math.sin on complex, imaginary part

Percentage Accurate: 54.4% → 99.0%

Time: 12.2s

Alternatives: 12

Speedup: 2.7×

• 49.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 50.4%

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

   1. /-rgt-identity 50.4%

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

   2. exp-0 50.4%

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

   3. associate-*l/ 50.4%

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

   4. cos-neg 50.4%

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

   5. associate-*l* 50.4%

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

   6. associate-*r/ 50.4%

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

   7. exp-0 50.4%

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

   8. /-rgt-identity 50.4%

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

   9. *-commutative 50.4%

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

   10. neg-sub0 50.4%

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

   11. cos-neg 50.4%

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

3. Simplified 50.4%

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

5. Taylor expanded in im around 0 56.3%

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

   1. log1p-expm1-u 98.9%

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

   2. *-commutative 98.9%

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

   3. associate-*l* 98.9%

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

7. Applied egg-rr 98.9%

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

8. Final simplification 98.9%

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

Alternative 2: 63.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 5.5 \cdot 10^{-7}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, -2, -0.3333333333333333 \cdot {im}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 5.5e-7)
   (* 0.5 (* (cos re) (* im -2.0)))
   (* 0.5 (fma im -2.0 (* -0.3333333333333333 (pow im 3.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 5.5e-7) {
		tmp = 0.5 * (cos(re) * (im * -2.0));
	} else {
		tmp = 0.5 * fma(im, -2.0, (-0.3333333333333333 * pow(im, 3.0)));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 5.5e-7)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im * -2.0)));
	else
		tmp = Float64(0.5 * fma(im, -2.0, Float64(-0.3333333333333333 * (im ^ 3.0))));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 5.5e-7], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * -2.0 + N[(-0.3333333333333333 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 5.5000000000000003e-7

   1. Initial program 36.6%

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

      1. /-rgt-identity 36.6%

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

      2. exp-0 36.6%

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

      3. associate-*l/ 36.6%

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

      4. cos-neg 36.6%

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

      5. associate-*l* 36.6%

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

      6. associate-*r/ 36.6%

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

      7. exp-0 36.6%

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

      8. /-rgt-identity 36.6%

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

      9. *-commutative 36.6%

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

      10. neg-sub0 36.6%

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

      11. cos-neg 36.6%

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

   3. Simplified 36.6%

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

   5. Taylor expanded in im around 0 70.0%

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

   if 5.5000000000000003e-7 < im

   1. Initial program 99.5%

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

      1. /-rgt-identity 99.5%

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

      2. exp-0 99.5%

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

      3. associate-*l/ 99.5%

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

      4. cos-neg 99.5%

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

      5. associate-*l* 99.5%

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

      6. associate-*r/ 99.5%

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

      7. exp-0 99.5%

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

      8. /-rgt-identity 99.5%

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

      9. *-commutative 99.5%

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

      10. neg-sub0 99.5%

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

      11. cos-neg 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in im around 0 61.8%

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

   6. Taylor expanded in re around 0 50.4%

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

      1. sub-neg 50.4%

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

      2. *-commutative 50.4%

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

      3. metadata-eval 50.4%

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

      4. +-commutative 50.4%

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

      5. distribute-lft-out 50.4%

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

      6. fma-define 50.4%

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

      7. associate-*r* 50.4%

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

      8. unpow2 50.4%

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

      9. cube-mult 50.4%

         \[\leadsto 0.5 \cdot \mathsf{fma}\left(im, -2, \color{blue}{{im}^{3}} \cdot -0.3333333333333333\right) \]

      10. *-commutative 50.4%

          \[\leadsto 0.5 \cdot \mathsf{fma}\left(im, -2, \color{blue}{-0.3333333333333333 \cdot {im}^{3}}\right) \]

   8. Simplified 50.4%

      \[\leadsto 0.5 \cdot \color{blue}{\mathsf{fma}\left(im, -2, -0.3333333333333333 \cdot {im}^{3}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 5.5 \cdot 10^{-7}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, -2, -0.3333333333333333 \cdot {im}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * (cos(re) * (im * (((-0.3333333333333333d0) * (im ** 2.0d0)) - 2.0d0)))
end function

Java

public static double code(double re, double im) {
	return 0.5 * (Math.cos(re) * (im * ((-0.3333333333333333 * Math.pow(im, 2.0)) - 2.0)));
}

Python

def code(re, im):
	return 0.5 * (math.cos(re) * (im * ((-0.3333333333333333 * math.pow(im, 2.0)) - 2.0)))

