math.sin on complex, imaginary part

Percentage Accurate: 53.8% → 99.7%

Time: 14.6s

Alternatives: 16

Speedup: 2.9×

• 49.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 53.8% 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: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot t_0\\ \mathbf{elif}\;t_0 \leq 0.02:\\ \;\;\;\;\cos re \cdot \left(\left(-0.16666666666666666 \cdot {im}^{3} + -0.008333333333333333 \cdot {im}^{5}\right) - im\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(e^{im} \cdot -0.5 + 0.5 \cdot \frac{1}{e^{im}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (<= t_0 (- INFINITY))
     (* (* 0.5 (cos re)) t_0)
     (if (<= t_0 0.02)
       (*
        (cos re)
        (-
         (+
          (* -0.16666666666666666 (pow im 3.0))
          (* -0.008333333333333333 (pow im 5.0)))
         im))
       (* (cos re) (+ (* (exp im) -0.5) (* 0.5 (/ 1.0 (exp im)))))))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (0.5 * cos(re)) * t_0;
	} else if (t_0 <= 0.02) {
		tmp = cos(re) * (((-0.16666666666666666 * pow(im, 3.0)) + (-0.008333333333333333 * pow(im, 5.0))) - im);
	} else {
		tmp = cos(re) * ((exp(im) * -0.5) + (0.5 * (1.0 / exp(im))));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(-im) - Math.exp(im);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = (0.5 * Math.cos(re)) * t_0;
	} else if (t_0 <= 0.02) {
		tmp = Math.cos(re) * (((-0.16666666666666666 * Math.pow(im, 3.0)) + (-0.008333333333333333 * Math.pow(im, 5.0))) - im);
	} else {
		tmp = Math.cos(re) * ((Math.exp(im) * -0.5) + (0.5 * (1.0 / Math.exp(im))));
	}
	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.cos(re)) * t_0
	elif t_0 <= 0.02:
		tmp = math.cos(re) * (((-0.16666666666666666 * math.pow(im, 3.0)) + (-0.008333333333333333 * math.pow(im, 5.0))) - im)
	else:
		tmp = math.cos(re) * ((math.exp(im) * -0.5) + (0.5 * (1.0 / math.exp(im))))
	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 * cos(re)) * t_0);
	elseif (t_0 <= 0.02)
		tmp = Float64(cos(re) * Float64(Float64(Float64(-0.16666666666666666 * (im ^ 3.0)) + Float64(-0.008333333333333333 * (im ^ 5.0))) - im));
	else
		tmp = Float64(cos(re) * Float64(Float64(exp(im) * -0.5) + Float64(0.5 * Float64(1.0 / exp(im)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = (0.5 * cos(re)) * t_0;
	elseif (t_0 <= 0.02)
		tmp = cos(re) * (((-0.16666666666666666 * (im ^ 3.0)) + (-0.008333333333333333 * (im ^ 5.0))) - im);
	else
		tmp = cos(re) * ((exp(im) * -0.5) + (0.5 * (1.0 / 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]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$0, 0.02], N[(N[Cos[re], $MachinePrecision] * N[(N[(N[(-0.16666666666666666 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision] + N[(-0.008333333333333333 * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(N[(N[Exp[im], $MachinePrecision] * -0.5), $MachinePrecision] + N[(0.5 * N[(1.0 / N[Exp[im], $MachinePrecision]), $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. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   if -inf.0 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 0.0200000000000000004

   1. Initial program 8.5%

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

      1. cos-neg 8.5%

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

      2. *-commutative 8.5%

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

      3. associate-*l* 8.5%

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

      4. sub-neg 8.5%

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

      5. neg-sub0 8.5%

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

      6. +-commutative 8.5%

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

      7. distribute-lft-in 8.5%

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

      8. cos-neg 8.5%

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

      9. distribute-lft-in 8.5%

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

      10. distribute-rgt-in 8.5%

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

      11. distribute-lft-neg-out 8.5%

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

      12. distribute-rgt-neg-in 8.5%

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

      13. metadata-eval 8.5%

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

      14. metadata-eval 8.5%

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

      15. fma-def 8.5%

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

      16. metadata-eval 8.5%

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

      17. neg-sub0 8.5%

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

      18. exp-diff 8.4%

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

   3. Simplified 8.4%

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

   4. Taylor expanded in im around 0 99.8%

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

   if 0.0200000000000000004 < (-.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. cos-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. +-commutative 100.0%

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

      7. distribute-lft-in 100.0%

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

      8. cos-neg 100.0%

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

      9. distribute-lft-in 100.0%

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

      10. distribute-rgt-in 100.0%

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

      11. distribute-lft-neg-out 100.0%

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

      12. distribute-rgt-neg-in 100.0%

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

      13. metadata-eval 100.0%

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

      14. metadata-eval 100.0%

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

      15. fma-def 100.0%

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

      16. metadata-eval 100.0%

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

      17. neg-sub0 100.0%

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

      18. exp-diff 100.0%

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

   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 im around inf 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -\infty:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;e^{-im} - e^{im} \leq 0.02:\\ \;\;\;\;\cos re \cdot \left(\left(-0.16666666666666666 \cdot {im}^{3} + -0.008333333333333333 \cdot {im}^{5}\right) - im\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(e^{im} \cdot -0.5 + 0.5 \cdot \frac{1}{e^{im}}\right)\\ \end{array} \]

Alternative 2: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot t_0\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(e^{im} \cdot -0.5 + 0.5 \cdot \frac{1}{e^{im}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (<= t_0 (- INFINITY))
     (* (* 0.5 (cos re)) t_0)
     (if (<= t_0 2e-15)
       (* (cos re) (- (* -0.16666666666666666 (pow im 3.0)) im))
       (* (cos re) (+ (* (exp im) -0.5) (* 0.5 (/ 1.0 (exp im)))))))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (0.5 * cos(re)) * t_0;
	} else if (t_0 <= 2e-15) {
		tmp = cos(re) * ((-0.16666666666666666 * pow(im, 3.0)) - im);
	} else {
		tmp = cos(re) * ((exp(im) * -0.5) + (0.5 * (1.0 / exp(im))));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(-im) - Math.exp(im);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = (0.5 * Math.cos(re)) * t_0;
	} else if (t_0 <= 2e-15) {
		tmp = Math.cos(re) * ((-0.16666666666666666 * Math.pow(im, 3.0)) - im);
	} else {
		tmp = Math.cos(re) * ((Math.exp(im) * -0.5) + (0.5 * (1.0 / Math.exp(im))));
	}
	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.cos(re)) * t_0
	elif t_0 <= 2e-15:
		tmp = math.cos(re) * ((-0.16666666666666666 * math.pow(im, 3.0)) - im)
	else:
		tmp = math.cos(re) * ((math.exp(im) * -0.5) + (0.5 * (1.0 / math.exp(im))))
	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 * cos(re)) * t_0);
	elseif (t_0 <= 2e-15)
		tmp = Float64(cos(re) * Float64(Float64(-0.16666666666666666 * (im ^ 3.0)) - im));
	else
		tmp = Float64(cos(re) * Float64(Float64(exp(im) * -0.5) + Float64(0.5 * Float64(1.0 / exp(im)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = (0.5 * cos(re)) * t_0;
	elseif (t_0 <= 2e-15)
		tmp = cos(re) * ((-0.16666666666666666 * (im ^ 3.0)) - im);
	else
		tmp = cos(re) * ((exp(im) * -0.5) + (0.5 * (1.0 / 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]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$0, 2e-15], 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[(N[(N[Exp[im], $MachinePrecision] * -0.5), $MachinePrecision] + N[(0.5 * N[(1.0 / N[Exp[im], $MachinePrecision]), $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. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   if -inf.0 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 2.0000000000000002e-15

   1. Initial program 7.8%

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

      1. cos-neg 7.8%

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

      2. *-commutative 7.8%

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

      3. associate-*l* 7.8%

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

      4. sub-neg 7.8%

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

      5. neg-sub0 7.8%

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

      6. +-commutative 7.8%

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

      7. distribute-lft-in 7.8%

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

      8. cos-neg 7.8%

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

      9. distribute-lft-in 7.8%

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

      10. distribute-rgt-in 7.8%

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

      11. distribute-lft-neg-out 7.8%

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

      12. distribute-rgt-neg-in 7.8%

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

      13. metadata-eval 7.8%

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

      14. metadata-eval 7.8%

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

      15. fma-def 7.8%

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

      16. metadata-eval 7.8%

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

      17. neg-sub0 7.8%

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

      18. exp-diff 7.8%

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

   3. Simplified 7.8%

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

   4. Taylor expanded in im around 0 99.8%

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

      1. associate-*r* 99.8%

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

      2. associate-*r* 99.8%

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

      3. distribute-rgt-out 99.8%

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

      4. *-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. neg-mul-1 99.8%

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

      7. unsub-neg 99.8%

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

   6. Simplified 99.8%

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

   if 2.0000000000000002e-15 < (-.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. cos-neg 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-*l* 99.9%

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

      4. sub-neg 99.9%

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

      5. neg-sub0 99.9%

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

      6. +-commutative 99.9%

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

      7. distribute-lft-in 99.9%

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

      8. cos-neg 99.9%

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

      9. distribute-lft-in 99.9%

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

      10. distribute-rgt-in 99.9%

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

      11. distribute-lft-neg-out 99.9%

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

      12. distribute-rgt-neg-in 99.9%

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

      13. metadata-eval 99.9%

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

      14. metadata-eval 99.9%

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

      15. fma-def 99.9%

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

      16. metadata-eval 99.9%

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

      17. neg-sub0 99.9%

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

      18. exp-diff 99.9%

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

   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 im around inf 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -\infty:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;e^{-im} - e^{im} \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(e^{im} \cdot -0.5 + 0.5 \cdot \frac{1}{e^{im}}\right)\\ \end{array} \]

