math.sin on complex, imaginary part

Percentage Accurate: 54.0% → 99.1%

Time: 11.6s

Alternatives: 10

Speedup: 2.8×

• 49.3% 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: 54.0% 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: 99.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ im \cdot \log \left(1 + \mathsf{expm1}\left(\cos re \cdot \mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)\right)\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (*
  im
  (log
   (+ 1.0 (expm1 (* (cos re) (fma (pow im 2.0) -0.16666666666666666 -1.0)))))))

C

double code(double re, double im) {
	return im * log((1.0 + expm1((cos(re) * fma(pow(im, 2.0), -0.16666666666666666, -1.0)))));
}

Julia

function code(re, im)
	return Float64(im * log(Float64(1.0 + expm1(Float64(cos(re) * fma((im ^ 2.0), -0.16666666666666666, -1.0))))))
end

Wolfram

code[re_, im_] := N[(im * N[Log[N[(1.0 + N[(Exp[N[(N[Cos[re], $MachinePrecision] * N[(N[Power[im, 2.0], $MachinePrecision] * -0.16666666666666666 + -1.0), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.0%

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

   1. /-rgt-identity 53.0%

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

   2. exp-0 53.0%

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

   3. associate-*l/ 53.0%

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

   4. cos-neg 53.0%

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

   5. associate-*l* 53.0%

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

   6. associate-*r/ 53.0%

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

   7. exp-0 53.0%

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

   8. /-rgt-identity 53.0%

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

   9. *-commutative 53.0%

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

   10. neg-sub0 53.0%

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

   11. cos-neg 53.0%

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

3. Simplified 53.0%

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

5. Taylor expanded in im around 0 92.2%

   \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-2 \cdot \cos re + {im}^{2} \cdot \left(-0.3333333333333333 \cdot \cos re + {im}^{2} \cdot \left(-0.016666666666666666 \cdot \cos re + -0.0003968253968253968 \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)\right)\right)} \]

6. Taylor expanded in im around 0 86.1%

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

   1. associate-*r* 86.1%

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

   2. distribute-rgt-out 86.1%

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

   3. *-commutative 86.1%

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

8. Simplified 86.1%

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

   1. log1p-expm1-u 98.6%

      \[\leadsto im \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos re \cdot \left(-1 + {im}^{2} \cdot -0.16666666666666666\right)\right)\right)} \]

   2. log1p-undefine 98.6%

      \[\leadsto im \cdot \color{blue}{\log \left(1 + \mathsf{expm1}\left(\cos re \cdot \left(-1 + {im}^{2} \cdot -0.16666666666666666\right)\right)\right)} \]

   3. +-commutative 98.6%

      \[\leadsto im \cdot \log \left(1 + \mathsf{expm1}\left(\cos re \cdot \color{blue}{\left({im}^{2} \cdot -0.16666666666666666 + -1\right)}\right)\right) \]

   4. fma-define 98.6%

      \[\leadsto im \cdot \log \left(1 + \mathsf{expm1}\left(\cos re \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)}\right)\right) \]

10. Applied egg-rr 98.6%

    \[\leadsto im \cdot \color{blue}{\log \left(1 + \mathsf{expm1}\left(\cos re \cdot \mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)\right)\right)} \]

11. Final simplification 98.6%

    \[\leadsto im \cdot \log \left(1 + \mathsf{expm1}\left(\cos re \cdot \mathsf{fma}\left({im}^{2}, -0.16666666666666666, -1\right)\right)\right) \]
12. Add Preprocessing

Alternative 2: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot \left(-\cos re\right)\right)\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (log1p (expm1 (* im (- (cos re))))))

C

double code(double re, double im) {
	return log1p(expm1((im * -cos(re))));
}

Java

public static double code(double re, double im) {
	return Math.log1p(Math.expm1((im * -Math.cos(re))));
}

Python

def code(re, im):
	return math.log1p(math.expm1((im * -math.cos(re))))

Julia

function code(re, im)
	return log1p(expm1(Float64(im * Float64(-cos(re)))))
end

Wolfram

code[re_, im_] := N[Log[1 + N[(Exp[N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 53.0%

