math.sin on complex, imaginary part

Percentage Accurate: 53.6% → 99.1%

Time: 12.2s

Alternatives: 9

Speedup: 2.7×

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

Specification

Math

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

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: 53.6% 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, 0.5× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(im \cdot \log \left({\left(e^{\cos re}\right)}^{\left(\mathsf{fma}\left(-0.3333333333333333, {im}^{2}, -2\right)\right)}\right)\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (*
  0.5
  (*
   im
   (log (pow (exp (cos re)) (fma -0.3333333333333333 (pow im 2.0) -2.0))))))

C

double code(double re, double im) {
	return 0.5 * (im * log(pow(exp(cos(re)), fma(-0.3333333333333333, pow(im, 2.0), -2.0))));
}

Julia

function code(re, im)
	return Float64(0.5 * Float64(im * log((exp(cos(re)) ^ fma(-0.3333333333333333, (im ^ 2.0), -2.0)))))
end

Wolfram

code[re_, im_] := N[(0.5 * N[(im * N[Log[N[Power[N[Exp[N[Cos[re], $MachinePrecision]], $MachinePrecision], N[(-0.3333333333333333 * N[Power[im, 2.0], $MachinePrecision] + -2.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.5%

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

   1. /-rgt-identity 57.5%

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

   2. exp-0 57.5%

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

   3. associate-*l/ 57.5%

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

   4. cos-neg 57.5%

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

   5. associate-*l* 57.5%

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

   6. associate-*r/ 57.5%

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

   7. exp-0 57.5%

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

   8. /-rgt-identity 57.5%

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

   9. *-commutative 57.5%

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

   10. neg-sub0 57.5%

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

   11. cos-neg 57.5%

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

3. Simplified 57.5%

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

5. Taylor expanded in im around 0 82.9%

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

   1. associate-*r* 82.9%

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

   2. distribute-rgt-out 82.9%

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

   3. +-commutative 82.9%

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

   4. metadata-eval 82.9%

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

   5. sub-neg 82.9%

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

   6. *-commutative 82.9%

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

   7. fma-neg 82.9%

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

   8. metadata-eval 82.9%

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

7. Simplified 82.9%

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

   1. metadata-eval 82.9%

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

   2. fma-neg 82.9%

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

   3. add-log-exp 82.7%

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

   4. log-pow 99.8%

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

   5. fma-neg 99.8%

      \[\leadsto 0.5 \cdot \left(im \cdot \log \left({\left(e^{\cos re}\right)}^{\color{blue}{\left(\mathsf{fma}\left(-0.3333333333333333, {im}^{2}, -2\right)\right)}}\right)\right) \]

   6. metadata-eval 99.8%

      \[\leadsto 0.5 \cdot \left(im \cdot \log \left({\left(e^{\cos re}\right)}^{\left(\mathsf{fma}\left(-0.3333333333333333, {im}^{2}, \color{blue}{-2}\right)\right)}\right)\right) \]

9. Applied egg-rr 99.8%

   \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{\log \left({\left(e^{\cos re}\right)}^{\left(\mathsf{fma}\left(-0.3333333333333333, {im}^{2}, -2\right)\right)}\right)}\right) \]
10. Add Preprocessing

Alternative 2: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot \left(\cos re \cdot -2\right)\right)\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* 0.5 (log1p (expm1 (* im (* (cos re) -2.0))))))

C

double code(double re, double im) {
	return 0.5 * log1p(expm1((im * (cos(re) * -2.0))));
}

Java

public static double code(double re, double im) {
	return 0.5 * Math.log1p(Math.expm1((im * (Math.cos(re) * -2.0))));
}

Python

def code(re, im):
	return 0.5 * math.log1p(math.expm1((im * (math.cos(re) * -2.0))))

Julia

function code(re, im)
	return Float64(0.5 * log1p(expm1(Float64(im * Float64(cos(re) * -2.0)))))
end

Wolfram

code[re_, im_] := N[(0.5 * N[Log[1 + N[(Exp[N[(im * N[(N[Cos[re], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.5%