Alternative 3: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 2 \cdot 10^{-15}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 2e-15)))
     (* (* 0.5 (cos re)) t_0)
     (* (cos re) (- (* -0.16666666666666666 (pow im 3.0)) im)))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || !(t_0 <= 2e-15)) {
		tmp = (0.5 * cos(re)) * t_0;
	} else {
		tmp = cos(re) * ((-0.16666666666666666 * pow(im, 3.0)) - im);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(-im) - Math.exp(im);
	double tmp;
	if ((t_0 <= -Double.POSITIVE_INFINITY) || !(t_0 <= 2e-15)) {
		tmp = (0.5 * Math.cos(re)) * t_0;
	} else {
		tmp = Math.cos(re) * ((-0.16666666666666666 * Math.pow(im, 3.0)) - im);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	tmp = 0
	if (t_0 <= -math.inf) or not (t_0 <= 2e-15):
		tmp = (0.5 * math.cos(re)) * t_0
	else:
		tmp = math.cos(re) * ((-0.16666666666666666 * math.pow(im, 3.0)) - im)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	tmp = 0.0
	if ((t_0 <= Float64(-Inf)) || !(t_0 <= 2e-15))
		tmp = Float64(Float64(0.5 * cos(re)) * t_0);
	else
		tmp = Float64(cos(re) * Float64(Float64(-0.16666666666666666 * (im ^ 3.0)) - im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	tmp = 0.0;
	if ((t_0 <= -Inf) || ~((t_0 <= 2e-15)))
		tmp = (0.5 * cos(re)) * t_0;
	else
		tmp = cos(re) * ((-0.16666666666666666 * (im ^ 3.0)) - im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, (-Infinity)], N[Not[LessEqual[t$95$0, 2e-15]], $MachinePrecision]], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(N[(-0.16666666666666666 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < -inf.0 or 2.0000000000000002e-15 < (-.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. neg-sub0 99.9%

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

   3. Simplified 99.9%

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

   if -inf.0 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 2.0000000000000002e-15

   1. Initial program 7.8%

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

      1. cos-neg 7.8%

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

      2. *-commutative 7.8%

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

      3. associate-*l* 7.8%

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

      4. sub-neg 7.8%

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

      5. neg-sub0 7.8%

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

      6. +-commutative 7.8%

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

      7. distribute-lft-in 7.8%

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

      8. cos-neg 7.8%

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

      9. distribute-lft-in 7.8%

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

      10. distribute-rgt-in 7.8%

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

      11. distribute-lft-neg-out 7.8%

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

      12. distribute-rgt-neg-in 7.8%

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

      13. metadata-eval 7.8%

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

      14. metadata-eval 7.8%

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

      15. fma-def 7.8%

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

      16. metadata-eval 7.8%

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

      17. neg-sub0 7.8%

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

      18. exp-diff 7.8%

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

   3. Simplified 7.8%

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

   4. Taylor expanded in im around 0 99.8%

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

      1. associate-*r* 99.8%

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

      2. associate-*r* 99.8%

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

      3. distribute-rgt-out 99.8%

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

      4. *-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. neg-mul-1 99.8%

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

      7. unsub-neg 99.8%

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

   6. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -\infty \lor \neg \left(e^{-im} - e^{im} \leq 2 \cdot 10^{-15}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \end{array} \]

Alternative 4: 97.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ t_1 := -0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right)\\ \mathbf{if}\;im \leq -4.5 \cdot 10^{+61}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.048:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.28:\\ \;\;\;\;\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \mathbf{elif}\;im \leq 4.4 \cdot 10^{+61}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (- (exp (- im)) (exp im)) (+ 0.5 (* (* re re) -0.25))))
        (t_1 (* -0.008333333333333333 (* (cos re) (pow im 5.0)))))
   (if (<= im -4.5e+61)
     t_1
     (if (<= im -0.048)
       t_0
       (if (<= im 0.28)
         (* (cos re) (- (* -0.16666666666666666 (pow im 3.0)) im))
         (if (<= im 4.4e+61) t_0 t_1))))))

C

double code(double re, double im) {
	double t_0 = (exp(-im) - exp(im)) * (0.5 + ((re * re) * -0.25));
	double t_1 = -0.008333333333333333 * (cos(re) * pow(im, 5.0));
	double tmp;
	if (im <= -4.5e+61) {
		tmp = t_1;
	} else if (im <= -0.048) {
		tmp = t_0;
	} else if (im <= 0.28) {
		tmp = cos(re) * ((-0.16666666666666666 * pow(im, 3.0)) - im);
	} else if (im <= 4.4e+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 = (exp(-im) - exp(im)) * (0.5d0 + ((re * re) * (-0.25d0)))
    t_1 = (-0.008333333333333333d0) * (cos(re) * (im ** 5.0d0))
    if (im <= (-4.5d+61)) then
        tmp = t_1
    else if (im <= (-0.048d0)) then
        tmp = t_0
    else if (im <= 0.28d0) then
        tmp = cos(re) * (((-0.16666666666666666d0) * (im ** 3.0d0)) - im)
    else if (im <= 4.4d+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 = (Math.exp(-im) - Math.exp(im)) * (0.5 + ((re * re) * -0.25));
	double t_1 = -0.008333333333333333 * (Math.cos(re) * Math.pow(im, 5.0));
	double tmp;
	if (im <= -4.5e+61) {
		tmp = t_1;
	} else if (im <= -0.048) {
		tmp = t_0;
	} else if (im <= 0.28) {
		tmp = Math.cos(re) * ((-0.16666666666666666 * Math.pow(im, 3.0)) - im);
	} else if (im <= 4.4e+61) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (math.exp(-im) - math.exp(im)) * (0.5 + ((re * re) * -0.25))
	t_1 = -0.008333333333333333 * (math.cos(re) * math.pow(im, 5.0))
	tmp = 0
	if im <= -4.5e+61:
		tmp = t_1
	elif im <= -0.048:
		tmp = t_0
	elif im <= 0.28:
		tmp = math.cos(re) * ((-0.16666666666666666 * math.pow(im, 3.0)) - im)
	elif im <= 4.4e+61:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(exp(Float64(-im)) - exp(im)) * Float64(0.5 + Float64(Float64(re * re) * -0.25)))
	t_1 = Float64(-0.008333333333333333 * Float64(cos(re) * (im ^ 5.0)))
	tmp = 0.0
	if (im <= -4.5e+61)
		tmp = t_1;
	elseif (im <= -0.048)
		tmp = t_0;
	elseif (im <= 0.28)
		tmp = Float64(cos(re) * Float64(Float64(-0.16666666666666666 * (im ^ 3.0)) - im));
	elseif (im <= 4.4e+61)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (exp(-im) - exp(im)) * (0.5 + ((re * re) * -0.25));
	t_1 = -0.008333333333333333 * (cos(re) * (im ^ 5.0));
	tmp = 0.0;
	if (im <= -4.5e+61)
		tmp = t_1;
	elseif (im <= -0.048)
		tmp = t_0;
	elseif (im <= 0.28)
		tmp = cos(re) * ((-0.16666666666666666 * (im ^ 3.0)) - im);
	elseif (im <= 4.4e+61)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-0.008333333333333333 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -4.5e+61], t$95$1, If[LessEqual[im, -0.048], t$95$0, If[LessEqual[im, 0.28], N[(N[Cos[re], $MachinePrecision] * N[(N[(-0.16666666666666666 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 4.4e+61], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -4.5e61 or 4.4000000000000001e61 < 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. cos-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. +-commutative 100.0%

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

      7. distribute-lft-in 100.0%

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

      8. cos-neg 100.0%

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

      9. distribute-lft-in 100.0%

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

      10. distribute-rgt-in 100.0%

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

      11. distribute-lft-neg-out 100.0%

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

      12. distribute-rgt-neg-in 100.0%

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

      13. metadata-eval 100.0%

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

      14. metadata-eval 100.0%

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

      15. fma-def 100.0%

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

      16. metadata-eval 100.0%

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

      17. neg-sub0 100.0%

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

      18. exp-diff 100.0%

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

   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 im around 0 100.0%

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

   5. Taylor expanded in im around inf 100.0%

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

   if -4.5e61 < im < -0.048000000000000001 or 0.28000000000000003 < im < 4.4000000000000001e61

   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. neg-sub0 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 4.1%

      \[\leadsto \color{blue}{-0.25 \cdot \left({re}^{2} \cdot \left(e^{-im} - e^{im}\right)\right) + 0.5 \cdot \left(e^{-im} - e^{im}\right)} \]
   5. Step-by-step derivation

      1. associate-*r* 4.1%

         \[\leadsto \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot \left(e^{-im} - e^{im}\right)} + 0.5 \cdot \left(e^{-im} - e^{im}\right) \]

      2. distribute-rgt-out 88.1%

         \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right)} \]

      3. +-commutative 88.1%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      4. *-commutative 88.1%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{{re}^{2} \cdot -0.25}\right) \]

      5. unpow2 88.1%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{\left(re \cdot re\right)} \cdot -0.25\right) \]

   6. Simplified 88.1%

      \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)} \]

   if -0.048000000000000001 < im < 0.28000000000000003

   1. Initial program 8.5%

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

      1. cos-neg 8.5%

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

      2. *-commutative 8.5%

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

      3. associate-*l* 8.5%

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

      4. sub-neg 8.5%

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

      5. neg-sub0 8.5%

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

      6. +-commutative 8.5%

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

      7. distribute-lft-in 8.5%

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

      8. cos-neg 8.5%

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

      9. distribute-lft-in 8.5%

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

      10. distribute-rgt-in 8.5%

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

      11. distribute-lft-neg-out 8.5%

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

      12. distribute-rgt-neg-in 8.5%

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

      13. metadata-eval 8.5%

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

      14. metadata-eval 8.5%

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

      15. fma-def 8.5%

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

      16. metadata-eval 8.5%

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

      17. neg-sub0 8.5%

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

      18. exp-diff 8.4%

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

   3. Simplified 8.4%

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

   4. Taylor expanded in im around 0 99.6%

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

      1. associate-*r* 99.6%

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

      2. associate-*r* 99.6%

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

      3. distribute-rgt-out 99.6%

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

      4. *-commutative 99.6%

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

      5. +-commutative 99.6%

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

      6. neg-mul-1 99.6%

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

      7. unsub-neg 99.6%

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

   6. Simplified 99.6%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -4.5 \cdot 10^{+61}:\\ \;\;\;\;-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right)\\ \mathbf{elif}\;im \leq -0.048:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \mathbf{elif}\;im \leq 0.28:\\ \;\;\;\;\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \mathbf{elif}\;im \leq 4.4 \cdot 10^{+61}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right)\\ \end{array} \]