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

   1. /-rgt-identity 53.0%

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

   2. exp-0 53.0%

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

   3. associate-*l/ 53.0%

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

   4. cos-neg 53.0%

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

   5. associate-*l* 53.0%

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

   6. associate-*r/ 53.0%

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

   7. exp-0 53.0%

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

   8. /-rgt-identity 53.0%

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

   9. *-commutative 53.0%

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

   10. neg-sub0 53.0%

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

   11. cos-neg 53.0%

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

3. Simplified 53.0%

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

5. Taylor expanded in im around 0 53.7%

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

   1. log1p-expm1-u 98.5%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(0.5 \cdot \left(\left(-2 \cdot im\right) \cdot \cos re\right)\right)\right)} \]

   2. associate-*r* 98.5%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left(0.5 \cdot \left(-2 \cdot im\right)\right) \cdot \cos re}\right)\right) \]

   3. associate-*r* 98.5%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left(\left(0.5 \cdot -2\right) \cdot im\right)} \cdot \cos re\right)\right) \]

   4. metadata-eval 98.5%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\left(\color{blue}{-1} \cdot im\right) \cdot \cos re\right)\right) \]

7. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(-1 \cdot im\right) \cdot \cos re\right)\right)} \]

8. Final simplification 98.5%

   \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot \left(-\cos re\right)\right)\right) \]
9. Add Preprocessing

Alternative 3: 90.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 500:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot \left({im}^{2} \cdot -0.3333333333333333\right) + im \cdot -2\right)\right)\\ \mathbf{elif}\;im \leq 2.25 \cdot 10^{+39}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(-im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 500.0)
   (*
    0.5
    (* (cos re) (+ (* im (* (pow im 2.0) -0.3333333333333333)) (* im -2.0))))
   (if (<= im 2.25e+39)
     (log1p (expm1 (- im)))
     (* -0.0001984126984126984 (* (cos re) (pow im 7.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 500.0) {
		tmp = 0.5 * (cos(re) * ((im * (pow(im, 2.0) * -0.3333333333333333)) + (im * -2.0)));
	} else if (im <= 2.25e+39) {
		tmp = log1p(expm1(-im));
	} else {
		tmp = -0.0001984126984126984 * (cos(re) * pow(im, 7.0));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 500.0) {
		tmp = 0.5 * (Math.cos(re) * ((im * (Math.pow(im, 2.0) * -0.3333333333333333)) + (im * -2.0)));
	} else if (im <= 2.25e+39) {
		tmp = Math.log1p(Math.expm1(-im));
	} else {
		tmp = -0.0001984126984126984 * (Math.cos(re) * Math.pow(im, 7.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 500.0:
		tmp = 0.5 * (math.cos(re) * ((im * (math.pow(im, 2.0) * -0.3333333333333333)) + (im * -2.0)))
	elif im <= 2.25e+39:
		tmp = math.log1p(math.expm1(-im))
	else:
		tmp = -0.0001984126984126984 * (math.cos(re) * math.pow(im, 7.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 500.0)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(Float64(im * Float64((im ^ 2.0) * -0.3333333333333333)) + Float64(im * -2.0))));
	elseif (im <= 2.25e+39)
		tmp = log1p(expm1(Float64(-im)));
	else
		tmp = Float64(-0.0001984126984126984 * Float64(cos(re) * (im ^ 7.0)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 500.0], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(N[(im * N[(N[Power[im, 2.0], $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision] + N[(im * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.25e+39], N[Log[1 + N[(Exp[(-im)] - 1), $MachinePrecision]], $MachinePrecision], N[(-0.0001984126984126984 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 500

   1. Initial program 38.0%

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

      1. /-rgt-identity 38.0%

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

      2. exp-0 38.0%

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

      3. associate-*l/ 38.0%

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

      4. cos-neg 38.0%

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

      5. associate-*l* 38.0%

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

      6. associate-*r/ 38.0%

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

      7. exp-0 38.0%

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

      8. /-rgt-identity 38.0%

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

      9. *-commutative 38.0%

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

      10. neg-sub0 38.0%

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

      11. cos-neg 38.0%

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

   3. Simplified 38.0%

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

   5. Taylor expanded in im around 0 89.1%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(im \cdot \left(-0.3333333333333333 \cdot {im}^{2} - 2\right)\right)} \cdot \cos re\right) \]
   6. Step-by-step derivation