Julia

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

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * (cos(re) * (im * ((-0.3333333333333333 * (im ^ 2.0)) - 2.0)));
end

Wolfram

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

Derivation

1. Initial program 50.4%

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

   1. /-rgt-identity 50.4%

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

   2. exp-0 50.4%

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

   3. associate-*l/ 50.4%

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

   4. cos-neg 50.4%

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

   5. associate-*l* 50.4%

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

   6. associate-*r/ 50.4%

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

   7. exp-0 50.4%

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

   8. /-rgt-identity 50.4%

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

   9. *-commutative 50.4%

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

   10. neg-sub0 50.4%

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

   11. cos-neg 50.4%

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

3. Simplified 50.4%

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

5. Taylor expanded in im around 0 83.7%

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

6. Final simplification 83.7%

   \[\leadsto 0.5 \cdot \left(\cos re \cdot \left(im \cdot \left(-0.3333333333333333 \cdot {im}^{2} - 2\right)\right)\right) \]
7. Add Preprocessing

Alternative 4: 56.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 5.8 \cdot 10^{+36}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{6.944444444444444 \cdot 10^{-5} \cdot {im}^{10}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 5.8e+36)
   (* 0.5 (* (cos re) (* im -2.0)))
   (sqrt (* 6.944444444444444e-5 (pow im 10.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 5.8e+36) {
		tmp = 0.5 * (cos(re) * (im * -2.0));
	} else {
		tmp = sqrt((6.944444444444444e-5 * pow(im, 10.0)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 5.8d+36) then
        tmp = 0.5d0 * (cos(re) * (im * (-2.0d0)))
    else
        tmp = sqrt((6.944444444444444d-5 * (im ** 10.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 5.8e+36) {
		tmp = 0.5 * (Math.cos(re) * (im * -2.0));
	} else {
		tmp = Math.sqrt((6.944444444444444e-5 * Math.pow(im, 10.0)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 5.8e+36:
		tmp = 0.5 * (math.cos(re) * (im * -2.0))
	else:
		tmp = math.sqrt((6.944444444444444e-5 * math.pow(im, 10.0)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 5.8e+36)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im * -2.0)));
	else
		tmp = sqrt(Float64(6.944444444444444e-5 * (im ^ 10.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 5.8e+36)
		tmp = 0.5 * (cos(re) * (im * -2.0));
	else
		tmp = sqrt((6.944444444444444e-5 * (im ^ 10.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 5.8e+36], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(6.944444444444444e-5 * N[Power[im, 10.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 5.8e36

   1. Initial program 38.9%

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

      1. /-rgt-identity 38.9%

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

      2. exp-0 38.9%

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

      3. associate-*l/ 38.9%

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

      4. cos-neg 38.9%

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

      5. associate-*l* 38.9%

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

      6. associate-*r/ 38.9%

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

      7. exp-0 38.9%

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

      8. /-rgt-identity 38.9%

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

      9. *-commutative 38.9%

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

      10. neg-sub0 38.9%

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

      11. cos-neg 38.9%

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

   3. Simplified 38.9%

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

   5. Taylor expanded in im around 0 68.0%

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

   if 5.8e36 < 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 92.2%

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

      1. distribute-lft-in 92.2%

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

      2. +-commutative 92.2%

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

      3. associate-*r* 92.2%

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

      4. *-commutative 92.2%

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

      5. fma-undefine 92.2%

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

   7. Simplified 92.2%

      \[\leadsto 0.5 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left({im}^{2}, -0.016666666666666666, -0.3333333333333333\right) \cdot \cos re, {im}^{3}, \left(-2 \cdot im\right) \cdot \cos re\right)} \]

   8. Taylor expanded in im around inf 92.2%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.016666666666666666 \cdot \left({im}^{5} \cdot \cos re\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 92.2%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(-0.016666666666666666 \cdot {im}^{5}\right) \cdot \cos re\right)} \]

      2. *-commutative 92.2%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\cos re \cdot \left(-0.016666666666666666 \cdot {im}^{5}\right)\right)} \]

   10. Simplified 92.2%

       \[\leadsto 0.5 \cdot \color{blue}{\left(\cos re \cdot \left(-0.016666666666666666 \cdot {im}^{5}\right)\right)} \]

   11. Taylor expanded in re around 0 69.2%

       \[\leadsto 0.5 \cdot \color{blue}{\left(-0.016666666666666666 \cdot {im}^{5}\right)} \]
   12. Step-by-step derivation

       1. add-sqr-sqrt 0.0%

          \[\leadsto \color{blue}{\sqrt{0.5 \cdot \left(-0.016666666666666666 \cdot {im}^{5}\right)} \cdot \sqrt{0.5 \cdot \left(-0.016666666666666666 \cdot {im}^{5}\right)}} \]