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

   1. /-rgt-identity 57.5%

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

   2. exp-0 57.5%

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

   3. associate-*l/ 57.5%

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

   4. cos-neg 57.5%

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

   5. associate-*l* 57.5%

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

   6. associate-*r/ 57.5%

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

   7. exp-0 57.5%

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

   8. /-rgt-identity 57.5%

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

   9. *-commutative 57.5%

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

   10. neg-sub0 57.5%

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

   11. cos-neg 57.5%

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

3. Simplified 57.5%

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

5. Taylor expanded in im around 0 49.0%

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

   1. log1p-expm1-u 99.5%

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

   2. *-commutative 99.5%

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

   3. associate-*l* 99.5%

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

7. Applied egg-rr 99.5%

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

8. Final simplification 99.5%

   \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot \left(\cos re \cdot -2\right)\right)\right) \]
9. Add Preprocessing

Alternative 3: 89.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 460:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(\cos re \cdot \left(-0.3333333333333333 \cdot {im}^{2} - 2\right)\right)\right)\\ \mathbf{elif}\;im \leq 5.8 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(-0.3333333333333333 \cdot {im}^{3}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 460.0)
   (* 0.5 (* im (* (cos re) (- (* -0.3333333333333333 (pow im 2.0)) 2.0))))
   (if (<= im 5.8e+102)
     (* 0.5 (log1p (expm1 (* im -2.0))))
     (* 0.5 (* (cos re) (* -0.3333333333333333 (pow im 3.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 460.0) {
		tmp = 0.5 * (im * (cos(re) * ((-0.3333333333333333 * pow(im, 2.0)) - 2.0)));
	} else if (im <= 5.8e+102) {
		tmp = 0.5 * log1p(expm1((im * -2.0)));
	} else {
		tmp = 0.5 * (cos(re) * (-0.3333333333333333 * pow(im, 3.0)));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 460.0) {
		tmp = 0.5 * (im * (Math.cos(re) * ((-0.3333333333333333 * Math.pow(im, 2.0)) - 2.0)));
	} else if (im <= 5.8e+102) {
		tmp = 0.5 * Math.log1p(Math.expm1((im * -2.0)));
	} else {
		tmp = 0.5 * (Math.cos(re) * (-0.3333333333333333 * Math.pow(im, 3.0)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 460.0:
		tmp = 0.5 * (im * (math.cos(re) * ((-0.3333333333333333 * math.pow(im, 2.0)) - 2.0)))
	elif im <= 5.8e+102:
		tmp = 0.5 * math.log1p(math.expm1((im * -2.0)))
	else:
		tmp = 0.5 * (math.cos(re) * (-0.3333333333333333 * math.pow(im, 3.0)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 460.0)
		tmp = Float64(0.5 * Float64(im * Float64(cos(re) * Float64(Float64(-0.3333333333333333 * (im ^ 2.0)) - 2.0))));
	elseif (im <= 5.8e+102)
		tmp = Float64(0.5 * log1p(expm1(Float64(im * -2.0))));
	else
		tmp = Float64(0.5 * Float64(cos(re) * Float64(-0.3333333333333333 * (im ^ 3.0))));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 460.0], N[(0.5 * N[(im * N[(N[Cos[re], $MachinePrecision] * N[(N[(-0.3333333333333333 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 5.8e+102], N[(0.5 * N[Log[1 + N[(Exp[N[(im * -2.0), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(-0.3333333333333333 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 460

