.7%

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

      15. associate-*l/ 10.7%

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

      16. exp-0 10.7%

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

      17. metadata-eval 10.7%

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

   3. Simplified 10.7%

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

   4. Taylor expanded in im around 0 97.5%

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

      1. neg-mul-1 97.5%

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

   6. Simplified 97.5%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3.4 \cdot 10^{+53}:\\ \;\;\;\;\cos re \cdot \left(-0.008333333333333333 \cdot {im}^{5}\right)\\ \mathbf{elif}\;im \leq -0.0038:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 165000:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(-0.008333333333333333 \cdot {im}^{5}\right)\\ \end{array} \]

Alternative 7: 97.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -3.4 \cdot 10^{+53}:\\ \;\;\;\;\cos re \cdot \left(-0.008333333333333333 \cdot {im}^{5}\right)\\ \mathbf{elif}\;im \leq -0.0225:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 2.1:\\ \;\;\;\;\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + e^{im} \cdot -0.5\right) \cdot \cos re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -3.4e+53)
   (* (cos re) (* -0.008333333333333333 (pow im 5.0)))
   (if (<= im -0.0225)
     (* 0.5 (- (exp (- im)) (exp im)))
     (if (<= im 2.1)
       (* (cos re) (- (* -0.16666666666666666 (pow im 3.0)) im))
       (* (+ 0.5 (* (exp im) -0.5)) (cos re))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -3.4e+53) {
		tmp = cos(re) * (-0.008333333333333333 * pow(im, 5.0));
	} else if (im <= -0.0225) {
		tmp = 0.5 * (exp(-im) - exp(im));
	} else if (im <= 2.1) {
		tmp = cos(re) * ((-0.16666666666666666 * pow(im, 3.0)) - im);
	} else {
		tmp = (0.5 + (exp(im) * -0.5)) * cos(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 <= (-3.4d+53)) then
        tmp = cos(re) * ((-0.008333333333333333d0) * (im ** 5.0d0))
    else if (im <= (-0.0225d0)) then
        tmp = 0.5d0 * (exp(-im) - exp(im))
    else if (im <= 2.1d0) then
        tmp = cos(re) * (((-0.16666666666666666d0) * (im ** 3.0d0)) - im)
    else
        tmp = (0.5d0 + (exp(im) * (-0.5d0))) * cos(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -3.4e+53) {
		tmp = Math.cos(re) * (-0.008333333333333333 * Math.pow(im, 5.0));
	} else if (im <= -0.0225) {
		tmp = 0.5 * (Math.exp(-im) - Math.exp(im));
	} else if (im <= 2.1) {
		tmp = Math.cos(re) * ((-0.16666666666666666 * Math.pow(im, 3.0)) - im);
	} else {
		tmp = (0.5 + (Math.exp(im) * -0.5)) * Math.cos(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -3.4e+53:
		tmp = math.cos(re) * (-0.008333333333333333 * math.pow(im, 5.0))
	elif im <= -0.0225:
		tmp = 0.5 * (math.exp(-im) - math.exp(im))
	elif im <= 2.1:
		tmp = math.cos(re) * ((-0.16666666666666666 * math.pow(im, 3.0)) - im)
	else:
		tmp = (0.5 + (math.exp(im) * -0.5)) * math.cos(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -3.4e+53)
		tmp = Float64(cos(re) * Float64(-0.008333333333333333 * (im ^ 5.0)));
	elseif (im <= -0.0225)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) - exp(im)));
	elseif (im <= 2.1)
		tmp = Float64(cos(re) * Float64(Float64(-0.16666666666666666 * (im ^ 3.0)) - im));
	else
		tmp = Float64(Float64(0.5 + Float64(exp(im) * -0.5)) * cos(re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -3.4e+53)
		tmp = cos(re) * (-0.008333333333333333 * (im ^ 5.0));
	elseif (im <= -0.0225)
		tmp = 0.5 * (exp(-im) - exp(im));
	elseif (im <= 2.1)
		tmp = cos(re) * ((-0.16666666666666666 * (im ^ 3.0)) - im);
	else
		tmp = (0.5 + (exp(im) * -0.5)) * cos(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -3.4e+53], N[(N[Cos[re], $MachinePrecision] * N[(-0.008333333333333333 * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, -0.0225], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.1], N[(N[Cos[re], $MachinePrecision] * N[(N[(-0.16666666666666666 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 + N[(N[Exp[im], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] * N[Cos[re], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < -3.39999999999999998e53

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-lft-in 100.0%

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

      7. distribute-rgt-in 100.0%

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

      8. distribute-lft-neg-out 100.0%

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

      9. distribute-rgt-neg-in 100.0%

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

      10. metadata-eval 100.0%

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

      11. metadata-eval 100.0%

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

      12. fma-def 100.0%

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

      13. metadata-eval 100.0%

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

      14. exp-diff 100.0%

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

      15. associate-*l/ 100.0%

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

      16. exp-0 100.0%

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

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r/ 100.0%

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

      3. metadata-eval 100.0%

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

      4. *-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in im around 0 98.5%

      \[\leadsto \color{blue}{\left(-0.008333333333333333 \cdot {im}^{5} + \left(-0.16666666666666666 \cdot {im}^{3} + -1 \cdot im\right)\right)} \cdot \cos re \]

   8. Taylor expanded in im around inf 98.5%

      \[\leadsto \color{blue}{\left(-0.008333333333333333 \cdot {im}^{5}\right)} \cdot \cos re \]

   if -3.39999999999999998e53 < im < -0.022499999999999999

   1. Initial program 99.8%

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

      1. sub0-neg 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in re around 0 79.7%

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

   if -0.022499999999999999 < im < 2.10000000000000009

   1. Initial program 9.3%

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

      1. *-commutative 9.3%

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

      2. associate-*l* 9.3%

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

      3. sub-neg 9.3%

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

      4. +-commutative 9.3%

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

      5. distribute-lft-in 9.3%

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

      6. distribute-lft-in 9.3%

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

      7. distribute-rgt-in 9.3%

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

      8. distribute-lft-neg-out 9.3%

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

      9. distribute-rgt-neg-in 9.3%

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

      10. metadata-eval 9.3%

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

      11. metadata-eval 9.3%

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

      12. fma-def 9.3%

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

      13. metadata-eval 9.3%

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

      14. exp-diff 9.3%

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

      15. associate-*l/ 9.3%

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

      16. exp-0 9.3%

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

      17. metadata-eval 9.3%

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

   3. Simplified 9.3%

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

   4. Taylor expanded in im around 0 99.6%

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

      1. neg-mul-1 99.6%

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

      2. unsub-neg 99.6%

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

   6. Simplified 99.6%

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

   if 2.10000000000000009 < im

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-lft-in 100.0%

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

      7. distribute-rgt-in 100.0%

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

      8. distribute-lft-neg-out 100.0%

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

      9. distribute-rgt-neg-in 100.0%

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

      10. metadata-eval 100.0%

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

      11. metadata-eval 100.0%

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

      12. fma-def 100.0%

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

      13. metadata-eval 100.0%

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

      14. exp-diff 100.0%

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

      15. associate-*l/ 100.0%

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

      16. exp-0 100.0%

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

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r/ 100.0%

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

      3. metadata-eval 100.0%

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

      4. *-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in im around 0 100.0%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3.4 \cdot 10^{+53}:\\ \;\;\;\;\cos re \cdot \left(-0.008333333333333333 \cdot {im}^{5}\right)\\ \mathbf{elif}\;im \leq -0.0225:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 2.1:\\ \;\;\;\;\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + e^{im} \cdot -0.5\right) \cdot \cos re\\ \end{array} \]

