math.sin on complex, imaginary part

Percentage Accurate: 54.1% → 99.0%

Time: 11.5s

Alternatives: 9

Speedup: 2.8×

• 49.5% 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: 54.1% 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 56.6%

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

   1. /-rgt-identity 56.6%

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

   2. exp-0 56.6%

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

   3. associate-*l/ 56.6%

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

   4. cos-neg 56.6%

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

   5. associate-*l* 56.6%

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

   6. associate-*r/ 56.6%

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

   7. exp-0 56.6%

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

   8. /-rgt-identity 56.6%

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

   9. *-commutative 56.6%

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

   10. neg-sub0 56.6%

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

   11. cos-neg 56.6%

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

3. Simplified 56.6%

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

5. Taylor expanded in im around 0 49.9%

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

   1. log1p-expm1-u 98.7%

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

   2. *-commutative 98.7%

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

   3. associate-*l* 98.7%

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

7. Applied egg-rr 98.7%

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

8. Final simplification 98.7%

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

Alternative 2: 87.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 450000000:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot \left({im}^{2} \cdot -0.3333333333333333\right) + im \cdot -2\right)\right)\\ \mathbf{elif}\;im \leq 2.9 \cdot 10^{+60}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + {re}^{2} \cdot \left(im + -0.08333333333333333 \cdot \left(im \cdot {re}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.016666666666666666 \cdot \left(\cos re \cdot {im}^{5}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 450000000.0)
   (*
    0.5
    (* (cos re) (+ (* im (* (pow im 2.0) -0.3333333333333333)) (* im -2.0))))
   (if (<= im 2.9e+60)
     (*
      0.5
      (+
       (* im -2.0)
       (* (pow re 2.0) (+ im (* -0.08333333333333333 (* im (pow re 2.0)))))))
     (* 0.5 (* -0.016666666666666666 (* (cos re) (pow im 5.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 450000000.0) {
		tmp = 0.5 * (cos(re) * ((im * (pow(im, 2.0) * -0.3333333333333333)) + (im * -2.0)));
	} else if (im <= 2.9e+60) {
		tmp = 0.5 * ((im * -2.0) + (pow(re, 2.0) * (im + (-0.08333333333333333 * (im * pow(re, 2.0))))));
	} else {
		tmp = 0.5 * (-0.016666666666666666 * (cos(re) * 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 <= 450000000.0d0) then
        tmp = 0.5d0 * (cos(re) * ((im * ((im ** 2.0d0) * (-0.3333333333333333d0))) + (im * (-2.0d0))))
    else if (im <= 2.9d+60) then
        tmp = 0.5d0 * ((im * (-2.0d0)) + ((re ** 2.0d0) * (im + ((-0.08333333333333333d0) * (im * (re ** 2.0d0))))))
    else
        tmp = 0.5d0 * ((-0.016666666666666666d0) * (cos(re) * (im ** 5.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 450000000.0) {
		tmp = 0.5 * (Math.cos(re) * ((im * (Math.pow(im, 2.0) * -0.3333333333333333)) + (im * -2.0)));
	} else if (im <= 2.9e+60) {
		tmp = 0.5 * ((im * -2.0) + (Math.pow(re, 2.0) * (im + (-0.08333333333333333 * (im * Math.pow(re, 2.0))))));
	} else {
		tmp = 0.5 * (-0.016666666666666666 * (Math.cos(re) * Math.pow(im, 5.0)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 450000000.0:
		tmp = 0.5 * (math.cos(re) * ((im * (math.pow(im, 2.0) * -0.3333333333333333)) + (im * -2.0)))
	elif im <= 2.9e+60:
		tmp = 0.5 * ((im * -2.0) + (math.pow(re, 2.0) * (im + (-0.08333333333333333 * (im * math.pow(re, 2.0))))))
	else:
		tmp = 0.5 * (-0.016666666666666666 * (math.cos(re) * math.pow(im, 5.0)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 450000000.0)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(Float64(im * Float64((im ^ 2.0) * -0.3333333333333333)) + Float64(im * -2.0))));
	elseif (im <= 2.9e+60)
		tmp = Float64(0.5 * Float64(Float64(im * -2.0) + Float64((re ^ 2.0) * Float64(im + Float64(-0.08333333333333333 * Float64(im * (re ^ 2.0)))))));
	else
		tmp = Float64(0.5 * Float64(-0.016666666666666666 * Float64(cos(re) * (im ^ 5.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 450000000.0)
		tmp = 0.5 * (cos(re) * ((im * ((im ^ 2.0) * -0.3333333333333333)) + (im * -2.0)));
	elseif (im <= 2.9e+60)
		tmp = 0.5 * ((im * -2.0) + ((re ^ 2.0) * (im + (-0.08333333333333333 * (im * (re ^ 2.0))))));
	else
		tmp = 0.5 * (-0.016666666666666666 * (cos(re) * (im ^ 5.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 450000000.0], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(N[(im * N[(N[Power[im, 2.0], $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision] + N[(im * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.9e+60], N[(0.5 * N[(N[(im * -2.0), $MachinePrecision] + N[(N[Power[re, 2.0], $MachinePrecision] * N[(im + N[(-0.08333333333333333 * N[(im * N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(-0.016666666666666666 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 4.5e8

