math.sin on complex, imaginary part

Percentage Accurate: 55.6% → 99.4%

Time: 12.0s

Alternatives: 8

Speedup: 2.8×

• 51.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.6% 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: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(im \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos re \cdot \mathsf{fma}\left({im}^{2}, \mathsf{fma}\left({im}^{2}, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (*
  0.5
  (*
   im
   (log1p
    (expm1
     (*
      (cos re)
      (fma
       (pow im 2.0)
       (fma (pow im 2.0) -0.016666666666666666 -0.3333333333333333)
       -2.0)))))))

C

double code(double re, double im) {
	return 0.5 * (im * log1p(expm1((cos(re) * fma(pow(im, 2.0), fma(pow(im, 2.0), -0.016666666666666666, -0.3333333333333333), -2.0)))));
}

Julia

function code(re, im)
	return Float64(0.5 * Float64(im * log1p(expm1(Float64(cos(re) * fma((im ^ 2.0), fma((im ^ 2.0), -0.016666666666666666, -0.3333333333333333), -2.0))))))
end

Wolfram

code[re_, im_] := N[(0.5 * N[(im * N[Log[1 + N[(Exp[N[(N[Cos[re], $MachinePrecision] * N[(N[Power[im, 2.0], $MachinePrecision] * N[(N[Power[im, 2.0], $MachinePrecision] * -0.016666666666666666 + -0.3333333333333333), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.8%

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

   1. /-rgt-identity 53.8%

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

   2. exp-0 53.8%

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

   3. associate-*l/ 53.8%

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

   4. cos-neg 53.8%

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

   5. associate-*l* 53.8%

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

   6. associate-*r/ 53.8%

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

   7. exp-0 53.8%

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

   8. /-rgt-identity 53.8%

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

   9. *-commutative 53.8%

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

   10. neg-sub0 53.8%

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

   11. cos-neg 53.8%

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

3. Simplified 53.8%

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

5. Taylor expanded in im around 0 92.1%

   \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-2 \cdot \cos re + {im}^{2} \cdot \left(-0.3333333333333333 \cdot \cos re + -0.016666666666666666 \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative 92.1%

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

   2. *-commutative 92.1%

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

   3. associate-*r* 92.1%

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

   4. distribute-rgt-out 92.1%

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

   5. +-commutative 92.1%

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

   6. metadata-eval 92.1%

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

   7. sub-neg 92.1%

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

   8. associate-*l* 92.1%

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

   9. *-commutative 92.1%

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

   10. distribute-lft-out 92.1%

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

   11. +-commutative 92.1%

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

   12. fma-define 92.1%

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

7. Simplified 92.1%

   \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(\cos re \cdot \mathsf{fma}\left({im}^{2}, \mathsf{fma}\left({im}^{2}, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right)\right)} \]
8. Step-by-step derivation

   1. log1p-expm1-u 99.5%

      \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos re \cdot \mathsf{fma}\left({im}^{2}, \mathsf{fma}\left({im}^{2}, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right)\right)}\right) \]

9. Applied egg-rr 99.5%

   \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos re \cdot \mathsf{fma}\left({im}^{2}, \mathsf{fma}\left({im}^{2}, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right)\right)}\right) \]
10. Add Preprocessing