Alternative 8: 97.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -3.4 \cdot 10^{+53}:\\ \;\;\;\;\cos re \cdot \left(-0.008333333333333333 \cdot {im}^{5}\right)\\ \mathbf{elif}\;im \leq -0.0042:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 1.25:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + e^{im} \cdot -0.5\right) \cdot \cos re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -3.4e+53)
   (* (cos re) (* -0.008333333333333333 (pow im 5.0)))
   (if (<= im -0.0042)
     (* 0.5 (- (exp (- im)) (exp im)))
     (if (<= im 1.25)
       (* im (- (cos re)))
       (* (+ 0.5 (* (exp im) -0.5)) (cos re))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -3.4e+53) {
		tmp = cos(re) * (-0.008333333333333333 * pow(im, 5.0));
	} else if (im <= -0.0042) {
		tmp = 0.5 * (exp(-im) - exp(im));
	} else if (im <= 1.25) {
		tmp = im * -cos(re);
	} else {
		tmp = (0.5 + (exp(im) * -0.5)) * cos(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 <= (-3.4d+53)) then
        tmp = cos(re) * ((-0.008333333333333333d0) * (im ** 5.0d0))
    else if (im <= (-0.0042d0)) then
        tmp = 0.5d0 * (exp(-im) - exp(im))
    else if (im <= 1.25d0) then
        tmp = im * -cos(re)
    else
        tmp = (0.5d0 + (exp(im) * (-0.5d0))) * cos(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -3.4e+53) {
		tmp = Math.cos(re) * (-0.008333333333333333 * Math.pow(im, 5.0));
	} else if (im <= -0.0042) {
		tmp = 0.5 * (Math.exp(-im) - Math.exp(im));
	} else if (im <= 1.25) {
		tmp = im * -Math.cos(re);
	} else {
		tmp = (0.5 + (Math.exp(im) * -0.5)) * Math.cos(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -3.4e+53:
		tmp = math.cos(re) * (-0.008333333333333333 * math.pow(im, 5.0))
	elif im <= -0.0042:
		tmp = 0.5 * (math.exp(-im) - math.exp(im))
	elif im <= 1.25:
		tmp = im * -math.cos(re)
	else:
		tmp = (0.5 + (math.exp(im) * -0.5)) * math.cos(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -3.4e+53)
		tmp = Float64(cos(re) * Float64(-0.008333333333333333 * (im ^ 5.0)));
	elseif (im <= -0.0042)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) - exp(im)));
	elseif (im <= 1.25)
		tmp = Float64(im * Float64(-cos(re)));
	else
		tmp = Float64(Float64(0.5 + Float64(exp(im) * -0.5)) * cos(re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -3.4e+53)
		tmp = cos(re) * (-0.008333333333333333 * (im ^ 5.0));
	elseif (im <= -0.0042)
		tmp = 0.5 * (exp(-im) - exp(im));
	elseif (im <= 1.25)
		tmp = im * -cos(re);
	else
		tmp = (0.5 + (exp(im) * -0.5)) * cos(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -3.4e+53], N[(N[Cos[re], $MachinePrecision] * N[(-0.008333333333333333 * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, -0.0042], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.25], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], N[(N[(0.5 + N[(N[Exp[im], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] * N[Cos[re], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < -3.39999999999999998e53

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-lft-in 100.0%

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

      7. distribute-rgt-in 100.0%

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

      8. distribute-lft-neg-out 100.0%

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

      9. distribute-rgt-neg-in 100.0%

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

      10. metadata-eval 100.0%

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

      11. metadata-eval 100.0%

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

      12. fma-def 100.0%

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

      13. metadata-eval 100.0%

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

      14. exp-diff 100.0%

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

      15. associate-*l/ 100.0%

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

      16. exp-0 100.0%

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

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r/ 100.0%

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

      3. metadata-eval 100.0%

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

      4. *-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in im around 0 98.5%

      \[\leadsto \color{blue}{\left(-0.008333333333333333 \cdot {im}^{5} + \left(-0.16666666666666666 \cdot {im}^{3} + -1 \cdot im\right)\right)} \cdot \cos re \]

   8. Taylor expanded in im around inf 98.5%

      \[\leadsto \color{blue}{\left(-0.008333333333333333 \cdot {im}^{5}\right)} \cdot \cos re \]

   if -3.39999999999999998e53 < im < -0.00419999999999999974

   1. Initial program 99.8%

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

      1. sub0-neg 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in re around 0 79.7%

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

   if -0.00419999999999999974 < im < 1.25

   1. Initial program 9.3%

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

      1. *-commutative 9.3%

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

      2. associate-*l* 9.3%

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

      3. sub-neg 9.3%

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

      4. +-commutative 9.3%

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

      5. distribute-lft-in 9.3%

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

      6. distribute-lft-in 9.3%

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

      7. distribute-rgt-in 9.3%

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

      8. distribute-lft-neg-out 9.3%

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

      9. distribute-rgt-neg-in 9.3%

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

      10. metadata-eval 9.3%

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

      11. metadata-eval 9.3%

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

      12. fma-def 9.3%

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

      13. metadata-eval 9.3%

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

      14. exp-diff 9.3%

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

      15. associate-*l/ 9.3%

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

      16. exp-0 9.3%

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

      17. metadata-eval 9.3%

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

   3. Simplified 9.3%

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

   4. Taylor expanded in im around 0 98.9%

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

      1. neg-mul-1 98.9%

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

   6. Simplified 98.9%

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

   if 1.25 < im

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-lft-in 100.0%

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

      7. distribute-rgt-in 100.0%

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

      8. distribute-lft-neg-out 100.0%

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

      9. distribute-rgt-neg-in 100.0%

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

      10. metadata-eval 100.0%

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

      11. metadata-eval 100.0%

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

      12. fma-def 100.0%

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

      13. metadata-eval 100.0%

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

      14. exp-diff 100.0%

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

      15. associate-*l/ 100.0%

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

      16. exp-0 100.0%

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

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r/ 100.0%

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

      3. metadata-eval 100.0%

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

      4. *-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in im around 0 100.0%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3.4 \cdot 10^{+53}:\\ \;\;\;\;\cos re \cdot \left(-0.008333333333333333 \cdot {im}^{5}\right)\\ \mathbf{elif}\;im \leq -0.0042:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 1.25:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + e^{im} \cdot -0.5\right) \cdot \cos re\\ \end{array} \]

Alternative 9: 86.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -0.007 \lor \neg \left(im \leq 165000\right):\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -0.007) (not (<= im 165000.0)))
   (* 0.5 (- (exp (- im)) (exp im)))
   (* im (- (cos re)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -0.007) || !(im <= 165000.0)) {
		tmp = 0.5 * (exp(-im) - exp(im));
	} else {
		tmp = im * -cos(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-0.007d0)) .or. (.not. (im <= 165000.0d0))) then
        tmp = 0.5d0 * (exp(-im) - exp(im))
    else
        tmp = im * -cos(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -0.007) || !(im <= 165000.0)) {
		tmp = 0.5 * (Math.exp(-im) - Math.exp(im));
	} else {
		tmp = im * -Math.cos(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -0.007) or not (im <= 165000.0):
		tmp = 0.5 * (math.exp(-im) - math.exp(im))
	else:
		tmp = im * -math.cos(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -0.007) || !(im <= 165000.0))
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) - exp(im)));
	else
		tmp = Float64(im * Float64(-cos(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -0.007) || ~((im <= 165000.0)))
		tmp = 0.5 * (exp(-im) - exp(im));
	else
		tmp = im * -cos(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -0.007], N[Not[LessEqual[im, 165000.0]], $MachinePrecision]], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -0.00700000000000000015 or 165000 < im