Alternative 5: 94.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(e^{-im} - e^{im}\right)\\ t_1 := -0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right)\\ \mathbf{if}\;im \leq -2.8 \cdot 10^{+78}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.0066:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 55000000000000:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 7.5 \cdot 10^{+48}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (- (exp (- im)) (exp im))))
        (t_1 (* -0.008333333333333333 (* (cos re) (pow im 5.0)))))
   (if (<= im -2.8e+78)
     t_1
     (if (<= im -0.0066)
       t_0
       (if (<= im 55000000000000.0)
         (* im (- (cos re)))
         (if (<= im 7.5e+48) t_0 t_1))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (exp(-im) - exp(im));
	double t_1 = -0.008333333333333333 * (cos(re) * pow(im, 5.0));
	double tmp;
	if (im <= -2.8e+78) {
		tmp = t_1;
	} else if (im <= -0.0066) {
		tmp = t_0;
	} else if (im <= 55000000000000.0) {
		tmp = im * -cos(re);
	} else if (im <= 7.5e+48) {
		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 = (-0.008333333333333333d0) * (cos(re) * (im ** 5.0d0))
    if (im <= (-2.8d+78)) then
        tmp = t_1
    else if (im <= (-0.0066d0)) then
        tmp = t_0
    else if (im <= 55000000000000.0d0) then
        tmp = im * -cos(re)
    else if (im <= 7.5d+48) 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 = -0.008333333333333333 * (Math.cos(re) * Math.pow(im, 5.0));
	double tmp;
	if (im <= -2.8e+78) {
		tmp = t_1;
	} else if (im <= -0.0066) {
		tmp = t_0;
	} else if (im <= 55000000000000.0) {
		tmp = im * -Math.cos(re);
	} else if (im <= 7.5e+48) {
		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 = -0.008333333333333333 * (math.cos(re) * math.pow(im, 5.0))
	tmp = 0
	if im <= -2.8e+78:
		tmp = t_1
	elif im <= -0.0066:
		tmp = t_0
	elif im <= 55000000000000.0:
		tmp = im * -math.cos(re)
	elif im <= 7.5e+48:
		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(-0.008333333333333333 * Float64(cos(re) * (im ^ 5.0)))
	tmp = 0.0
	if (im <= -2.8e+78)
		tmp = t_1;
	elseif (im <= -0.0066)
		tmp = t_0;
	elseif (im <= 55000000000000.0)
		tmp = Float64(im * Float64(-cos(re)));
	elseif (im <= 7.5e+48)
		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 = -0.008333333333333333 * (cos(re) * (im ^ 5.0));
	tmp = 0.0;
	if (im <= -2.8e+78)
		tmp = t_1;
	elseif (im <= -0.0066)
		tmp = t_0;
	elseif (im <= 55000000000000.0)
		tmp = im * -cos(re);
	elseif (im <= 7.5e+48)
		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[(-0.008333333333333333 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -2.8e+78], t$95$1, If[LessEqual[im, -0.0066], t$95$0, If[LessEqual[im, 55000000000000.0], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im, 7.5e+48], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -2.8000000000000001e78 or 7.5000000000000006e48 < 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. cos-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. +-commutative 100.0%

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

      7. distribute-lft-in 100.0%

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

      8. cos-neg 100.0%

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

      9. distribute-lft-in 100.0%

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

      10. distribute-rgt-in 100.0%

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

      11. distribute-lft-neg-out 100.0%

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

      12. distribute-rgt-neg-in 100.0%

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

      13. metadata-eval 100.0%

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

      14. metadata-eval 100.0%

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

      15. fma-def 100.0%

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

      16. metadata-eval 100.0%

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

      17. neg-sub0 100.0%

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

      18. exp-diff 100.0%

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

   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 im around 0 98.1%

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

   5. Taylor expanded in im around inf 98.1%

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

   if -2.8000000000000001e78 < im < -0.0066 or 5.5e13 < im < 7.5000000000000006e48

   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. neg-sub0 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 80.0%

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

   if -0.0066 < im < 5.5e13

   1. Initial program 9.2%

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

      1. cos-neg 9.2%

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

      2. *-commutative 9.2%

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

      3. associate-*l* 9.2%

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

      4. sub-neg 9.2%

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

      5. neg-sub0 9.2%

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

      6. +-commutative 9.2%

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

      7. distribute-lft-in 9.2%

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

      8. cos-neg 9.2%

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

      9. distribute-lft-in 9.2%

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

      10. distribute-rgt-in 9.2%

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

      11. distribute-lft-neg-out 9.2%

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

      12. distribute-rgt-neg-in 9.2%

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

      13. metadata-eval 9.2%

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

      14. metadata-eval 9.2%

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

      15. fma-def 9.2%

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

      16. metadata-eval 9.2%

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

      17. neg-sub0 9.2%

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

      18. exp-diff 9.1%

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

   3. Simplified 9.1%

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

   4. Taylor expanded in im around 0 98.3%

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

      1. associate-*r* 98.3%

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

      2. neg-mul-1 98.3%

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

   6. Simplified 98.3%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.8 \cdot 10^{+78}:\\ \;\;\;\;-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right)\\ \mathbf{elif}\;im \leq -0.0066:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 55000000000000:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 7.5 \cdot 10^{+48}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right)\\ \end{array} \]

Alternative 6: 95.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(e^{-im} - e^{im}\right)\\ t_1 := -0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right)\\ \mathbf{if}\;im \leq -2.8 \cdot 10^{+78}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.048:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 55000000000000:\\ \;\;\;\;\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \mathbf{elif}\;im \leq 7.5 \cdot 10^{+48}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (- (exp (- im)) (exp im))))
        (t_1 (* -0.008333333333333333 (* (cos re) (pow im 5.0)))))
   (if (<= im -2.8e+78)
     t_1
     (if (<= im -0.048)
       t_0
       (if (<= im 55000000000000.0)
         (* (cos re) (- (* -0.16666666666666666 (pow im 3.0)) im))
         (if (<= im 7.5e+48) t_0 t_1))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (exp(-im) - exp(im));
	double t_1 = -0.008333333333333333 * (cos(re) * pow(im, 5.0));
	double tmp;
	if (im <= -2.8e+78) {
		tmp = t_1;
	} else if (im <= -0.048) {
		tmp = t_0;
	} else if (im <= 55000000000000.0) {
		tmp = cos(re) * ((-0.16666666666666666 * pow(im, 3.0)) - im);
	} else if (im <= 7.5e+48) {
		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 = (-0.008333333333333333d0) * (cos(re) * (im ** 5.0d0))
    if (im <= (-2.8d+78)) then
        tmp = t_1
    else if (im <= (-0.048d0)) then
        tmp = t_0
    else if (im <= 55000000000000.0d0) then
        tmp = cos(re) * (((-0.16666666666666666d0) * (im ** 3.0d0)) - im)
    else if (im <= 7.5d+48) 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 = -0.008333333333333333 * (Math.cos(re) * Math.pow(im, 5.0));
	double tmp;
	if (im <= -2.8e+78) {
		tmp = t_1;
	} else if (im <= -0.048) {
		tmp = t_0;
	} else if (im <= 55000000000000.0) {
		tmp = Math.cos(re) * ((-0.16666666666666666 * Math.pow(im, 3.0)) - im);
	} else if (im <= 7.5e+48) {
		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 = -0.008333333333333333 * (math.cos(re) * math.pow(im, 5.0))
	tmp = 0
	if im <= -2.8e+78:
		tmp = t_1
	elif im <= -0.048:
		tmp = t_0
	elif im <= 55000000000000.0:
		tmp = math.cos(re) * ((-0.16666666666666666 * math.pow(im, 3.0)) - im)
	elif im <= 7.5e+48:
		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(-0.008333333333333333 * Float64(cos(re) * (im ^ 5.0)))
	tmp = 0.0
	if (im <= -2.8e+78)
		tmp = t_1;
	elseif (im <= -0.048)
		tmp = t_0;
	elseif (im <= 55000000000000.0)
		tmp = Float64(cos(re) * Float64(Float64(-0.16666666666666666 * (im ^ 3.0)) - im));
	elseif (im <= 7.5e+48)
		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 = -0.008333333333333333 * (cos(re) * (im ^ 5.0));
	tmp = 0.0;
	if (im <= -2.8e+78)
		tmp = t_1;
	elseif (im <= -0.048)
		tmp = t_0;
	elseif (im <= 55000000000000.0)
		tmp = cos(re) * ((-0.16666666666666666 * (im ^ 3.0)) - im);
	elseif (im <= 7.5e+48)
		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[(-0.008333333333333333 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -2.8e+78], t$95$1, If[LessEqual[im, -0.048], t$95$0, If[LessEqual[im, 55000000000000.0], N[(N[Cos[re], $MachinePrecision] * N[(N[(-0.16666666666666666 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 7.5e+48], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -2.8000000000000001e78 or 7.5000000000000006e48 < 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. cos-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. +-commutative 100.0%

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

      7. distribute-lft-in 100.0%

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

      8. cos-neg 100.0%

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

      9. distribute-lft-in 100.0%

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

      10. distribute-rgt-in 100.0%

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

      11. distribute-lft-neg-out 100.0%

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

      12. distribute-rgt-neg-in 100.0%

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

      13. metadata-eval 100.0%

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

      14. metadata-eval 100.0%

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

      15. fma-def 100.0%

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

      16. metadata-eval 100.0%

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

      17. neg-sub0 100.0%

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

      18. exp-diff 100.0%

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

   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 im around 0 98.1%

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

   5. Taylor expanded in im around inf 98.1%

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

   if -2.8000000000000001e78 < im < -0.048000000000000001 or 5.5e13 < im < 7.5000000000000006e48

   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. neg-sub0 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 80.0%

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

   if -0.048000000000000001 < im < 5.5e13

   1. Initial program 9.2%

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

      1. cos-neg 9.2%

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

      2. *-commutative 9.2%

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

      3. associate-*l* 9.2%

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

      4. sub-neg 9.2%

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

      5. neg-sub0 9.2%

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

      6. +-commutative 9.2%

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

      7. distribute-lft-in 9.2%

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

      8. cos-neg 9.2%

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

      9. distribute-lft-in 9.2%

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

      10. distribute-rgt-in 9.2%

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

      11. distribute-lft-neg-out 9.2%

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

      12. distribute-rgt-neg-in 9.2%

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

      13. metadata-eval 9.2%

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

      14. metadata-eval 9.2%

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

      15. fma-def 9.2%

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

      16. metadata-eval 9.2%

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

      17. neg-sub0 9.2%

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

      18. exp-diff 9.1%

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

   3. Simplified 9.1%

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

   4. Taylor expanded in im around 0 98.9%

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

      1. associate-*r* 98.9%

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

      2. associate-*r* 98.9%

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

      3. distribute-rgt-out 98.9%

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

      4. *-commutative 98.9%

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

      5. +-commutative 98.9%

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

      6. neg-mul-1 98.9%

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

      7. unsub-neg 98.9%

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

   6. Simplified 98.9%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.8 \cdot 10^{+78}:\\ \;\;\;\;-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right)\\ \mathbf{elif}\;im \leq -0.048:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 55000000000000:\\ \;\;\;\;\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \mathbf{elif}\;im \leq 7.5 \cdot 10^{+48}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right)\\ \end{array} \]