   1. Initial program 45.4%

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

      1. /-rgt-identity 45.4%

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

      2. exp-0 45.4%

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

      3. associate-*l/ 45.4%

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

      4. cos-neg 45.4%

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

      5. associate-*l* 45.4%

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

      6. associate-*r/ 45.4%

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

      7. exp-0 45.4%

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

      8. /-rgt-identity 45.4%

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

      9. *-commutative 45.4%

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

      10. neg-sub0 45.4%

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

      11. cos-neg 45.4%

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

   3. Simplified 45.4%

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

   5. Taylor expanded in im around 0 87.0%

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

      1. associate-*r* 87.0%

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

      2. distribute-rgt-out 87.0%

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

      3. +-commutative 87.0%

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

      4. metadata-eval 87.0%

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

      5. sub-neg 87.0%

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

      6. *-commutative 87.0%

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

      7. fma-neg 87.0%

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

      8. metadata-eval 87.0%

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

   7. Simplified 87.0%

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

   8. Taylor expanded in im around 0 87.0%

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

   if 460 < im < 5.8000000000000005e102

   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 3.6%

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

      1. add-sqr-sqrt 1.1%

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

      2. pow2 1.1%

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

      3. *-commutative 1.1%

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

      4. associate-*l* 1.1%

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

   7. Applied egg-rr 1.1%

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

      1. log1p-expm1-u 31.6%

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

      2. unpow2 31.6%

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

      3. add-sqr-sqrt 100.0%

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

      4. associate-*r* 100.0%

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

      5. *-commutative 100.0%

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in re around 0 68.4%

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

   if 5.8000000000000005e102 < 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%

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

      1. associate-*r* 100.0%

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

      2. distribute-rgt-out 100.0%

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

      3. +-commutative 100.0%

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

      4. metadata-eval 100.0%

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

      5. sub-neg 100.0%

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

      6. *-commutative 100.0%

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

      7. fma-neg 100.0%

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

      8. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around inf 100.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.3333333333333333 \cdot \left({im}^{3} \cdot \cos re\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 100.0%

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

   10. Simplified 100.0%

       \[\leadsto 0.5 \cdot \color{blue}{\left(\left(-0.3333333333333333 \cdot {im}^{3}\right) \cdot \cos re\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 460:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(\cos re \cdot \left(-0.3333333333333333 \cdot {im}^{2} - 2\right)\right)\right)\\ \mathbf{elif}\;im \leq 5.8 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(-0.3333333333333333 \cdot {im}^{3}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 430:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;im \leq 5.8 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(-0.3333333333333333 \cdot {im}^{3}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 430.0)
   (* 0.5 (* (cos re) (* im -2.0)))
   (if (<= im 5.8e+102)
     (* 0.5 (log1p (expm1 (* im -2.0))))
     (* 0.5 (* (cos re) (* -0.3333333333333333 (pow im 3.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 430.0) {
		tmp = 0.5 * (cos(re) * (im * -2.0));
	} else if (im <= 5.8e+102) {
		tmp = 0.5 * log1p(expm1((im * -2.0)));
	} else {
		tmp = 0.5 * (cos(re) * (-0.3333333333333333 * pow(im, 3.0)));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 430.0) {
		tmp = 0.5 * (Math.cos(re) * (im * -2.0));
	} else if (im <= 5.8e+102) {
		tmp = 0.5 * Math.log1p(Math.expm1((im * -2.0)));
	} else {
		tmp = 0.5 * (Math.cos(re) * (-0.3333333333333333 * Math.pow(im, 3.0)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 430.0:
		tmp = 0.5 * (math.cos(re) * (im * -2.0))
	elif im <= 5.8e+102:
		tmp = 0.5 * math.log1p(math.expm1((im * -2.0)))
	else:
		tmp = 0.5 * (math.cos(re) * (-0.3333333333333333 * math.pow(im, 3.0)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 430.0)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im * -2.0)));
	elseif (im <= 5.8e+102)
		tmp = Float64(0.5 * log1p(expm1(Float64(im * -2.0))));
	else
		tmp = Float64(0.5 * Float64(cos(re) * Float64(-0.3333333333333333 * (im ^ 3.0))));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 430.0], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 5.8e+102], N[(0.5 * N[Log[1 + N[(Exp[N[(im * -2.0), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(-0.3333333333333333 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 430

   1. Initial program 45.4%

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

      1. /-rgt-identity 45.4%

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

      2. exp-0 45.4%

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

      3. associate-*l/ 45.4%

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

      4. cos-neg 45.4%

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

      5. associate-*l* 45.4%

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

      6. associate-*r/ 45.4%

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

      7. exp-0 45.4%

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

      8. /-rgt-identity 45.4%

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

      9. *-commutative 45.4%

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

      10. neg-sub0 45.4%

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

      11. cos-neg 45.4%

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

   3. Simplified 45.4%

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

   5. Taylor expanded in im around 0 61.4%

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

   if 430 < im < 5.8000000000000005e102

   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 3.6%

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

      1. add-sqr-sqrt 1.1%

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

      2. pow2 1.1%

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

      3. *-commutative 1.1%

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

      4. associate-*l* 1.1%

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

   7. Applied egg-rr 1.1%

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

      1. log1p-expm1-u 31.6%

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

      2. unpow2 31.6%

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

      3. add-sqr-sqrt 100.0%

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

      4. associate-*r* 100.0%

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

      5. *-commutative 100.0%

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in re around 0 68.4%

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

   if 5.8000000000000005e102 < 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%