   1. Initial program 100.0%

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

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 68.1%

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

   if -0.00700000000000000015 < im < 165000

   1. Initial program 10.7%

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

      1. *-commutative 10.7%

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

      2. associate-*l* 10.7%

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

      3. sub-neg 10.7%

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

      4. +-commutative 10.7%

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

      5. distribute-lft-in 10.7%

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

      6. distribute-lft-in 10.7%

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

      7. distribute-rgt-in 10.7%

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

      8. distribute-lft-neg-out 10.7%

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

      9. distribute-rgt-neg-in 10.7%

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

      10. metadata-eval 10.7%

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

      11. metadata-eval 10.7%

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

      12. fma-def 10.7%

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

      13. metadata-eval 10.7%

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

      14. exp-diff 10.7%

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

      15. associate-*l/ 10.7%

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

      16. exp-0 10.7%

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

      17. metadata-eval 10.7%

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

   3. Simplified 10.7%

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

   4. Taylor expanded in im around 0 97.5%

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

      1. neg-mul-1 97.5%

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

   6. Simplified 97.5%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -0.007 \lor \neg \left(im \leq 165000\right):\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \]

Alternative 10: 39.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.01:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right) - im\\ \mathbf{else}:\\ \;\;\;\;-im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) -0.01) (- (* im (* 0.5 (* re re))) im) (- im)))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= -0.01) {
		tmp = (im * (0.5 * (re * re))) - im;
	} else {
		tmp = -im;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (cos(re) <= (-0.01d0)) then
        tmp = (im * (0.5d0 * (re * re))) - im
    else
        tmp = -im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.cos(re) <= -0.01) {
		tmp = (im * (0.5 * (re * re))) - im;
	} else {
		tmp = -im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.cos(re) <= -0.01:
		tmp = (im * (0.5 * (re * re))) - im
	else:
		tmp = -im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= -0.01)
		tmp = Float64(Float64(im * Float64(0.5 * Float64(re * re))) - im);
	else
		tmp = Float64(-im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (cos(re) <= -0.01)
		tmp = (im * (0.5 * (re * re))) - im;
	else
		tmp = -im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.01], N[(N[(im * N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], (-im)]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -0.0100000000000000002

   1. Initial program 57.3%

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

      1. *-commutative 57.3%

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

      2. associate-*l* 57.3%

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

      3. sub-neg 57.3%

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

      4. +-commutative 57.3%

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

      5. distribute-lft-in 57.3%

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

      6. distribute-lft-in 57.3%

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

      7. distribute-rgt-in 57.3%

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

      8. distribute-lft-neg-out 57.3%

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

      9. distribute-rgt-neg-in 57.3%

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

      10. metadata-eval 57.3%

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

      11. metadata-eval 57.3%

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

      12. fma-def 57.3%

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

      13. metadata-eval 57.3%

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

      14. exp-diff 57.2%

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

      15. associate-*l/ 57.2%

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

      16. exp-0 57.2%

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

      17. metadata-eval 57.2%

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

   3. Simplified 57.2%

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

   4. Taylor expanded in re around inf 57.2%

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

      1. *-commutative 57.2%

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

      2. associate-*r/ 57.2%

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

      3. metadata-eval 57.2%

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

      4. *-commutative 57.2%

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

   6. Simplified 57.2%

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

   7. Taylor expanded in im around 0 50.2%

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

      1. neg-mul-1 50.2%

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

   9. Simplified 50.2%

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

   10. Taylor expanded in re around 0 36.1%

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

       1. +-commutative 36.1%

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

       2. mul-1-neg 36.1%

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

       3. unsub-neg 36.1%

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

       4. *-commutative 36.1%

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

       5. *-commutative 36.1%

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

       6. associate-*l* 36.1%

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

       7. unpow2 36.1%

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

   12. Simplified 36.1%

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

   if -0.0100000000000000002 < (cos.f64 re)

   1. Initial program 53.0%

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

      1. *-commutative 53.0%

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

      2. associate-*l* 53.0%

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

      3. sub-neg 53.0%

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

      4. +-commutative 53.0%

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

      5. distribute-lft-in 53.0%

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

      6. distribute-lft-in 53.0%

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

      7. distribute-rgt-in 53.0%

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

      8. distribute-lft-neg-out 53.0%

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

      9. distribute-rgt-neg-in 53.0%

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

      10. metadata-eval 53.0%

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

      11. metadata-eval 53.0%

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

      12. fma-def 53.0%

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

      13. metadata-eval 53.0%

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

      14. exp-diff 53.0%

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

      15. associate-*l/ 53.0%

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

      16. exp-0 53.0%

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

      17. metadata-eval 53.0%

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

   3. Simplified 53.0%

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

   4. Taylor expanded in re around inf 53.0%

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

      1. *-commutative 53.0%

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

      2. associate-*r/ 53.0%

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

      3. metadata-eval 53.0%

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

      4. *-commutative 53.0%

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

   6. Simplified 53.0%

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

   7. Taylor expanded in im around 0 53.7%

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

      1. neg-mul-1 53.7%

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

   9. Simplified 53.7%

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

   10. Taylor expanded in re around 0 38.4%

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

       1. +-commutative 38.4%

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

       2. mul-1-neg 38.4%

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

       3. unsub-neg 38.4%

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

       4. *-commutative 38.4%

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

       5. *-commutative 38.4%

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

       6. associate-*l* 38.4%

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

       7. unpow2 38.4%

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

   12. Simplified 38.4%

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

   13. Taylor expanded in re around 0 42.1%

       \[\leadsto \color{blue}{-1 \cdot im} \]
   14. Step-by-step derivation

       1. neg-mul-1 42.1%

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

   15. Simplified 42.1%

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

4. Final simplification 40.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq -0.01:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right) - im\\ \mathbf{else}:\\ \;\;\;\;-im\\ \end{array} \]

Alternative 11: 75.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -4.3 \cdot 10^{+64} \lor \neg \left(im \leq 8 \cdot 10^{+54}\right):\\ \;\;\;\;-0.16666666666666666 \cdot {im}^{3} - im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -4.3e+64) (not (<= im 8e+54)))
   (- (* -0.16666666666666666 (pow im 3.0)) im)
   (* im (- (cos re)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -4.3e+64) || !(im <= 8e+54)) {
		tmp = (-0.16666666666666666 * pow(im, 3.0)) - im;
	} else {
		tmp = im * -cos(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 <= (-4.3d+64)) .or. (.not. (im <= 8d+54))) then
        tmp = ((-0.16666666666666666d0) * (im ** 3.0d0)) - im
    else
        tmp = im * -cos(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -4.3e+64) || !(im <= 8e+54)) {
		tmp = (-0.16666666666666666 * Math.pow(im, 3.0)) - im;
	} else {
		tmp = im * -Math.cos(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -4.3e+64) or not (im <= 8e+54):
		tmp = (-0.16666666666666666 * math.pow(im, 3.0)) - im
	else:
		tmp = im * -math.cos(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -4.3e+64) || !(im <= 8e+54))
		tmp = Float64(Float64(-0.16666666666666666 * (im ^ 3.0)) - im);
	else
		tmp = Float64(im * Float64(-cos(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -4.3e+64) || ~((im <= 8e+54)))
		tmp = (-0.16666666666666666 * (im ^ 3.0)) - im;
	else
		tmp = im * -cos(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -4.3e+64], N[Not[LessEqual[im, 8e+54]], $MachinePrecision]], N[(N[(-0.16666666666666666 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -4.2999999999999998e64 or 8.0000000000000006e54 < im