      1. sub-neg 89.1%

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

      2. metadata-eval 89.1%

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

      3. distribute-rgt-in 89.1%

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

      4. *-commutative 89.1%

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

      5. *-commutative 89.1%

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

   7. Applied egg-rr 89.1%

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

   if 500 < im < 2.24999999999999998e39

   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. /-rgt-identity 100.0%

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

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

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

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 3.3%

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

      1. log1p-expm1-u 100.0%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(0.5 \cdot \left(\left(-2 \cdot im\right) \cdot \cos re\right)\right)\right)} \]

      2. associate-*r* 100.0%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left(0.5 \cdot \left(-2 \cdot im\right)\right) \cdot \cos re}\right)\right) \]

      3. associate-*r* 100.0%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left(\left(0.5 \cdot -2\right) \cdot im\right)} \cdot \cos re\right)\right) \]

      4. metadata-eval 100.0%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\left(\color{blue}{-1} \cdot im\right) \cdot \cos re\right)\right) \]

   7. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(-1 \cdot im\right) \cdot \cos re\right)\right)} \]

   8. Taylor expanded in re around 0 85.7%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{e^{-1 \cdot im} - 1}\right) \]
   9. Step-by-step derivation

      1. expm1-define 85.7%

         \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(-1 \cdot im\right)}\right) \]

      2. mul-1-neg 85.7%

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

   10. Simplified 85.7%

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

   if 2.24999999999999998e39 < 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. /-rgt-identity 100.0%

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

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

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

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-2 \cdot \cos re + {im}^{2} \cdot \left(-0.3333333333333333 \cdot \cos re + {im}^{2} \cdot \left(-0.016666666666666666 \cdot \cos re + -0.0003968253968253968 \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)\right)\right)} \]

   6. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 500:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot \left({im}^{2} \cdot -0.3333333333333333\right) + im \cdot -2\right)\right)\\ \mathbf{elif}\;im \leq 2.25 \cdot 10^{+39}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(-im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 460:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 2.25 \cdot 10^{+39}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(-im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 460.0)
   (* im (- (cos re)))
   (if (<= im 2.25e+39)
     (log1p (expm1 (- im)))
     (* -0.0001984126984126984 (* (cos re) (pow im 7.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 460.0) {
		tmp = im * -cos(re);
	} else if (im <= 2.25e+39) {
		tmp = log1p(expm1(-im));
	} else {
		tmp = -0.0001984126984126984 * (cos(re) * pow(im, 7.0));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 460.0) {
		tmp = im * -Math.cos(re);
	} else if (im <= 2.25e+39) {
		tmp = Math.log1p(Math.expm1(-im));
	} else {
		tmp = -0.0001984126984126984 * (Math.cos(re) * Math.pow(im, 7.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 460.0:
		tmp = im * -math.cos(re)
	elif im <= 2.25e+39:
		tmp = math.log1p(math.expm1(-im))
	else:
		tmp = -0.0001984126984126984 * (math.cos(re) * math.pow(im, 7.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 460.0)
		tmp = Float64(im * Float64(-cos(re)));
	elseif (im <= 2.25e+39)
		tmp = log1p(expm1(Float64(-im)));
	else
		tmp = Float64(-0.0001984126984126984 * Float64(cos(re) * (im ^ 7.0)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 460.0], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im, 2.25e+39], N[Log[1 + N[(Exp[(-im)] - 1), $MachinePrecision]], $MachinePrecision], N[(-0.0001984126984126984 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 460

   1. Initial program 38.0%

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

      1. /-rgt-identity 38.0%

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

      2. exp-0 38.0%

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

      3. associate-*l/ 38.0%

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

      4. cos-neg 38.0%

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

      5. associate-*l* 38.0%

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

      6. associate-*r/ 38.0%

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

      7. exp-0 38.0%

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

      8. /-rgt-identity 38.0%

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

      9. *-commutative 38.0%

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

      10. neg-sub0 38.0%

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

      11. cos-neg 38.0%

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

   3. Simplified 38.0%

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

   5. Taylor expanded in im around 0 93.2%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-2 \cdot \cos re + {im}^{2} \cdot \left(-0.3333333333333333 \cdot \cos re + {im}^{2} \cdot \left(-0.016666666666666666 \cdot \cos re + -0.0003968253968253968 \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)\right)\right)} \]