   1. Initial program 42.1%

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

      1. /-rgt-identity 42.1%

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

      2. exp-0 42.1%

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

      3. associate-*l/ 42.1%

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

      4. cos-neg 42.1%

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

      5. associate-*l* 42.1%

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

      6. associate-*r/ 42.1%

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

      7. exp-0 42.1%

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

      8. /-rgt-identity 42.1%

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

      9. *-commutative 42.1%

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

      10. neg-sub0 42.1%

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

      11. cos-neg 42.1%

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

   3. Simplified 42.1%

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

   5. Taylor expanded in im around 0 88.4%

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

      1. sub-neg 88.4%

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

      2. metadata-eval 88.4%

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

      3. distribute-rgt-in 88.5%

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

      4. *-commutative 88.5%

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

      5. *-commutative 88.5%

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

   7. Applied egg-rr 88.5%

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

   if 4.5e8 < im < 2.9e60

   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 3.4%

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

   6. Taylor expanded in re around 0 31.4%

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

   if 2.9e60 < 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. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. *-commutative 100.0%

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

      5. fma-undefine 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around inf 100.0%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 450000000:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot \left({im}^{2} \cdot -0.3333333333333333\right) + im \cdot -2\right)\right)\\ \mathbf{elif}\;im \leq 2.9 \cdot 10^{+60}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + {re}^{2} \cdot \left(im + -0.08333333333333333 \cdot \left(im \cdot {re}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.016666666666666666 \cdot \left(\cos re \cdot {im}^{5}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 450000000:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot \left({im}^{2} \cdot -0.3333333333333333\right) + im \cdot -2\right)\right)\\ \mathbf{elif}\;im \leq 2.85 \cdot 10^{+60}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + {re}^{4} \cdot \left(im \cdot -0.08333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.016666666666666666 \cdot \left(\cos re \cdot {im}^{5}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 450000000.0)
   (*
    0.5
    (* (cos re) (+ (* im (* (pow im 2.0) -0.3333333333333333)) (* im -2.0))))
   (if (<= im 2.85e+60)
     (* 0.5 (+ (* im -2.0) (* (pow re 4.0) (* im -0.08333333333333333))))
     (* 0.5 (* -0.016666666666666666 (* (cos re) (pow im 5.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 450000000.0) {
		tmp = 0.5 * (cos(re) * ((im * (pow(im, 2.0) * -0.3333333333333333)) + (im * -2.0)));
	} else if (im <= 2.85e+60) {
		tmp = 0.5 * ((im * -2.0) + (pow(re, 4.0) * (im * -0.08333333333333333)));
	} else {
		tmp = 0.5 * (-0.016666666666666666 * (cos(re) * 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 <= 450000000.0d0) then
        tmp = 0.5d0 * (cos(re) * ((im * ((im ** 2.0d0) * (-0.3333333333333333d0))) + (im * (-2.0d0))))
    else if (im <= 2.85d+60) then
        tmp = 0.5d0 * ((im * (-2.0d0)) + ((re ** 4.0d0) * (im * (-0.08333333333333333d0))))
    else
