math.sin on complex, imaginary part

Percentage Accurate: 54.4% → 99.6%

Time: 9.4s

Alternatives: 11

Speedup: 2.8×

• 49.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{+53} \lor \neg \left(t_0 \leq 2 \cdot 10^{-10}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(\left({im}^{3} \cdot -0.16666666666666666 + {im}^{5} \cdot -0.008333333333333333\right) - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 -1e+53) (not (<= t_0 2e-10)))
     (* (* 0.5 (cos re)) t_0)
     (*
      (cos re)
      (-
       (+
        (* (pow im 3.0) -0.16666666666666666)
        (* (pow im 5.0) -0.008333333333333333))
       im)))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if ((t_0 <= -1e+53) || !(t_0 <= 2e-10)) {
		tmp = (0.5 * cos(re)) * t_0;
	} else {
		tmp = cos(re) * (((pow(im, 3.0) * -0.16666666666666666) + (pow(im, 5.0) * -0.008333333333333333)) - im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-im) - exp(im)
    if ((t_0 <= (-1d+53)) .or. (.not. (t_0 <= 2d-10))) then
        tmp = (0.5d0 * cos(re)) * t_0
    else
        tmp = cos(re) * ((((im ** 3.0d0) * (-0.16666666666666666d0)) + ((im ** 5.0d0) * (-0.008333333333333333d0))) - im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(-im) - Math.exp(im);
	double tmp;
	if ((t_0 <= -1e+53) || !(t_0 <= 2e-10)) {
		tmp = (0.5 * Math.cos(re)) * t_0;
	} else {
		tmp = Math.cos(re) * (((Math.pow(im, 3.0) * -0.16666666666666666) + (Math.pow(im, 5.0) * -0.008333333333333333)) - im);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	tmp = 0
	if (t_0 <= -1e+53) or not (t_0 <= 2e-10):
		tmp = (0.5 * math.cos(re)) * t_0
	else:
		tmp = math.cos(re) * (((math.pow(im, 3.0) * -0.16666666666666666) + (math.pow(im, 5.0) * -0.008333333333333333)) - im)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	tmp = 0.0
	if ((t_0 <= -1e+53) || !(t_0 <= 2e-10))
		tmp = Float64(Float64(0.5 * cos(re)) * t_0);
	else
		tmp = Float64(cos(re) * Float64(Float64(Float64((im ^ 3.0) * -0.16666666666666666) + Float64((im ^ 5.0) * -0.008333333333333333)) - im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	tmp = 0.0;
	if ((t_0 <= -1e+53) || ~((t_0 <= 2e-10)))
		tmp = (0.5 * cos(re)) * t_0;
	else
		tmp = cos(re) * ((((im ^ 3.0) * -0.16666666666666666) + ((im ^ 5.0) * -0.008333333333333333)) - im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -1e+53], N[Not[LessEqual[t$95$0, 2e-10]], $MachinePrecision]], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + N[(N[Power[im, 5.0], $MachinePrecision] * -0.008333333333333333), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < -9.9999999999999999e52 or 2.00000000000000007e-10 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 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. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   if -9.9999999999999999e52 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 2.00000000000000007e-10

   1. Initial program 8.1%

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

      1. neg-sub0 8.1%

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

   3. Simplified 8.1%

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

   4. Taylor expanded in im around 0 99.8%

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

      1. distribute-lft-in 99.8%

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

      2. fma-def 99.8%

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

      3. *-commutative 99.8%

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

      4. *-commutative 99.8%

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

      5. *-commutative 99.8%

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

      6. associate-*l* 99.8%

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

      7. *-commutative 99.8%

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

      8. fma-def 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in re around inf 99.8%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \cos re\right) + 0.5 \cdot \left(\cos re \cdot \left(-0.3333333333333333 \cdot {im}^{3} + -0.016666666666666666 \cdot {im}^{5}\right)\right)} \]

   9. Simplified 99.8%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left({im}^{5} \cdot -0.008333333333333333 + {im}^{3} \cdot -0.16666666666666666\right) - im\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -1 \cdot 10^{+53} \lor \neg \left(e^{-im} - e^{im} \leq 2 \cdot 10^{-10}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(\left({im}^{3} \cdot -0.16666666666666666 + {im}^{5} \cdot -0.008333333333333333\right) - im\right)\\ \end{array} \]