       2. sqrt-unprod 25.0%

          \[\leadsto \color{blue}{\sqrt{\left(0.5 \cdot \left(-0.016666666666666666 \cdot {im}^{5}\right)\right) \cdot \left(0.5 \cdot \left(-0.016666666666666666 \cdot {im}^{5}\right)\right)}} \]

       3. associate-*r* 25.0%

          \[\leadsto \sqrt{\color{blue}{\left(\left(0.5 \cdot -0.016666666666666666\right) \cdot {im}^{5}\right)} \cdot \left(0.5 \cdot \left(-0.016666666666666666 \cdot {im}^{5}\right)\right)} \]

       4. associate-*r* 25.0%

          \[\leadsto \sqrt{\left(\left(0.5 \cdot -0.016666666666666666\right) \cdot {im}^{5}\right) \cdot \color{blue}{\left(\left(0.5 \cdot -0.016666666666666666\right) \cdot {im}^{5}\right)}} \]

       5. swap-sqr 25.0%

          \[\leadsto \sqrt{\color{blue}{\left(\left(0.5 \cdot -0.016666666666666666\right) \cdot \left(0.5 \cdot -0.016666666666666666\right)\right) \cdot \left({im}^{5} \cdot {im}^{5}\right)}} \]

       6. metadata-eval 25.0%

          \[\leadsto \sqrt{\left(\color{blue}{-0.008333333333333333} \cdot \left(0.5 \cdot -0.016666666666666666\right)\right) \cdot \left({im}^{5} \cdot {im}^{5}\right)} \]

       7. metadata-eval 25.0%

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

       8. metadata-eval 25.0%

          \[\leadsto \sqrt{\color{blue}{6.944444444444444 \cdot 10^{-5}} \cdot \left({im}^{5} \cdot {im}^{5}\right)} \]

       9. pow-prod-up 25.0%

          \[\leadsto \sqrt{6.944444444444444 \cdot 10^{-5} \cdot \color{blue}{{im}^{\left(5 + 5\right)}}} \]

       10. metadata-eval 25.0%

           \[\leadsto \sqrt{6.944444444444444 \cdot 10^{-5} \cdot {im}^{\color{blue}{10}}} \]

   13. Applied egg-rr 25.0%

       \[\leadsto \color{blue}{\sqrt{6.944444444444444 \cdot 10^{-5} \cdot {im}^{10}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 5.8 \cdot 10^{+36}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{6.944444444444444 \cdot 10^{-5} \cdot {im}^{10}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 65.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 50.4%

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

   1. /-rgt-identity 50.4%

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

   2. exp-0 50.4%

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

   3. associate-*l/ 50.4%

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

   4. cos-neg 50.4%

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

   5. associate-*l* 50.4%

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

   6. associate-*r/ 50.4%

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

   7. exp-0 50.4%

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

   8. /-rgt-identity 50.4%

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

   9. *-commutative 50.4%

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

   10. neg-sub0 50.4%

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

   11. cos-neg 50.4%

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

3. Simplified 50.4%

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

5. Taylor expanded in im around 0 56.3%

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

   1. log1p-expm1-u 98.9%

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

   2. *-commutative 98.9%

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

   3. associate-*l* 98.9%

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

7. Applied egg-rr 98.9%

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

8. Taylor expanded in re around 0 66.5%

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

9. Final simplification 66.5%

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

Alternative 6: 63.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 5.5 \cdot 10^{-7}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(-0.3333333333333333 \cdot {im}^{2} - 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 5.5e-7)
   (* 0.5 (* (cos re) (* im -2.0)))
   (* 0.5 (* im (- (* -0.3333333333333333 (pow im 2.0)) 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 5.5e-7) {
		tmp = 0.5 * (cos(re) * (im * -2.0));
	} else {
		tmp = 0.5 * (im * ((-0.3333333333333333 * pow(im, 2.0)) - 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 (im <= 5.5d-7) then
        tmp = 0.5d0 * (cos(re) * (im * (-2.0d0)))
    else
        tmp = 0.5d0 * (im * (((-0.3333333333333333d0) * (im ** 2.0d0)) - 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 5.5e-7) {
		tmp = 0.5 * (Math.cos(re) * (im * -2.0));
	} else {
		tmp = 0.5 * (im * ((-0.3333333333333333 * Math.pow(im, 2.0)) - 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 5.5e-7:
		tmp = 0.5 * (math.cos(re) * (im * -2.0))
	else:
		tmp = 0.5 * (im * ((-0.3333333333333333 * math.pow(im, 2.0)) - 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 5.5e-7)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im * -2.0)));
	else
		tmp = Float64(0.5 * Float64(im * Float64(Float64(-0.3333333333333333 * (im ^ 2.0)) - 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 5.5e-7)
		tmp = 0.5 * (cos(re) * (im * -2.0));
	else
		tmp = 0.5 * (im * ((-0.3333333333333333 * (im ^ 2.0)) - 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 5.5e-7], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[(N[(-0.3333333333333333 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 5.5000000000000003e-7