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-lft-in 100.0%

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

      7. distribute-rgt-in 100.0%

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

      8. distribute-lft-neg-out 100.0%

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

      9. distribute-rgt-neg-in 100.0%

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

      10. metadata-eval 100.0%

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

      11. metadata-eval 100.0%

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

      12. fma-def 100.0%

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

      13. metadata-eval 100.0%

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

      14. exp-diff 100.0%

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

      15. associate-*l/ 100.0%

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

      16. exp-0 100.0%

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

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r/ 100.0%

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

      3. metadata-eval 100.0%

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

      4. *-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in im around 0 87.3%

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

      1. neg-mul-1 87.3%

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

      2. unsub-neg 87.3%

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

   9. Simplified 87.3%

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

   10. Taylor expanded in re around 0 57.7%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3} - im} \]

   if -4.2999999999999998e64 < im < 8.0000000000000006e54

   1. Initial program 24.1%

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

      1. *-commutative 24.1%

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

      2. associate-*l* 24.1%

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

      3. sub-neg 24.1%

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

      4. +-commutative 24.1%

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

      5. distribute-lft-in 24.1%

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

      6. distribute-lft-in 24.1%

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

      7. distribute-rgt-in 24.1%

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

      8. distribute-lft-neg-out 24.1%

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

      9. distribute-rgt-neg-in 24.1%

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

      10. metadata-eval 24.1%

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

      11. metadata-eval 24.1%

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

      12. fma-def 24.1%

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

      13. metadata-eval 24.1%

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

      14. exp-diff 24.0%

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

      15. associate-*l/ 24.0%

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

      16. exp-0 24.0%

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

      17. metadata-eval 24.0%

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

   3. Simplified 24.0%

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

   4. Taylor expanded in im around 0 83.4%

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

      1. neg-mul-1 83.4%

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

   6. Simplified 83.4%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -4.3 \cdot 10^{+64} \lor \neg \left(im \leq 8 \cdot 10^{+54}\right):\\ \;\;\;\;-0.16666666666666666 \cdot {im}^{3} - im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \]

Alternative 12: 60.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -4.4 \cdot 10^{+39} \lor \neg \left(im \leq 600\right):\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right) - im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -4.4e+39) (not (<= im 600.0)))
   (- (* im (* 0.5 (* re re))) im)
   (* im (- (cos re)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -4.4e+39) || !(im <= 600.0)) {
		tmp = (im * (0.5 * (re * re))) - im;
	} else {
		tmp = im * -cos(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 <= (-4.4d+39)) .or. (.not. (im <= 600.0d0))) then
        tmp = (im * (0.5d0 * (re * re))) - im
    else
        tmp = im * -cos(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -4.4e+39) || !(im <= 600.0)) {
		tmp = (im * (0.5 * (re * re))) - im;
	} else {
		tmp = im * -Math.cos(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -4.4e+39) or not (im <= 600.0):
		tmp = (im * (0.5 * (re * re))) - im
	else:
		tmp = im * -math.cos(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -4.4e+39) || !(im <= 600.0))
		tmp = Float64(Float64(im * Float64(0.5 * Float64(re * re))) - im);
	else
		tmp = Float64(im * Float64(-cos(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -4.4e+39) || ~((im <= 600.0)))
		tmp = (im * (0.5 * (re * re))) - im;
	else
		tmp = im * -cos(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -4.4e+39], N[Not[LessEqual[im, 600.0]], $MachinePrecision]], N[(N[(im * N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -4.4000000000000003e39 or 600 < im

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-lft-in 100.0%

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

      7. distribute-rgt-in 100.0%

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

      8. distribute-lft-neg-out 100.0%

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

      9. distribute-rgt-neg-in 100.0%

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

      10. metadata-eval 100.0%

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

      11. metadata-eval 100.0%

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

      12. fma-def 100.0%

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

      13. metadata-eval 100.0%

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

      14. exp-diff 100.0%

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

      15. associate-*l/ 100.0%

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

      16. exp-0 100.0%

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

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r/ 100.0%

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

      3. metadata-eval 100.0%

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

      4. *-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in im around 0 5.6%

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

      1. neg-mul-1 5.6%

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

   9. Simplified 5.6%

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

   10. Taylor expanded in re around 0 24.7%

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

       1. +-commutative 24.7%

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

       2. mul-1-neg 24.7%

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

       3. unsub-neg 24.7%

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

       4. *-commutative 24.7%

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

       5. *-commutative 24.7%

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

       6. associate-*l* 24.7%

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

       7. unpow2 24.7%

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

   12. Simplified 24.7%

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

   if -4.4000000000000003e39 < im < 600

   1. Initial program 12.7%

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

      1. *-commutative 12.7%

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

      2. associate-*l* 12.7%

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

      3. sub-neg 12.7%

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

      4. +-commutative 12.7%

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

      5. distribute-lft-in 12.7%

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

      6. distribute-lft-in 12.7%

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

      7. distribute-rgt-in 12.7%

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

      8. distribute-lft-neg-out 12.7%

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

      9. distribute-rgt-neg-in 12.7%

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

      10. metadata-eval 12.7%

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

      11. metadata-eval 12.7%

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

      12. fma-def 12.7%

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

      13. metadata-eval 12.7%

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

      14. exp-diff 12.7%

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

      15. associate-*l/ 12.7%

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

      16. exp-0 12.7%

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

      17. metadata-eval 12.7%

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

   3. Simplified 12.7%

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

   4. Taylor expanded in im around 0 95.4%

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

      1. neg-mul-1 95.4%

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

   6. Simplified 95.4%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -4.4 \cdot 10^{+39} \lor \neg \left(im \leq 600\right):\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right) - im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \]

Alternative 13: 35.3% accurate, 33.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -680 \lor \neg \left(im \leq 520\right):\\ \;\;\;\;re \cdot \left(re \cdot 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;-im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -680.0) (not (<= im 520.0))) (* re (* re 1.5)) (- im)))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -680.0) || !(im <= 520.0)) {
		tmp = re * (re * 1.5);
	} else {
		tmp = -im;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-680.0d0)) .or. (.not. (im <= 520.0d0))) then
        tmp = re * (re * 1.5d0)
    else
        tmp = -im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -680.0) || !(im <= 520.0)) {
		tmp = re * (re * 1.5);
	} else {
		tmp = -im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -680.0) or not (im <= 520.0):
		tmp = re * (re * 1.5)
	else:
		tmp = -im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -680.0) || !(im <= 520.0))
		tmp = Float64(re * Float64(re * 1.5));
	else
		tmp = Float64(-im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -680.0) || ~((im <= 520.0)))
		tmp = re * (re * 1.5);
	else
		tmp = -im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -680.0], N[Not[LessEqual[im, 520.0]], $MachinePrecision]], N[(re * N[(re * 1.5), $MachinePrecision]), $MachinePrecision], (-im)]