Alternative 7: 89.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -3.3 \lor \neg \left(im \leq 3.4\right):\\ \;\;\;\;-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -3.3) (not (<= im 3.4)))
   (* -0.008333333333333333 (* (cos re) (pow im 5.0)))
   (* im (- (cos re)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -3.3) || !(im <= 3.4)) {
		tmp = -0.008333333333333333 * (cos(re) * pow(im, 5.0));
	} 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 <= (-3.3d0)) .or. (.not. (im <= 3.4d0))) then
        tmp = (-0.008333333333333333d0) * (cos(re) * (im ** 5.0d0))
    else
        tmp = im * -cos(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -3.3) || !(im <= 3.4)) {
		tmp = -0.008333333333333333 * (Math.cos(re) * Math.pow(im, 5.0));
	} else {
		tmp = im * -Math.cos(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -3.3) or not (im <= 3.4):
		tmp = -0.008333333333333333 * (math.cos(re) * math.pow(im, 5.0))
	else:
		tmp = im * -math.cos(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -3.3) || !(im <= 3.4))
		tmp = Float64(-0.008333333333333333 * Float64(cos(re) * (im ^ 5.0)));
	else
		tmp = Float64(im * Float64(-cos(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -3.3) || ~((im <= 3.4)))
		tmp = -0.008333333333333333 * (cos(re) * (im ^ 5.0));
	else
		tmp = im * -cos(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -3.3], N[Not[LessEqual[im, 3.4]], $MachinePrecision]], N[(-0.008333333333333333 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -3.2999999999999998 or 3.39999999999999991 < 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. cos-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. +-commutative 100.0%

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

      7. distribute-lft-in 100.0%

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

      8. cos-neg 100.0%

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

      9. distribute-lft-in 100.0%

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

      10. distribute-rgt-in 100.0%

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

      11. distribute-lft-neg-out 100.0%

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

      12. distribute-rgt-neg-in 100.0%

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

      13. metadata-eval 100.0%

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

      14. metadata-eval 100.0%

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

      15. fma-def 100.0%

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

      16. metadata-eval 100.0%

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

      17. neg-sub0 100.0%

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

      18. exp-diff 100.0%

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

   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 im around 0 81.2%

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

   5. Taylor expanded in im around inf 81.2%

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

   if -3.2999999999999998 < im < 3.39999999999999991

   1. Initial program 9.2%

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

      1. cos-neg 9.2%

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

      2. *-commutative 9.2%

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

      3. associate-*l* 9.2%

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

      4. sub-neg 9.2%

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

      5. neg-sub0 9.2%

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

      6. +-commutative 9.2%

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

      7. distribute-lft-in 9.2%

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

      8. cos-neg 9.2%

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

      9. distribute-lft-in 9.2%

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

      10. distribute-rgt-in 9.2%

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

      11. distribute-lft-neg-out 9.2%

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

      12. distribute-rgt-neg-in 9.2%

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

      13. metadata-eval 9.2%

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

      14. metadata-eval 9.2%

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

      15. fma-def 9.2%

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

      16. metadata-eval 9.2%

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

      17. neg-sub0 9.2%

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

      18. exp-diff 9.1%

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

   3. Simplified 9.1%

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

   4. Taylor expanded in im around 0 98.5%

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

      1. associate-*r* 98.5%

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

      2. neg-mul-1 98.5%

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

   6. Simplified 98.5%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3.3 \lor \neg \left(im \leq 3.4\right):\\ \;\;\;\;-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \]

Alternative 8: 79.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.16666666666666666 \cdot {im}^{3}\\ t_1 := 1 + -0.5 \cdot \left(re \cdot re\right)\\ \mathbf{if}\;im \leq -2.5 \cdot 10^{+102}:\\ \;\;\;\;t_0 \cdot t_1\\ \mathbf{elif}\;im \leq -4.4 \cdot 10^{+37}:\\ \;\;\;\;-0.008333333333333333 \cdot {im}^{5}\\ \mathbf{elif}\;im \leq -0.07 \lor \neg \left(im \leq 480\right):\\ \;\;\;\;\left(t_0 - im\right) \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* -0.16666666666666666 (pow im 3.0)))
        (t_1 (+ 1.0 (* -0.5 (* re re)))))
   (if (<= im -2.5e+102)
     (* t_0 t_1)
     (if (<= im -4.4e+37)
       (* -0.008333333333333333 (pow im 5.0))
       (if (or (<= im -0.07) (not (<= im 480.0)))
         (* (- t_0 im) t_1)
         (* im (- (cos re))))))))

C

double code(double re, double im) {
	double t_0 = -0.16666666666666666 * pow(im, 3.0);
	double t_1 = 1.0 + (-0.5 * (re * re));
	double tmp;
	if (im <= -2.5e+102) {
		tmp = t_0 * t_1;
	} else if (im <= -4.4e+37) {
		tmp = -0.008333333333333333 * pow(im, 5.0);
	} else if ((im <= -0.07) || !(im <= 480.0)) {
		tmp = (t_0 - im) * t_1;
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (-0.16666666666666666d0) * (im ** 3.0d0)
    t_1 = 1.0d0 + ((-0.5d0) * (re * re))
    if (im <= (-2.5d+102)) then
        tmp = t_0 * t_1
    else if (im <= (-4.4d+37)) then
        tmp = (-0.008333333333333333d0) * (im ** 5.0d0)
    else if ((im <= (-0.07d0)) .or. (.not. (im <= 480.0d0))) then
        tmp = (t_0 - im) * t_1
    else
        tmp = im * -cos(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = -0.16666666666666666 * Math.pow(im, 3.0);
	double t_1 = 1.0 + (-0.5 * (re * re));
	double tmp;
	if (im <= -2.5e+102) {
		tmp = t_0 * t_1;
	} else if (im <= -4.4e+37) {
		tmp = -0.008333333333333333 * Math.pow(im, 5.0);
	} else if ((im <= -0.07) || !(im <= 480.0)) {
		tmp = (t_0 - im) * t_1;
	} else {
		tmp = im * -Math.cos(re);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = -0.16666666666666666 * math.pow(im, 3.0)
	t_1 = 1.0 + (-0.5 * (re * re))
	tmp = 0
	if im <= -2.5e+102:
		tmp = t_0 * t_1
	elif im <= -4.4e+37:
		tmp = -0.008333333333333333 * math.pow(im, 5.0)
	elif (im <= -0.07) or not (im <= 480.0):
		tmp = (t_0 - im) * t_1
	else:
		tmp = im * -math.cos(re)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(-0.16666666666666666 * (im ^ 3.0))
	t_1 = Float64(1.0 + Float64(-0.5 * Float64(re * re)))
	tmp = 0.0
	if (im <= -2.5e+102)
		tmp = Float64(t_0 * t_1);
	elseif (im <= -4.4e+37)
		tmp = Float64(-0.008333333333333333 * (im ^ 5.0));
	elseif ((im <= -0.07) || !(im <= 480.0))
		tmp = Float64(Float64(t_0 - im) * t_1);
	else
		tmp = Float64(im * Float64(-cos(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = -0.16666666666666666 * (im ^ 3.0);
	t_1 = 1.0 + (-0.5 * (re * re));
	tmp = 0.0;
	if (im <= -2.5e+102)
		tmp = t_0 * t_1;
	elseif (im <= -4.4e+37)
		tmp = -0.008333333333333333 * (im ^ 5.0);
	elseif ((im <= -0.07) || ~((im <= 480.0)))
		tmp = (t_0 - im) * t_1;
	else
		tmp = im * -cos(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(-0.16666666666666666 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -2.5e+102], N[(t$95$0 * t$95$1), $MachinePrecision], If[LessEqual[im, -4.4e+37], N[(-0.008333333333333333 * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[im, -0.07], N[Not[LessEqual[im, 480.0]], $MachinePrecision]], N[(N[(t$95$0 - im), $MachinePrecision] * t$95$1), $MachinePrecision], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -2.5e102

   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. cos-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. +-commutative 100.0%

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

      7. distribute-lft-in 100.0%

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

      8. cos-neg 100.0%

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

      9. distribute-lft-in 100.0%

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

      10. distribute-rgt-in 100.0%

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

      11. distribute-lft-neg-out 100.0%

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

      12. distribute-rgt-neg-in 100.0%

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

      13. metadata-eval 100.0%

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

      14. metadata-eval 100.0%

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

      15. fma-def 100.0%

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

      16. metadata-eval 100.0%

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

      17. neg-sub0 100.0%

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

      18. exp-diff 100.0%

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

   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 im around 0 100.0%

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

      1. associate-*r* 100.0%

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

      2. associate-*r* 100.0%

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

      3. distribute-rgt-out 100.0%

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

      4. *-commutative 100.0%

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

      5. +-commutative 100.0%

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

      6. neg-mul-1 100.0%

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

      7. unsub-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around 0 0.0%

      \[\leadsto \color{blue}{\left(-0.5 \cdot \left({re}^{2} \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\right) + -0.16666666666666666 \cdot {im}^{3}\right) - im} \]
   8. Step-by-step derivation

      1. associate--l+ 0.0%

         \[\leadsto \color{blue}{-0.5 \cdot \left({re}^{2} \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\right) + \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} \]

      2. associate-*r* 0.0%

         \[\leadsto \color{blue}{\left(-0.5 \cdot {re}^{2}\right) \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} + \left(-0.16666666666666666 \cdot {im}^{3} - im\right) \]

      3. distribute-lft1-in 91.2%

         \[\leadsto \color{blue}{\left(-0.5 \cdot {re}^{2} + 1\right) \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} \]

