math.sin on complex, imaginary part

Percentage Accurate: 54.7% → 98.9%

Time: 10.5s

Alternatives: 11

Speedup: 2.8×

• 50.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.7% 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: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 57.0%

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

   1. sub-neg 57.0%

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

   2. neg-sub0 57.0%

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

   3. remove-double-neg 57.0%

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

   4. remove-double-neg 57.0%

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

   5. sub0-neg 57.0%

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

   6. distribute-neg-in 57.0%

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

   7. +-commutative 57.0%

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

   8. sub-neg 57.0%

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

   9. cos-neg 57.0%

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

   10. associate-*l* 57.0%

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

   11. distribute-rgt-neg-in 57.0%

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

   12. *-commutative 57.0%

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

3. Simplified 57.0%

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

5. Taylor expanded in im around 0 50.6%

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

   1. log1p-expm1-u 99.0%

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

   2. associate-*r* 99.0%

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

   3. *-commutative 99.0%

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

   4. associate-*r* 99.0%

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

   5. metadata-eval 99.0%

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

7. Applied egg-rr 99.0%

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

8. Taylor expanded in re around inf 56.4%

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

   1. expm1-def 99.0%

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

   2. associate-*r* 99.0%

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

   3. mul-1-neg 99.0%

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

10. Simplified 99.0%

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

11. Final simplification 99.0%

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

Alternative 2: 90.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.0095:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2 + -0.3333333333333333 \cdot {im}^{3}\right)\right)\\ \mathbf{elif}\;im \leq 1.06 \cdot 10^{+44}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(-0.0003968253968253968 \cdot {im}^{7}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.0095)
   (* 0.5 (* (cos re) (+ (* im -2.0) (* -0.3333333333333333 (pow im 3.0)))))
   (if (<= im 1.06e+44)
     (* 0.5 (- (exp (- im)) (exp im)))
     (* 0.5 (* (cos re) (* -0.0003968253968253968 (pow im 7.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.0095) {
		tmp = 0.5 * (cos(re) * ((im * -2.0) + (-0.3333333333333333 * pow(im, 3.0))));
	} else if (im <= 1.06e+44) {
		tmp = 0.5 * (exp(-im) - exp(im));
	} else {
		tmp = 0.5 * (cos(re) * (-0.0003968253968253968 * pow(im, 7.0)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.0095d0) then
        tmp = 0.5d0 * (cos(re) * ((im * (-2.0d0)) + ((-0.3333333333333333d0) * (im ** 3.0d0))))
    else if (im <= 1.06d+44) then
        tmp = 0.5d0 * (exp(-im) - exp(im))
    else
        tmp = 0.5d0 * (cos(re) * ((-0.0003968253968253968d0) * (im ** 7.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.0095) {
		tmp = 0.5 * (Math.cos(re) * ((im * -2.0) + (-0.3333333333333333 * Math.pow(im, 3.0))));
	} else if (im <= 1.06e+44) {
		tmp = 0.5 * (Math.exp(-im) - Math.exp(im));
	} else {
		tmp = 0.5 * (Math.cos(re) * (-0.0003968253968253968 * Math.pow(im, 7.0)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.0095:
		tmp = 0.5 * (math.cos(re) * ((im * -2.0) + (-0.3333333333333333 * math.pow(im, 3.0))))
	elif im <= 1.06e+44:
		tmp = 0.5 * (math.exp(-im) - math.exp(im))
	else:
		tmp = 0.5 * (math.cos(re) * (-0.0003968253968253968 * math.pow(im, 7.0)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.0095)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(Float64(im * -2.0) + Float64(-0.3333333333333333 * (im ^ 3.0)))));
	elseif (im <= 1.06e+44)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) - exp(im)));
	else
		tmp = Float64(0.5 * Float64(cos(re) * Float64(-0.0003968253968253968 * (im ^ 7.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.0095)
		tmp = 0.5 * (cos(re) * ((im * -2.0) + (-0.3333333333333333 * (im ^ 3.0))));
	elseif (im <= 1.06e+44)
		tmp = 0.5 * (exp(-im) - exp(im));
	else
		tmp = 0.5 * (cos(re) * (-0.0003968253968253968 * (im ^ 7.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.0095], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(N[(im * -2.0), $MachinePrecision] + N[(-0.3333333333333333 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.06e+44], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(-0.0003968253968253968 * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.00949999999999999976