   6. Taylor expanded in im around 0 68.7%

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

      1. associate-*r* 68.7%

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

      2. *-commutative 68.7%

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

      3. mul-1-neg 68.7%

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

   8. Simplified 68.7%

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

   if 460 < im < 2.24999999999999998e39

   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. /-rgt-identity 100.0%

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

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

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

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 3.3%

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

      1. log1p-expm1-u 100.0%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(0.5 \cdot \left(\left(-2 \cdot im\right) \cdot \cos re\right)\right)\right)} \]

      2. associate-*r* 100.0%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left(0.5 \cdot \left(-2 \cdot im\right)\right) \cdot \cos re}\right)\right) \]

      3. associate-*r* 100.0%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left(\left(0.5 \cdot -2\right) \cdot im\right)} \cdot \cos re\right)\right) \]

      4. metadata-eval 100.0%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\left(\color{blue}{-1} \cdot im\right) \cdot \cos re\right)\right) \]

   7. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(-1 \cdot im\right) \cdot \cos re\right)\right)} \]

   8. Taylor expanded in re around 0 85.7%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{e^{-1 \cdot im} - 1}\right) \]
   9. Step-by-step derivation

      1. expm1-define 85.7%

         \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(-1 \cdot im\right)}\right) \]

      2. mul-1-neg 85.7%

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

   10. Simplified 85.7%

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

   if 2.24999999999999998e39 < 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. /-rgt-identity 100.0%

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

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

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

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-2 \cdot \cos re + {im}^{2} \cdot \left(-0.3333333333333333 \cdot \cos re + {im}^{2} \cdot \left(-0.016666666666666666 \cdot \cos re + -0.0003968253968253968 \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)\right)\right)} \]

   6. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 460:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 2.25 \cdot 10^{+39}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(-im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 410:\\ \;\;\;\;im \cdot \left(\cos re \cdot \left(-1 + {im}^{2} \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;im \leq 2.25 \cdot 10^{+39}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(-im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 410.0)
   (* im (* (cos re) (+ -1.0 (* (pow im 2.0) -0.16666666666666666))))
   (if (<= im 2.25e+39)
     (log1p (expm1 (- im)))
     (* -0.0001984126984126984 (* (cos re) (pow im 7.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 410.0) {
		tmp = im * (cos(re) * (-1.0 + (pow(im, 2.0) * -0.16666666666666666)));
	} else if (im <= 2.25e+39) {
		tmp = log1p(expm1(-im));
	} else {
		tmp = -0.0001984126984126984 * (cos(re) * pow(im, 7.0));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 410.0) {
		tmp = im * (Math.cos(re) * (-1.0 + (Math.pow(im, 2.0) * -0.16666666666666666)));
	} else if (im <= 2.25e+39) {
		tmp = Math.log1p(Math.expm1(-im));
	} else {
		tmp = -0.0001984126984126984 * (Math.cos(re) * Math.pow(im, 7.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 410.0:
		tmp = im * (math.cos(re) * (-1.0 + (math.pow(im, 2.0) * -0.16666666666666666)))
	elif im <= 2.25e+39:
		tmp = math.log1p(math.expm1(-im))
	else:
		tmp = -0.0001984126984126984 * (math.cos(re) * math.pow(im, 7.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 410.0)
		tmp = Float64(im * Float64(cos(re) * Float64(-1.0 + Float64((im ^ 2.0) * -0.16666666666666666))));
	elseif (im <= 2.25e+39)
		tmp = log1p(expm1(Float64(-im)));
	else
		tmp = Float64(-0.0001984126984126984 * Float64(cos(re) * (im ^ 7.0)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 410.0], N[(im * N[(N[Cos[re], $MachinePrecision] * N[(-1.0 + N[(N[Power[im, 2.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.25e+39], N[Log[1 + N[(Exp[(-im)] - 1), $MachinePrecision]], $MachinePrecision], N[(-0.0001984126984126984 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 410