Alternative 2: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -0.004 \lor \neg \left(t_0 \leq 2 \cdot 10^{-10}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 -0.004) (not (<= t_0 2e-10)))
     (* (* 0.5 (cos re)) t_0)
     (* (cos re) (- (* (pow im 3.0) -0.16666666666666666) im)))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if ((t_0 <= -0.004) || !(t_0 <= 2e-10)) {
		tmp = (0.5 * cos(re)) * t_0;
	} else {
		tmp = cos(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-im) - exp(im)
    if ((t_0 <= (-0.004d0)) .or. (.not. (t_0 <= 2d-10))) then
        tmp = (0.5d0 * cos(re)) * t_0
    else
        tmp = cos(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(-im) - Math.exp(im);
	double tmp;
	if ((t_0 <= -0.004) || !(t_0 <= 2e-10)) {
		tmp = (0.5 * Math.cos(re)) * t_0;
	} else {
		tmp = Math.cos(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	tmp = 0
	if (t_0 <= -0.004) or not (t_0 <= 2e-10):
		tmp = (0.5 * math.cos(re)) * t_0
	else:
		tmp = math.cos(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	tmp = 0.0
	if ((t_0 <= -0.004) || !(t_0 <= 2e-10))
		tmp = Float64(Float64(0.5 * cos(re)) * t_0);
	else
		tmp = Float64(cos(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	tmp = 0.0;
	if ((t_0 <= -0.004) || ~((t_0 <= 2e-10)))
		tmp = (0.5 * cos(re)) * t_0;
	else
		tmp = cos(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -0.004], N[Not[LessEqual[t$95$0, 2e-10]], $MachinePrecision]], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < -0.0040000000000000001 or 2.00000000000000007e-10 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im))

   1. Initial program 99.8%

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

      1. neg-sub0 99.8%

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

   3. Simplified 99.8%

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

   if -0.0040000000000000001 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 2.00000000000000007e-10

   1. Initial program 6.8%

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

      1. neg-sub0 6.8%

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

   3. Simplified 6.8%

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

   4. Taylor expanded in im around 0 99.8%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \cos re\right) + -0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right)} \]
   5. Step-by-step derivation

      1. +-commutative 99.8%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) + -1 \cdot \left(im \cdot \cos re\right)} \]

      2. mul-1-neg 99.8%

         \[\leadsto -0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) + \color{blue}{\left(-im \cdot \cos re\right)} \]

      3. unsub-neg 99.8%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) - im \cdot \cos re} \]

      4. associate-*r* 99.8%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \cos re} - im \cdot \cos re \]

      5. distribute-rgt-out-- 99.8%

         \[\leadsto \color{blue}{\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} \]

      6. *-commutative 99.8%

         \[\leadsto \cos re \cdot \left(\color{blue}{{im}^{3} \cdot -0.16666666666666666} - im\right) \]

   6. Simplified 99.8%

      \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -0.004 \lor \neg \left(e^{-im} - e^{im} \leq 2 \cdot 10^{-10}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternative 3: 93.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos re \cdot \left({im}^{5} \cdot -0.008333333333333333\right)\\ \mathbf{if}\;im \leq -3.9 \cdot 10^{+62}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -0.106:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+19}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 2.45 \cdot 10^{+58}:\\ \;\;\;\;{re}^{2} \cdot \left(im \cdot 0.5\right) - im\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (cos re) (* (pow im 5.0) -0.008333333333333333))))
   (if (<= im -3.9e+62)
     t_0
     (if (<= im -0.106)
       (* 0.5 (- (exp (- im)) (exp im)))
       (if (<= im 1.1e+19)
         (* im (- (cos re)))
         (if (<= im 2.45e+58) (- (* (pow re 2.0) (* im 0.5)) im) t_0))))))