Alternative 2: 99.1% 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 53.8%

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

   1. /-rgt-identity 53.8%

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

   2. exp-0 53.8%

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

   3. associate-*l/ 53.8%

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

   4. cos-neg 53.8%

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

   5. associate-*l* 53.8%

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

   6. associate-*r/ 53.8%

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

   7. exp-0 53.8%

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

   8. /-rgt-identity 53.8%

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

   9. *-commutative 53.8%

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

   10. neg-sub0 53.8%

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

   11. cos-neg 53.8%

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

3. Simplified 53.8%

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

5. Taylor expanded in im around 0 52.7%

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

   1. log1p-expm1-u 99.3%

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

   2. *-commutative 99.3%

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

   3. associate-*l* 99.3%

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

7. Applied egg-rr 99.3%

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

8. Final simplification 99.3%

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

Alternative 3: 90.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 480:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(\cos re \cdot \left({im}^{2} \cdot -0.3333333333333333 - 2\right)\right)\right)\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{5} \cdot -0.008333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 480.0)
   (* 0.5 (* im (* (cos re) (- (* (pow im 2.0) -0.3333333333333333) 2.0))))
   (if (<= im 4.5e+61)
     (* 0.5 (log1p (expm1 (* im -2.0))))
     (* (cos re) (* (pow im 5.0) -0.008333333333333333)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 480.0) {
		tmp = 0.5 * (im * (cos(re) * ((pow(im, 2.0) * -0.3333333333333333) - 2.0)));
	} else if (im <= 4.5e+61) {
		tmp = 0.5 * log1p(expm1((im * -2.0)));
	} else {
		tmp = cos(re) * (pow(im, 5.0) * -0.008333333333333333);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 480.0) {
		tmp = 0.5 * (im * (Math.cos(re) * ((Math.pow(im, 2.0) * -0.3333333333333333) - 2.0)));
	} else if (im <= 4.5e+61) {
		tmp = 0.5 * Math.log1p(Math.expm1((im * -2.0)));
	} else {
		tmp = Math.cos(re) * (Math.pow(im, 5.0) * -0.008333333333333333);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 480.0:
		tmp = 0.5 * (im * (math.cos(re) * ((math.pow(im, 2.0) * -0.3333333333333333) - 2.0)))
	elif im <= 4.5e+61:
		tmp = 0.5 * math.log1p(math.expm1((im * -2.0)))
	else:
		tmp = math.cos(re) * (math.pow(im, 5.0) * -0.008333333333333333)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 480.0)
		tmp = Float64(0.5 * Float64(im * Float64(cos(re) * Float64(Float64((im ^ 2.0) * -0.3333333333333333) - 2.0))));
	elseif (im <= 4.5e+61)
		tmp = Float64(0.5 * log1p(expm1(Float64(im * -2.0))));
	else
		tmp = Float64(cos(re) * Float64((im ^ 5.0) * -0.008333333333333333));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 480.0], N[(0.5 * N[(im * N[(N[Cos[re], $MachinePrecision] * N[(N[(N[Power[im, 2.0], $MachinePrecision] * -0.3333333333333333), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 4.5e+61], N[(0.5 * N[Log[1 + N[(Exp[N[(im * -2.0), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(N[Power[im, 5.0], $MachinePrecision] * -0.008333333333333333), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 480

   1. Initial program 39.1%

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

      1. /-rgt-identity 39.1%

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

      2. exp-0 39.1%

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

      3. associate-*l/ 39.1%

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

      4. cos-neg 39.1%

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

      5. associate-*l* 39.1%

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

      6. associate-*r/ 39.1%

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

      7. exp-0 39.1%

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

      8. /-rgt-identity 39.1%

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

      9. *-commutative 39.1%

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

      10. neg-sub0 39.1%

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

      11. cos-neg 39.1%

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

   3. Simplified 39.1%

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

   5. Taylor expanded in im around 0 95.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-2 \cdot \cos re + {im}^{2} \cdot \left(-0.3333333333333333 \cdot \cos re + -0.016666666666666666 \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 95.0%

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

      2. *-commutative 95.0%

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

      3. associate-*r* 95.0%

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

      4. distribute-rgt-out 95.0%

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

      5. +-commutative 95.0%

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

      6. metadata-eval 95.0%

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

      7. sub-neg 95.0%

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

      8. associate-*l* 95.0%

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

      9. *-commutative 95.0%

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

      10. distribute-lft-out 95.0%

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

      11. +-commutative 95.0%

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

      12. fma-define 95.0%

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

   7. Simplified 95.0%

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

   8. Taylor expanded in im around 0 92.0%

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

   if 480 < im < 4.5e61

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

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

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

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

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 3.3%

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

      1. add-sqr-sqrt 0.9%

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

      2. pow2 0.9%

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

      3. *-commutative 0.9%

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

      4. associate-*l* 0.9%

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

   7. Applied egg-rr 0.9%

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

      1. log1p-expm1-u 27.3%

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

      2. unpow2 27.3%

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

      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. *-commutative 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-*l* 100.0%