   1. Initial program 38.0%

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

      1. /-rgt-identity 38.0%

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

      2. exp-0 38.0%

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

      3. associate-*l/ 38.0%

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

      4. cos-neg 38.0%

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

      5. associate-*l* 38.0%

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

      6. associate-*r/ 38.0%

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

      7. exp-0 38.0%

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

      8. /-rgt-identity 38.0%

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

      9. *-commutative 38.0%

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

      10. neg-sub0 38.0%

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

      11. cos-neg 38.0%

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

   3. Simplified 38.0%

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

   5. Taylor expanded in im around 0 93.2%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-2 \cdot \cos re + {im}^{2} \cdot \left(-0.3333333333333333 \cdot \cos re + {im}^{2} \cdot \left(-0.016666666666666666 \cdot \cos re + -0.0003968253968253968 \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)\right)\right)} \]

   6. Taylor expanded in im around 0 89.1%

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

      1. associate-*r* 89.1%

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

      2. distribute-rgt-out 89.1%

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

      3. *-commutative 89.1%

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

   8. Simplified 89.1%

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

   if 410 < im < 2.24999999999999998e39

   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. /-rgt-identity 100.0%

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

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

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

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 3.3%

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

      1. log1p-expm1-u 100.0%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(0.5 \cdot \left(\left(-2 \cdot im\right) \cdot \cos re\right)\right)\right)} \]

      2. associate-*r* 100.0%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left(0.5 \cdot \left(-2 \cdot im\right)\right) \cdot \cos re}\right)\right) \]

      3. associate-*r* 100.0%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left(\left(0.5 \cdot -2\right) \cdot im\right)} \cdot \cos re\right)\right) \]

      4. metadata-eval 100.0%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\left(\color{blue}{-1} \cdot im\right) \cdot \cos re\right)\right) \]

   7. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(-1 \cdot im\right) \cdot \cos re\right)\right)} \]

   8. Taylor expanded in re around 0 85.7%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{e^{-1 \cdot im} - 1}\right) \]
   9. Step-by-step derivation

      1. expm1-define 85.7%

         \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(-1 \cdot im\right)}\right) \]

      2. mul-1-neg 85.7%

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

   10. Simplified 85.7%

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

   if 2.24999999999999998e39 < 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. /-rgt-identity 100.0%

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

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

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

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-2 \cdot \cos re + {im}^{2} \cdot \left(-0.3333333333333333 \cdot \cos re + {im}^{2} \cdot \left(-0.016666666666666666 \cdot \cos re + -0.0003968253968253968 \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)\right)\right)} \]

   6. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 410:\\ \;\;\;\;im \cdot \left(\cos re \cdot \left(-1 + {im}^{2} \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;im \leq 2.25 \cdot 10^{+39}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(-im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 410:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(-im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 410.0) (* im (- (cos re))) (log1p (expm1 (- im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 410.0) {
		tmp = im * -cos(re);
	} else {
		tmp = log1p(expm1(-im));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 410.0) {
		tmp = im * -Math.cos(re);
	} else {
		tmp = Math.log1p(Math.expm1(-im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 410.0:
		tmp = im * -math.cos(re)
	else:
		tmp = math.log1p(math.expm1(-im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 410.0)
		tmp = Float64(im * Float64(-cos(re)));
	else
		tmp = log1p(expm1(Float64(-im)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 410.0], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], N[Log[1 + N[(Exp[(-im)] - 1), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 410

   1. Initial program 38.0%

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

      1. /-rgt-identity 38.0%

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

      2. exp-0 38.0%

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

      3. associate-*l/ 38.0%

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

      4. cos-neg 38.0%

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

      5. associate-*l* 38.0%

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

      6. associate-*r/ 38.0%

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

      7. exp-0 38.0%

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

      8. /-rgt-identity 38.0%

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

      9. *-commutative 38.0%

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

      10. neg-sub0 38.0%

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

      11. cos-neg 38.0%

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

   3. Simplified 38.0%

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

   5. Taylor expanded in im around 0 93.2%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-2 \cdot \cos re + {im}^{2} \cdot \left(-0.3333333333333333 \cdot \cos re + {im}^{2} \cdot \left(-0.016666666666666666 \cdot \cos re + -0.0003968253968253968 \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)\right)\right)} \]

   6. Taylor expanded in im around 0 68.7%

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

      1. associate-*r* 68.7%

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

      2. *-commutative 68.7%

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

      3. mul-1-neg 68.7%

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

   8. Simplified 68.7%

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

   if 410 < 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. /-rgt-identity 100.0%