C

double code(double re, double im) {
	double t_0 = cos(re) * (pow(im, 5.0) * -0.008333333333333333);
	double tmp;
	if (im <= -3.9e+62) {
		tmp = t_0;
	} else if (im <= -0.106) {
		tmp = 0.5 * (exp(-im) - exp(im));
	} else if (im <= 1.1e+19) {
		tmp = im * -cos(re);
	} else if (im <= 2.45e+58) {
		tmp = (pow(re, 2.0) * (im * 0.5)) - im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(re) * ((im ** 5.0d0) * (-0.008333333333333333d0))
    if (im <= (-3.9d+62)) then
        tmp = t_0
    else if (im <= (-0.106d0)) then
        tmp = 0.5d0 * (exp(-im) - exp(im))
    else if (im <= 1.1d+19) then
        tmp = im * -cos(re)
    else if (im <= 2.45d+58) then
        tmp = ((re ** 2.0d0) * (im * 0.5d0)) - im
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.cos(re) * (Math.pow(im, 5.0) * -0.008333333333333333);
	double tmp;
	if (im <= -3.9e+62) {
		tmp = t_0;
	} else if (im <= -0.106) {
		tmp = 0.5 * (Math.exp(-im) - Math.exp(im));
	} else if (im <= 1.1e+19) {
		tmp = im * -Math.cos(re);
	} else if (im <= 2.45e+58) {
		tmp = (Math.pow(re, 2.0) * (im * 0.5)) - im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.cos(re) * (math.pow(im, 5.0) * -0.008333333333333333)
	tmp = 0
	if im <= -3.9e+62:
		tmp = t_0
	elif im <= -0.106:
		tmp = 0.5 * (math.exp(-im) - math.exp(im))
	elif im <= 1.1e+19:
		tmp = im * -math.cos(re)
	elif im <= 2.45e+58:
		tmp = (math.pow(re, 2.0) * (im * 0.5)) - im
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(cos(re) * Float64((im ^ 5.0) * -0.008333333333333333))
	tmp = 0.0
	if (im <= -3.9e+62)
		tmp = t_0;
	elseif (im <= -0.106)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) - exp(im)));
	elseif (im <= 1.1e+19)
		tmp = Float64(im * Float64(-cos(re)));
	elseif (im <= 2.45e+58)
		tmp = Float64(Float64((re ^ 2.0) * Float64(im * 0.5)) - im);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = cos(re) * ((im ^ 5.0) * -0.008333333333333333);
	tmp = 0.0;
	if (im <= -3.9e+62)
		tmp = t_0;
	elseif (im <= -0.106)
		tmp = 0.5 * (exp(-im) - exp(im));
	elseif (im <= 1.1e+19)
		tmp = im * -cos(re);
	elseif (im <= 2.45e+58)
		tmp = ((re ^ 2.0) * (im * 0.5)) - im;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Cos[re], $MachinePrecision] * N[(N[Power[im, 5.0], $MachinePrecision] * -0.008333333333333333), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -3.9e+62], t$95$0, If[LessEqual[im, -0.106], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.1e+19], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im, 2.45e+58], N[(N[(N[Power[re, 2.0], $MachinePrecision] * N[(im * 0.5), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -3.9e62 or 2.45000000000000009e58 < 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. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 99.0%

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

   5. Taylor expanded in im around inf 99.0%

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

      1. associate-*r* 99.0%

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

   7. Simplified 99.0%

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

   if -3.9e62 < im < -0.105999999999999997

   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. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 80.0%

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

   if -0.105999999999999997 < im < 1.1e19

   1. Initial program 11.3%

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

      1. neg-sub0 11.3%

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

   3. Simplified 11.3%

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

   4. Taylor expanded in im around 0 95.8%

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

      1. associate-*r* 95.8%

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

      2. neg-mul-1 95.8%

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

   6. Simplified 95.8%

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

   if 1.1e19 < im < 2.45000000000000009e58

   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. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 3.4%

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

      1. associate-*r* 3.4%

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

      2. neg-mul-1 3.4%

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

   6. Simplified 3.4%

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

   7. Taylor expanded in re around 0 75.8%

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

      1. mul-1-neg 75.8%

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

      2. +-commutative 75.8%

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

      3. unsub-neg 75.8%

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

      4. *-commutative 75.8%

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

      5. *-commutative 75.8%

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

      6. associate-*l* 75.8%

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

   9. Simplified 75.8%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3.9 \cdot 10^{+62}:\\ \;\;\;\;\cos re \cdot \left({im}^{5} \cdot -0.008333333333333333\right)\\ \mathbf{elif}\;im \leq -0.106:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+19}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 2.45 \cdot 10^{+58}:\\ \;\;\;\;{re}^{2} \cdot \left(im \cdot 0.5\right) - im\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{5} \cdot -0.008333333333333333\right)\\ \end{array} \]

Alternative 4: 93.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos re \cdot \left({im}^{5} \cdot -0.008333333333333333\right)\\ \mathbf{if}\;im \leq -3.9 \cdot 10^{+62}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -0.12:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+19}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{+58}:\\ \;\;\;\;{re}^{2} \cdot \left(im \cdot 0.5\right) - im\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (cos re) (* (pow im 5.0) -0.008333333333333333))))
   (if (<= im -3.9e+62)
     t_0
     (if (<= im -0.12)
       (* 0.5 (- (exp (- im)) (exp im)))
       (if (<= im 1.1e+19)
         (* (cos re) (- (* (pow im 3.0) -0.16666666666666666) im))
         (if (<= im 4.2e+58) (- (* (pow re 2.0) (* im 0.5)) im) t_0))))))