        tmp = 0.5d0 * ((-0.016666666666666666d0) * (cos(re) * (im ** 5.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 450000000.0) {
		tmp = 0.5 * (Math.cos(re) * ((im * (Math.pow(im, 2.0) * -0.3333333333333333)) + (im * -2.0)));
	} else if (im <= 2.85e+60) {
		tmp = 0.5 * ((im * -2.0) + (Math.pow(re, 4.0) * (im * -0.08333333333333333)));
	} else {
		tmp = 0.5 * (-0.016666666666666666 * (Math.cos(re) * Math.pow(im, 5.0)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 450000000.0:
		tmp = 0.5 * (math.cos(re) * ((im * (math.pow(im, 2.0) * -0.3333333333333333)) + (im * -2.0)))
	elif im <= 2.85e+60:
		tmp = 0.5 * ((im * -2.0) + (math.pow(re, 4.0) * (im * -0.08333333333333333)))
	else:
		tmp = 0.5 * (-0.016666666666666666 * (math.cos(re) * math.pow(im, 5.0)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 450000000.0)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(Float64(im * Float64((im ^ 2.0) * -0.3333333333333333)) + Float64(im * -2.0))));
	elseif (im <= 2.85e+60)
		tmp = Float64(0.5 * Float64(Float64(im * -2.0) + Float64((re ^ 4.0) * Float64(im * -0.08333333333333333))));
	else
		tmp = Float64(0.5 * Float64(-0.016666666666666666 * Float64(cos(re) * (im ^ 5.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 450000000.0)
		tmp = 0.5 * (cos(re) * ((im * ((im ^ 2.0) * -0.3333333333333333)) + (im * -2.0)));
	elseif (im <= 2.85e+60)
		tmp = 0.5 * ((im * -2.0) + ((re ^ 4.0) * (im * -0.08333333333333333)));
	else
		tmp = 0.5 * (-0.016666666666666666 * (cos(re) * (im ^ 5.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 450000000.0], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(N[(im * N[(N[Power[im, 2.0], $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision] + N[(im * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.85e+60], N[(0.5 * N[(N[(im * -2.0), $MachinePrecision] + N[(N[Power[re, 4.0], $MachinePrecision] * N[(im * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(-0.016666666666666666 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 4.5e8

   1. Initial program 42.1%

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

      1. /-rgt-identity 42.1%

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

      2. exp-0 42.1%

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

      3. associate-*l/ 42.1%

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

      4. cos-neg 42.1%

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

      5. associate-*l* 42.1%

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

      6. associate-*r/ 42.1%

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

      7. exp-0 42.1%

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

      8. /-rgt-identity 42.1%

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

      9. *-commutative 42.1%

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

      10. neg-sub0 42.1%

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

      11. cos-neg 42.1%

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

   3. Simplified 42.1%

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

   5. Taylor expanded in im around 0 88.4%

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

      1. sub-neg 88.4%

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

      2. metadata-eval 88.4%

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

      3. distribute-rgt-in 88.5%

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

      4. *-commutative 88.5%

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

      5. *-commutative 88.5%

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

   7. Applied egg-rr 88.5%

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

   if 4.5e8 < im < 2.84999999999999989e60

   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 3.4%

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

   6. Taylor expanded in re around 0 31.4%

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

   7. Taylor expanded in re around inf 31.4%

      \[\leadsto 0.5 \cdot \left(-2 \cdot im + \color{blue}{-0.08333333333333333 \cdot \left(im \cdot {re}^{4}\right)}\right) \]
   8. Step-by-step derivation