Derivation

1. Split input into 2 regimes

2. if im < -680 or 520 < im

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. distribute-lft-in 100.0%

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

      7. distribute-rgt-in 100.0%

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

      8. distribute-lft-neg-out 100.0%

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

      9. distribute-rgt-neg-in 100.0%

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

      10. metadata-eval 100.0%

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

      11. metadata-eval 100.0%

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

      12. fma-def 100.0%

          \[\leadsto \cos re \cdot \color{blue}{\mathsf{fma}\left(e^{im}, 0.5 \cdot -1, e^{0 - im} \cdot 0.5\right)} \]

      13. metadata-eval 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(e^{im}, \color{blue}{-0.5}, e^{0 - im} \cdot 0.5\right) \]

      14. exp-diff 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(e^{im}, -0.5, \color{blue}{\frac{e^{0}}{e^{im}}} \cdot 0.5\right) \]

      15. associate-*l/ 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(e^{im}, -0.5, \color{blue}{\frac{e^{0} \cdot 0.5}{e^{im}}}\right) \]

      16. exp-0 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(e^{im}, -0.5, \frac{\color{blue}{1} \cdot 0.5}{e^{im}}\right) \]

      17. metadata-eval 100.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(e^{im}, -0.5, \frac{\color{blue}{0.5}}{e^{im}}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(e^{im}, -0.5, \frac{0.5}{e^{im}}\right)} \]

   5. Applied egg-rr 1.8%

      \[\leadsto \cos re \cdot \color{blue}{-3} \]

   6. Taylor expanded in re around 0 15.8%

      \[\leadsto \color{blue}{1.5 \cdot {re}^{2} - 3} \]
   7. Step-by-step derivation

      1. *-commutative 15.8%

         \[\leadsto \color{blue}{{re}^{2} \cdot 1.5} - 3 \]

      2. unpow2 15.8%

         \[\leadsto \color{blue}{\left(re \cdot re\right)} \cdot 1.5 - 3 \]

      3. associate-*l* 15.8%

         \[\leadsto \color{blue}{re \cdot \left(re \cdot 1.5\right)} - 3 \]

      4. fma-neg 15.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(re, re \cdot 1.5, -3\right)} \]

      5. metadata-eval 15.8%

         \[\leadsto \mathsf{fma}\left(re, re \cdot 1.5, \color{blue}{-3}\right) \]

   8. Simplified 15.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, re \cdot 1.5, -3\right)} \]

   9. Taylor expanded in re around inf 15.9%

      \[\leadsto \color{blue}{1.5 \cdot {re}^{2}} \]
   10. Step-by-step derivation

       1. *-commutative 15.9%

          \[\leadsto \color{blue}{{re}^{2} \cdot 1.5} \]

       2. unpow2 15.9%

          \[\leadsto \color{blue}{\left(re \cdot re\right)} \cdot 1.5 \]

       3. associate-*r* 15.9%

          \[\leadsto \color{blue}{re \cdot \left(re \cdot 1.5\right)} \]

   11. Simplified 15.9%

       \[\leadsto \color{blue}{re \cdot \left(re \cdot 1.5\right)} \]

   if -680 < im < 520

   1. Initial program 10.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. *-commutative 10.0%

         \[\leadsto \color{blue}{\left(\cos re \cdot 0.5\right)} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. associate-*l* 10.0%

         \[\leadsto \color{blue}{\cos re \cdot \left(0.5 \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]

      3. sub-neg 10.0%

         \[\leadsto \cos re \cdot \left(0.5 \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im}\right)\right)}\right) \]

      4. +-commutative 10.0%

         \[\leadsto \cos re \cdot \left(0.5 \cdot \color{blue}{\left(\left(-e^{im}\right) + e^{0 - im}\right)}\right) \]

      5. distribute-lft-in 10.0%

         \[\leadsto \cos re \cdot \color{blue}{\left(0.5 \cdot \left(-e^{im}\right) + 0.5 \cdot e^{0 - im}\right)} \]

      6. distribute-lft-in 10.0%

         \[\leadsto \cos re \cdot \color{blue}{\left(0.5 \cdot \left(\left(-e^{im}\right) + e^{0 - im}\right)\right)} \]

      7. distribute-rgt-in 10.0%

         \[\leadsto \cos re \cdot \color{blue}{\left(\left(-e^{im}\right) \cdot 0.5 + e^{0 - im} \cdot 0.5\right)} \]

      8. distribute-lft-neg-out 10.0%

         \[\leadsto \cos re \cdot \left(\color{blue}{\left(-e^{im} \cdot 0.5\right)} + e^{0 - im} \cdot 0.5\right) \]

      9. distribute-rgt-neg-in 10.0%

         \[\leadsto \cos re \cdot \left(\color{blue}{e^{im} \cdot \left(-0.5\right)} + e^{0 - im} \cdot 0.5\right) \]

      10. metadata-eval 10.0%

          \[\leadsto \cos re \cdot \left(e^{im} \cdot \color{blue}{-0.5} + e^{0 - im} \cdot 0.5\right) \]

      11. metadata-eval 10.0%

          \[\leadsto \cos re \cdot \left(e^{im} \cdot \color{blue}{\left(0.5 \cdot -1\right)} + e^{0 - im} \cdot 0.5\right) \]

      12. fma-def 10.0%

          \[\leadsto \cos re \cdot \color{blue}{\mathsf{fma}\left(e^{im}, 0.5 \cdot -1, e^{0 - im} \cdot 0.5\right)} \]

      13. metadata-eval 10.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(e^{im}, \color{blue}{-0.5}, e^{0 - im} \cdot 0.5\right) \]

      14. exp-diff 10.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(e^{im}, -0.5, \color{blue}{\frac{e^{0}}{e^{im}}} \cdot 0.5\right) \]

      15. associate-*l/ 10.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(e^{im}, -0.5, \color{blue}{\frac{e^{0} \cdot 0.5}{e^{im}}}\right) \]

      16. exp-0 10.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(e^{im}, -0.5, \frac{\color{blue}{1} \cdot 0.5}{e^{im}}\right) \]

      17. metadata-eval 10.0%

          \[\leadsto \cos re \cdot \mathsf{fma}\left(e^{im}, -0.5, \frac{\color{blue}{0.5}}{e^{im}}\right) \]

   3. Simplified 10.0%

      \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(e^{im}, -0.5, \frac{0.5}{e^{im}}\right)} \]

   4. Taylor expanded in re around inf 10.0%

      \[\leadsto \color{blue}{\cos re \cdot \left(0.5 \cdot \frac{1}{e^{im}} + -0.5 \cdot e^{im}\right)} \]
   5. Step-by-step derivation

      1. *-commutative 10.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \frac{1}{e^{im}} + -0.5 \cdot e^{im}\right) \cdot \cos re} \]

      2. associate-*r/ 10.0%

         \[\leadsto \left(\color{blue}{\frac{0.5 \cdot 1}{e^{im}}} + -0.5 \cdot e^{im}\right) \cdot \cos re \]

      3. metadata-eval 10.0%

         \[\leadsto \left(\frac{\color{blue}{0.5}}{e^{im}} + -0.5 \cdot e^{im}\right) \cdot \cos re \]

      4. *-commutative 10.0%

         \[\leadsto \left(\frac{0.5}{e^{im}} + \color{blue}{e^{im} \cdot -0.5}\right) \cdot \cos re \]

   6. Simplified 10.0%

      \[\leadsto \color{blue}{\left(\frac{0.5}{e^{im}} + e^{im} \cdot -0.5\right) \cdot \cos re} \]

   7. Taylor expanded in im around 0 98.2%

      \[\leadsto \color{blue}{\left(-1 \cdot im\right)} \cdot \cos re \]
   8. Step-by-step derivation