C

double code(double re, double im) {
	double t_0 = cos(re) * (pow(im, 5.0) * -0.008333333333333333);
	double tmp;
	if (im <= -3.9e+62) {
		tmp = t_0;
	} else if (im <= -0.12) {
		tmp = 0.5 * (exp(-im) - exp(im));
	} else if (im <= 1.1e+19) {
		tmp = cos(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	} else if (im <= 4.2e+58) {
		tmp = (pow(re, 2.0) * (im * 0.5)) - im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(re) * ((im ** 5.0d0) * (-0.008333333333333333d0))
    if (im <= (-3.9d+62)) then
        tmp = t_0
    else if (im <= (-0.12d0)) then
        tmp = 0.5d0 * (exp(-im) - exp(im))
    else if (im <= 1.1d+19) then
        tmp = cos(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    else if (im <= 4.2d+58) then
        tmp = ((re ** 2.0d0) * (im * 0.5d0)) - im
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.cos(re) * (Math.pow(im, 5.0) * -0.008333333333333333);
	double tmp;
	if (im <= -3.9e+62) {
		tmp = t_0;
	} else if (im <= -0.12) {
		tmp = 0.5 * (Math.exp(-im) - Math.exp(im));
	} else if (im <= 1.1e+19) {
		tmp = Math.cos(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	} else if (im <= 4.2e+58) {
		tmp = (Math.pow(re, 2.0) * (im * 0.5)) - im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.cos(re) * (math.pow(im, 5.0) * -0.008333333333333333)
	tmp = 0
	if im <= -3.9e+62:
		tmp = t_0
	elif im <= -0.12:
		tmp = 0.5 * (math.exp(-im) - math.exp(im))
	elif im <= 1.1e+19:
		tmp = math.cos(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	elif im <= 4.2e+58:
		tmp = (math.pow(re, 2.0) * (im * 0.5)) - im
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(cos(re) * Float64((im ^ 5.0) * -0.008333333333333333))
	tmp = 0.0
	if (im <= -3.9e+62)
		tmp = t_0;
	elseif (im <= -0.12)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) - exp(im)));
	elseif (im <= 1.1e+19)
		tmp = Float64(cos(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	elseif (im <= 4.2e+58)
		tmp = Float64(Float64((re ^ 2.0) * Float64(im * 0.5)) - im);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = cos(re) * ((im ^ 5.0) * -0.008333333333333333);
	tmp = 0.0;
	if (im <= -3.9e+62)
		tmp = t_0;
	elseif (im <= -0.12)
		tmp = 0.5 * (exp(-im) - exp(im));
	elseif (im <= 1.1e+19)
		tmp = cos(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	elseif (im <= 4.2e+58)
		tmp = ((re ^ 2.0) * (im * 0.5)) - im;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Cos[re], $MachinePrecision] * N[(N[Power[im, 5.0], $MachinePrecision] * -0.008333333333333333), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -3.9e+62], t$95$0, If[LessEqual[im, -0.12], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.1e+19], N[(N[Cos[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 4.2e+58], N[(N[(N[Power[re, 2.0], $MachinePrecision] * N[(im * 0.5), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -3.9e62 or 4.20000000000000024e58 < 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. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 99.0%

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

   5. Taylor expanded in im around inf 99.0%

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

      1. associate-*r* 99.0%

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

   7. Simplified 99.0%

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

   if -3.9e62 < im < -0.12

   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. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 80.0%

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

   if -0.12 < im < 1.1e19

   1. Initial program 11.3%

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

      1. neg-sub0 11.3%

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

   3. Simplified 11.3%

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

   4. Taylor expanded in im around 0 96.5%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \cos re\right) + -0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right)} \]
   5. Step-by-step derivation

      1. +-commutative 96.5%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) + -1 \cdot \left(im \cdot \cos re\right)} \]

      2. mul-1-neg 96.5%

         \[\leadsto -0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) + \color{blue}{\left(-im \cdot \cos re\right)} \]

      3. unsub-neg 96.5%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) - im \cdot \cos re} \]

      4. associate-*r* 96.5%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \cos re} - im \cdot \cos re \]

      5. distribute-rgt-out-- 96.5%

         \[\leadsto \color{blue}{\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} \]

      6. *-commutative 96.5%

         \[\leadsto \cos re \cdot \left(\color{blue}{{im}^{3} \cdot -0.16666666666666666} - im\right) \]

   6. Simplified 96.5%

      \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]

   if 1.1e19 < im < 4.20000000000000024e58

   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. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 3.4%

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

      1. associate-*r* 3.4%

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

      2. neg-mul-1 3.4%

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

   6. Simplified 3.4%

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

   7. Taylor expanded in re around 0 75.8%

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

      1. mul-1-neg 75.8%

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

      2. +-commutative 75.8%

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

      3. unsub-neg 75.8%

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

      4. *-commutative 75.8%

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

      5. *-commutative 75.8%

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

      6. associate-*l* 75.8%

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

   9. Simplified 75.8%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3.9 \cdot 10^{+62}:\\ \;\;\;\;\cos re \cdot \left({im}^{5} \cdot -0.008333333333333333\right)\\ \mathbf{elif}\;im \leq -0.12:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+19}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{+58}:\\ \;\;\;\;{re}^{2} \cdot \left(im \cdot 0.5\right) - im\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{5} \cdot -0.008333333333333333\right)\\ \end{array} \]