   1. Initial program 40.3%

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

      1. sub-neg 40.3%

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

      2. neg-sub0 40.3%

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

      3. remove-double-neg 40.3%

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

      4. remove-double-neg 40.3%

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

      5. sub0-neg 40.3%

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

      6. distribute-neg-in 40.3%

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

      7. +-commutative 40.3%

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

      8. sub-neg 40.3%

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

      9. cos-neg 40.3%

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

      10. associate-*l* 40.3%

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

      11. distribute-rgt-neg-in 40.3%

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

      12. *-commutative 40.3%

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

   3. Simplified 40.3%

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

   5. Taylor expanded in im around 0 90.9%

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

   if 0.00949999999999999976 < im < 1.06e44

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. neg-sub0 99.6%

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

      3. remove-double-neg 99.6%

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

      4. remove-double-neg 99.6%

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

      5. sub0-neg 99.6%

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

      6. distribute-neg-in 99.6%

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

      7. +-commutative 99.6%

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

      8. sub-neg 99.6%

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

      9. cos-neg 99.6%

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

      10. associate-*l* 99.6%

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

      11. distribute-rgt-neg-in 99.6%

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

      12. *-commutative 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in re around 0 63.2%

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

   if 1.06e44 < 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. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. remove-double-neg 100.0%

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

      4. remove-double-neg 100.0%

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

      5. sub0-neg 100.0%

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

      6. distribute-neg-in 100.0%

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

      7. +-commutative 100.0%

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

      8. sub-neg 100.0%

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

      9. cos-neg 100.0%

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

      10. associate-*l* 100.0%

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

      11. distribute-rgt-neg-in 100.0%

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

      12. *-commutative 100.0%

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

   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%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(-2 \cdot im + \left(-0.3333333333333333 \cdot {im}^{3} + \left(-0.016666666666666666 \cdot {im}^{5} + -0.0003968253968253968 \cdot {im}^{7}\right)\right)\right)} \cdot \cos re\right) \]

   6. Taylor expanded in im around inf 100.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.0003968253968253968 \cdot \left({im}^{7} \cdot \cos re\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(-0.0003968253968253968 \cdot {im}^{7}\right) \cdot \cos re\right)} \]

      2. *-commutative 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\cos re \cdot \left(-0.0003968253968253968 \cdot {im}^{7}\right)\right)} \]

   8. Simplified 100.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\cos re \cdot \left(-0.0003968253968253968 \cdot {im}^{7}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.0095:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2 + -0.3333333333333333 \cdot {im}^{3}\right)\right)\\ \mathbf{elif}\;im \leq 1.06 \cdot 10^{+44}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(-0.0003968253968253968 \cdot {im}^{7}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 460:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 1.06 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(-im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(-0.0003968253968253968 \cdot {im}^{7}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 460.0)
   (* im (- (cos re)))
   (if (<= im 1.06e+44)
     (log1p (expm1 (- im)))
     (* 0.5 (* (cos re) (* -0.0003968253968253968 (pow im 7.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 460.0) {
		tmp = im * -cos(re);
	} else if (im <= 1.06e+44) {
		tmp = log1p(expm1(-im));
	} else {
		tmp = 0.5 * (cos(re) * (-0.0003968253968253968 * pow(im, 7.0)));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 460.0) {
		tmp = im * -Math.cos(re);
	} else if (im <= 1.06e+44) {
		tmp = Math.log1p(Math.expm1(-im));
	} else {
		tmp = 0.5 * (Math.cos(re) * (-0.0003968253968253968 * Math.pow(im, 7.0)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 460.0:
		tmp = im * -math.cos(re)
	elif im <= 1.06e+44:
		tmp = math.log1p(math.expm1(-im))
	else:
		tmp = 0.5 * (math.cos(re) * (-0.0003968253968253968 * math.pow(im, 7.0)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 460.0)
		tmp = Float64(im * Float64(-cos(re)));
	elseif (im <= 1.06e+44)
		tmp = log1p(expm1(Float64(-im)));
	else
		tmp = Float64(0.5 * Float64(cos(re) * Float64(-0.0003968253968253968 * (im ^ 7.0))));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 460.0], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im, 1.06e+44], N[Log[1 + N[(Exp[(-im)] - 1), $MachinePrecision]], $MachinePrecision], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(-0.0003968253968253968 * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 460