   1. Initial program 36.6%

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

      1. /-rgt-identity 36.6%

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

      2. exp-0 36.6%

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

      3. associate-*l/ 36.6%

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

      4. cos-neg 36.6%

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

      5. associate-*l* 36.6%

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

      6. associate-*r/ 36.6%

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

      7. exp-0 36.6%

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

      8. /-rgt-identity 36.6%

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

      9. *-commutative 36.6%

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

      10. neg-sub0 36.6%

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

      11. cos-neg 36.6%

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

   3. Simplified 36.6%

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

   5. Taylor expanded in im around 0 70.0%

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

   if 5.5000000000000003e-7 < im

   1. Initial program 99.5%

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

      1. /-rgt-identity 99.5%

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

      2. exp-0 99.5%

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

      3. associate-*l/ 99.5%

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

      4. cos-neg 99.5%

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

      5. associate-*l* 99.5%

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

      6. associate-*r/ 99.5%

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

      7. exp-0 99.5%

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

      8. /-rgt-identity 99.5%

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

      9. *-commutative 99.5%

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

      10. neg-sub0 99.5%

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

      11. cos-neg 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in im around 0 61.8%

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

   6. Taylor expanded in re around 0 50.4%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-0.3333333333333333 \cdot {im}^{2} - 2\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 5.5 \cdot 10^{-7}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(-0.3333333333333333 \cdot {im}^{2} - 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 55.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.22 \cdot 10^{+17}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.22e+17)
   (* 0.5 (* (cos re) (* im -2.0)))
   (* 0.5 (* im (+ -2.0 (pow re 2.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.22e+17) {
		tmp = 0.5 * (cos(re) * (im * -2.0));
	} else {
		tmp = 0.5 * (im * (-2.0 + pow(re, 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 (im <= 1.22d+17) then
        tmp = 0.5d0 * (cos(re) * (im * (-2.0d0)))
    else
        tmp = 0.5d0 * (im * ((-2.0d0) + (re ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.22e+17) {
		tmp = 0.5 * (Math.cos(re) * (im * -2.0));
	} else {
		tmp = 0.5 * (im * (-2.0 + Math.pow(re, 2.0)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.22e+17:
		tmp = 0.5 * (math.cos(re) * (im * -2.0))
	else:
		tmp = 0.5 * (im * (-2.0 + math.pow(re, 2.0)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.22e+17)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im * -2.0)));
	else
		tmp = Float64(0.5 * Float64(im * Float64(-2.0 + (re ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.22e+17)
		tmp = 0.5 * (cos(re) * (im * -2.0));
	else
		tmp = 0.5 * (im * (-2.0 + (re ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.22e+17], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[(-2.0 + N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.22e17

   1. Initial program 38.0%

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

      1. /-rgt-identity 38.0%

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

      2. exp-0 38.0%

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

      3. associate-*l/ 38.0%

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

      4. cos-neg 38.0%

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

      5. associate-*l* 38.0%

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

      6. associate-*r/ 38.0%

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

      7. exp-0 38.0%

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

      8. /-rgt-identity 38.0%

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

      9. *-commutative 38.0%

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

      10. neg-sub0 38.0%

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

      11. cos-neg 38.0%

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

   3. Simplified 38.0%

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

   5. Taylor expanded in im around 0 69.0%

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

   if 1.22e17 < 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. Taylor expanded in re around 0 17.6%