Alternative 5: 82.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -0.106:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+19}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 1.8 \cdot 10^{+97}:\\ \;\;\;\;{re}^{2} \cdot \left(im \cdot 0.5\right) - im\\ \mathbf{else}:\\ \;\;\;\;{im}^{5} \cdot -0.008333333333333333 - im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -0.106)
   (* 0.5 (- (exp (- im)) (exp im)))
   (if (<= im 1.1e+19)
     (* im (- (cos re)))
     (if (<= im 1.8e+97)
       (- (* (pow re 2.0) (* im 0.5)) im)
       (- (* (pow im 5.0) -0.008333333333333333) im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -0.106) {
		tmp = 0.5 * (exp(-im) - exp(im));
	} else if (im <= 1.1e+19) {
		tmp = im * -cos(re);
	} else if (im <= 1.8e+97) {
		tmp = (pow(re, 2.0) * (im * 0.5)) - im;
	} else {
		tmp = (pow(im, 5.0) * -0.008333333333333333) - 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 <= (-0.106d0)) then
        tmp = 0.5d0 * (exp(-im) - exp(im))
    else if (im <= 1.1d+19) then
        tmp = im * -cos(re)
    else if (im <= 1.8d+97) then
        tmp = ((re ** 2.0d0) * (im * 0.5d0)) - im
    else
        tmp = ((im ** 5.0d0) * (-0.008333333333333333d0)) - im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -0.106) {
		tmp = 0.5 * (Math.exp(-im) - Math.exp(im));
	} else if (im <= 1.1e+19) {
		tmp = im * -Math.cos(re);
	} else if (im <= 1.8e+97) {
		tmp = (Math.pow(re, 2.0) * (im * 0.5)) - im;
	} else {
		tmp = (Math.pow(im, 5.0) * -0.008333333333333333) - im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -0.106:
		tmp = 0.5 * (math.exp(-im) - math.exp(im))
	elif im <= 1.1e+19:
		tmp = im * -math.cos(re)
	elif im <= 1.8e+97:
		tmp = (math.pow(re, 2.0) * (im * 0.5)) - im
	else:
		tmp = (math.pow(im, 5.0) * -0.008333333333333333) - im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -0.106)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) - exp(im)));
	elseif (im <= 1.1e+19)
		tmp = Float64(im * Float64(-cos(re)));
	elseif (im <= 1.8e+97)
		tmp = Float64(Float64((re ^ 2.0) * Float64(im * 0.5)) - im);
	else
		tmp = Float64(Float64((im ^ 5.0) * -0.008333333333333333) - im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -0.106)
		tmp = 0.5 * (exp(-im) - exp(im));
	elseif (im <= 1.1e+19)
		tmp = im * -cos(re);
	elseif (im <= 1.8e+97)
		tmp = ((re ^ 2.0) * (im * 0.5)) - im;
	else
		tmp = ((im ^ 5.0) * -0.008333333333333333) - im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -0.106], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.1e+19], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im, 1.8e+97], N[(N[(N[Power[re, 2.0], $MachinePrecision] * N[(im * 0.5), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], N[(N[(N[Power[im, 5.0], $MachinePrecision] * -0.008333333333333333), $MachinePrecision] - im), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < -0.105999999999999997

   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. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 73.8%

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

   if -0.105999999999999997 < im < 1.1e19

   1. Initial program 11.3%

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

      1. neg-sub0 11.3%

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

   3. Simplified 11.3%

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

   4. Taylor expanded in im around 0 95.8%

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

      1. associate-*r* 95.8%

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

      2. neg-mul-1 95.8%

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

   6. Simplified 95.8%

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

   if 1.1e19 < im < 1.79999999999999983e97

   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. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 3.6%

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

      1. associate-*r* 3.6%

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

      2. neg-mul-1 3.6%

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

   6. Simplified 3.6%

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

   7. Taylor expanded in re around 0 77.7%

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

      1. mul-1-neg 77.7%

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

      2. +-commutative 77.7%

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

      3. unsub-neg 77.7%

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

      4. *-commutative 77.7%

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

      5. *-commutative 77.7%

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

      6. associate-*l* 77.7%

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

   9. Simplified 77.7%

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

   if 1.79999999999999983e97 < 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. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

   5. Taylor expanded in im around inf 100.0%

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

   6. Taylor expanded in re around inf 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos re \cdot \left(-2 \cdot im + -0.016666666666666666 \cdot {im}^{5}\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. associate-*r* 100.0%

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

      6. metadata-eval 100.0%

         \[\leadsto \cos re \cdot \left(\color{blue}{-1} \cdot im + 0.5 \cdot \left(-0.016666666666666666 \cdot {im}^{5}\right)\right) \]

      7. associate-*r* 100.0%

         \[\leadsto \cos re \cdot \left(-1 \cdot im + \color{blue}{\left(0.5 \cdot -0.016666666666666666\right) \cdot {im}^{5}}\right) \]