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

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

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

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 6.8%

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

      1. log1p-expm1-u 100.0%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(0.5 \cdot \left(\left(-2 \cdot im\right) \cdot \cos re\right)\right)\right)} \]

      2. associate-*r* 100.0%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left(0.5 \cdot \left(-2 \cdot im\right)\right) \cdot \cos re}\right)\right) \]

      3. associate-*r* 100.0%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left(\left(0.5 \cdot -2\right) \cdot im\right)} \cdot \cos re\right)\right) \]

      4. metadata-eval 100.0%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\left(\color{blue}{-1} \cdot im\right) \cdot \cos re\right)\right) \]

   7. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(-1 \cdot im\right) \cdot \cos re\right)\right)} \]

   8. Taylor expanded in re around 0 83.9%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{e^{-1 \cdot im} - 1}\right) \]
   9. Step-by-step derivation

      1. expm1-define 83.9%

         \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(-1 \cdot im\right)}\right) \]

      2. mul-1-neg 83.9%

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

   10. Simplified 83.9%

       \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(-im\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 410:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(-im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 44.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.00082:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot {im}^{7}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.00082) (- im) (* -0.0001984126984126984 (pow im 7.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.00082) {
		tmp = -im;
	} else {
		tmp = -0.0001984126984126984 * pow(im, 7.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.00082d0) then
        tmp = -im
    else
        tmp = (-0.0001984126984126984d0) * (im ** 7.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.00082) {
		tmp = -im;
	} else {
		tmp = -0.0001984126984126984 * Math.pow(im, 7.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.00082:
		tmp = -im
	else:
		tmp = -0.0001984126984126984 * math.pow(im, 7.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.00082)
		tmp = Float64(-im);
	else
		tmp = Float64(-0.0001984126984126984 * (im ^ 7.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.00082)
		tmp = -im;
	else
		tmp = -0.0001984126984126984 * (im ^ 7.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.00082], (-im), N[(-0.0001984126984126984 * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 8.1999999999999998e-4

   1. Initial program 37.0%

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

      1. /-rgt-identity 37.0%

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

      2. exp-0 37.0%

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

      3. associate-*l/ 37.0%

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

      4. cos-neg 37.0%

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

      5. associate-*l* 37.0%

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

      6. associate-*r/ 37.0%

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

      7. exp-0 37.0%

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

      8. /-rgt-identity 37.0%

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

      9. *-commutative 37.0%

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

      10. neg-sub0 37.0%

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

      11. cos-neg 37.0%

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

   3. Simplified 37.0%

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

   5. Taylor expanded in im around 0 69.5%

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

   6. Taylor expanded in re around 0 43.9%

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

      1. mul-1-neg 43.9%

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

   8. Simplified 43.9%

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

   if 8.1999999999999998e-4 < 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. /-rgt-identity 99.9%

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

      2. exp-0 99.9%

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

      3. associate-*l/ 99.9%

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

      4. cos-neg 99.9%

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

      5. associate-*l* 99.9%

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

      6. associate-*r/ 99.9%

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

      7. exp-0 99.9%

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

      8. /-rgt-identity 99.9%

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

      9. *-commutative 99.9%

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

      10. neg-sub0 99.9%

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

      11. cos-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in im around 0 87.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-2 \cdot \cos re + {im}^{2} \cdot \left(-0.3333333333333333 \cdot \cos re + {im}^{2} \cdot \left(-0.016666666666666666 \cdot \cos re + -0.0003968253968253968 \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)\right)\right)} \]

   6. Taylor expanded in im around inf 85.6%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)} \]

   7. Taylor expanded in re around 0 71.3%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot {im}^{7}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.00082:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot {im}^{7}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.6 \cdot 10^{+19}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot {im}^{7}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.6e+19)
   (* im (- (cos re)))
   (* -0.0001984126984126984 (pow im 7.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.6e+19) {
		tmp = im * -cos(re);
	} else {
		tmp = -0.0001984126984126984 * pow(im, 7.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 3.6d+19) then
        tmp = im * -cos(re)
    else
        tmp = (-0.0001984126984126984d0) * (im ** 7.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3.6e+19) {
		tmp = im * -Math.cos(re);
	} else {
		tmp = -0.0001984126984126984 * Math.pow(im, 7.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.6e+19:
		tmp = im * -math.cos(re)
	else:
		tmp = -0.0001984126984126984 * math.pow(im, 7.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.6e+19)
		tmp = Float64(im * Float64(-cos(re)));
	else
		tmp = Float64(-0.0001984126984126984 * (im ^ 7.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3.6e+19)
		tmp = im * -cos(re);
	else
		tmp = -0.0001984126984126984 * (im ^ 7.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.6e+19], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], N[(-0.0001984126984126984 * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 3.6e19