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

      7. *-commutative 100.0%

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

   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 72.7%

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

   if 4.5e61 < im

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

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

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

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

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-2 \cdot \cos re + {im}^{2} \cdot \left(-0.3333333333333333 \cdot \cos re + -0.016666666666666666 \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. distribute-rgt-out 100.0%

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

      5. +-commutative 100.0%

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

      6. metadata-eval 100.0%

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

      7. sub-neg 100.0%

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

      8. associate-*l* 100.0%

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

      9. *-commutative 100.0%

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

      10. distribute-lft-out 100.0%

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

      11. +-commutative 100.0%

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

      12. fma-define 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around inf 100.0%

      \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{\left(-0.016666666666666666 \cdot \left({im}^{4} \cdot \cos re\right)\right)}\right) \]
   9. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{\left(\left(-0.016666666666666666 \cdot {im}^{4}\right) \cdot \cos re\right)}\right) \]

   10. Simplified 100.0%

       \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{\left(\left(-0.016666666666666666 \cdot {im}^{4}\right) \cdot \cos re\right)}\right) \]

   11. Taylor expanded in im around 0 100.0%

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

       1. associate-*r* 100.0%

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

       2. *-commutative 100.0%

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

   13. Simplified 100.0%

       \[\leadsto \color{blue}{\left({im}^{5} \cdot -0.008333333333333333\right) \cdot \cos re} \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 480:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(\cos re \cdot \left({im}^{2} \cdot -0.3333333333333333 - 2\right)\right)\right)\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{5} \cdot -0.008333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 410:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{5} \cdot -0.008333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 410.0)
   (* 0.5 (* (cos re) (* im -2.0)))
   (if (<= im 4.5e+61)
     (* 0.5 (log1p (expm1 (* im -2.0))))
     (* (cos re) (* (pow im 5.0) -0.008333333333333333)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 410.0) {
		tmp = 0.5 * (cos(re) * (im * -2.0));
	} else if (im <= 4.5e+61) {
		tmp = 0.5 * log1p(expm1((im * -2.0)));
	} else {
		tmp = cos(re) * (pow(im, 5.0) * -0.008333333333333333);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 410.0) {
		tmp = 0.5 * (Math.cos(re) * (im * -2.0));
	} else if (im <= 4.5e+61) {
		tmp = 0.5 * Math.log1p(Math.expm1((im * -2.0)));
	} else {
		tmp = Math.cos(re) * (Math.pow(im, 5.0) * -0.008333333333333333);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 410.0:
		tmp = 0.5 * (math.cos(re) * (im * -2.0))
	elif im <= 4.5e+61:
		tmp = 0.5 * math.log1p(math.expm1((im * -2.0)))
	else:
		tmp = math.cos(re) * (math.pow(im, 5.0) * -0.008333333333333333)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 410.0)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im * -2.0)));
	elseif (im <= 4.5e+61)
		tmp = Float64(0.5 * log1p(expm1(Float64(im * -2.0))));
	else
		tmp = Float64(cos(re) * Float64((im ^ 5.0) * -0.008333333333333333));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 410.0], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 4.5e+61], N[(0.5 * N[Log[1 + N[(Exp[N[(im * -2.0), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(N[Power[im, 5.0], $MachinePrecision] * -0.008333333333333333), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 410