      1. associate-*r* 31.4%

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

      2. *-commutative 31.4%

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

      3. *-commutative 31.4%

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

   9. Simplified 31.4%

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

   if 2.84999999999999989e60 < 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. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. *-commutative 100.0%

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

      5. fma-undefine 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around inf 100.0%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 450000000:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot \left({im}^{2} \cdot -0.3333333333333333\right) + im \cdot -2\right)\right)\\ \mathbf{elif}\;im \leq 2.85 \cdot 10^{+60}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + {re}^{4} \cdot \left(im \cdot -0.08333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.016666666666666666 \cdot \left(\cos re \cdot {im}^{5}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.3:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;im \leq 1.3 \cdot 10^{+60}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + {re}^{4} \cdot \left(im \cdot -0.08333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.016666666666666666 \cdot \left(\cos re \cdot {im}^{5}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.3)
   (* 0.5 (* (cos re) (* im -2.0)))
   (if (<= im 1.3e+60)
     (* 0.5 (+ (* im -2.0) (* (pow re 4.0) (* im -0.08333333333333333))))
     (* 0.5 (* -0.016666666666666666 (* (cos re) (pow im 5.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.3) {
		tmp = 0.5 * (cos(re) * (im * -2.0));
	} else if (im <= 1.3e+60) {
		tmp = 0.5 * ((im * -2.0) + (pow(re, 4.0) * (im * -0.08333333333333333)));
	} else {
		tmp = 0.5 * (-0.016666666666666666 * (cos(re) * 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 <= 0.3d0) then
        tmp = 0.5d0 * (cos(re) * (im * (-2.0d0)))
    else if (im <= 1.3d+60) then
        tmp = 0.5d0 * ((im * (-2.0d0)) + ((re ** 4.0d0) * (im * (-0.08333333333333333d0))))
    else
        tmp = 0.5d0 * ((-0.016666666666666666d0) * (cos(re) * (im ** 5.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.3) {
		tmp = 0.5 * (Math.cos(re) * (im * -2.0));
	} else if (im <= 1.3e+60) {
		tmp = 0.5 * ((im * -2.0) + (Math.pow(re, 4.0) * (im * -0.08333333333333333)));
	} else {
		tmp = 0.5 * (-0.016666666666666666 * (Math.cos(re) * Math.pow(im, 5.0)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.3:
		tmp = 0.5 * (math.cos(re) * (im * -2.0))
	elif im <= 1.3e+60:
		tmp = 0.5 * ((im * -2.0) + (math.pow(re, 4.0) * (im * -0.08333333333333333)))
	else:
		tmp = 0.5 * (-0.016666666666666666 * (math.cos(re) * math.pow(im, 5.0)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.3)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im * -2.0)));
	elseif (im <= 1.3e+60)
		tmp = Float64(0.5 * Float64(Float64(im * -2.0) + Float64((re ^ 4.0) * Float64(im * -0.08333333333333333))));
	else
		tmp = Float64(0.5 * Float64(-0.016666666666666666 * Float64(cos(re) * (im ^ 5.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.3)
		tmp = 0.5 * (cos(re) * (im * -2.0));
	elseif (im <= 1.3e+60)
		tmp = 0.5 * ((im * -2.0) + ((re ^ 4.0) * (im * -0.08333333333333333)));
	else
		tmp = 0.5 * (-0.016666666666666666 * (cos(re) * (im ^ 5.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.3], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.3e+60], N[(0.5 * N[(N[(im * -2.0), $MachinePrecision] + N[(N[Power[re, 4.0], $MachinePrecision] * N[(im * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(-0.016666666666666666 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.299999999999999989