      4. unpow2 91.2%

         \[\leadsto \left(-0.5 \cdot \color{blue}{\left(re \cdot re\right)} + 1\right) \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right) \]

   9. Simplified 91.2%

      \[\leadsto \color{blue}{\left(-0.5 \cdot \left(re \cdot re\right) + 1\right) \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} \]

   10. Taylor expanded in im around inf 91.2%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot \left(1 + -0.5 \cdot {re}^{2}\right)\right)} \]
   11. Step-by-step derivation

       1. associate-*r* 91.2%

          \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \left(1 + -0.5 \cdot {re}^{2}\right)} \]

       2. unpow2 91.2%

          \[\leadsto \left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \left(1 + -0.5 \cdot \color{blue}{\left(re \cdot re\right)}\right) \]

   12. Simplified 91.2%

       \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \left(1 + -0.5 \cdot \left(re \cdot re\right)\right)} \]

   if -2.5e102 < im < -4.4000000000000001e37

   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. cos-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. +-commutative 100.0%

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

      7. distribute-lft-in 100.0%

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

      8. cos-neg 100.0%

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

      9. distribute-lft-in 100.0%

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

      10. distribute-rgt-in 100.0%

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

      11. distribute-lft-neg-out 100.0%

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

      12. distribute-rgt-neg-in 100.0%

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

      13. metadata-eval 100.0%

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

      14. metadata-eval 100.0%

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

      15. fma-def 100.0%

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

      16. metadata-eval 100.0%

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

      17. neg-sub0 100.0%

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

      18. exp-diff 100.0%

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

   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 im around 0 56.7%

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

   5. Taylor expanded in im around inf 56.7%

      \[\leadsto \color{blue}{-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 56.7%

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

      2. *-commutative 56.7%

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

      3. associate-*l* 56.7%

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

   7. Simplified 56.7%

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

   8. Taylor expanded in re around 0 49.1%

      \[\leadsto {im}^{5} \cdot \color{blue}{-0.008333333333333333} \]

   if -4.4000000000000001e37 < im < -0.070000000000000007 or 480 < 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. cos-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. +-commutative 100.0%

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

      7. distribute-lft-in 100.0%

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

      8. cos-neg 100.0%

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

      9. distribute-lft-in 100.0%

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

      10. distribute-rgt-in 100.0%

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

      11. distribute-lft-neg-out 100.0%

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

      12. distribute-rgt-neg-in 100.0%

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

      13. metadata-eval 100.0%

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

      14. metadata-eval 100.0%

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

      15. fma-def 100.0%

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

      16. metadata-eval 100.0%

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

      17. neg-sub0 100.0%

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

      18. exp-diff 100.0%

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

   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 im around 0 63.9%

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

      1. associate-*r* 63.9%

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

      2. associate-*r* 63.9%

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

      3. distribute-rgt-out 63.9%

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

      4. *-commutative 63.9%

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

      5. +-commutative 63.9%

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

      6. neg-mul-1 63.9%

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

      7. unsub-neg 63.9%

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

   6. Simplified 63.9%

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

   7. Taylor expanded in re around 0 12.5%

      \[\leadsto \color{blue}{\left(-0.5 \cdot \left({re}^{2} \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\right) + -0.16666666666666666 \cdot {im}^{3}\right) - im} \]
   8. Step-by-step derivation

      1. associate--l+ 12.5%

         \[\leadsto \color{blue}{-0.5 \cdot \left({re}^{2} \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\right) + \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} \]

      2. associate-*r* 12.5%

         \[\leadsto \color{blue}{\left(-0.5 \cdot {re}^{2}\right) \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} + \left(-0.16666666666666666 \cdot {im}^{3} - im\right) \]

      3. distribute-lft1-in 59.2%

         \[\leadsto \color{blue}{\left(-0.5 \cdot {re}^{2} + 1\right) \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} \]

      4. unpow2 59.2%

         \[\leadsto \left(-0.5 \cdot \color{blue}{\left(re \cdot re\right)} + 1\right) \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right) \]

   9. Simplified 59.2%

      \[\leadsto \color{blue}{\left(-0.5 \cdot \left(re \cdot re\right) + 1\right) \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} \]

   if -0.070000000000000007 < im < 480

   1. Initial program 8.5%

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

      1. cos-neg 8.5%

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

      2. *-commutative 8.5%

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

      3. associate-*l* 8.5%

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

      4. sub-neg 8.5%

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

      5. neg-sub0 8.5%

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

      6. +-commutative 8.5%

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

      7. distribute-lft-in 8.5%

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

      8. cos-neg 8.5%

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

      9. distribute-lft-in 8.5%

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

      10. distribute-rgt-in 8.5%

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

      11. distribute-lft-neg-out 8.5%

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

      12. distribute-rgt-neg-in 8.5%

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

      13. metadata-eval 8.5%

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

      14. metadata-eval 8.5%

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

      15. fma-def 8.5%

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

      16. metadata-eval 8.5%

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

      17. neg-sub0 8.5%

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

      18. exp-diff 8.4%

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

   3. Simplified 8.4%

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

   4. Taylor expanded in im around 0 99.0%

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

      1. associate-*r* 99.0%

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

      2. neg-mul-1 99.0%

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

   6. Simplified 99.0%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.5 \cdot 10^{+102}:\\ \;\;\;\;\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \left(1 + -0.5 \cdot \left(re \cdot re\right)\right)\\ \mathbf{elif}\;im \leq -4.4 \cdot 10^{+37}:\\ \;\;\;\;-0.008333333333333333 \cdot {im}^{5}\\ \mathbf{elif}\;im \leq -0.07 \lor \neg \left(im \leq 480\right):\\ \;\;\;\;\left(-0.16666666666666666 \cdot {im}^{3} - im\right) \cdot \left(1 + -0.5 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \]

Alternative 9: 79.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \left(1 + -0.5 \cdot \left(re \cdot re\right)\right)\\ \mathbf{if}\;im \leq -2 \cdot 10^{+103}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -4.4 \cdot 10^{+37}:\\ \;\;\;\;-0.008333333333333333 \cdot {im}^{5}\\ \mathbf{elif}\;im \leq -550 \lor \neg \left(im \leq 560\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (* (* -0.16666666666666666 (pow im 3.0)) (+ 1.0 (* -0.5 (* re re))))))
   (if (<= im -2e+103)
     t_0
     (if (<= im -4.4e+37)
       (* -0.008333333333333333 (pow im 5.0))
       (if (or (<= im -550.0) (not (<= im 560.0))) t_0 (* im (- (cos re))))))))

C

double code(double re, double im) {
	double t_0 = (-0.16666666666666666 * pow(im, 3.0)) * (1.0 + (-0.5 * (re * re)));
	double tmp;
	if (im <= -2e+103) {
		tmp = t_0;
	} else if (im <= -4.4e+37) {
		tmp = -0.008333333333333333 * pow(im, 5.0);
	} else if ((im <= -550.0) || !(im <= 560.0)) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = ((-0.16666666666666666d0) * (im ** 3.0d0)) * (1.0d0 + ((-0.5d0) * (re * re)))
    if (im <= (-2d+103)) then
        tmp = t_0
    else if (im <= (-4.4d+37)) then
        tmp = (-0.008333333333333333d0) * (im ** 5.0d0)
    else if ((im <= (-550.0d0)) .or. (.not. (im <= 560.0d0))) then
        tmp = t_0
    else
        tmp = im * -cos(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (-0.16666666666666666 * Math.pow(im, 3.0)) * (1.0 + (-0.5 * (re * re)));
	double tmp;
	if (im <= -2e+103) {
		tmp = t_0;
	} else if (im <= -4.4e+37) {
		tmp = -0.008333333333333333 * Math.pow(im, 5.0);
	} else if ((im <= -550.0) || !(im <= 560.0)) {
		tmp = t_0;
	} else {
		tmp = im * -Math.cos(re);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (-0.16666666666666666 * math.pow(im, 3.0)) * (1.0 + (-0.5 * (re * re)))
	tmp = 0
	if im <= -2e+103:
		tmp = t_0
	elif im <= -4.4e+37:
		tmp = -0.008333333333333333 * math.pow(im, 5.0)
	elif (im <= -550.0) or not (im <= 560.0):
		tmp = t_0
	else:
		tmp = im * -math.cos(re)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(-0.16666666666666666 * (im ^ 3.0)) * Float64(1.0 + Float64(-0.5 * Float64(re * re))))
	tmp = 0.0
	if (im <= -2e+103)
		tmp = t_0;
	elseif (im <= -4.4e+37)
		tmp = Float64(-0.008333333333333333 * (im ^ 5.0));
	elseif ((im <= -550.0) || !(im <= 560.0))
		tmp = t_0;
	else
		tmp = Float64(im * Float64(-cos(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (-0.16666666666666666 * (im ^ 3.0)) * (1.0 + (-0.5 * (re * re)));
	tmp = 0.0;
	if (im <= -2e+103)
		tmp = t_0;
	elseif (im <= -4.4e+37)
		tmp = -0.008333333333333333 * (im ^ 5.0);
	elseif ((im <= -550.0) || ~((im <= 560.0)))
		tmp = t_0;
	else
		tmp = im * -cos(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(-0.16666666666666666 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -2e+103], t$95$0, If[LessEqual[im, -4.4e+37], N[(-0.008333333333333333 * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[im, -550.0], N[Not[LessEqual[im, 560.0]], $MachinePrecision]], t$95$0, N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -2e103 or -4.4000000000000001e37 < im < -550 or 560 < 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. cos-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. +-commutative 100.0%

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

      7. distribute-lft-in 100.0%

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

      8. cos-neg 100.0%

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

      9. distribute-lft-in 100.0%

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

      10. distribute-rgt-in 100.0%

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

      11. distribute-lft-neg-out 100.0%

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

      12. distribute-rgt-neg-in 100.0%

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

      13. metadata-eval 100.0%

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

      14. metadata-eval 100.0%

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

      15. fma-def 100.0%

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

      16. metadata-eval 100.0%

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

      17. neg-sub0 100.0%

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

      18. exp-diff 100.0%

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

   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 im around 0 76.0%

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

      1. associate-*r* 76.0%

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

      2. associate-*r* 76.0%

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

      3. distribute-rgt-out 76.0%

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

      4. *-commutative 76.0%

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

      5. +-commutative 76.0%

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

      6. neg-mul-1 76.0%

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

      7. unsub-neg 76.0%

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

   6. Simplified 76.0%

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

   7. Taylor expanded in re around 0 8.2%

      \[\leadsto \color{blue}{\left(-0.5 \cdot \left({re}^{2} \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\right) + -0.16666666666666666 \cdot {im}^{3}\right) - im} \]
   8. Step-by-step derivation