   1. Initial program 39.1%

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

      1. /-rgt-identity 39.1%

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

      2. exp-0 39.1%

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

      3. associate-*l/ 39.1%

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

      4. cos-neg 39.1%

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

      5. associate-*l* 39.1%

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

      6. associate-*r/ 39.1%

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

      7. exp-0 39.1%

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

      8. /-rgt-identity 39.1%

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

      9. *-commutative 39.1%

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

      10. neg-sub0 39.1%

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

      11. cos-neg 39.1%

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

   3. Simplified 39.1%

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

   5. Taylor expanded in im around 0 67.8%

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

   if 410 < im < 4.5e61

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

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

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

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

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 3.3%

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

      1. add-sqr-sqrt 0.9%

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

      2. pow2 0.9%

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

      3. *-commutative 0.9%

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

      4. associate-*l* 0.9%

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

   7. Applied egg-rr 0.9%

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

      1. log1p-expm1-u 27.3%

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

      2. unpow2 27.3%

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

      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. *-commutative 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-*l* 100.0%

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

      7. *-commutative 100.0%

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

   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 72.7%

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

   if 4.5e61 < im

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

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

      2. exp-0 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-0 100.0%

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

      8. /-rgt-identity 100.0%

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

      9. *-commutative 100.0%

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

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 100.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-2 \cdot \cos re + {im}^{2} \cdot \left(-0.3333333333333333 \cdot \cos re + -0.016666666666666666 \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. distribute-rgt-out 100.0%

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

      5. +-commutative 100.0%

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

      6. metadata-eval 100.0%

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

      7. sub-neg 100.0%

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

      8. associate-*l* 100.0%

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

      9. *-commutative 100.0%

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

      10. distribute-lft-out 100.0%

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

      11. +-commutative 100.0%

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

      12. fma-define 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around inf 100.0%

      \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{\left(-0.016666666666666666 \cdot \left({im}^{4} \cdot \cos re\right)\right)}\right) \]
   9. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{\left(\left(-0.016666666666666666 \cdot {im}^{4}\right) \cdot \cos re\right)}\right) \]

   10. Simplified 100.0%

       \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{\left(\left(-0.016666666666666666 \cdot {im}^{4}\right) \cdot \cos re\right)}\right) \]

   11. Taylor expanded in im around 0 100.0%

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

       1. associate-*r* 100.0%

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

       2. *-commutative 100.0%

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

   13. Simplified 100.0%

       \[\leadsto \color{blue}{\left({im}^{5} \cdot -0.008333333333333333\right) \cdot \cos re} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 410:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{5} \cdot -0.008333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 480:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot -2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 480.0)
   (* 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 <= 480.0) {
		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 <= 480.0) {
		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 <= 480.0:
		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 <= 480.0)
		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, 480.0], 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 < 480

   1. Initial program 39.1%

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

      1. /-rgt-identity 39.1%

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

      2. exp-0 39.1%

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

      3. associate-*l/ 39.1%

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

      4. cos-neg 39.1%

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

      5. associate-*l* 39.1%

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

      6. associate-*r/ 39.1%

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

      7. exp-0 39.1%

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

      8. /-rgt-identity 39.1%

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

      9. *-commutative 39.1%

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

      10. neg-sub0 39.1%

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

      11. cos-neg 39.1%

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

   3. Simplified 39.1%

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

   5. Taylor expanded in im around 0 67.8%

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

   if 480 < im

   1. Initial program 100.0%

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

      1. /-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 5.3%

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

      1. add-sqr-sqrt 1.4%

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

      2. pow2 1.4%

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

      3. *-commutative 1.4%

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

      4. associate-*l* 1.4%

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

   7. Applied egg-rr 1.4%

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

      1. log1p-expm1-u 24.2%

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

      2. unpow2 24.2%

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

      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. *-commutative 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-*l* 100.0%