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

      1. associate-*r* 100.0%

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

      2. distribute-rgt-out 100.0%

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

      3. +-commutative 100.0%

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

      4. metadata-eval 100.0%

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

      5. sub-neg 100.0%

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

      6. *-commutative 100.0%

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

      7. fma-neg 100.0%

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

      8. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around inf 100.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.3333333333333333 \cdot \left({im}^{3} \cdot \cos re\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 100.0%

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

   10. Simplified 100.0%

       \[\leadsto 0.5 \cdot \color{blue}{\left(\left(-0.3333333333333333 \cdot {im}^{3}\right) \cdot \cos re\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 430:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;im \leq 5.8 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(-0.3333333333333333 \cdot {im}^{3}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 420:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot -2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 420.0)
   (* 0.5 (* (cos re) (* im -2.0)))
   (* 0.5 (log1p (expm1 (* im -2.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 420.0) {
		tmp = 0.5 * (cos(re) * (im * -2.0));
	} else {
		tmp = 0.5 * log1p(expm1((im * -2.0)));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 420.0) {
		tmp = 0.5 * (Math.cos(re) * (im * -2.0));
	} else {
		tmp = 0.5 * Math.log1p(Math.expm1((im * -2.0)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 420.0:
		tmp = 0.5 * (math.cos(re) * (im * -2.0))
	else:
		tmp = 0.5 * math.log1p(math.expm1((im * -2.0)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 420.0)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im * -2.0)));
	else
		tmp = Float64(0.5 * log1p(expm1(Float64(im * -2.0))));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 420.0], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Log[1 + N[(Exp[N[(im * -2.0), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 420