   1. Initial program 40.6%

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

      1. sub-neg 40.6%

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

      2. neg-sub0 40.6%

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

      3. remove-double-neg 40.6%

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

      4. remove-double-neg 40.6%

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

      5. sub0-neg 40.6%

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

      6. distribute-neg-in 40.6%

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

      7. +-commutative 40.6%

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

      8. sub-neg 40.6%

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

      9. cos-neg 40.6%

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

      10. associate-*l* 40.6%

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

      11. distribute-rgt-neg-in 40.6%

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

      12. *-commutative 40.6%

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

   3. Simplified 40.6%

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

   5. Taylor expanded in im around 0 67.4%

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

   6. Taylor expanded in im around 0 67.4%

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

      1. mul-1-neg 67.4%

         \[\leadsto \color{blue}{-im \cdot \cos re} \]

      2. distribute-rgt-neg-in 67.4%

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

   8. Simplified 67.4%

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

   if 460 < im < 1.06e44

   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. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. remove-double-neg 100.0%

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

      4. remove-double-neg 100.0%

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

      5. sub0-neg 100.0%

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

      6. distribute-neg-in 100.0%

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

      7. +-commutative 100.0%

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

      8. sub-neg 100.0%

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

      9. cos-neg 100.0%

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

      10. associate-*l* 100.0%

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

      11. distribute-rgt-neg-in 100.0%

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

      12. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 3.3%

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

      1. log1p-expm1-u 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*r* 100.0%

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

      5. metadata-eval 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in re around 0 60.0%

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

      1. expm1-def 60.0%

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

      2. mul-1-neg 60.0%

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

   10. Simplified 60.0%

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

   if 1.06e44 < 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. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. remove-double-neg 100.0%

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

      4. remove-double-neg 100.0%

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

      5. sub0-neg 100.0%

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

      6. distribute-neg-in 100.0%

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

      7. +-commutative 100.0%

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

      8. sub-neg 100.0%

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

      9. cos-neg 100.0%

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

      10. associate-*l* 100.0%

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

      11. distribute-rgt-neg-in 100.0%

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

      12. *-commutative 100.0%

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

   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%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(-2 \cdot im + \left(-0.3333333333333333 \cdot {im}^{3} + \left(-0.016666666666666666 \cdot {im}^{5} + -0.0003968253968253968 \cdot {im}^{7}\right)\right)\right)} \cdot \cos re\right) \]

   6. Taylor expanded in im around inf 100.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.0003968253968253968 \cdot \left({im}^{7} \cdot \cos re\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(-0.0003968253968253968 \cdot {im}^{7}\right) \cdot \cos re\right)} \]

      2. *-commutative 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\cos re \cdot \left(-0.0003968253968253968 \cdot {im}^{7}\right)\right)} \]