   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 65.8%

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

   if 0.299999999999999989 < im < 1.30000000000000004e60

   1. Initial program 99.9%

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

      1. /-rgt-identity 99.9%

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

      2. exp-0 99.9%

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

      3. associate-*l/ 99.9%

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

      4. cos-neg 99.9%

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

      5. associate-*l* 99.9%

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

      6. associate-*r/ 99.9%

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

      7. exp-0 99.9%

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

      8. /-rgt-identity 99.9%

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

      9. *-commutative 99.9%

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

      10. neg-sub0 99.9%

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

      11. cos-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in im around 0 7.3%

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

   6. Taylor expanded in re around 0 27.3%

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

   7. Taylor expanded in re around inf 27.3%

      \[\leadsto 0.5 \cdot \left(-2 \cdot im + \color{blue}{-0.08333333333333333 \cdot \left(im \cdot {re}^{4}\right)}\right) \]
   8. Step-by-step derivation

      1. associate-*r* 27.3%

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

      2. *-commutative 27.3%

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

      3. *-commutative 27.3%

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

   9. Simplified 27.3%

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

   if 1.30000000000000004e60 < 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. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. *-commutative 100.0%

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

      5. fma-undefine 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around inf 100.0%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.3:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;im \leq 1.3 \cdot 10^{+60}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + {re}^{4} \cdot \left(im \cdot -0.08333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.016666666666666666 \cdot \left(\cos re \cdot {im}^{5}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 450000000:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(\cos re \cdot \left({im}^{2} \cdot -0.3333333333333333 - 2\right)\right)\right)\\ \mathbf{elif}\;im \leq 2.9 \cdot 10^{+60}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + {re}^{4} \cdot \left(im \cdot -0.08333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.016666666666666666 \cdot \left(\cos re \cdot {im}^{5}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 450000000.0)
   (* 0.5 (* im (* (cos re) (- (* (pow im 2.0) -0.3333333333333333) 2.0))))
   (if (<= im 2.9e+60)
     (* 0.5 (+ (* im -2.0) (* (pow re 4.0) (* im -0.08333333333333333))))
     (* 0.5 (* -0.016666666666666666 (* (cos re) (pow im 5.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 450000000.0) {
		tmp = 0.5 * (im * (cos(re) * ((pow(im, 2.0) * -0.3333333333333333) - 2.0)));
	} else if (im <= 2.9e+60) {
		tmp = 0.5 * ((im * -2.0) + (pow(re, 4.0) * (im * -0.08333333333333333)));
	} else {
		tmp = 0.5 * (-0.016666666666666666 * (cos(re) * 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 <= 450000000.0d0) then
        tmp = 0.5d0 * (im * (cos(re) * (((im ** 2.0d0) * (-0.3333333333333333d0)) - 2.0d0)))
    else if (im <= 2.9d+60) then
        tmp = 0.5d0 * ((im * (-2.0d0)) + ((re ** 4.0d0) * (im * (-0.08333333333333333d0))))
    else
        tmp = 0.5d0 * ((-0.016666666666666666d0) * (cos(re) * (im ** 5.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 450000000.0) {
		tmp = 0.5 * (im * (Math.cos(re) * ((Math.pow(im, 2.0) * -0.3333333333333333) - 2.0)));
	} else if (im <= 2.9e+60) {
		tmp = 0.5 * ((im * -2.0) + (Math.pow(re, 4.0) * (im * -0.08333333333333333)));
	} else {
		tmp = 0.5 * (-0.016666666666666666 * (Math.cos(re) * Math.pow(im, 5.0)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 450000000.0:
		tmp = 0.5 * (im * (math.cos(re) * ((math.pow(im, 2.0) * -0.3333333333333333) - 2.0)))
	elif im <= 2.9e+60:
		tmp = 0.5 * ((im * -2.0) + (math.pow(re, 4.0) * (im * -0.08333333333333333)))
	else:
		tmp = 0.5 * (-0.016666666666666666 * (math.cos(re) * math.pow(im, 5.0)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 450000000.0)
		tmp = Float64(0.5 * Float64(im * Float64(cos(re) * Float64(Float64((im ^ 2.0) * -0.3333333333333333) - 2.0))));
	elseif (im <= 2.9e+60)
		tmp = Float64(0.5 * Float64(Float64(im * -2.0) + Float64((re ^ 4.0) * Float64(im * -0.08333333333333333))));
	else
		tmp = Float64(0.5 * Float64(-0.016666666666666666 * Float64(cos(re) * (im ^ 5.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 450000000.0)
		tmp = 0.5 * (im * (cos(re) * (((im ^ 2.0) * -0.3333333333333333) - 2.0)));
	elseif (im <= 2.9e+60)
		tmp = 0.5 * ((im * -2.0) + ((re ^ 4.0) * (im * -0.08333333333333333)));
	else
		tmp = 0.5 * (-0.016666666666666666 * (cos(re) * (im ^ 5.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 450000000.0], 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, 2.9e+60], N[(0.5 * N[(N[(im * -2.0), $MachinePrecision] + N[(N[Power[re, 4.0], $MachinePrecision] * N[(im * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(-0.016666666666666666 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 4.5e8