      8. metadata-eval 100.0%

         \[\leadsto \cos re \cdot \left(-1 \cdot im + \color{blue}{-0.008333333333333333} \cdot {im}^{5}\right) \]

      9. mul-1-neg 100.0%

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

      10. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in re around 0 80.5%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -0.106:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+19}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 1.8 \cdot 10^{+97}:\\ \;\;\;\;{re}^{2} \cdot \left(im \cdot 0.5\right) - im\\ \mathbf{else}:\\ \;\;\;\;{im}^{5} \cdot -0.008333333333333333 - im\\ \end{array} \]

Alternative 6: 90.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \cos re \cdot \left({im}^{5} \cdot -0.008333333333333333 - im\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (cos re) (- (* (pow im 5.0) -0.008333333333333333) im)))

C

double code(double re, double im) {
	return cos(re) * ((pow(im, 5.0) * -0.008333333333333333) - im);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = cos(re) * (((im ** 5.0d0) * (-0.008333333333333333d0)) - im)
end function

Java

public static double code(double re, double im) {
	return Math.cos(re) * ((Math.pow(im, 5.0) * -0.008333333333333333) - im);
}

Python

def code(re, im):
	return math.cos(re) * ((math.pow(im, 5.0) * -0.008333333333333333) - im)

Julia

function code(re, im)
	return Float64(cos(re) * Float64(Float64((im ^ 5.0) * -0.008333333333333333) - im))
end

MATLAB

function tmp = code(re, im)
	tmp = cos(re) * (((im ^ 5.0) * -0.008333333333333333) - im);
end

Wolfram

code[re_, im_] := N[(N[Cos[re], $MachinePrecision] * N[(N[(N[Power[im, 5.0], $MachinePrecision] * -0.008333333333333333), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.1%

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

   1. neg-sub0 51.1%

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

3. Simplified 51.1%

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

4. Taylor expanded in im around 0 89.8%

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

5. Taylor expanded in im around inf 89.2%

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

6. Taylor expanded in re around inf 89.2%

   \[\leadsto \color{blue}{0.5 \cdot \left(\cos re \cdot \left(-2 \cdot im + -0.016666666666666666 \cdot {im}^{5}\right)\right)} \]
7. Step-by-step derivation

   1. *-commutative 89.2%

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

   2. associate-*r* 89.2%

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

   3. *-commutative 89.2%

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

   4. distribute-lft-in 89.2%

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

   5. associate-*r* 89.2%

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

   6. metadata-eval 89.2%

      \[\leadsto \cos re \cdot \left(\color{blue}{-1} \cdot im + 0.5 \cdot \left(-0.016666666666666666 \cdot {im}^{5}\right)\right) \]

   7. associate-*r* 89.2%

      \[\leadsto \cos re \cdot \left(-1 \cdot im + \color{blue}{\left(0.5 \cdot -0.016666666666666666\right) \cdot {im}^{5}}\right) \]

   8. metadata-eval 89.2%

      \[\leadsto \cos re \cdot \left(-1 \cdot im + \color{blue}{-0.008333333333333333} \cdot {im}^{5}\right) \]

   9. mul-1-neg 89.2%

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

   10. *-commutative 89.2%

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

8. Simplified 89.2%

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

9. Taylor expanded in re around inf 89.2%

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

10. Final simplification 89.2%

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

Alternative 7: 79.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{5} \cdot -0.008333333333333333 - im\\ \mathbf{if}\;im \leq -1.6 \cdot 10^{+23}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 620:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 1.7 \cdot 10^{+97}:\\ \;\;\;\;{re}^{2} \cdot \left(im \cdot 0.5\right) - im\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (* (pow im 5.0) -0.008333333333333333) im)))
   (if (<= im -1.6e+23)
     t_0
     (if (<= im 620.0)
       (* im (- (cos re)))
       (if (<= im 1.7e+97) (- (* (pow re 2.0) (* im 0.5)) im) t_0)))))