   1. Initial program 45.4%

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

      1. /-rgt-identity 45.4%

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

      2. exp-0 45.4%

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

      3. associate-*l/ 45.4%

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

      4. cos-neg 45.4%

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

      5. associate-*l* 45.4%

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

      6. associate-*r/ 45.4%

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

      7. exp-0 45.4%

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

      8. /-rgt-identity 45.4%

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

      9. *-commutative 45.4%

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

      10. neg-sub0 45.4%

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

      11. cos-neg 45.4%

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

   3. Simplified 45.4%

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

   5. Taylor expanded in im around 0 61.4%

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

   if 420 < 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 5.4%

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

      1. add-sqr-sqrt 1.6%

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

      2. pow2 1.6%

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

      3. *-commutative 1.6%

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

      4. associate-*l* 1.6%

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

   7. Applied egg-rr 1.6%

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

      1. log1p-expm1-u 28.1%

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

      2. unpow2 28.1%

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

      3. add-sqr-sqrt 100.0%

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

      4. associate-*r* 100.0%

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

      5. *-commutative 100.0%

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in re around 0 71.9%

       \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{-2 \cdot im}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 420:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 29000000000000:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;im \leq 1.05 \cdot 10^{+85}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + re \cdot \left(im \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.3333333333333333 \cdot {im}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 29000000000000.0)
   (* 0.5 (* (cos re) (* im -2.0)))
   (if (<= im 1.05e+85)
     (* 0.5 (+ (* im -2.0) (* re (* im re))))
     (* 0.5 (* -0.3333333333333333 (pow im 3.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 29000000000000.0) {
		tmp = 0.5 * (cos(re) * (im * -2.0));
	} else if (im <= 1.05e+85) {
		tmp = 0.5 * ((im * -2.0) + (re * (im * re)));
	} else {
		tmp = 0.5 * (-0.3333333333333333 * pow(im, 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 29000000000000.0d0) then
        tmp = 0.5d0 * (cos(re) * (im * (-2.0d0)))
    else if (im <= 1.05d+85) then
        tmp = 0.5d0 * ((im * (-2.0d0)) + (re * (im * re)))
    else
        tmp = 0.5d0 * ((-0.3333333333333333d0) * (im ** 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 29000000000000.0) {
		tmp = 0.5 * (Math.cos(re) * (im * -2.0));
	} else if (im <= 1.05e+85) {
		tmp = 0.5 * ((im * -2.0) + (re * (im * re)));
	} else {
		tmp = 0.5 * (-0.3333333333333333 * Math.pow(im, 3.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 29000000000000.0:
		tmp = 0.5 * (math.cos(re) * (im * -2.0))
	elif im <= 1.05e+85:
		tmp = 0.5 * ((im * -2.0) + (re * (im * re)))
	else:
		tmp = 0.5 * (-0.3333333333333333 * math.pow(im, 3.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 29000000000000.0)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im * -2.0)));
	elseif (im <= 1.05e+85)
		tmp = Float64(0.5 * Float64(Float64(im * -2.0) + Float64(re * Float64(im * re))));
	else
		tmp = Float64(0.5 * Float64(-0.3333333333333333 * (im ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 29000000000000.0)
		tmp = 0.5 * (cos(re) * (im * -2.0));
	elseif (im <= 1.05e+85)
		tmp = 0.5 * ((im * -2.0) + (re * (im * re)));
	else
		tmp = 0.5 * (-0.3333333333333333 * (im ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 29000000000000.0], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.05e+85], N[(0.5 * N[(N[(im * -2.0), $MachinePrecision] + N[(re * N[(im * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(-0.3333333333333333 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 2.9e13

   1. Initial program 45.9%

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

      1. /-rgt-identity 45.9%

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

      2. exp-0 45.9%

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

      3. associate-*l/ 45.9%

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

      4. cos-neg 45.9%

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

      5. associate-*l* 45.9%

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

      6. associate-*r/ 45.9%

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

      7. exp-0 45.9%

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

      8. /-rgt-identity 45.9%

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

      9. *-commutative 45.9%

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

      10. neg-sub0 45.9%

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

      11. cos-neg 45.9%

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

   3. Simplified 45.9%

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

   5. Taylor expanded in im around 0 60.9%

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

   if 2.9e13 < im < 1.05000000000000005e85

   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 3.5%

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

   6. Taylor expanded in re around 0 25.8%

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

      1. add-sqr-sqrt 25.8%

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

      2. pow2 25.8%

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

      3. *-commutative 25.8%

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

      4. sqrt-prod 25.8%

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

      5. sqrt-pow1 25.8%

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

      6. metadata-eval 25.8%

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

      7. pow1 25.8%

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

   8. Applied egg-rr 25.8%

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

      1. unpow2 25.8%

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

      2. *-commutative 25.8%

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

      3. *-commutative 25.8%

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

      4. swap-sqr 25.8%

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

      5. add-sqr-sqrt 25.8%

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

      6. associate-*r* 25.8%

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

   10. Applied egg-rr 25.8%

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

   if 1.05000000000000005e85 < 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 91.3%