   1. Initial program 42.1%

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

      1. /-rgt-identity 42.1%

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

      2. exp-0 42.1%

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

      3. associate-*l/ 42.1%

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

      4. cos-neg 42.1%

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

      5. associate-*l* 42.1%

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

      6. associate-*r/ 42.1%

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

      7. exp-0 42.1%

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

      8. /-rgt-identity 42.1%

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

      9. *-commutative 42.1%

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

      10. neg-sub0 42.1%

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

      11. cos-neg 42.1%

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

   3. Simplified 42.1%

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

   5. Taylor expanded in im around 0 88.4%

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

   6. Taylor expanded in re around inf 88.4%

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

   if 4.5e8 < im < 2.9e60

   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 3.4%

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

   6. Taylor expanded in re around 0 31.4%

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

   7. Taylor expanded in re around inf 31.4%

      \[\leadsto 0.5 \cdot \left(-2 \cdot im + \color{blue}{-0.08333333333333333 \cdot \left(im \cdot {re}^{4}\right)}\right) \]
   8. Step-by-step derivation

      1. associate-*r* 31.4%

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

      2. *-commutative 31.4%

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

      3. *-commutative 31.4%

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

   9. Simplified 31.4%

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

   if 2.9e60 < 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. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. *-commutative 100.0%