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

      7. *-commutative 100.0%

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

   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 75.8%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 480:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 5.6 \cdot 10^{+20}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{5} \cdot -0.008333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 5.6e+20)
   (* 0.5 (* (cos re) (* im -2.0)))
   (* (pow im 5.0) -0.008333333333333333)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 5.6e+20) {
		tmp = 0.5 * (cos(re) * (im * -2.0));
	} else {
		tmp = pow(im, 5.0) * -0.008333333333333333;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 5.6d+20) then
        tmp = 0.5d0 * (cos(re) * (im * (-2.0d0)))
    else
        tmp = (im ** 5.0d0) * (-0.008333333333333333d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 5.6e+20) {
		tmp = 0.5 * (Math.cos(re) * (im * -2.0));
	} else {
		tmp = Math.pow(im, 5.0) * -0.008333333333333333;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 5.6e+20:
		tmp = 0.5 * (math.cos(re) * (im * -2.0))
	else:
		tmp = math.pow(im, 5.0) * -0.008333333333333333
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 5.6e+20)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im * -2.0)));
	else
		tmp = Float64((im ^ 5.0) * -0.008333333333333333);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 5.6e+20)
		tmp = 0.5 * (cos(re) * (im * -2.0));
	else
		tmp = (im ^ 5.0) * -0.008333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 5.6e+20], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[im, 5.0], $MachinePrecision] * -0.008333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 5.6e20

   1. Initial program 40.9%

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

      1. /-rgt-identity 40.9%

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

      2. exp-0 40.9%

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

      3. associate-*l/ 40.9%

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

      4. cos-neg 40.9%

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

      5. associate-*l* 40.9%

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

      6. associate-*r/ 40.9%

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

      7. exp-0 40.9%

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

      8. /-rgt-identity 40.9%

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

      9. *-commutative 40.9%

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

      10. neg-sub0 40.9%

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

      11. cos-neg 40.9%

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

   3. Simplified 40.9%

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

   5. Taylor expanded in im around 0 65.9%

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

   if 5.6e20 < 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 91.6%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-2 \cdot \cos re + {im}^{2} \cdot \left(-0.3333333333333333 \cdot \cos re + -0.016666666666666666 \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 91.6%

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

      2. *-commutative 91.6%

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

      3. associate-*r* 91.6%

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

      4. distribute-rgt-out 91.6%

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

      5. +-commutative 91.6%

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

      6. metadata-eval 91.6%

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

      7. sub-neg 91.6%

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

      8. associate-*l* 91.6%

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

      9. *-commutative 91.6%

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

      10. distribute-lft-out 91.6%

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

      11. +-commutative 91.6%

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

      12. fma-define 91.6%

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

   7. Simplified 91.6%

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

   8. Taylor expanded in im around inf 91.6%

      \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{\left(-0.016666666666666666 \cdot \left({im}^{4} \cdot \cos re\right)\right)}\right) \]
   9. Step-by-step derivation

      1. associate-*r* 91.6%

         \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{\left(\left(-0.016666666666666666 \cdot {im}^{4}\right) \cdot \cos re\right)}\right) \]

   10. Simplified 91.6%

       \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{\left(\left(-0.016666666666666666 \cdot {im}^{4}\right) \cdot \cos re\right)}\right) \]

   11. Taylor expanded in re around 0 70.1%

       \[\leadsto \color{blue}{-0.008333333333333333 \cdot {im}^{5}} \]
   12. Step-by-step derivation

       1. *-commutative 70.1%

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

   13. Simplified 70.1%

       \[\leadsto \color{blue}{{im}^{5} \cdot -0.008333333333333333} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 5.6 \cdot 10^{+20}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{5} \cdot -0.008333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 43.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 6.5 \cdot 10^{-8}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{5} \cdot -0.008333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 6.5e-8)
   (* 0.5 (* im -2.0))
   (* (pow im 5.0) -0.008333333333333333)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 6.5e-8) {
		tmp = 0.5 * (im * -2.0);
	} else {
		tmp = pow(im, 5.0) * -0.008333333333333333;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 6.5d-8) then
        tmp = 0.5d0 * (im * (-2.0d0))
    else
        tmp = (im ** 5.0d0) * (-0.008333333333333333d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 6.5e-8) {
		tmp = 0.5 * (im * -2.0);
	} else {
		tmp = Math.pow(im, 5.0) * -0.008333333333333333;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 6.5e-8:
		tmp = 0.5 * (im * -2.0)
	else:
		tmp = math.pow(im, 5.0) * -0.008333333333333333
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 6.5e-8)
		tmp = Float64(0.5 * Float64(im * -2.0));
	else
		tmp = Float64((im ^ 5.0) * -0.008333333333333333);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 6.5e-8)
		tmp = 0.5 * (im * -2.0);
	else
		tmp = (im ^ 5.0) * -0.008333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 6.5e-8], N[(0.5 * N[(im * -2.0), $MachinePrecision]), $MachinePrecision], N[(N[Power[im, 5.0], $MachinePrecision] * -0.008333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 6.49999999999999997e-8