   8. Simplified 100.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\cos re \cdot \left(-0.0003968253968253968 \cdot {im}^{7}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 460:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 1.06 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(-im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(-0.0003968253968253968 \cdot {im}^{7}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.009:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 1.06 \cdot 10^{+44}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(-0.0003968253968253968 \cdot {im}^{7}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.009)
   (* im (- (cos re)))
   (if (<= im 1.06e+44)
     (* 0.5 (- (exp (- im)) (exp im)))
     (* 0.5 (* (cos re) (* -0.0003968253968253968 (pow im 7.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.009) {
		tmp = im * -cos(re);
	} else if (im <= 1.06e+44) {
		tmp = 0.5 * (exp(-im) - exp(im));
	} else {
		tmp = 0.5 * (cos(re) * (-0.0003968253968253968 * pow(im, 7.0)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.009d0) then
        tmp = im * -cos(re)
    else if (im <= 1.06d+44) then
        tmp = 0.5d0 * (exp(-im) - exp(im))
    else
        tmp = 0.5d0 * (cos(re) * ((-0.0003968253968253968d0) * (im ** 7.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.009) {
		tmp = im * -Math.cos(re);
	} else if (im <= 1.06e+44) {
		tmp = 0.5 * (Math.exp(-im) - Math.exp(im));
	} else {
		tmp = 0.5 * (Math.cos(re) * (-0.0003968253968253968 * Math.pow(im, 7.0)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.009:
		tmp = im * -math.cos(re)
	elif im <= 1.06e+44:
		tmp = 0.5 * (math.exp(-im) - math.exp(im))
	else:
		tmp = 0.5 * (math.cos(re) * (-0.0003968253968253968 * math.pow(im, 7.0)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.009)
		tmp = Float64(im * Float64(-cos(re)));
	elseif (im <= 1.06e+44)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) - exp(im)));
	else
		tmp = Float64(0.5 * Float64(cos(re) * Float64(-0.0003968253968253968 * (im ^ 7.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.009)
		tmp = im * -cos(re);
	elseif (im <= 1.06e+44)
		tmp = 0.5 * (exp(-im) - exp(im));
	else
		tmp = 0.5 * (cos(re) * (-0.0003968253968253968 * (im ^ 7.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.009], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im, 1.06e+44], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(-0.0003968253968253968 * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.00899999999999999932

   1. Initial program 40.3%

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

      1. sub-neg 40.3%

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

      2. neg-sub0 40.3%

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

      3. remove-double-neg 40.3%

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

      4. remove-double-neg 40.3%

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

      5. sub0-neg 40.3%

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

      6. distribute-neg-in 40.3%

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

      7. +-commutative 40.3%

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

      8. sub-neg 40.3%

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

      9. cos-neg 40.3%

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

      10. associate-*l* 40.3%

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

      11. distribute-rgt-neg-in 40.3%

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

      12. *-commutative 40.3%

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

   3. Simplified 40.3%

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

   5. Taylor expanded in im around 0 67.6%

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

   6. Taylor expanded in im around 0 67.6%

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

      1. mul-1-neg 67.6%

         \[\leadsto \color{blue}{-im \cdot \cos re} \]

      2. distribute-rgt-neg-in 67.6%

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

   8. Simplified 67.6%

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

   if 0.00899999999999999932 < im < 1.06e44

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. neg-sub0 99.6%

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

      3. remove-double-neg 99.6%

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

      4. remove-double-neg 99.6%

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

      5. sub0-neg 99.6%

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

      6. distribute-neg-in 99.6%

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

      7. +-commutative 99.6%

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

      8. sub-neg 99.6%

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

      9. cos-neg 99.6%

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

      10. associate-*l* 99.6%

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

      11. distribute-rgt-neg-in 99.6%

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

      12. *-commutative 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in re around 0 63.2%

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

   if 1.06e44 < 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. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. remove-double-neg 100.0%

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

      4. remove-double-neg 100.0%

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

      5. sub0-neg 100.0%

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

      6. distribute-neg-in 100.0%

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

      7. +-commutative 100.0%

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

      8. sub-neg 100.0%

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

      9. cos-neg 100.0%

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

      10. associate-*l* 100.0%

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

      11. distribute-rgt-neg-in 100.0%

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

      12. *-commutative 100.0%

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

   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%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(-2 \cdot im + \left(-0.3333333333333333 \cdot {im}^{3} + \left(-0.016666666666666666 \cdot {im}^{5} + -0.0003968253968253968 \cdot {im}^{7}\right)\right)\right)} \cdot \cos re\right) \]