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

      1. *-commutative 17.6%

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

      2. distribute-lft-out 17.6%

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

   8. Simplified 17.6%

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

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.22 \cdot 10^{+17}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 720:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left({re}^{4} \cdot \left(im \cdot -0.08333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 720.0)
   (* 0.5 (* (cos re) (* im -2.0)))
   (* 0.5 (* (pow re 4.0) (* im -0.08333333333333333)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 720.0) {
		tmp = 0.5 * (cos(re) * (im * -2.0));
	} else {
		tmp = 0.5 * (pow(re, 4.0) * (im * -0.08333333333333333));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 720.0d0) then
        tmp = 0.5d0 * (cos(re) * (im * (-2.0d0)))
    else
        tmp = 0.5d0 * ((re ** 4.0d0) * (im * (-0.08333333333333333d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 720.0) {
		tmp = 0.5 * (Math.cos(re) * (im * -2.0));
	} else {
		tmp = 0.5 * (Math.pow(re, 4.0) * (im * -0.08333333333333333));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 720.0:
		tmp = 0.5 * (math.cos(re) * (im * -2.0))
	else:
		tmp = 0.5 * (math.pow(re, 4.0) * (im * -0.08333333333333333))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 720.0)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im * -2.0)));
	else
		tmp = Float64(0.5 * Float64((re ^ 4.0) * Float64(im * -0.08333333333333333)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 720.0)
		tmp = 0.5 * (cos(re) * (im * -2.0));
	else
		tmp = 0.5 * ((re ^ 4.0) * (im * -0.08333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 720.0], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Power[re, 4.0], $MachinePrecision] * N[(im * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 720

   1. Initial program 37.4%

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

      1. /-rgt-identity 37.4%

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

      2. exp-0 37.4%

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

      3. associate-*l/ 37.4%

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

      4. cos-neg 37.4%

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

      5. associate-*l* 37.4%

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

      6. associate-*r/ 37.4%

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

      7. exp-0 37.4%

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

      8. /-rgt-identity 37.4%

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

      9. *-commutative 37.4%

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

      10. neg-sub0 37.4%

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

      11. cos-neg 37.4%

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

   3. Simplified 37.4%

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

   5. Taylor expanded in im around 0 69.6%

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

   if 720 < 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. Taylor expanded in re around 0 25.8%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-2 \cdot im + {re}^{2} \cdot \left(im + -0.08333333333333333 \cdot \left(im \cdot {re}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. distribute-rgt-in 6.9%

         \[\leadsto 0.5 \cdot \left(-2 \cdot im + \color{blue}{\left(im \cdot {re}^{2} + \left(-0.08333333333333333 \cdot \left(im \cdot {re}^{2}\right)\right) \cdot {re}^{2}\right)}\right) \]

      2. associate-+r+ 6.9%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(-2 \cdot im + im \cdot {re}^{2}\right) + \left(-0.08333333333333333 \cdot \left(im \cdot {re}^{2}\right)\right) \cdot {re}^{2}\right)} \]

      3. *-commutative 6.9%

         \[\leadsto 0.5 \cdot \left(\left(\color{blue}{im \cdot -2} + im \cdot {re}^{2}\right) + \left(-0.08333333333333333 \cdot \left(im \cdot {re}^{2}\right)\right) \cdot {re}^{2}\right) \]

      4. distribute-lft-out 6.9%

         \[\leadsto 0.5 \cdot \left(\color{blue}{im \cdot \left(-2 + {re}^{2}\right)} + \left(-0.08333333333333333 \cdot \left(im \cdot {re}^{2}\right)\right) \cdot {re}^{2}\right) \]

      5. associate-*r* 6.9%

         \[\leadsto 0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right) + \color{blue}{\left(\left(-0.08333333333333333 \cdot im\right) \cdot {re}^{2}\right)} \cdot {re}^{2}\right) \]

      6. associate-*l* 6.9%

         \[\leadsto 0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right) + \color{blue}{\left(-0.08333333333333333 \cdot im\right) \cdot \left({re}^{2} \cdot {re}^{2}\right)}\right) \]

      7. *-commutative 6.9%

         \[\leadsto 0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right) + \color{blue}{\left(im \cdot -0.08333333333333333\right)} \cdot \left({re}^{2} \cdot {re}^{2}\right)\right) \]

      8. pow-sqr 6.9%

         \[\leadsto 0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right) + \left(im \cdot -0.08333333333333333\right) \cdot \color{blue}{{re}^{\left(2 \cdot 2\right)}}\right) \]

      9. metadata-eval 6.9%

         \[\leadsto 0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right) + \left(im \cdot -0.08333333333333333\right) \cdot {re}^{\color{blue}{4}}\right) \]

   8. Simplified 6.9%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-2 + {re}^{2}\right) + \left(im \cdot -0.08333333333333333\right) \cdot {re}^{4}\right)} \]

   9. Taylor expanded in re around inf 23.9%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.08333333333333333 \cdot \left(im \cdot {re}^{4}\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative 23.9%

          \[\leadsto 0.5 \cdot \color{blue}{\left(\left(im \cdot {re}^{4}\right) \cdot -0.08333333333333333\right)} \]