   1. Initial program 39.2%

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

      1. /-rgt-identity 39.2%

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

      2. exp-0 39.2%

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

      3. associate-*l/ 39.2%

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

      4. cos-neg 39.2%

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

      5. associate-*l* 39.2%

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

      6. associate-*r/ 39.2%

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

      7. exp-0 39.2%

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

      8. /-rgt-identity 39.2%

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

      9. *-commutative 39.2%

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

      10. neg-sub0 39.2%

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

      11. cos-neg 39.2%

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

   3. Simplified 39.2%

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

   5. Taylor expanded in im around 0 91.4%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-2 \cdot \cos re + {im}^{2} \cdot \left(-0.3333333333333333 \cdot \cos re + {im}^{2} \cdot \left(-0.016666666666666666 \cdot \cos re + -0.0003968253968253968 \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)\right)\right)} \]

   6. Taylor expanded in im around 0 67.4%

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

      1. associate-*r* 67.4%

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

      2. *-commutative 67.4%

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

      3. mul-1-neg 67.4%

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

   8. Simplified 67.4%

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

   if 3.6e19 < 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. /-rgt-identity 100.0%

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

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

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

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 95.1%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-2 \cdot \cos re + {im}^{2} \cdot \left(-0.3333333333333333 \cdot \cos re + {im}^{2} \cdot \left(-0.016666666666666666 \cdot \cos re + -0.0003968253968253968 \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)\right)\right)} \]

   6. Taylor expanded in im around inf 95.1%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left({im}^{7} \cdot \cos re\right)} \]

   7. Taylor expanded in re around 0 79.6%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot {im}^{7}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.6 \cdot 10^{+19}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot {im}^{7}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 29.7% accurate, 61.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.0%

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

   1. /-rgt-identity 53.0%

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

   2. exp-0 53.0%

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

   3. associate-*l/ 53.0%

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

   4. cos-neg 53.0%

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

   5. associate-*l* 53.0%

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

   6. associate-*r/ 53.0%

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

   7. exp-0 53.0%

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

   8. /-rgt-identity 53.0%

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

   9. *-commutative 53.0%

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

   10. neg-sub0 53.0%

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

   11. cos-neg 53.0%

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

3. Simplified 53.0%

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

5. Taylor expanded in im around 0 53.7%

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

6. Taylor expanded in re around 0 34.2%

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

   1. *-commutative 34.2%

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

8. Simplified 34.2%

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

9. Final simplification 34.2%

   \[\leadsto 0.5 \cdot \left(im \cdot -2\right) \]
10. Add Preprocessing

Alternative 10: 29.6% accurate, 154.5× speedup

Math

\[\begin{array}{l} \\ -im \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return -im

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := (-im)

Derivation

1. Initial program 53.0%

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

   1. /-rgt-identity 53.0%

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

   2. exp-0 53.0%

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

   3. associate-*l/ 53.0%

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

   4. cos-neg 53.0%

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

   5. associate-*l* 53.0%

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

   6. associate-*r/ 53.0%

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

   7. exp-0 53.0%

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

   8. /-rgt-identity 53.0%

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

   9. *-commutative 53.0%

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

   10. neg-sub0 53.0%

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

   11. cos-neg 53.0%

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

3. Simplified 53.0%

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

5. Taylor expanded in im around 0 53.7%

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

6. Taylor expanded in re around 0 33.9%

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

   1. mul-1-neg 33.9%

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

8. Simplified 33.9%

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

9. Final simplification 33.9%

   \[\leadsto -im \]
10. Add Preprocessing

Developer target: 99.8% accurate, 0.7× 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 2024078 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"
  :precision binary64

  :alt
  (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))))