      1. associate--l+ 8.2%

         \[\leadsto \color{blue}{-0.5 \cdot \left({re}^{2} \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\right) + \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} \]

      2. associate-*r* 8.2%

         \[\leadsto \color{blue}{\left(-0.5 \cdot {re}^{2}\right) \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} + \left(-0.16666666666666666 \cdot {im}^{3} - im\right) \]

      3. distribute-lft1-in 69.9%

         \[\leadsto \color{blue}{\left(-0.5 \cdot {re}^{2} + 1\right) \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} \]

      4. unpow2 69.9%

         \[\leadsto \left(-0.5 \cdot \color{blue}{\left(re \cdot re\right)} + 1\right) \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right) \]

   9. Simplified 69.9%

      \[\leadsto \color{blue}{\left(-0.5 \cdot \left(re \cdot re\right) + 1\right) \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} \]

   10. Taylor expanded in im around inf 69.9%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot \left(1 + -0.5 \cdot {re}^{2}\right)\right)} \]
   11. Step-by-step derivation

       1. associate-*r* 69.9%

          \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \left(1 + -0.5 \cdot {re}^{2}\right)} \]

       2. unpow2 69.9%

          \[\leadsto \left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \left(1 + -0.5 \cdot \color{blue}{\left(re \cdot re\right)}\right) \]

   12. Simplified 69.9%

       \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \left(1 + -0.5 \cdot \left(re \cdot re\right)\right)} \]

   if -2e103 < im < -4.4000000000000001e37

   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. cos-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. +-commutative 100.0%

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

      7. distribute-lft-in 100.0%

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

      8. cos-neg 100.0%

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

      9. distribute-lft-in 100.0%

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

      10. distribute-rgt-in 100.0%

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

      11. distribute-lft-neg-out 100.0%

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

      12. distribute-rgt-neg-in 100.0%

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

      13. metadata-eval 100.0%

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

      14. metadata-eval 100.0%

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

      15. fma-def 100.0%

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

      16. metadata-eval 100.0%

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

      17. neg-sub0 100.0%

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

      18. exp-diff 100.0%

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

   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 im around 0 56.7%

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

   5. Taylor expanded in im around inf 56.7%

      \[\leadsto \color{blue}{-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 56.7%

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

      2. *-commutative 56.7%

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

      3. associate-*l* 56.7%

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

   7. Simplified 56.7%

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

   8. Taylor expanded in re around 0 49.1%

      \[\leadsto {im}^{5} \cdot \color{blue}{-0.008333333333333333} \]

   if -550 < im < 560

   1. Initial program 9.8%

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

      1. cos-neg 9.8%

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

      2. *-commutative 9.8%

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

      3. associate-*l* 9.8%

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

      4. sub-neg 9.8%

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

      5. neg-sub0 9.8%

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

      6. +-commutative 9.8%

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

      7. distribute-lft-in 9.8%

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

      8. cos-neg 9.8%

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

      9. distribute-lft-in 9.8%

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

      10. distribute-rgt-in 9.8%

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

      11. distribute-lft-neg-out 9.8%

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

      12. distribute-rgt-neg-in 9.8%

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

      13. metadata-eval 9.8%

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

      14. metadata-eval 9.8%

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

      15. fma-def 9.8%

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

      16. metadata-eval 9.8%

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

      17. neg-sub0 9.8%

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

      18. exp-diff 9.8%

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

   3. Simplified 9.8%

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

   4. Taylor expanded in im around 0 97.8%

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

      1. associate-*r* 97.8%

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

      2. neg-mul-1 97.8%

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

   6. Simplified 97.8%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2 \cdot 10^{+103}:\\ \;\;\;\;\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \left(1 + -0.5 \cdot \left(re \cdot re\right)\right)\\ \mathbf{elif}\;im \leq -4.4 \cdot 10^{+37}:\\ \;\;\;\;-0.008333333333333333 \cdot {im}^{5}\\ \mathbf{elif}\;im \leq -550 \lor \neg \left(im \leq 560\right):\\ \;\;\;\;\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \left(1 + -0.5 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \]

Alternative 10: 39.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.002:\\ \;\;\;\;im \cdot \left(-1 - -0.5 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) -0.002) (* im (- -1.0 (* -0.5 (* re re)))) (- im)))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= -0.002) {
		tmp = im * (-1.0 - (-0.5 * (re * re)));
	} 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) <= (-0.002d0)) then
        tmp = im * ((-1.0d0) - ((-0.5d0) * (re * re)))
    else
        tmp = -im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.cos(re) <= -0.002) {
		tmp = im * (-1.0 - (-0.5 * (re * re)));
	} else {
		tmp = -im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.cos(re) <= -0.002:
		tmp = im * (-1.0 - (-0.5 * (re * re)))
	else:
		tmp = -im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= -0.002)
		tmp = Float64(im * Float64(-1.0 - Float64(-0.5 * Float64(re * re))));
	else
		tmp = Float64(-im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (cos(re) <= -0.002)
		tmp = im * (-1.0 - (-0.5 * (re * re)));
	else
		tmp = -im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.002], N[(im * N[(-1.0 - N[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-im)]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -2e-3

   1. Initial program 54.5%

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

      1. cos-neg 54.5%

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

      2. *-commutative 54.5%

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

      3. associate-*l* 54.5%

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

      4. sub-neg 54.5%

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

      5. neg-sub0 54.5%

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

      6. +-commutative 54.5%

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

      7. distribute-lft-in 54.5%

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

      8. cos-neg 54.5%

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

      9. distribute-lft-in 54.5%

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

      10. distribute-rgt-in 54.5%

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

      11. distribute-lft-neg-out 54.5%

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

      12. distribute-rgt-neg-in 54.5%

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

      13. metadata-eval 54.5%

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

      14. metadata-eval 54.5%

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

      15. fma-def 54.5%

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

      16. metadata-eval 54.5%

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

      17. neg-sub0 54.5%

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

      18. exp-diff 54.6%

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

   3. Simplified 54.6%

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

   4. Taylor expanded in im around 0 83.5%

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

      1. associate-*r* 83.5%

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

      2. associate-*r* 83.5%

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

      3. distribute-rgt-out 83.5%

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

      4. *-commutative 83.5%

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

      5. +-commutative 83.5%

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

      6. neg-mul-1 83.5%

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

      7. unsub-neg 83.5%

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

   6. Simplified 83.5%

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

   7. Taylor expanded in re around 0 14.6%

      \[\leadsto \color{blue}{\left(-0.5 \cdot \left({re}^{2} \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\right) + -0.16666666666666666 \cdot {im}^{3}\right) - im} \]
   8. Step-by-step derivation

      1. associate--l+ 14.6%

         \[\leadsto \color{blue}{-0.5 \cdot \left({re}^{2} \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\right) + \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} \]

      2. associate-*r* 14.6%

         \[\leadsto \color{blue}{\left(-0.5 \cdot {re}^{2}\right) \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} + \left(-0.16666666666666666 \cdot {im}^{3} - im\right) \]

      3. distribute-lft1-in 48.4%

         \[\leadsto \color{blue}{\left(-0.5 \cdot {re}^{2} + 1\right) \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} \]

      4. unpow2 48.4%

         \[\leadsto \left(-0.5 \cdot \color{blue}{\left(re \cdot re\right)} + 1\right) \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right) \]

   9. Simplified 48.4%

      \[\leadsto \color{blue}{\left(-0.5 \cdot \left(re \cdot re\right) + 1\right) \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} \]

   10. Taylor expanded in im around 0 41.7%

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

       1. mul-1-neg 41.7%

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

   12. Simplified 41.7%

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

   if -2e-3 < (cos.f64 re)

   1. Initial program 51.1%

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

      1. cos-neg 51.1%

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

      2. *-commutative 51.1%

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

      3. associate-*l* 51.1%

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

      4. sub-neg 51.1%

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

      5. neg-sub0 51.1%

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

      6. +-commutative 51.1%

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

      7. distribute-lft-in 51.1%

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

      8. cos-neg 51.1%

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

      9. distribute-lft-in 51.1%

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

      10. distribute-rgt-in 51.1%

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

      11. distribute-lft-neg-out 51.1%

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

      12. distribute-rgt-neg-in 51.1%

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

      13. metadata-eval 51.1%

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

      14. metadata-eval 51.1%

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

      15. fma-def 51.1%

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

      16. metadata-eval 51.1%

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

      17. neg-sub0 51.1%

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

      18. exp-diff 51.1%

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

   3. Simplified 51.1%

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

   4. Taylor expanded in im around 0 91.6%

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

   5. Taylor expanded in im around inf 91.0%

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

   6. Taylor expanded in re around 0 74.7%

      \[\leadsto \color{blue}{-1 \cdot im + -0.008333333333333333 \cdot {im}^{5}} \]

   7. Taylor expanded in im around 0 39.3%

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

      1. mul-1-neg 39.3%

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

   9. Simplified 39.3%

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

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq -0.002:\\ \;\;\;\;im \cdot \left(-1 - -0.5 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-im\\ \end{array} \]

Alternative 11: 79.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.008333333333333333 \cdot {im}^{5}\\ \mathbf{if}\;im \leq -2 \cdot 10^{+36}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+58}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 - im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* -0.008333333333333333 (pow im 5.0))))
   (if (<= im -2e+36)
     t_0
     (if (<= im 1.15e+58) (* im (- (cos re))) (- t_0 im)))))

C

double code(double re, double im) {
	double t_0 = -0.008333333333333333 * pow(im, 5.0);
	double tmp;
	if (im <= -2e+36) {
		tmp = t_0;
	} else if (im <= 1.15e+58) {
		tmp = im * -cos(re);
	} else {
		tmp = t_0 - im;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-0.008333333333333333d0) * (im ** 5.0d0)
    if (im <= (-2d+36)) then
        tmp = t_0
    else if (im <= 1.15d+58) then
        tmp = im * -cos(re)
    else
        tmp = t_0 - im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = -0.008333333333333333 * Math.pow(im, 5.0);
	double tmp;
	if (im <= -2e+36) {
		tmp = t_0;
	} else if (im <= 1.15e+58) {
		tmp = im * -Math.cos(re);
	} else {
		tmp = t_0 - im;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = -0.008333333333333333 * math.pow(im, 5.0)
	tmp = 0
	if im <= -2e+36:
		tmp = t_0
	elif im <= 1.15e+58:
		tmp = im * -math.cos(re)
	else:
		tmp = t_0 - im
	return tmp