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

      1. associate-*r* 91.3%

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

      2. distribute-rgt-out 91.3%

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

      3. +-commutative 91.3%

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

      4. metadata-eval 91.3%

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

      5. sub-neg 91.3%

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

      6. *-commutative 91.3%

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

      7. fma-neg 91.3%

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

      8. metadata-eval 91.3%

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

   7. Simplified 91.3%

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

   8. Taylor expanded in re around 0 67.5%

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

   9. Taylor expanded in im around inf 67.5%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.3333333333333333 \cdot {im}^{3}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 29000000000000:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;im \leq 1.05 \cdot 10^{+85}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + re \cdot \left(im \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.3333333333333333 \cdot {im}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 42.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 29000000000000:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2\right)\\ \mathbf{elif}\;im \leq 9 \cdot 10^{+84}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + re \cdot \left(im \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.3333333333333333 \cdot {im}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 29000000000000.0)
   (* 0.5 (* im -2.0))
   (if (<= im 9e+84)
     (* 0.5 (+ (* im -2.0) (* re (* im re))))
     (* 0.5 (* -0.3333333333333333 (pow im 3.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 29000000000000.0) {
		tmp = 0.5 * (im * -2.0);
	} else if (im <= 9e+84) {
		tmp = 0.5 * ((im * -2.0) + (re * (im * re)));
	} else {
		tmp = 0.5 * (-0.3333333333333333 * pow(im, 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 29000000000000.0d0) then
        tmp = 0.5d0 * (im * (-2.0d0))
    else if (im <= 9d+84) then
        tmp = 0.5d0 * ((im * (-2.0d0)) + (re * (im * re)))
    else
        tmp = 0.5d0 * ((-0.3333333333333333d0) * (im ** 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 29000000000000.0) {
		tmp = 0.5 * (im * -2.0);
	} else if (im <= 9e+84) {
		tmp = 0.5 * ((im * -2.0) + (re * (im * re)));
	} else {
		tmp = 0.5 * (-0.3333333333333333 * Math.pow(im, 3.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 29000000000000.0:
		tmp = 0.5 * (im * -2.0)
	elif im <= 9e+84:
		tmp = 0.5 * ((im * -2.0) + (re * (im * re)))
	else:
		tmp = 0.5 * (-0.3333333333333333 * math.pow(im, 3.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 29000000000000.0)
		tmp = Float64(0.5 * Float64(im * -2.0));
	elseif (im <= 9e+84)
		tmp = Float64(0.5 * Float64(Float64(im * -2.0) + Float64(re * Float64(im * re))));
	else
		tmp = Float64(0.5 * Float64(-0.3333333333333333 * (im ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 29000000000000.0)
		tmp = 0.5 * (im * -2.0);
	elseif (im <= 9e+84)
		tmp = 0.5 * ((im * -2.0) + (re * (im * re)));
	else
		tmp = 0.5 * (-0.3333333333333333 * (im ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 29000000000000.0], N[(0.5 * N[(im * -2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 9e+84], N[(0.5 * N[(N[(im * -2.0), $MachinePrecision] + N[(re * N[(im * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(-0.3333333333333333 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 2.9e13

   1. Initial program 45.9%

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

      1. /-rgt-identity 45.9%

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

      2. exp-0 45.9%

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

      3. associate-*l/ 45.9%

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

      4. cos-neg 45.9%

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

      5. associate-*l* 45.9%

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

      6. associate-*r/ 45.9%

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

      7. exp-0 45.9%

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

      8. /-rgt-identity 45.9%

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

      9. *-commutative 45.9%

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

      10. neg-sub0 45.9%

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

      11. cos-neg 45.9%

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

   3. Simplified 45.9%

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

   5. Taylor expanded in im around 0 60.9%

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

   6. Taylor expanded in re around 0 33.3%

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

   if 2.9e13 < im < 8.9999999999999994e84

   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 3.5%

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

   6. Taylor expanded in re around 0 25.8%

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

      1. add-sqr-sqrt 25.8%

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

      2. pow2 25.8%

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

      3. *-commutative 25.8%

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

      4. sqrt-prod 25.8%

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

      5. sqrt-pow1 25.8%

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

      6. metadata-eval 25.8%

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

      7. pow1 25.8%

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

   8. Applied egg-rr 25.8%

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

      1. unpow2 25.8%

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

      2. *-commutative 25.8%

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

      3. *-commutative 25.8%

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

      4. swap-sqr 25.8%

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

      5. add-sqr-sqrt 25.8%

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

      6. associate-*r* 25.8%

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

   10. Applied egg-rr 25.8%

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

   if 8.9999999999999994e84 < 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 91.3%