      1. neg-mul-1 98.2%

         \[\leadsto \color{blue}{\left(-im\right)} \cdot \cos re \]

   9. Simplified 98.2%

      \[\leadsto \color{blue}{\left(-im\right)} \cdot \cos re \]

   10. Taylor expanded in re around 0 51.0%

       \[\leadsto \color{blue}{-1 \cdot im + 0.5 \cdot \left({re}^{2} \cdot im\right)} \]
   11. Step-by-step derivation

       1. +-commutative 51.0%

          \[\leadsto \color{blue}{0.5 \cdot \left({re}^{2} \cdot im\right) + -1 \cdot im} \]

       2. mul-1-neg 51.0%

          \[\leadsto 0.5 \cdot \left({re}^{2} \cdot im\right) + \color{blue}{\left(-im\right)} \]

       3. unsub-neg 51.0%

          \[\leadsto \color{blue}{0.5 \cdot \left({re}^{2} \cdot im\right) - im} \]

       4. *-commutative 51.0%

          \[\leadsto \color{blue}{\left({re}^{2} \cdot im\right) \cdot 0.5} - im \]

       5. *-commutative 51.0%

          \[\leadsto \color{blue}{\left(im \cdot {re}^{2}\right)} \cdot 0.5 - im \]

       6. associate-*l* 51.0%

          \[\leadsto \color{blue}{im \cdot \left({re}^{2} \cdot 0.5\right)} - im \]

       7. unpow2 51.0%

          \[\leadsto im \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot 0.5\right) - im \]

   12. Simplified 51.0%

       \[\leadsto \color{blue}{im \cdot \left(\left(re \cdot re\right) \cdot 0.5\right) - im} \]

   13. Taylor expanded in re around 0 55.0%

       \[\leadsto \color{blue}{-1 \cdot im} \]
   14. Step-by-step derivation

       1. neg-mul-1 55.0%

          \[\leadsto \color{blue}{-im} \]

   15. Simplified 55.0%

       \[\leadsto \color{blue}{-im} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -680 \lor \neg \left(im \leq 520\right):\\ \;\;\;\;re \cdot \left(re \cdot 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;-im\\ \end{array} \]

Alternative 14: 30.6% accurate, 154.5× speedup

Math

\[\begin{array}{l} \\ -im \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (- im))

C

double code(double re, double im) {
	return -im;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = -im
end function

Java

public static double code(double re, double im) {
	return -im;
}

Python

def code(re, im):
	return -im

Julia

function code(re, im)
	return Float64(-im)
end

MATLAB

function tmp = code(re, im)
	tmp = -im;
end

Wolfram

code[re_, im_] := (-im)

Derivation

1. Initial program 54.3%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
2. Step-by-step derivation

   1. *-commutative 54.3%

      \[\leadsto \color{blue}{\left(\cos re \cdot 0.5\right)} \cdot \left(e^{0 - im} - e^{im}\right) \]

   2. associate-*l* 54.3%

      \[\leadsto \color{blue}{\cos re \cdot \left(0.5 \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]

   3. sub-neg 54.3%

      \[\leadsto \cos re \cdot \left(0.5 \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im}\right)\right)}\right) \]

   4. +-commutative 54.3%

      \[\leadsto \cos re \cdot \left(0.5 \cdot \color{blue}{\left(\left(-e^{im}\right) + e^{0 - im}\right)}\right) \]

   5. distribute-lft-in 54.3%

      \[\leadsto \cos re \cdot \color{blue}{\left(0.5 \cdot \left(-e^{im}\right) + 0.5 \cdot e^{0 - im}\right)} \]

   6. distribute-lft-in 54.3%

      \[\leadsto \cos re \cdot \color{blue}{\left(0.5 \cdot \left(\left(-e^{im}\right) + e^{0 - im}\right)\right)} \]

   7. distribute-rgt-in 54.3%

      \[\leadsto \cos re \cdot \color{blue}{\left(\left(-e^{im}\right) \cdot 0.5 + e^{0 - im} \cdot 0.5\right)} \]

   8. distribute-lft-neg-out 54.3%

      \[\leadsto \cos re \cdot \left(\color{blue}{\left(-e^{im} \cdot 0.5\right)} + e^{0 - im} \cdot 0.5\right) \]

   9. distribute-rgt-neg-in 54.3%

      \[\leadsto \cos re \cdot \left(\color{blue}{e^{im} \cdot \left(-0.5\right)} + e^{0 - im} \cdot 0.5\right) \]

   10. metadata-eval 54.3%

       \[\leadsto \cos re \cdot \left(e^{im} \cdot \color{blue}{-0.5} + e^{0 - im} \cdot 0.5\right) \]

   11. metadata-eval 54.3%

       \[\leadsto \cos re \cdot \left(e^{im} \cdot \color{blue}{\left(0.5 \cdot -1\right)} + e^{0 - im} \cdot 0.5\right) \]

   12. fma-def 54.3%

       \[\leadsto \cos re \cdot \color{blue}{\mathsf{fma}\left(e^{im}, 0.5 \cdot -1, e^{0 - im} \cdot 0.5\right)} \]

   13. metadata-eval 54.3%

       \[\leadsto \cos re \cdot \mathsf{fma}\left(e^{im}, \color{blue}{-0.5}, e^{0 - im} \cdot 0.5\right) \]

   14. exp-diff 54.3%

       \[\leadsto \cos re \cdot \mathsf{fma}\left(e^{im}, -0.5, \color{blue}{\frac{e^{0}}{e^{im}}} \cdot 0.5\right) \]

   15. associate-*l/ 54.3%

       \[\leadsto \cos re \cdot \mathsf{fma}\left(e^{im}, -0.5, \color{blue}{\frac{e^{0} \cdot 0.5}{e^{im}}}\right) \]

   16. exp-0 54.3%

       \[\leadsto \cos re \cdot \mathsf{fma}\left(e^{im}, -0.5, \frac{\color{blue}{1} \cdot 0.5}{e^{im}}\right) \]

   17. metadata-eval 54.3%

       \[\leadsto \cos re \cdot \mathsf{fma}\left(e^{im}, -0.5, \frac{\color{blue}{0.5}}{e^{im}}\right) \]

3. Simplified 54.3%

   \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(e^{im}, -0.5, \frac{0.5}{e^{im}}\right)} \]

4. Taylor expanded in re around inf 54.3%

   \[\leadsto \color{blue}{\cos re \cdot \left(0.5 \cdot \frac{1}{e^{im}} + -0.5 \cdot e^{im}\right)} \]
5. Step-by-step derivation

   1. *-commutative 54.3%

      \[\leadsto \color{blue}{\left(0.5 \cdot \frac{1}{e^{im}} + -0.5 \cdot e^{im}\right) \cdot \cos re} \]

   2. associate-*r/ 54.3%

      \[\leadsto \left(\color{blue}{\frac{0.5 \cdot 1}{e^{im}}} + -0.5 \cdot e^{im}\right) \cdot \cos re \]

   3. metadata-eval 54.3%

      \[\leadsto \left(\frac{\color{blue}{0.5}}{e^{im}} + -0.5 \cdot e^{im}\right) \cdot \cos re \]

   4. *-commutative 54.3%

      \[\leadsto \left(\frac{0.5}{e^{im}} + \color{blue}{e^{im} \cdot -0.5}\right) \cdot \cos re \]