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

      5. fma-undefine 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around inf 100.0%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 450000000:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(\cos re \cdot \left({im}^{2} \cdot -0.3333333333333333 - 2\right)\right)\right)\\ \mathbf{elif}\;im \leq 2.9 \cdot 10^{+60}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + {re}^{4} \cdot \left(im \cdot -0.08333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.016666666666666666 \cdot \left(\cos re \cdot {im}^{5}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.3:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;im \leq 2.9 \cdot 10^{+60}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + {re}^{4} \cdot \left(im \cdot -0.08333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.016666666666666666 \cdot {im}^{5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.3)
   (* 0.5 (* (cos re) (* im -2.0)))
   (if (<= im 2.9e+60)
     (* 0.5 (+ (* im -2.0) (* (pow re 4.0) (* im -0.08333333333333333))))
     (* 0.5 (* -0.016666666666666666 (pow im 5.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.3) {
		tmp = 0.5 * (cos(re) * (im * -2.0));
	} else if (im <= 2.9e+60) {
		tmp = 0.5 * ((im * -2.0) + (pow(re, 4.0) * (im * -0.08333333333333333)));
	} 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 <= 0.3d0) then
        tmp = 0.5d0 * (cos(re) * (im * (-2.0d0)))
    else if (im <= 2.9d+60) then
        tmp = 0.5d0 * ((im * (-2.0d0)) + ((re ** 4.0d0) * (im * (-0.08333333333333333d0))))
    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 <= 0.3) {
		tmp = 0.5 * (Math.cos(re) * (im * -2.0));
	} else if (im <= 2.9e+60) {
		tmp = 0.5 * ((im * -2.0) + (Math.pow(re, 4.0) * (im * -0.08333333333333333)));
	} else {
		tmp = 0.5 * (-0.016666666666666666 * Math.pow(im, 5.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.3:
		tmp = 0.5 * (math.cos(re) * (im * -2.0))
	elif im <= 2.9e+60:
		tmp = 0.5 * ((im * -2.0) + (math.pow(re, 4.0) * (im * -0.08333333333333333)))
	else:
		tmp = 0.5 * (-0.016666666666666666 * math.pow(im, 5.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.3)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im * -2.0)));
	elseif (im <= 2.9e+60)
		tmp = Float64(0.5 * Float64(Float64(im * -2.0) + Float64((re ^ 4.0) * Float64(im * -0.08333333333333333))));
	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 <= 0.3)
		tmp = 0.5 * (cos(re) * (im * -2.0));
	elseif (im <= 2.9e+60)
		tmp = 0.5 * ((im * -2.0) + ((re ^ 4.0) * (im * -0.08333333333333333)));
	else
		tmp = 0.5 * (-0.016666666666666666 * (im ^ 5.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.3], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.9e+60], N[(0.5 * N[(N[(im * -2.0), $MachinePrecision] + N[(N[Power[re, 4.0], $MachinePrecision] * N[(im * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(-0.016666666666666666 * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.299999999999999989

   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 65.8%

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

   if 0.299999999999999989 < im < 2.9e60

   1. Initial program 99.9%

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

      1. /-rgt-identity 99.9%

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

      2. exp-0 99.9%

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

      3. associate-*l/ 99.9%

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

      4. cos-neg 99.9%

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

      5. associate-*l* 99.9%

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

      6. associate-*r/ 99.9%

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

      7. exp-0 99.9%

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

      8. /-rgt-identity 99.9%

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

      9. *-commutative 99.9%

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

      10. neg-sub0 99.9%

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

      11. cos-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in im around 0 7.3%

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

   6. Taylor expanded in re around 0 27.3%

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

   7. Taylor expanded in re around inf 27.3%

      \[\leadsto 0.5 \cdot \left(-2 \cdot im + \color{blue}{-0.08333333333333333 \cdot \left(im \cdot {re}^{4}\right)}\right) \]
   8. Step-by-step derivation

      1. associate-*r* 27.3%

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

      2. *-commutative 27.3%

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

      3. *-commutative 27.3%

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

   9. Simplified 27.3%

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

   if 2.9e60 < 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. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. *-commutative 100.0%

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

      5. fma-undefine 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around inf 100.0%

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

   9. Taylor expanded in re around 0 74.1%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.3:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;im \leq 2.9 \cdot 10^{+60}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + {re}^{4} \cdot \left(im \cdot -0.08333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.016666666666666666 \cdot {im}^{5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 9.8 \cdot 10^{+24}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.016666666666666666 \cdot {im}^{5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 9.8e+24)
   (* 0.5 (* (cos re) (* im -2.0)))
   (* 0.5 (* -0.016666666666666666 (pow im 5.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 9.8e+24) {
		tmp = 0.5 * (cos(re) * (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 <= 9.8d+24) then
        tmp = 0.5d0 * (cos(re) * (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 <= 9.8e+24) {
		tmp = 0.5 * (Math.cos(re) * (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 <= 9.8e+24:
		tmp = 0.5 * (math.cos(re) * (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 <= 9.8e+24)
		tmp = Float64(0.5 * Float64(cos(re) * 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 <= 9.8e+24)
		tmp = 0.5 * (cos(re) * (im * -2.0));
	else
		tmp = 0.5 * (-0.016666666666666666 * (im ^ 5.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 9.8e+24], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im * -2.0), $MachinePrecision]), $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 < 9.80000000000000059e24