   6. Taylor expanded in im around inf 100.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.0003968253968253968 \cdot \left({im}^{7} \cdot \cos re\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(-0.0003968253968253968 \cdot {im}^{7}\right) \cdot \cos re\right)} \]

      2. *-commutative 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\cos re \cdot \left(-0.0003968253968253968 \cdot {im}^{7}\right)\right)} \]

   8. Simplified 100.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\cos re \cdot \left(-0.0003968253968253968 \cdot {im}^{7}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.009:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 1.06 \cdot 10^{+44}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(-0.0003968253968253968 \cdot {im}^{7}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.0095:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(-im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.0095) (* im (- (cos re))) (log1p (expm1 (- im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.0095) {
		tmp = im * -cos(re);
	} else {
		tmp = log1p(expm1(-im));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.0095) {
		tmp = im * -Math.cos(re);
	} else {
		tmp = Math.log1p(Math.expm1(-im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.0095:
		tmp = im * -math.cos(re)
	else:
		tmp = math.log1p(math.expm1(-im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.0095)
		tmp = Float64(im * Float64(-cos(re)));
	else
		tmp = log1p(expm1(Float64(-im)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.0095], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], N[Log[1 + N[(Exp[(-im)] - 1), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 0.00949999999999999976

   1. Initial program 40.3%

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

      1. sub-neg 40.3%

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

      2. neg-sub0 40.3%

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

      3. remove-double-neg 40.3%

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

      4. remove-double-neg 40.3%

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

      5. sub0-neg 40.3%

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

      6. distribute-neg-in 40.3%

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

      7. +-commutative 40.3%

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

      8. sub-neg 40.3%

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

      9. cos-neg 40.3%

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

      10. associate-*l* 40.3%

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

      11. distribute-rgt-neg-in 40.3%

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

      12. *-commutative 40.3%

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

   3. Simplified 40.3%

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

   5. Taylor expanded in im around 0 67.6%

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

   6. Taylor expanded in im around 0 67.6%

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

      1. mul-1-neg 67.6%

         \[\leadsto \color{blue}{-im \cdot \cos re} \]