6. Simplified 54.3%

   \[\leadsto \color{blue}{\left(\frac{0.5}{e^{im}} + e^{im} \cdot -0.5\right) \cdot \cos re} \]

7. Taylor expanded in im around 0 52.6%

   \[\leadsto \color{blue}{\left(-1 \cdot im\right)} \cdot \cos re \]
8. Step-by-step derivation

   1. neg-mul-1 52.6%

      \[\leadsto \color{blue}{\left(-im\right)} \cdot \cos re \]

9. Simplified 52.6%

   \[\leadsto \color{blue}{\left(-im\right)} \cdot \cos re \]

10. Taylor expanded in re around 0 37.7%

    \[\leadsto \color{blue}{-1 \cdot im + 0.5 \cdot \left({re}^{2} \cdot im\right)} \]
11. Step-by-step derivation

    1. +-commutative 37.7%

       \[\leadsto \color{blue}{0.5 \cdot \left({re}^{2} \cdot im\right) + -1 \cdot im} \]

    2. mul-1-neg 37.7%

       \[\leadsto 0.5 \cdot \left({re}^{2} \cdot im\right) + \color{blue}{\left(-im\right)} \]

    3. unsub-neg 37.7%

       \[\leadsto \color{blue}{0.5 \cdot \left({re}^{2} \cdot im\right) - im} \]

    4. *-commutative 37.7%

       \[\leadsto \color{blue}{\left({re}^{2} \cdot im\right) \cdot 0.5} - im \]

    5. *-commutative 37.7%

       \[\leadsto \color{blue}{\left(im \cdot {re}^{2}\right)} \cdot 0.5 - im \]

    6. associate-*l* 37.7%

       \[\leadsto \color{blue}{im \cdot \left({re}^{2} \cdot 0.5\right)} - im \]

    7. unpow2 37.7%

       \[\leadsto im \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot 0.5\right) - im \]

12. Simplified 37.7%

    \[\leadsto \color{blue}{im \cdot \left(\left(re \cdot re\right) \cdot 0.5\right) - im} \]

13. Taylor expanded in re around 0 29.8%

    \[\leadsto \color{blue}{-1 \cdot im} \]
14. Step-by-step derivation

    1. neg-mul-1 29.8%

       \[\leadsto \color{blue}{-im} \]

15. Simplified 29.8%

    \[\leadsto \color{blue}{-im} \]

16. Final simplification 29.8%

    \[\leadsto -im \]

Alternative 15: 2.9% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ -3 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 -3.0)

C

double code(double re, double im) {
	return -3.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = -3.0d0
end function

Java

public static double code(double re, double im) {
	return -3.0;
}

Python

def code(re, im):
	return -3.0

Julia

function code(re, im)
	return -3.0
end

MATLAB

function tmp = code(re, im)
	tmp = -3.0;
end

Wolfram

code[re_, im_] := -3.0

Derivation

1. Initial program 54.3%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
2. Step-by-step derivation

   1. *-commutative 54.3%

      \[\leadsto \color{blue}{\left(\cos re \cdot 0.5\right)} \cdot \left(e^{0 - im} - e^{im}\right) \]

   2. associate-*l* 54.3%

      \[\leadsto \color{blue}{\cos re \cdot \left(0.5 \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]

   3. sub-neg 54.3%

      \[\leadsto \cos re \cdot \left(0.5 \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im}\right)\right)}\right) \]

   4. +-commutative 54.3%

      \[\leadsto \cos re \cdot \left(0.5 \cdot \color{blue}{\left(\left(-e^{im}\right) + e^{0 - im}\right)}\right) \]

   5. distribute-lft-in 54.3%

      \[\leadsto \cos re \cdot \color{blue}{\left(0.5 \cdot \left(-e^{im}\right) + 0.5 \cdot e^{0 - im}\right)} \]

   6. distribute-lft-in 54.3%

      \[\leadsto \cos re \cdot \color{blue}{\left(0.5 \cdot \left(\left(-e^{im}\right) + e^{0 - im}\right)\right)} \]

   7. distribute-rgt-in 54.3%

      \[\leadsto \cos re \cdot \color{blue}{\left(\left(-e^{im}\right) \cdot 0.5 + e^{0 - im} \cdot 0.5\right)} \]

   8. distribute-lft-neg-out 54.3%

      \[\leadsto \cos re \cdot \left(\color{blue}{\left(-e^{im} \cdot 0.5\right)} + e^{0 - im} \cdot 0.5\right) \]

   9. distribute-rgt-neg-in 54.3%

      \[\leadsto \cos re \cdot \left(\color{blue}{e^{im} \cdot \left(-0.5\right)} + e^{0 - im} \cdot 0.5\right) \]

   10. metadata-eval 54.3%

       \[\leadsto \cos re \cdot \left(e^{im} \cdot \color{blue}{-0.5} + e^{0 - im} \cdot 0.5\right) \]

   11. metadata-eval 54.3%

       \[\leadsto \cos re \cdot \left(e^{im} \cdot \color{blue}{\left(0.5 \cdot -1\right)} + e^{0 - im} \cdot 0.5\right) \]

   12. fma-def 54.3%

       \[\leadsto \cos re \cdot \color{blue}{\mathsf{fma}\left(e^{im}, 0.5 \cdot -1, e^{0 - im} \cdot 0.5\right)} \]

   13. metadata-eval 54.3%

       \[\leadsto \cos re \cdot \mathsf{fma}\left(e^{im}, \color{blue}{-0.5}, e^{0 - im} \cdot 0.5\right) \]

   14. exp-diff 54.3%

       \[\leadsto \cos re \cdot \mathsf{fma}\left(e^{im}, -0.5, \color{blue}{\frac{e^{0}}{e^{im}}} \cdot 0.5\right) \]

   15. associate-*l/ 54.3%

       \[\leadsto \cos re \cdot \mathsf{fma}\left(e^{im}, -0.5, \color{blue}{\frac{e^{0} \cdot 0.5}{e^{im}}}\right) \]

   16. exp-0 54.3%

       \[\leadsto \cos re \cdot \mathsf{fma}\left(e^{im}, -0.5, \frac{\color{blue}{1} \cdot 0.5}{e^{im}}\right) \]

   17. metadata-eval 54.3%

       \[\leadsto \cos re \cdot \mathsf{fma}\left(e^{im}, -0.5, \frac{\color{blue}{0.5}}{e^{im}}\right) \]

3. Simplified 54.3%

   \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(e^{im}, -0.5, \frac{0.5}{e^{im}}\right)} \]

5. Applied egg-rr 2.8%

   \[\leadsto \cos re \cdot \color{blue}{-3} \]

6. Taylor expanded in re around 0 2.8%

   \[\leadsto \color{blue}{-3} \]

7. Final simplification 2.8%

   \[\leadsto -3 \]

Alternative 16: 3.5% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.0)

C

double code(double re, double im) {
	return 0.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.0d0
end function

Java

public static double code(double re, double im) {
	return 0.0;
}

Python

def code(re, im):
	return 0.0

Julia

function code(re, im)
	return 0.0
end

MATLAB

function tmp = code(re, im)
	tmp = 0.0;
end

Wolfram

code[re_, im_] := 0.0

Derivation

1. Initial program 54.3%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
2. Step-by-step derivation

   1. *-commutative 54.3%

      \[\leadsto \color{blue}{\left(\cos re \cdot 0.5\right)} \cdot \left(e^{0 - im} - e^{im}\right) \]

   2. associate-*l* 54.3%

      \[\leadsto \color{blue}{\cos re \cdot \left(0.5 \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]