Julia

function code(re, im)
	t_0 = Float64(-0.008333333333333333 * (im ^ 5.0))
	tmp = 0.0
	if (im <= -2e+36)
		tmp = t_0;
	elseif (im <= 1.15e+58)
		tmp = Float64(im * Float64(-cos(re)));
	else
		tmp = Float64(t_0 - im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = -0.008333333333333333 * (im ^ 5.0);
	tmp = 0.0;
	if (im <= -2e+36)
		tmp = t_0;
	elseif (im <= 1.15e+58)
		tmp = im * -cos(re);
	else
		tmp = t_0 - im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(-0.008333333333333333 * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -2e+36], t$95$0, If[LessEqual[im, 1.15e+58], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], N[(t$95$0 - im), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < -2.00000000000000008e36

   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. cos-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. +-commutative 100.0%

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

      7. distribute-lft-in 100.0%

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

      8. cos-neg 100.0%

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

      9. distribute-lft-in 100.0%

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

      10. distribute-rgt-in 100.0%

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

      11. distribute-lft-neg-out 100.0%

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

      12. distribute-rgt-neg-in 100.0%

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

      13. metadata-eval 100.0%

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

      14. metadata-eval 100.0%

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

      15. fma-def 100.0%

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

      16. metadata-eval 100.0%

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

      17. neg-sub0 100.0%

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

      18. exp-diff 100.0%

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

   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 im around 0 88.0%

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

   5. Taylor expanded in im around inf 88.0%

      \[\leadsto \color{blue}{-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 88.0%

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

      2. *-commutative 88.0%

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

      3. associate-*l* 88.0%

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

   7. Simplified 88.0%

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

   8. Taylor expanded in re around 0 60.4%

      \[\leadsto {im}^{5} \cdot \color{blue}{-0.008333333333333333} \]

   if -2.00000000000000008e36 < im < 1.15000000000000001e58

   1. Initial program 19.8%

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

      1. cos-neg 19.8%

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

      2. *-commutative 19.8%

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

      3. associate-*l* 19.8%

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

      4. sub-neg 19.8%

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

      5. neg-sub0 19.8%

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

      6. +-commutative 19.8%

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

      7. distribute-lft-in 19.8%

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

      8. cos-neg 19.8%

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

      9. distribute-lft-in 19.8%

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

      10. distribute-rgt-in 19.8%

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

      11. distribute-lft-neg-out 19.8%

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

      12. distribute-rgt-neg-in 19.8%

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

      13. metadata-eval 19.8%

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

      14. metadata-eval 19.8%

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

      15. fma-def 19.8%

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

      16. metadata-eval 19.8%

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

      17. neg-sub0 19.8%

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

      18. exp-diff 19.8%

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

   3. Simplified 19.8%

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

   4. Taylor expanded in im around 0 87.3%

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

      1. associate-*r* 87.3%

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

      2. neg-mul-1 87.3%

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

   6. Simplified 87.3%

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

   if 1.15000000000000001e58 < 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. cos-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. +-commutative 100.0%

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

      7. distribute-lft-in 100.0%

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

      8. cos-neg 100.0%

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

      9. distribute-lft-in 100.0%

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

      10. distribute-rgt-in 100.0%

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

      11. distribute-lft-neg-out 100.0%

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

      12. distribute-rgt-neg-in 100.0%

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

      13. metadata-eval 100.0%

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

      14. metadata-eval 100.0%

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

      15. fma-def 100.0%

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

      16. metadata-eval 100.0%

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

      17. neg-sub0 100.0%

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

      18. exp-diff 100.0%

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

   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 im around 0 100.0%

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

   5. Taylor expanded in im around inf 100.0%

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

   6. Taylor expanded in re around 0 73.2%

      \[\leadsto \color{blue}{-1 \cdot im + -0.008333333333333333 \cdot {im}^{5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2 \cdot 10^{+36}:\\ \;\;\;\;-0.008333333333333333 \cdot {im}^{5}\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+58}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;-0.008333333333333333 \cdot {im}^{5} - im\\ \end{array} \]

Alternative 12: 60.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1320 \lor \neg \left(im \leq 59000\right):\\ \;\;\;\;im \cdot \left(-1 - -0.5 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -1320.0) (not (<= im 59000.0)))
   (* im (- -1.0 (* -0.5 (* re re))))
   (* im (- (cos re)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -1320.0) || !(im <= 59000.0)) {
		tmp = im * (-1.0 - (-0.5 * (re * re)));
	} 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 <= (-1320.0d0)) .or. (.not. (im <= 59000.0d0))) then
        tmp = im * ((-1.0d0) - ((-0.5d0) * (re * re)))
    else
        tmp = im * -cos(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -1320.0) || !(im <= 59000.0)) {
		tmp = im * (-1.0 - (-0.5 * (re * re)));
	} else {
		tmp = im * -Math.cos(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -1320.0) or not (im <= 59000.0):
		tmp = im * (-1.0 - (-0.5 * (re * re)))
	else:
		tmp = im * -math.cos(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -1320.0) || !(im <= 59000.0))
		tmp = Float64(im * Float64(-1.0 - Float64(-0.5 * Float64(re * re))));
	else
		tmp = Float64(im * Float64(-cos(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -1320.0) || ~((im <= 59000.0)))
		tmp = im * (-1.0 - (-0.5 * (re * re)));
	else
		tmp = im * -cos(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -1320.0], N[Not[LessEqual[im, 59000.0]], $MachinePrecision]], N[(im * N[(-1.0 - N[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -1320 or 59000 < 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. cos-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. +-commutative 100.0%

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

      7. distribute-lft-in 100.0%

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

      8. cos-neg 100.0%

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

      9. distribute-lft-in 100.0%

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

      10. distribute-rgt-in 100.0%

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

      11. distribute-lft-neg-out 100.0%

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

      12. distribute-rgt-neg-in 100.0%

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

      13. metadata-eval 100.0%

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

      14. metadata-eval 100.0%

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

      15. fma-def 100.0%

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

      16. metadata-eval 100.0%

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

      17. neg-sub0 100.0%

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

      18. exp-diff 100.0%

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

   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 im around 0 69.0%

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

      1. associate-*r* 69.0%

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

      2. associate-*r* 69.0%

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

      3. distribute-rgt-out 69.0%

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

      4. *-commutative 69.0%

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

      5. +-commutative 69.0%

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

      6. neg-mul-1 69.0%

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

      7. unsub-neg 69.0%

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

   6. Simplified 69.0%

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

   7. Taylor expanded in re around 0 8.7%

      \[\leadsto \color{blue}{\left(-0.5 \cdot \left({re}^{2} \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\right) + -0.16666666666666666 \cdot {im}^{3}\right) - im} \]
   8. Step-by-step derivation

      1. associate--l+ 8.7%

         \[\leadsto \color{blue}{-0.5 \cdot \left({re}^{2} \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\right) + \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} \]

      2. associate-*r* 8.7%

         \[\leadsto \color{blue}{\left(-0.5 \cdot {re}^{2}\right) \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} + \left(-0.16666666666666666 \cdot {im}^{3} - im\right) \]

      3. distribute-lft1-in 64.1%

         \[\leadsto \color{blue}{\left(-0.5 \cdot {re}^{2} + 1\right) \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} \]

      4. unpow2 64.1%

         \[\leadsto \left(-0.5 \cdot \color{blue}{\left(re \cdot re\right)} + 1\right) \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right) \]

   9. Simplified 64.1%

      \[\leadsto \color{blue}{\left(-0.5 \cdot \left(re \cdot re\right) + 1\right) \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} \]

   10. Taylor expanded in im around 0 26.7%

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

       1. mul-1-neg 26.7%

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

   12. Simplified 26.7%

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

   if -1320 < im < 59000

   1. Initial program 10.5%

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

      1. cos-neg 10.5%

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

      2. *-commutative 10.5%

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

      3. associate-*l* 10.5%

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

      4. sub-neg 10.5%

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

      5. neg-sub0 10.5%

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

      6. +-commutative 10.5%

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

      7. distribute-lft-in 10.5%

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

      8. cos-neg 10.5%

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

      9. distribute-lft-in 10.5%

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

      10. distribute-rgt-in 10.5%

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

      11. distribute-lft-neg-out 10.5%

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

      12. distribute-rgt-neg-in 10.5%

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

      13. metadata-eval 10.5%

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

      14. metadata-eval 10.5%

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

      15. fma-def 10.5%

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

      16. metadata-eval 10.5%

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

      17. neg-sub0 10.5%

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

      18. exp-diff 10.4%

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

   3. Simplified 10.4%

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

   4. Taylor expanded in im around 0 97.1%

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

      1. associate-*r* 97.1%

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

      2. neg-mul-1 97.1%

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

   6. Simplified 97.1%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1320 \lor \neg \left(im \leq 59000\right):\\ \;\;\;\;im \cdot \left(-1 - -0.5 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \]

Alternative 13: 79.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -2.35 \cdot 10^{+36} \lor \neg \left(im \leq 1.15 \cdot 10^{+58}\right):\\ \;\;\;\;-0.008333333333333333 \cdot {im}^{5}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -2.35e+36) (not (<= im 1.15e+58)))
   (* -0.008333333333333333 (pow im 5.0))
   (* im (- (cos re)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -2.35e+36) || !(im <= 1.15e+58)) {
		tmp = -0.008333333333333333 * pow(im, 5.0);
	} 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 <= (-2.35d+36)) .or. (.not. (im <= 1.15d+58))) then
        tmp = (-0.008333333333333333d0) * (im ** 5.0d0)
    else
        tmp = im * -cos(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -2.35e+36) || !(im <= 1.15e+58)) {
		tmp = -0.008333333333333333 * Math.pow(im, 5.0);
	} else {
		tmp = im * -Math.cos(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -2.35e+36) or not (im <= 1.15e+58):
		tmp = -0.008333333333333333 * math.pow(im, 5.0)
	else:
		tmp = im * -math.cos(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -2.35e+36) || !(im <= 1.15e+58))
		tmp = Float64(-0.008333333333333333 * (im ^ 5.0));
	else
		tmp = Float64(im * Float64(-cos(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -2.35e+36) || ~((im <= 1.15e+58)))
		tmp = -0.008333333333333333 * (im ^ 5.0);
	else
		tmp = im * -cos(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -2.35e+36], N[Not[LessEqual[im, 1.15e+58]], $MachinePrecision]], N[(-0.008333333333333333 * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -2.34999999999999994e36 or 1.15000000000000001e58 < 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. cos-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