   1. Initial program 42.7%

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

      1. /-rgt-identity 42.7%

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

      2. exp-0 42.7%

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

      3. associate-*l/ 42.7%

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

      4. cos-neg 42.7%

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

      5. associate-*l* 42.7%

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

      6. associate-*r/ 42.7%

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

      7. exp-0 42.7%

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

      8. /-rgt-identity 42.7%

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

      9. *-commutative 42.7%

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

      10. neg-sub0 42.7%

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

      11. cos-neg 42.7%

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

   3. Simplified 42.7%

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

   5. Taylor expanded in im around 0 64.1%

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

   if 9.80000000000000059e24 < 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 88.0%

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

      1. distribute-lft-in 88.0%

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

      2. +-commutative 88.0%

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

      3. associate-*r* 88.0%

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

      4. *-commutative 88.0%

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

      5. fma-undefine 88.0%

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

   7. Simplified 88.0%

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

   8. Taylor expanded in im around inf 88.0%

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

   9. Taylor expanded in re around 0 65.2%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 9.8 \cdot 10^{+24}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.016666666666666666 \cdot {im}^{5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 43.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.3:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.016666666666666666 \cdot {im}^{5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.3)
   (* 0.5 (* im -2.0))
   (* 0.5 (* -0.016666666666666666 (pow im 5.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.3) {
		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 <= 3.3d0) 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 <= 3.3) {
		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 <= 3.3:
		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 <= 3.3)
		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 <= 3.3)
		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, 3.3], 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 < 3.2999999999999998

   1. Initial program 41.5%

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

      1. /-rgt-identity 41.5%

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

      2. exp-0 41.5%

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

      3. associate-*l/ 41.5%

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

      4. cos-neg 41.5%

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

      5. associate-*l* 41.5%

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

      6. associate-*r/ 41.5%

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

      7. exp-0 41.5%

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

      8. /-rgt-identity 41.5%

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

      9. *-commutative 41.5%

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

      10. neg-sub0 41.5%

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

      11. cos-neg 41.5%

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

   3. Simplified 41.5%

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

   5. Taylor expanded in im around 0 65.3%

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

   6. Taylor expanded in re around 0 38.0%

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

      1. *-commutative 38.0%

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

   8. Simplified 38.0%

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

   if 3.2999999999999998 < 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 83.0%

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

      1. distribute-lft-in 83.0%

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

      2. +-commutative 83.0%

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

      3. associate-*r* 83.0%

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

      4. *-commutative 83.0%

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

      5. fma-undefine 83.0%

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

   7. Simplified 83.0%

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

   8. Taylor expanded in im around inf 83.0%

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

   9. Taylor expanded in re around 0 61.6%

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

4. Final simplification 44.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.3:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-0.016666666666666666 \cdot {im}^{5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 29.1% 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 56.6%

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

   1. /-rgt-identity 56.6%

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

   2. exp-0 56.6%

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

   3. associate-*l/ 56.6%

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

   4. cos-neg 56.6%

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

   5. associate-*l* 56.6%

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

   6. associate-*r/ 56.6%

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

   7. exp-0 56.6%

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

   8. /-rgt-identity 56.6%

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

   9. *-commutative 56.6%

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

   10. neg-sub0 56.6%

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

   11. cos-neg 56.6%

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

3. Simplified 56.6%

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

5. Taylor expanded in im around 0 49.9%

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

6. Taylor expanded in re around 0 29.3%

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

   1. *-commutative 29.3%

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

8. Simplified 29.3%

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

9. Final simplification 29.3%

   \[\leadsto 0.5 \cdot \left(im \cdot -2\right) \]
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 2024055 
(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))))