C

double code(double re, double im) {
	double t_0 = (pow(im, 5.0) * -0.008333333333333333) - im;
	double tmp;
	if (im <= -1.6e+23) {
		tmp = t_0;
	} else if (im <= 620.0) {
		tmp = im * -cos(re);
	} else if (im <= 1.7e+97) {
		tmp = (pow(re, 2.0) * (im * 0.5)) - im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((im ** 5.0d0) * (-0.008333333333333333d0)) - im
    if (im <= (-1.6d+23)) then
        tmp = t_0
    else if (im <= 620.0d0) then
        tmp = im * -cos(re)
    else if (im <= 1.7d+97) then
        tmp = ((re ** 2.0d0) * (im * 0.5d0)) - im
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (Math.pow(im, 5.0) * -0.008333333333333333) - im;
	double tmp;
	if (im <= -1.6e+23) {
		tmp = t_0;
	} else if (im <= 620.0) {
		tmp = im * -Math.cos(re);
	} else if (im <= 1.7e+97) {
		tmp = (Math.pow(re, 2.0) * (im * 0.5)) - im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (math.pow(im, 5.0) * -0.008333333333333333) - im
	tmp = 0
	if im <= -1.6e+23:
		tmp = t_0
	elif im <= 620.0:
		tmp = im * -math.cos(re)
	elif im <= 1.7e+97:
		tmp = (math.pow(re, 2.0) * (im * 0.5)) - im
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64((im ^ 5.0) * -0.008333333333333333) - im)
	tmp = 0.0
	if (im <= -1.6e+23)
		tmp = t_0;
	elseif (im <= 620.0)
		tmp = Float64(im * Float64(-cos(re)));
	elseif (im <= 1.7e+97)
		tmp = Float64(Float64((re ^ 2.0) * Float64(im * 0.5)) - im);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = ((im ^ 5.0) * -0.008333333333333333) - im;
	tmp = 0.0;
	if (im <= -1.6e+23)
		tmp = t_0;
	elseif (im <= 620.0)
		tmp = im * -cos(re);
	elseif (im <= 1.7e+97)
		tmp = ((re ^ 2.0) * (im * 0.5)) - im;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Power[im, 5.0], $MachinePrecision] * -0.008333333333333333), $MachinePrecision] - im), $MachinePrecision]}, If[LessEqual[im, -1.6e+23], t$95$0, If[LessEqual[im, 620.0], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im, 1.7e+97], N[(N[(N[Power[re, 2.0], $MachinePrecision] * N[(im * 0.5), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.6e23 or 1.70000000000000005e97 < 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. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 93.1%

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

   5. Taylor expanded in im around inf 93.1%

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

   6. Taylor expanded in re around inf 93.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos re \cdot \left(-2 \cdot im + -0.016666666666666666 \cdot {im}^{5}\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative 93.1%

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

      2. associate-*r* 93.1%

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

      3. *-commutative 93.1%

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

      4. distribute-lft-in 93.1%

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

      5. associate-*r* 93.1%

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

      6. metadata-eval 93.1%

         \[\leadsto \cos re \cdot \left(\color{blue}{-1} \cdot im + 0.5 \cdot \left(-0.016666666666666666 \cdot {im}^{5}\right)\right) \]

      7. associate-*r* 93.1%

         \[\leadsto \cos re \cdot \left(-1 \cdot im + \color{blue}{\left(0.5 \cdot -0.016666666666666666\right) \cdot {im}^{5}}\right) \]

      8. metadata-eval 93.1%

         \[\leadsto \cos re \cdot \left(-1 \cdot im + \color{blue}{-0.008333333333333333} \cdot {im}^{5}\right) \]

      9. mul-1-neg 93.1%

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

      10. *-commutative 93.1%

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

   8. Simplified 93.1%

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

   9. Taylor expanded in re around 0 71.0%

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

   if -1.6e23 < im < 620

   1. Initial program 13.7%

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

      1. neg-sub0 13.7%

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

   3. Simplified 13.7%

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

   4. Taylor expanded in im around 0 93.2%

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

      1. associate-*r* 93.2%

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

      2. neg-mul-1 93.2%

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

   6. Simplified 93.2%

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

   if 620 < im < 1.70000000000000005e97

   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. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 3.5%

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

      1. associate-*r* 3.5%

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

      2. neg-mul-1 3.5%

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

   6. Simplified 3.5%

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

   7. Taylor expanded in re around 0 63.7%

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

      1. mul-1-neg 63.7%

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

      2. +-commutative 63.7%

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

      3. unsub-neg 63.7%

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

      4. *-commutative 63.7%

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

      5. *-commutative 63.7%

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

      6. associate-*l* 63.7%

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

   9. Simplified 63.7%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.6 \cdot 10^{+23}:\\ \;\;\;\;{im}^{5} \cdot -0.008333333333333333 - im\\ \mathbf{elif}\;im \leq 620:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 1.7 \cdot 10^{+97}:\\ \;\;\;\;{re}^{2} \cdot \left(im \cdot 0.5\right) - im\\ \mathbf{else}:\\ \;\;\;\;{im}^{5} \cdot -0.008333333333333333 - im\\ \end{array} \]