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

      1. associate-*r* 91.3%

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

      2. distribute-rgt-out 91.3%

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

      3. +-commutative 91.3%

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

      4. metadata-eval 91.3%

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

      5. sub-neg 91.3%

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

      6. *-commutative 91.3%

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

      7. fma-neg 91.3%

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

      8. metadata-eval 91.3%

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

   7. Simplified 91.3%

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

   8. Taylor expanded in re around 0 67.5%

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

   9. Taylor expanded in im around inf 67.5%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.3333333333333333 \cdot {im}^{3}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 38.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 29000000000000:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2\right)\\ \mathbf{elif}\;im \leq 9 \cdot 10^{+84}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + re \cdot \left(im \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.3333333333333333 \cdot {im}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 39.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.01:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + re \cdot \left(im \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) -0.01)
   (* 0.5 (+ (* im -2.0) (* re (* im re))))
   (* 0.5 (* im -2.0))))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= -0.01) {
		tmp = 0.5 * ((im * -2.0) + (re * (im * re)));
	} else {
		tmp = 0.5 * (im * -2.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (cos(re) <= (-0.01d0)) then
        tmp = 0.5d0 * ((im * (-2.0d0)) + (re * (im * re)))
    else
        tmp = 0.5d0 * (im * (-2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.cos(re) <= -0.01) {
		tmp = 0.5 * ((im * -2.0) + (re * (im * re)));
	} else {
		tmp = 0.5 * (im * -2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.cos(re) <= -0.01:
		tmp = 0.5 * ((im * -2.0) + (re * (im * re)))
	else:
		tmp = 0.5 * (im * -2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= -0.01)
		tmp = Float64(0.5 * Float64(Float64(im * -2.0) + Float64(re * Float64(im * re))));
	else
		tmp = Float64(0.5 * Float64(im * -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (cos(re) <= -0.01)
		tmp = 0.5 * ((im * -2.0) + (re * (im * re)));
	else
		tmp = 0.5 * (im * -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.01], N[(0.5 * N[(N[(im * -2.0), $MachinePrecision] + N[(re * N[(im * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * -2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 57.2%

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

      1. /-rgt-identity 57.2%

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

      2. exp-0 57.2%

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

      3. associate-*l/ 57.2%

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

      4. cos-neg 57.2%

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

      5. associate-*l* 57.2%

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

      6. associate-*r/ 57.2%

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

      7. exp-0 57.2%

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

      8. /-rgt-identity 57.2%

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

      9. *-commutative 57.2%

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

      10. neg-sub0 57.2%

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

      11. cos-neg 57.2%

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

   3. Simplified 57.2%

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

   5. Taylor expanded in im around 0 48.7%

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

   6. Taylor expanded in re around 0 46.7%

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

      1. add-sqr-sqrt 18.5%

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

      2. pow2 18.5%

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

      3. *-commutative 18.5%

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

      4. sqrt-prod 18.5%

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

      5. sqrt-pow1 18.5%

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

      6. metadata-eval 18.5%

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

      7. pow1 18.5%

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

   8. Applied egg-rr 18.5%

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

      1. unpow2 18.5%

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

      2. *-commutative 18.5%

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

      3. *-commutative 18.5%

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

      4. swap-sqr 18.5%

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

      5. add-sqr-sqrt 46.7%

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

      6. associate-*r* 46.8%

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

   10. Applied egg-rr 46.8%

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

   if -0.0100000000000000002 < (cos.f64 re)

   1. Initial program 57.7%

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

      1. /-rgt-identity 57.7%

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

      2. exp-0 57.7%

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

      3. associate-*l/ 57.7%

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

      4. cos-neg 57.7%

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

      5. associate-*l* 57.7%

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

      6. associate-*r/ 57.7%

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

      7. exp-0 57.7%

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

      8. /-rgt-identity 57.7%

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

      9. *-commutative 57.7%

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

      10. neg-sub0 57.7%

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

      11. cos-neg 57.7%

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

   3. Simplified 57.7%

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

   5. Taylor expanded in im around 0 49.0%

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

   6. Taylor expanded in re around 0 36.4%

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

4. Final simplification 39.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq -0.01:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + re \cdot \left(im \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 29.9% 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 57.5%

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

   1. /-rgt-identity 57.5%

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

   2. exp-0 57.5%

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

   3. associate-*l/ 57.5%

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

   4. cos-neg 57.5%

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

   5. associate-*l* 57.5%

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

   6. associate-*r/ 57.5%

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

   7. exp-0 57.5%

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

   8. /-rgt-identity 57.5%

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

   9. *-commutative 57.5%

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

   10. neg-sub0 57.5%

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

   11. cos-neg 57.5%

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

3. Simplified 57.5%

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

5. Taylor expanded in im around 0 49.0%

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

6. Taylor expanded in re around 0 27.0%

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

7. Final simplification 27.0%

   \[\leadsto 0.5 \cdot \left(im \cdot -2\right) \]
8. 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 2024087 
(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))))