       2. *-commutative 23.9%

          \[\leadsto 0.5 \cdot \left(\color{blue}{\left({re}^{4} \cdot im\right)} \cdot -0.08333333333333333\right) \]

       3. associate-*r* 23.9%

          \[\leadsto 0.5 \cdot \color{blue}{\left({re}^{4} \cdot \left(im \cdot -0.08333333333333333\right)\right)} \]

   11. Simplified 23.9%

       \[\leadsto 0.5 \cdot \color{blue}{\left({re}^{4} \cdot \left(im \cdot -0.08333333333333333\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 720:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left({re}^{4} \cdot \left(im \cdot -0.08333333333333333\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 44.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.15:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.016666666666666666 \cdot {im}^{5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.15)
   (* 0.5 (* im -2.0))
   (* 0.5 (* -0.016666666666666666 (pow im 5.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.15) {
		tmp = 0.5 * (im * -2.0);
	} else {
		tmp = 0.5 * (-0.016666666666666666 * pow(im, 5.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 3.15d0) then
        tmp = 0.5d0 * (im * (-2.0d0))
    else
        tmp = 0.5d0 * ((-0.016666666666666666d0) * (im ** 5.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3.15) {
		tmp = 0.5 * (im * -2.0);
	} else {
		tmp = 0.5 * (-0.016666666666666666 * Math.pow(im, 5.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.15:
		tmp = 0.5 * (im * -2.0)
	else:
		tmp = 0.5 * (-0.016666666666666666 * math.pow(im, 5.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.15)
		tmp = Float64(0.5 * Float64(im * -2.0));
	else
		tmp = Float64(0.5 * Float64(-0.016666666666666666 * (im ^ 5.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3.15)
		tmp = 0.5 * (im * -2.0);
	else
		tmp = 0.5 * (-0.016666666666666666 * (im ^ 5.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.15], N[(0.5 * N[(im * -2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(-0.016666666666666666 * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 3.14999999999999991

   1. Initial program 37.1%

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

      1. /-rgt-identity 37.1%

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

      2. exp-0 37.1%

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

      3. associate-*l/ 37.1%

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

      4. cos-neg 37.1%

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

      5. associate-*l* 37.1%

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

      6. associate-*r/ 37.1%

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

      7. exp-0 37.1%

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

      8. /-rgt-identity 37.1%

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

      9. *-commutative 37.1%

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

      10. neg-sub0 37.1%

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

      11. cos-neg 37.1%

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

   3. Simplified 37.1%

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

   5. Taylor expanded in im around 0 69.9%

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

   6. Taylor expanded in re around 0 41.5%

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

      1. *-commutative 41.5%

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

   8. Simplified 41.5%

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

   if 3.14999999999999991 < 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 82.8%

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

      1. distribute-lft-in 82.8%

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

      2. +-commutative 82.8%

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

      3. associate-*r* 82.8%

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

      4. *-commutative 82.8%

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

      5. fma-undefine 82.8%

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

   7. Simplified 82.8%

      \[\leadsto 0.5 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left({im}^{2}, -0.016666666666666666, -0.3333333333333333\right) \cdot \cos re, {im}^{3}, \left(-2 \cdot im\right) \cdot \cos re\right)} \]

   8. Taylor expanded in im around inf 82.7%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.016666666666666666 \cdot \left({im}^{5} \cdot \cos re\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 82.7%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(-0.016666666666666666 \cdot {im}^{5}\right) \cdot \cos re\right)} \]

      2. *-commutative 82.7%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\cos re \cdot \left(-0.016666666666666666 \cdot {im}^{5}\right)\right)} \]

   10. Simplified 82.7%

       \[\leadsto 0.5 \cdot \color{blue}{\left(\cos re \cdot \left(-0.016666666666666666 \cdot {im}^{5}\right)\right)} \]