      2. distribute-rgt-neg-in 67.6%

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

   8. Simplified 67.6%

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

   if 0.00949999999999999976 < im

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. neg-sub0 99.9%

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

      3. remove-double-neg 99.9%

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

      4. remove-double-neg 99.9%

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

      5. sub0-neg 99.9%

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

      6. distribute-neg-in 99.9%

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

      7. +-commutative 99.9%

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

      8. sub-neg 99.9%

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

      9. cos-neg 99.9%

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

      10. associate-*l* 99.9%

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

      11. distribute-rgt-neg-in 99.9%

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

      12. *-commutative 99.9%

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

   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. Step-by-step derivation

      1. log1p-expm1-u 99.2%

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

      2. associate-*r* 99.2%

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

      3. *-commutative 99.2%

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

      4. associate-*r* 99.2%

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

      5. metadata-eval 99.2%

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

   7. Applied egg-rr 99.2%

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

   8. Taylor expanded in re around 0 70.0%

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

      1. expm1-def 70.0%

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

      2. mul-1-neg 70.0%

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

   10. Simplified 70.0%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.0095:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(-im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 580:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 3.6 \cdot 10^{+84}:\\ \;\;\;\;{re}^{2} \cdot \left(im \cdot 0.5\right) - im\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + -0.3333333333333333 \cdot {im}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 580.0)
   (* im (- (cos re)))
   (if (<= im 3.6e+84)
     (- (* (pow re 2.0) (* im 0.5)) im)
     (* 0.5 (+ (* im -2.0) (* -0.3333333333333333 (pow im 3.0)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 580.0) {
		tmp = im * -cos(re);
	} else if (im <= 3.6e+84) {
		tmp = (pow(re, 2.0) * (im * 0.5)) - im;
	} else {
		tmp = 0.5 * ((im * -2.0) + (-0.3333333333333333 * pow(im, 3.0)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 580.0d0) then
        tmp = im * -cos(re)
    else if (im <= 3.6d+84) then
        tmp = ((re ** 2.0d0) * (im * 0.5d0)) - im
    else
        tmp = 0.5d0 * ((im * (-2.0d0)) + ((-0.3333333333333333d0) * (im ** 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 580.0) {
		tmp = im * -Math.cos(re);
	} else if (im <= 3.6e+84) {
		tmp = (Math.pow(re, 2.0) * (im * 0.5)) - im;
	} else {
		tmp = 0.5 * ((im * -2.0) + (-0.3333333333333333 * Math.pow(im, 3.0)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 580.0:
		tmp = im * -math.cos(re)
	elif im <= 3.6e+84:
		tmp = (math.pow(re, 2.0) * (im * 0.5)) - im
	else:
		tmp = 0.5 * ((im * -2.0) + (-0.3333333333333333 * math.pow(im, 3.0)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 580.0)
		tmp = Float64(im * Float64(-cos(re)));
	elseif (im <= 3.6e+84)
		tmp = Float64(Float64((re ^ 2.0) * Float64(im * 0.5)) - im);
	else
		tmp = Float64(0.5 * Float64(Float64(im * -2.0) + Float64(-0.3333333333333333 * (im ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 580.0)
		tmp = im * -cos(re);
	elseif (im <= 3.6e+84)
		tmp = ((re ^ 2.0) * (im * 0.5)) - im;
	else
		tmp = 0.5 * ((im * -2.0) + (-0.3333333333333333 * (im ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 580.0], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im, 3.6e+84], N[(N[(N[Power[re, 2.0], $MachinePrecision] * N[(im * 0.5), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], N[(0.5 * N[(N[(im * -2.0), $MachinePrecision] + N[(-0.3333333333333333 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 580

   1. Initial program 40.6%

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

      1. sub-neg 40.6%

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

      2. neg-sub0 40.6%

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

      3. remove-double-neg 40.6%

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

      4. remove-double-neg 40.6%

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

      5. sub0-neg 40.6%

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

      6. distribute-neg-in 40.6%

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

      7. +-commutative 40.6%

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

      8. sub-neg 40.6%

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

      9. cos-neg 40.6%

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

      10. associate-*l* 40.6%

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

      11. distribute-rgt-neg-in 40.6%

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

      12. *-commutative 40.6%

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

   3. Simplified 40.6%

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

   5. Taylor expanded in im around 0 67.4%

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

   6. Taylor expanded in im around 0 67.4%

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

      1. mul-1-neg 67.4%

         \[\leadsto \color{blue}{-im \cdot \cos re} \]

      2. distribute-rgt-neg-in 67.4%

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

   8. Simplified 67.4%

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

   if 580 < im < 3.5999999999999999e84

   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. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. remove-double-neg 100.0%

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

      4. remove-double-neg 100.0%

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

      5. sub0-neg 100.0%

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

      6. distribute-neg-in 100.0%

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

      7. +-commutative 100.0%

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

      8. sub-neg 100.0%

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

      9. cos-neg 100.0%

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

      10. associate-*l* 100.0%

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

      11. distribute-rgt-neg-in 100.0%

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

      12. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 3.5%

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

   6. Taylor expanded in re around 0 25.9%

      \[\leadsto \color{blue}{-1 \cdot im + 0.5 \cdot \left(im \cdot {re}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative 25.9%

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

      2. mul-1-neg 25.9%

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

      3. unsub-neg 25.9%

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

      4. *-commutative 25.9%

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

      5. *-commutative 25.9%

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

      6. associate-*l* 25.9%

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

   8. Simplified 25.9%

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

   if 3.5999999999999999e84 < 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. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. remove-double-neg 100.0%