   1. Initial program 38.6%

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

      1. /-rgt-identity 38.6%

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

      2. exp-0 38.6%

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

      3. associate-*l/ 38.6%

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

      4. cos-neg 38.6%

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

      5. associate-*l* 38.6%

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

      6. associate-*r/ 38.6%

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

      7. exp-0 38.6%

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

      8. /-rgt-identity 38.6%

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

      9. *-commutative 38.6%

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

      10. neg-sub0 38.6%

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

      11. cos-neg 38.6%

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

   3. Simplified 38.6%

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

   5. Taylor expanded in im around 0 68.0%

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

   6. Taylor expanded in re around 0 35.6%

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

      1. *-commutative 35.6%

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

   8. Simplified 35.6%

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

   if 6.49999999999999997e-8 < im

   1. Initial program 99.4%

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

      1. /-rgt-identity 99.4%

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

      2. exp-0 99.4%

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

      3. associate-*l/ 99.4%

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

      4. cos-neg 99.4%

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

      5. associate-*l* 99.4%

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

      6. associate-*r/ 99.4%

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

      7. exp-0 99.4%

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

      8. /-rgt-identity 99.4%

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

      9. *-commutative 99.4%

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

      10. neg-sub0 99.4%

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

      11. cos-neg 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in im around 0 82.2%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-2 \cdot \cos re + {im}^{2} \cdot \left(-0.3333333333333333 \cdot \cos re + -0.016666666666666666 \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 82.2%

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

      2. *-commutative 82.2%

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

      3. associate-*r* 82.2%

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

      4. distribute-rgt-out 82.2%

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

      5. +-commutative 82.2%

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

      6. metadata-eval 82.2%

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

      7. sub-neg 82.2%

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

      8. associate-*l* 82.2%

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

      9. *-commutative 82.2%

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

      10. distribute-lft-out 82.2%

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

      11. +-commutative 82.2%

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

      12. fma-define 82.2%

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

   7. Simplified 82.2%

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

   8. Taylor expanded in im around inf 80.8%

      \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{\left(-0.016666666666666666 \cdot \left({im}^{4} \cdot \cos re\right)\right)}\right) \]
   9. Step-by-step derivation

      1. associate-*r* 80.8%

         \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{\left(\left(-0.016666666666666666 \cdot {im}^{4}\right) \cdot \cos re\right)}\right) \]

   10. Simplified 80.8%

       \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{\left(\left(-0.016666666666666666 \cdot {im}^{4}\right) \cdot \cos re\right)}\right) \]

   11. Taylor expanded in re around 0 61.7%

       \[\leadsto \color{blue}{-0.008333333333333333 \cdot {im}^{5}} \]
   12. Step-by-step derivation

       1. *-commutative 61.7%

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

   13. Simplified 61.7%

       \[\leadsto \color{blue}{{im}^{5} \cdot -0.008333333333333333} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 29.2% accurate, 61.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.8%

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

   1. /-rgt-identity 53.8%

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

   2. exp-0 53.8%

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

   3. associate-*l/ 53.8%

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

   4. cos-neg 53.8%

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

   5. associate-*l* 53.8%

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

   6. associate-*r/ 53.8%

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

   7. exp-0 53.8%

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

   8. /-rgt-identity 53.8%

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

   9. *-commutative 53.8%

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

   10. neg-sub0 53.8%

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

   11. cos-neg 53.8%

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

3. Simplified 53.8%

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

5. Taylor expanded in im around 0 52.7%

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

6. Taylor expanded in re around 0 27.7%

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

   1. *-commutative 27.7%

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

8. Simplified 27.7%

   \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot -2\right)} \]
9. 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 2024096 
(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))))