   11. Taylor expanded in re around 0 62.2%

       \[\leadsto 0.5 \cdot \color{blue}{\left(-0.016666666666666666 \cdot {im}^{5}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.15:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.016666666666666666 \cdot {im}^{5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 34.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 550:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;{re}^{4} \cdot \left(im \cdot -0.041666666666666664\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 550.0)
   (* 0.5 (* im -2.0))
   (* (pow re 4.0) (* im -0.041666666666666664))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 550.0) {
		tmp = 0.5 * (im * -2.0);
	} else {
		tmp = pow(re, 4.0) * (im * -0.041666666666666664);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 550.0d0) then
        tmp = 0.5d0 * (im * (-2.0d0))
    else
        tmp = (re ** 4.0d0) * (im * (-0.041666666666666664d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 550.0) {
		tmp = 0.5 * (im * -2.0);
	} else {
		tmp = Math.pow(re, 4.0) * (im * -0.041666666666666664);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 550.0:
		tmp = 0.5 * (im * -2.0)
	else:
		tmp = math.pow(re, 4.0) * (im * -0.041666666666666664)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 550.0)
		tmp = Float64(0.5 * Float64(im * -2.0));
	else
		tmp = Float64((re ^ 4.0) * Float64(im * -0.041666666666666664));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 550.0)
		tmp = 0.5 * (im * -2.0);
	else
		tmp = (re ^ 4.0) * (im * -0.041666666666666664);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 550.0], N[(0.5 * N[(im * -2.0), $MachinePrecision]), $MachinePrecision], N[(N[Power[re, 4.0], $MachinePrecision] * N[(im * -0.041666666666666664), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 550

   1. Initial program 37.4%

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

      1. /-rgt-identity 37.4%

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

      2. exp-0 37.4%

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

      3. associate-*l/ 37.4%

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

      4. cos-neg 37.4%

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

      5. associate-*l* 37.4%

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

      6. associate-*r/ 37.4%

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

      7. exp-0 37.4%

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

      8. /-rgt-identity 37.4%

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

      9. *-commutative 37.4%

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

      10. neg-sub0 37.4%

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

      11. cos-neg 37.4%

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

   3. Simplified 37.4%

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

   5. Taylor expanded in im around 0 69.6%

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

   6. Taylor expanded in re around 0 41.3%

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

      1. *-commutative 41.3%

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

   8. Simplified 41.3%

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

   if 550 < 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. Taylor expanded in re around 0 25.8%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-2 \cdot im + {re}^{2} \cdot \left(im + -0.08333333333333333 \cdot \left(im \cdot {re}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. distribute-rgt-in 6.9%

         \[\leadsto 0.5 \cdot \left(-2 \cdot im + \color{blue}{\left(im \cdot {re}^{2} + \left(-0.08333333333333333 \cdot \left(im \cdot {re}^{2}\right)\right) \cdot {re}^{2}\right)}\right) \]

      2. associate-+r+ 6.9%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(-2 \cdot im + im \cdot {re}^{2}\right) + \left(-0.08333333333333333 \cdot \left(im \cdot {re}^{2}\right)\right) \cdot {re}^{2}\right)} \]

      3. *-commutative 6.9%

         \[\leadsto 0.5 \cdot \left(\left(\color{blue}{im \cdot -2} + im \cdot {re}^{2}\right) + \left(-0.08333333333333333 \cdot \left(im \cdot {re}^{2}\right)\right) \cdot {re}^{2}\right) \]

      4. distribute-lft-out 6.9%

         \[\leadsto 0.5 \cdot \left(\color{blue}{im \cdot \left(-2 + {re}^{2}\right)} + \left(-0.08333333333333333 \cdot \left(im \cdot {re}^{2}\right)\right) \cdot {re}^{2}\right) \]

      5. associate-*r* 6.9%

         \[\leadsto 0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right) + \color{blue}{\left(\left(-0.08333333333333333 \cdot im\right) \cdot {re}^{2}\right)} \cdot {re}^{2}\right) \]

      6. associate-*l* 6.9%

         \[\leadsto 0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right) + \color{blue}{\left(-0.08333333333333333 \cdot im\right) \cdot \left({re}^{2} \cdot {re}^{2}\right)}\right) \]

      7. *-commutative 6.9%

         \[\leadsto 0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right) + \color{blue}{\left(im \cdot -0.08333333333333333\right)} \cdot \left({re}^{2} \cdot {re}^{2}\right)\right) \]

      8. pow-sqr 6.9%

         \[\leadsto 0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right) + \left(im \cdot -0.08333333333333333\right) \cdot \color{blue}{{re}^{\left(2 \cdot 2\right)}}\right) \]

      9. metadata-eval 6.9%

         \[\leadsto 0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right) + \left(im \cdot -0.08333333333333333\right) \cdot {re}^{\color{blue}{4}}\right) \]

   8. Simplified 6.9%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-2 + {re}^{2}\right) + \left(im \cdot -0.08333333333333333\right) \cdot {re}^{4}\right)} \]

   9. Taylor expanded in re around inf 23.9%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.08333333333333333 \cdot \left(im \cdot {re}^{4}\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative 23.9%

          \[\leadsto 0.5 \cdot \color{blue}{\left(\left(im \cdot {re}^{4}\right) \cdot -0.08333333333333333\right)} \]