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

      4. remove-double-neg 100.0%

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

      5. sub0-neg 100.0%

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

      6. distribute-neg-in 100.0%

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

      7. +-commutative 100.0%

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

      8. sub-neg 100.0%

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

      9. cos-neg 100.0%

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

      10. associate-*l* 100.0%

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

      11. distribute-rgt-neg-in 100.0%

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

      12. *-commutative 100.0%

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

   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%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(-2 \cdot im + \left(-0.3333333333333333 \cdot {im}^{3} + -0.016666666666666666 \cdot {im}^{5}\right)\right)} \cdot \cos re\right) \]

   6. Taylor expanded in im around 0 94.9%

      \[\leadsto 0.5 \cdot \left(\left(-2 \cdot im + \color{blue}{-0.3333333333333333 \cdot {im}^{3}}\right) \cdot \cos re\right) \]
   7. Step-by-step derivation

      1. *-commutative 94.9%

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

   8. Simplified 94.9%

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

   9. Taylor expanded in re around 0 70.9%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 580:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 3.6 \cdot 10^{+84}:\\ \;\;\;\;{re}^{2} \cdot \left(im \cdot 0.5\right) - im\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + -0.3333333333333333 \cdot {im}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 55.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.0095:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;{re}^{2} \cdot \left(im \cdot 0.5\right) - im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.0095) (* im (- (cos re))) (- (* (pow re 2.0) (* im 0.5)) im)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.0095) {
		tmp = im * -cos(re);
	} else {
		tmp = (pow(re, 2.0) * (im * 0.5)) - im;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.0095d0) then
        tmp = im * -cos(re)
    else
        tmp = ((re ** 2.0d0) * (im * 0.5d0)) - im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.0095) {
		tmp = im * -Math.cos(re);
	} else {
		tmp = (Math.pow(re, 2.0) * (im * 0.5)) - im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.0095:
		tmp = im * -math.cos(re)
	else:
		tmp = (math.pow(re, 2.0) * (im * 0.5)) - im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.0095)
		tmp = Float64(im * Float64(-cos(re)));
	else
		tmp = Float64(Float64((re ^ 2.0) * Float64(im * 0.5)) - im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.0095)
		tmp = im * -cos(re);
	else
		tmp = ((re ^ 2.0) * (im * 0.5)) - im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.0095], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], N[(N[(N[Power[re, 2.0], $MachinePrecision] * N[(im * 0.5), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 0.00949999999999999976

   1. Initial program 40.3%

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

      1. sub-neg 40.3%

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

      2. neg-sub0 40.3%

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

      3. remove-double-neg 40.3%

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

      4. remove-double-neg 40.3%

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

      5. sub0-neg 40.3%

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

      6. distribute-neg-in 40.3%

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

      7. +-commutative 40.3%

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

      8. sub-neg 40.3%

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

      9. cos-neg 40.3%

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

      10. associate-*l* 40.3%

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

      11. distribute-rgt-neg-in 40.3%

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

      12. *-commutative 40.3%

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

   3. Simplified 40.3%

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

   5. Taylor expanded in im around 0 67.6%

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

   6. Taylor expanded in im around 0 67.6%

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

      1. mul-1-neg 67.6%

         \[\leadsto \color{blue}{-im \cdot \cos re} \]

      2. distribute-rgt-neg-in 67.6%

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

   8. Simplified 67.6%

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

   if 0.00949999999999999976 < im

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. neg-sub0 99.9%

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

      3. remove-double-neg 99.9%

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

      4. remove-double-neg 99.9%

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

      5. sub0-neg 99.9%

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

      6. distribute-neg-in 99.9%

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

      7. +-commutative 99.9%

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

      8. sub-neg 99.9%

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

      9. cos-neg 99.9%

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

      10. associate-*l* 99.9%

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

      11. distribute-rgt-neg-in 99.9%

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

      12. *-commutative 99.9%

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

   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 23.3%

      \[\leadsto \color{blue}{-1 \cdot im + 0.5 \cdot \left(im \cdot {re}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative 23.3%