   3. sub-neg 54.3%

      \[\leadsto \cos re \cdot \left(0.5 \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im}\right)\right)}\right) \]

   4. +-commutative 54.3%

      \[\leadsto \cos re \cdot \left(0.5 \cdot \color{blue}{\left(\left(-e^{im}\right) + e^{0 - im}\right)}\right) \]

   5. distribute-lft-in 54.3%

      \[\leadsto \cos re \cdot \color{blue}{\left(0.5 \cdot \left(-e^{im}\right) + 0.5 \cdot e^{0 - im}\right)} \]

   6. distribute-lft-in 54.3%

      \[\leadsto \cos re \cdot \color{blue}{\left(0.5 \cdot \left(\left(-e^{im}\right) + e^{0 - im}\right)\right)} \]

   7. distribute-rgt-in 54.3%

      \[\leadsto \cos re \cdot \color{blue}{\left(\left(-e^{im}\right) \cdot 0.5 + e^{0 - im} \cdot 0.5\right)} \]

   8. distribute-lft-neg-out 54.3%

      \[\leadsto \cos re \cdot \left(\color{blue}{\left(-e^{im} \cdot 0.5\right)} + e^{0 - im} \cdot 0.5\right) \]

   9. distribute-rgt-neg-in 54.3%

      \[\leadsto \cos re \cdot \left(\color{blue}{e^{im} \cdot \left(-0.5\right)} + e^{0 - im} \cdot 0.5\right) \]

   10. metadata-eval 54.3%

       \[\leadsto \cos re \cdot \left(e^{im} \cdot \color{blue}{-0.5} + e^{0 - im} \cdot 0.5\right) \]

   11. metadata-eval 54.3%

       \[\leadsto \cos re \cdot \left(e^{im} \cdot \color{blue}{\left(0.5 \cdot -1\right)} + e^{0 - im} \cdot 0.5\right) \]

   12. fma-def 54.3%

       \[\leadsto \cos re \cdot \color{blue}{\mathsf{fma}\left(e^{im}, 0.5 \cdot -1, e^{0 - im} \cdot 0.5\right)} \]

   13. metadata-eval 54.3%

       \[\leadsto \cos re \cdot \mathsf{fma}\left(e^{im}, \color{blue}{-0.5}, e^{0 - im} \cdot 0.5\right) \]

   14. exp-diff 54.3%

       \[\leadsto \cos re \cdot \mathsf{fma}\left(e^{im}, -0.5, \color{blue}{\frac{e^{0}}{e^{im}}} \cdot 0.5\right) \]

   15. associate-*l/ 54.3%

       \[\leadsto \cos re \cdot \mathsf{fma}\left(e^{im}, -0.5, \color{blue}{\frac{e^{0} \cdot 0.5}{e^{im}}}\right) \]

   16. exp-0 54.3%

       \[\leadsto \cos re \cdot \mathsf{fma}\left(e^{im}, -0.5, \frac{\color{blue}{1} \cdot 0.5}{e^{im}}\right) \]

   17. metadata-eval 54.3%

       \[\leadsto \cos re \cdot \mathsf{fma}\left(e^{im}, -0.5, \frac{\color{blue}{0.5}}{e^{im}}\right) \]

3. Simplified 54.3%

   \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(e^{im}, -0.5, \frac{0.5}{e^{im}}\right)} \]

4. Taylor expanded in re around inf 54.3%

   \[\leadsto \color{blue}{\cos re \cdot \left(0.5 \cdot \frac{1}{e^{im}} + -0.5 \cdot e^{im}\right)} \]
5. Step-by-step derivation

   1. *-commutative 54.3%

      \[\leadsto \color{blue}{\left(0.5 \cdot \frac{1}{e^{im}} + -0.5 \cdot e^{im}\right) \cdot \cos re} \]

   2. associate-*r/ 54.3%

      \[\leadsto \left(\color{blue}{\frac{0.5 \cdot 1}{e^{im}}} + -0.5 \cdot e^{im}\right) \cdot \cos re \]

   3. metadata-eval 54.3%

      \[\leadsto \left(\frac{\color{blue}{0.5}}{e^{im}} + -0.5 \cdot e^{im}\right) \cdot \cos re \]

   4. *-commutative 54.3%

      \[\leadsto \left(\frac{0.5}{e^{im}} + \color{blue}{e^{im} \cdot -0.5}\right) \cdot \cos re \]

6. Simplified 54.3%

   \[\leadsto \color{blue}{\left(\frac{0.5}{e^{im}} + e^{im} \cdot -0.5\right) \cdot \cos re} \]
7. Step-by-step derivation

   1. *-commutative 54.3%

      \[\leadsto \color{blue}{\cos re \cdot \left(\frac{0.5}{e^{im}} + e^{im} \cdot -0.5\right)} \]

   2. distribute-rgt-in 54.3%

      \[\leadsto \color{blue}{\frac{0.5}{e^{im}} \cdot \cos re + \left(e^{im} \cdot -0.5\right) \cdot \cos re} \]

8. Applied egg-rr 54.3%

   \[\leadsto \color{blue}{\frac{0.5}{e^{im}} \cdot \cos re + \left(e^{im} \cdot -0.5\right) \cdot \cos re} \]

9. Taylor expanded in im around 0 3.6%

   \[\leadsto \color{blue}{0.5 \cdot \cos re + -0.5 \cdot \cos re} \]
10. Step-by-step derivation

    1. distribute-rgt-out 3.6%

       \[\leadsto \color{blue}{\cos re \cdot \left(0.5 + -0.5\right)} \]

    2. metadata-eval 3.6%

       \[\leadsto \cos re \cdot \color{blue}{0} \]

    3. mul0-rgt 3.6%

       \[\leadsto \color{blue}{0} \]

11. Simplified 3.6%

    \[\leadsto \color{blue}{0} \]

12. Final simplification 3.6%

    \[\leadsto 0 \]

Developer target: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|im\right| < 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(0.16666666666666666 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (< (fabs im) 1.0)
   (-
    (*
     (cos re)
     (+
      (+ im (* (* (* 0.16666666666666666 im) im) im))
      (* (* (* (* (* 0.008333333333333333 im) im) im) im) im))))
   (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if (fabs(im) < 1.0) {
		tmp = -(cos(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * cos(re)) * (exp((0.0 - 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 = -(cos(re) * ((im + (((0.16666666666666666d0 * im) * im) * im)) + (((((0.008333333333333333d0 * im) * im) * im) * im) * im)))
    else
        tmp = (0.5d0 * cos(re)) * (exp((0.0d0 - 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.cos(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * Math.cos(re)) * (Math.exp((0.0 - im)) - Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.fabs(im) < 1.0:
		tmp = -(math.cos(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)))
	else:
		tmp = (0.5 * math.cos(re)) * (math.exp((0.0 - im)) - math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (abs(im) < 1.0)
		tmp = Float64(-Float64(cos(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 * cos(re)) * Float64(exp(Float64(0.0 - im)) - exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (abs(im) < 1.0)
		tmp = -(cos(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	else
		tmp = (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Less[N[Abs[im], $MachinePrecision], 1.0], (-N[(N[Cos[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[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2023173 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"
  :precision binary64

  :herbie-target
  (if (< (fabs im) 1.0) (- (* (cos re) (+ (+ im (* (* (* 0.16666666666666666 im) im) im)) (* (* (* (* (* 0.008333333333333333 im) im) im) im) im)))) (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))

  (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))