       2. *-commutative 23.9%

          \[\leadsto 0.5 \cdot \left(\color{blue}{\left({re}^{4} \cdot im\right)} \cdot -0.08333333333333333\right) \]

       3. associate-*r* 23.9%

          \[\leadsto 0.5 \cdot \color{blue}{\left({re}^{4} \cdot \left(im \cdot -0.08333333333333333\right)\right)} \]

   11. Simplified 23.9%

       \[\leadsto 0.5 \cdot \color{blue}{\left({re}^{4} \cdot \left(im \cdot -0.08333333333333333\right)\right)} \]

   12. Taylor expanded in re around 0 23.9%

       \[\leadsto \color{blue}{-0.041666666666666664 \cdot \left(im \cdot {re}^{4}\right)} \]
   13. Step-by-step derivation

       1. *-commutative 23.9%

          \[\leadsto \color{blue}{\left(im \cdot {re}^{4}\right) \cdot -0.041666666666666664} \]

       2. *-commutative 23.9%

          \[\leadsto \color{blue}{\left({re}^{4} \cdot im\right)} \cdot -0.041666666666666664 \]

       3. associate-*l* 23.9%

          \[\leadsto \color{blue}{{re}^{4} \cdot \left(im \cdot -0.041666666666666664\right)} \]

   14. Simplified 23.9%

       \[\leadsto \color{blue}{{re}^{4} \cdot \left(im \cdot -0.041666666666666664\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 550:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;{re}^{4} \cdot \left(im \cdot -0.041666666666666664\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 52.1% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.4%

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

   1. /-rgt-identity 50.4%

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

   2. exp-0 50.4%

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

   3. associate-*l/ 50.4%

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

   4. cos-neg 50.4%

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

   5. associate-*l* 50.4%

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

   6. associate-*r/ 50.4%

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

   7. exp-0 50.4%

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

   8. /-rgt-identity 50.4%

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

   9. *-commutative 50.4%

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

   10. neg-sub0 50.4%

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

   11. cos-neg 50.4%

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

3. Simplified 50.4%

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

5. Taylor expanded in im around 0 56.3%

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

6. Final simplification 56.3%

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

Alternative 12: 29.8% 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 50.4%

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

   1. /-rgt-identity 50.4%

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

   2. exp-0 50.4%

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

   3. associate-*l/ 50.4%

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

   4. cos-neg 50.4%

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

   5. associate-*l* 50.4%

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

   6. associate-*r/ 50.4%

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

   7. exp-0 50.4%

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

   8. /-rgt-identity 50.4%

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

   9. *-commutative 50.4%

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

   10. neg-sub0 50.4%

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

   11. cos-neg 50.4%

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

3. Simplified 50.4%

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

5. Taylor expanded in im around 0 56.3%

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

6. Taylor expanded in re around 0 33.7%

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

   1. *-commutative 33.7%

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

8. Simplified 33.7%

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

9. Final simplification 33.7%

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

Developer target: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|im\right| < 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(0.16666666666666666 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (< (fabs im) 1.0)
   (-
    (*
     (cos re)
     (+
      (+ im (* (* (* 0.16666666666666666 im) im) im))
      (* (* (* (* (* 0.008333333333333333 im) im) im) im) im))))
   (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if (fabs(im) < 1.0) {
		tmp = -(cos(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (abs(im) < 1.0d0) then
        tmp = -(cos(re) * ((im + (((0.16666666666666666d0 * im) * im) * im)) + (((((0.008333333333333333d0 * im) * im) * im) * im) * im)))
    else
        tmp = (0.5d0 * cos(re)) * (exp((0.0d0 - im)) - exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.abs(im) < 1.0) {
		tmp = -(Math.cos(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * Math.cos(re)) * (Math.exp((0.0 - im)) - Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.fabs(im) < 1.0:
		tmp = -(math.cos(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)))
	else:
		tmp = (0.5 * math.cos(re)) * (math.exp((0.0 - im)) - math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (abs(im) < 1.0)
		tmp = Float64(-Float64(cos(re) * Float64(Float64(im + Float64(Float64(Float64(0.16666666666666666 * im) * im) * im)) + Float64(Float64(Float64(Float64(Float64(0.008333333333333333 * im) * im) * im) * im) * im))));
	else
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - im)) - exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (abs(im) < 1.0)
		tmp = -(cos(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	else
		tmp = (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Less[N[Abs[im], $MachinePrecision], 1.0], (-N[(N[Cos[re], $MachinePrecision] * N[(N[(im + N[(N[(N[(0.16666666666666666 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[(0.008333333333333333 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024066 
(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))))