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

      2. mul-1-neg 23.3%

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

      3. unsub-neg 23.3%

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

      4. *-commutative 23.3%

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

      5. *-commutative 23.3%

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

      6. associate-*l* 23.3%

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

   8. Simplified 23.3%

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

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.0095:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;{re}^{2} \cdot \left(im \cdot 0.5\right) - im\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.7% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.0%

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

   1. sub-neg 57.0%

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

   2. neg-sub0 57.0%

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

   3. remove-double-neg 57.0%

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

   4. remove-double-neg 57.0%

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

   5. sub0-neg 57.0%

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

   6. distribute-neg-in 57.0%

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

   7. +-commutative 57.0%

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

   8. sub-neg 57.0%

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

   9. cos-neg 57.0%

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

   10. associate-*l* 57.0%

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

   11. distribute-rgt-neg-in 57.0%

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

   12. *-commutative 57.0%

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

3. Simplified 57.0%

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

5. Taylor expanded in im around 0 50.6%

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

6. Final simplification 50.6%

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

Alternative 9: 51.6% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.0%

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

   1. sub-neg 57.0%

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

   2. neg-sub0 57.0%

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

   3. remove-double-neg 57.0%

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

   4. remove-double-neg 57.0%

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

   5. sub0-neg 57.0%

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

   6. distribute-neg-in 57.0%

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

   7. +-commutative 57.0%

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

   8. sub-neg 57.0%

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

   9. cos-neg 57.0%

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

   10. associate-*l* 57.0%

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

   11. distribute-rgt-neg-in 57.0%

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

   12. *-commutative 57.0%

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

3. Simplified 57.0%

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

5. Taylor expanded in im around 0 50.6%

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

6. Taylor expanded in im around 0 50.3%

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

   1. mul-1-neg 50.3%

      \[\leadsto \color{blue}{-im \cdot \cos re} \]

   2. distribute-rgt-neg-in 50.3%

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

8. Simplified 50.3%

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

9. Final simplification 50.3%

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

Alternative 10: 29.7% accurate, 61.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.0%

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

   1. sub-neg 57.0%

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

   2. neg-sub0 57.0%

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

   3. remove-double-neg 57.0%

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

   4. remove-double-neg 57.0%

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

   5. sub0-neg 57.0%

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

   6. distribute-neg-in 57.0%

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

   7. +-commutative 57.0%

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

   8. sub-neg 57.0%

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

   9. cos-neg 57.0%

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

   10. associate-*l* 57.0%

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

   11. distribute-rgt-neg-in 57.0%

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

   12. *-commutative 57.0%

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

3. Simplified 57.0%

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

5. Taylor expanded in im around 0 50.6%

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

6. Taylor expanded in re around 0 26.2%

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

7. Final simplification 26.2%

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

Alternative 11: 29.7% accurate, 154.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return -im

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.0%

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

   1. sub-neg 57.0%

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

   2. neg-sub0 57.0%

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

   3. remove-double-neg 57.0%

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

   4. remove-double-neg 57.0%

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

   5. sub0-neg 57.0%

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

   6. distribute-neg-in 57.0%

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

   7. +-commutative 57.0%

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

   8. sub-neg 57.0%

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

   9. cos-neg 57.0%

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

   10. associate-*l* 57.0%

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

   11. distribute-rgt-neg-in 57.0%

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

   12. *-commutative 57.0%

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

3. Simplified 57.0%

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

5. Taylor expanded in im around 0 50.6%

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

6. Taylor expanded in re around 0 25.9%

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

   1. mul-1-neg 25.9%

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

8. Simplified 25.9%

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

9. Final simplification 25.9%

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

Developer target: 99.8% accurate, 0.8× 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 2024010 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"
  :precision binary64

  :herbie-target
  (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))))