      4. sub-neg 100.0%

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

      5. neg-sub0 100.0%

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

      6. +-commutative 100.0%

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

      7. distribute-lft-in 100.0%

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

      8. cos-neg 100.0%

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

      9. distribute-lft-in 100.0%

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

      10. distribute-rgt-in 100.0%

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

      11. distribute-lft-neg-out 100.0%

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

      12. distribute-rgt-neg-in 100.0%

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

      13. metadata-eval 100.0%

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

      14. metadata-eval 100.0%

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

      15. fma-def 100.0%

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

      16. metadata-eval 100.0%

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

      17. neg-sub0 100.0%

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

      18. exp-diff 100.0%

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

   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 im around 0 94.5%

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

   5. Taylor expanded in im around inf 94.5%

      \[\leadsto \color{blue}{-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 94.5%

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

      2. *-commutative 94.5%

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

      3. associate-*l* 94.5%

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

   7. Simplified 94.5%

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

   8. Taylor expanded in re around 0 67.4%

      \[\leadsto {im}^{5} \cdot \color{blue}{-0.008333333333333333} \]

   if -2.34999999999999994e36 < im < 1.15000000000000001e58

   1. Initial program 19.8%

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

      1. cos-neg 19.8%

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

      2. *-commutative 19.8%

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

      3. associate-*l* 19.8%

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

      4. sub-neg 19.8%

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

      5. neg-sub0 19.8%

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

      6. +-commutative 19.8%

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

      7. distribute-lft-in 19.8%

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

      8. cos-neg 19.8%

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

      9. distribute-lft-in 19.8%

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

      10. distribute-rgt-in 19.8%

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

      11. distribute-lft-neg-out 19.8%

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

      12. distribute-rgt-neg-in 19.8%

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

      13. metadata-eval 19.8%

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

      14. metadata-eval 19.8%

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

      15. fma-def 19.8%

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

      16. metadata-eval 19.8%

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

      17. neg-sub0 19.8%

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

      18. exp-diff 19.8%

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

   3. Simplified 19.8%

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

   4. Taylor expanded in im around 0 87.3%

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

      1. associate-*r* 87.3%

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

      2. neg-mul-1 87.3%

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

   6. Simplified 87.3%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.35 \cdot 10^{+36} \lor \neg \left(im \leq 1.15 \cdot 10^{+58}\right):\\ \;\;\;\;-0.008333333333333333 \cdot {im}^{5}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \]

Alternative 14: 31.9% accurate, 34.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.38 \cdot 10^{+153}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;27 + re \cdot \left(re \cdot -13.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.38e+153) (- im) (+ 27.0 (* re (* re -13.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.38e+153) {
		tmp = -im;
	} else {
		tmp = 27.0 + (re * (re * -13.5));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 1.38d+153) then
        tmp = -im
    else
        tmp = 27.0d0 + (re * (re * (-13.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.38e+153) {
		tmp = -im;
	} else {
		tmp = 27.0 + (re * (re * -13.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.38e+153:
		tmp = -im
	else:
		tmp = 27.0 + (re * (re * -13.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.38e+153)
		tmp = Float64(-im);
	else
		tmp = Float64(27.0 + Float64(re * Float64(re * -13.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.38e+153)
		tmp = -im;
	else
		tmp = 27.0 + (re * (re * -13.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.38e+153], (-im), N[(27.0 + N[(re * N[(re * -13.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.38e153

   1. Initial program 53.1%

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

      1. cos-neg 53.1%

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

      2. *-commutative 53.1%

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

      3. associate-*l* 53.1%

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

      4. sub-neg 53.1%

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

      5. neg-sub0 53.1%

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

      6. +-commutative 53.1%

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

      7. distribute-lft-in 53.1%

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

      8. cos-neg 53.1%

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

      9. distribute-lft-in 53.1%

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

      10. distribute-rgt-in 53.1%

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

      11. distribute-lft-neg-out 53.1%

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

      12. distribute-rgt-neg-in 53.1%

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

      13. metadata-eval 53.1%

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

      14. metadata-eval 53.1%

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

      15. fma-def 53.1%

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

      16. metadata-eval 53.1%

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

      17. neg-sub0 53.1%

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

      18. exp-diff 53.1%

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

   3. Simplified 53.1%

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

   4. Taylor expanded in im around 0 91.0%

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

   5. Taylor expanded in im around inf 90.4%

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

   6. Taylor expanded in re around 0 60.1%

      \[\leadsto \color{blue}{-1 \cdot im + -0.008333333333333333 \cdot {im}^{5}} \]

   7. Taylor expanded in im around 0 32.5%

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

      1. mul-1-neg 32.5%

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

   9. Simplified 32.5%

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

   if 1.38e153 < re

   1. Initial program 46.5%

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

      1. cos-neg 46.5%

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

      2. *-commutative 46.5%

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

      3. associate-*l* 46.5%

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

      4. sub-neg 46.5%

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

      5. neg-sub0 46.5%

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

      6. +-commutative 46.5%

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

      7. distribute-lft-in 46.5%

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

      8. cos-neg 46.5%

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

      9. distribute-lft-in 46.5%

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

      10. distribute-rgt-in 46.5%

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

      11. distribute-lft-neg-out 46.5%

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

      12. distribute-rgt-neg-in 46.5%

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

      13. metadata-eval 46.5%

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

      14. metadata-eval 46.5%

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

      15. fma-def 46.5%

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

      16. metadata-eval 46.5%

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

      17. neg-sub0 46.5%

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

      18. exp-diff 46.4%

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

   3. Simplified 46.4%

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

   5. Applied egg-rr 2.8%

      \[\leadsto \cos re \cdot \color{blue}{27} \]

   6. Taylor expanded in re around 0 16.1%

      \[\leadsto \color{blue}{27 + -13.5 \cdot {re}^{2}} \]
   7. Step-by-step derivation

      1. *-commutative 16.1%

         \[\leadsto 27 + \color{blue}{{re}^{2} \cdot -13.5} \]

      2. unpow2 16.1%

         \[\leadsto 27 + \color{blue}{\left(re \cdot re\right)} \cdot -13.5 \]

   8. Simplified 16.1%

      \[\leadsto \color{blue}{27 + \left(re \cdot re\right) \cdot -13.5} \]

   9. Taylor expanded in re around 0 16.1%

      \[\leadsto 27 + \color{blue}{-13.5 \cdot {re}^{2}} \]
   10. Step-by-step derivation

       1. *-commutative 16.1%

          \[\leadsto 27 + \color{blue}{{re}^{2} \cdot -13.5} \]

       2. unpow2 16.1%

          \[\leadsto 27 + \color{blue}{\left(re \cdot re\right)} \cdot -13.5 \]

       3. associate-*r* 16.1%

          \[\leadsto 27 + \color{blue}{re \cdot \left(re \cdot -13.5\right)} \]

   11. Simplified 16.1%

       \[\leadsto 27 + \color{blue}{re \cdot \left(re \cdot -13.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.38 \cdot 10^{+153}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;27 + re \cdot \left(re \cdot -13.5\right)\\ \end{array} \]

Alternative 15: 29.8% 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 52.1%

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

   1. cos-neg 52.1%

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

   2. *-commutative 52.1%

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

   3. associate-*l* 52.1%

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

   4. sub-neg 52.1%

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

   5. neg-sub0 52.1%

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

   6. +-commutative 52.1%

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

   7. distribute-lft-in 52.1%

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

   8. cos-neg 52.1%

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

   9. distribute-lft-in 52.1%

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

   10. distribute-rgt-in 52.1%

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

   11. distribute-lft-neg-out 52.1%

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

   12. distribute-rgt-neg-in 52.1%

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

   13. metadata-eval 52.1%

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

   14. metadata-eval 52.1%

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

   15. fma-def 52.1%

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

   16. metadata-eval 52.1%

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

   17. neg-sub0 52.1%

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

   18. exp-diff 52.1%

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

3. Simplified 52.1%

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

4. Taylor expanded in im around 0 90.9%

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

5. Taylor expanded in im around inf 90.3%

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

6. Taylor expanded in re around 0 54.5%

   \[\leadsto \color{blue}{-1 \cdot im + -0.008333333333333333 \cdot {im}^{5}} \]

7. Taylor expanded in im around 0 29.0%

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

   1. mul-1-neg 29.0%

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

9. Simplified 29.0%

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

10. Final simplification 29.0%

    \[\leadsto -im \]

Alternative 16: 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 52.1%

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

   1. cos-neg 52.1%

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

   2. *-commutative 52.1%

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

   3. associate-*l* 52.1%

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

   4. sub-neg 52.1%

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

   5. neg-sub0 52.1%

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

   6. +-commutative 52.1%

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

   7. distribute-lft-in 52.1%

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

   8. cos-neg 52.1%

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

   9. distribute-lft-in 52.1%

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

   10. distribute-rgt-in 52.1%

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

   11. distribute-lft-neg-out 52.1%

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

   12. distribute-rgt-neg-in 52.1%

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

   13. metadata-eval 52.1%

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

   14. metadata-eval 52.1%

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

   15. fma-def 52.1%

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

   16. metadata-eval 52.1%

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

   17. neg-sub0 52.1%

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

   18. exp-diff 52.1%

       \[\leadsto \cos re \cdot \mathsf{fma}\left(e^{im}, -0.5, \color{blue}{\frac{e^{0}}{e^{im}}} \cdot 0.5\right) \]

3. Simplified 52.1%

   \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(e^{im}, -0.5, \frac{0.5}{e^{im}}\right)} \]

5. Applied egg-rr 3.0%

   \[\leadsto \cos re \cdot \color{blue}{-3} \]

6. Taylor expanded in re around 0 3.0%

   \[\leadsto \color{blue}{-3} \]

7. Final simplification 3.0%

   \[\leadsto -3 \]

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 2023283 
(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))))