Alternative 8: 75.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -4.2 \cdot 10^{+46} \lor \neg \left(im \leq 1.7 \cdot 10^{+97}\right):\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -4.2e+46) (not (<= im 1.7e+97)))
   (- (* (pow im 3.0) -0.16666666666666666) im)
   (* im (- (cos re)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -4.2e+46) || !(im <= 1.7e+97)) {
		tmp = (pow(im, 3.0) * -0.16666666666666666) - im;
	} else {
		tmp = im * -cos(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-4.2d+46)) .or. (.not. (im <= 1.7d+97))) then
        tmp = ((im ** 3.0d0) * (-0.16666666666666666d0)) - im
    else
        tmp = im * -cos(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -4.2e+46) || !(im <= 1.7e+97)) {
		tmp = (Math.pow(im, 3.0) * -0.16666666666666666) - im;
	} else {
		tmp = im * -Math.cos(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -4.2e+46) or not (im <= 1.7e+97):
		tmp = (math.pow(im, 3.0) * -0.16666666666666666) - im
	else:
		tmp = im * -math.cos(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -4.2e+46) || !(im <= 1.7e+97))
		tmp = Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im);
	else
		tmp = Float64(im * Float64(-cos(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -4.2e+46) || ~((im <= 1.7e+97)))
		tmp = ((im ^ 3.0) * -0.16666666666666666) - im;
	else
		tmp = im * -cos(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -4.2e+46], N[Not[LessEqual[im, 1.7e+97]], $MachinePrecision]], N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -4.2e46 or 1.70000000000000005e97 < 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. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 90.7%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \cos re\right) + -0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right)} \]
   5. Step-by-step derivation

      1. +-commutative 90.7%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) + -1 \cdot \left(im \cdot \cos re\right)} \]

      2. mul-1-neg 90.7%

         \[\leadsto -0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) + \color{blue}{\left(-im \cdot \cos re\right)} \]

      3. unsub-neg 90.7%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) - im \cdot \cos re} \]

      4. associate-*r* 90.7%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \cos re} - im \cdot \cos re \]

      5. distribute-rgt-out-- 90.7%

         \[\leadsto \color{blue}{\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} \]

      6. *-commutative 90.7%

         \[\leadsto \cos re \cdot \left(\color{blue}{{im}^{3} \cdot -0.16666666666666666} - im\right) \]

   6. Simplified 90.7%

      \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]

   7. Taylor expanded in re around 0 69.5%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3} - im} \]

   if -4.2e46 < im < 1.70000000000000005e97

   1. Initial program 24.7%

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

      1. neg-sub0 24.7%

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

   3. Simplified 24.7%

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

   4. Taylor expanded in im around 0 81.9%

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

      1. associate-*r* 81.9%

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

      2. neg-mul-1 81.9%

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

   6. Simplified 81.9%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -4.2 \cdot 10^{+46} \lor \neg \left(im \leq 1.7 \cdot 10^{+97}\right):\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \]

Alternative 9: 77.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1.48 \cdot 10^{+23} \lor \neg \left(im \leq 1.7 \cdot 10^{+97}\right):\\ \;\;\;\;{im}^{5} \cdot -0.008333333333333333 - im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -1.48e+23) (not (<= im 1.7e+97)))
   (- (* (pow im 5.0) -0.008333333333333333) im)
   (* im (- (cos re)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -1.48e+23) || !(im <= 1.7e+97)) {
		tmp = (pow(im, 5.0) * -0.008333333333333333) - im;
	} else {
		tmp = im * -cos(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-1.48d+23)) .or. (.not. (im <= 1.7d+97))) then
        tmp = ((im ** 5.0d0) * (-0.008333333333333333d0)) - im
    else
        tmp = im * -cos(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -1.48e+23) || !(im <= 1.7e+97)) {
		tmp = (Math.pow(im, 5.0) * -0.008333333333333333) - im;
	} else {
		tmp = im * -Math.cos(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -1.48e+23) or not (im <= 1.7e+97):
		tmp = (math.pow(im, 5.0) * -0.008333333333333333) - im
	else:
		tmp = im * -math.cos(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -1.48e+23) || !(im <= 1.7e+97))
		tmp = Float64(Float64((im ^ 5.0) * -0.008333333333333333) - im);
	else
		tmp = Float64(im * Float64(-cos(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -1.48e+23) || ~((im <= 1.7e+97)))
		tmp = ((im ^ 5.0) * -0.008333333333333333) - im;
	else
		tmp = im * -cos(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -1.48e+23], N[Not[LessEqual[im, 1.7e+97]], $MachinePrecision]], N[(N[(N[Power[im, 5.0], $MachinePrecision] * -0.008333333333333333), $MachinePrecision] - im), $MachinePrecision], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -1.4799999999999999e23 or 1.70000000000000005e97 < 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. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 93.1%

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

   5. Taylor expanded in im around inf 93.1%

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

   6. Taylor expanded in re around inf 93.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(\cos re \cdot \left(-2 \cdot im + -0.016666666666666666 \cdot {im}^{5}\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative 93.1%

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

      2. associate-*r* 93.1%

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

      3. *-commutative 93.1%

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

      4. distribute-lft-in 93.1%

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

      5. associate-*r* 93.1%

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

      6. metadata-eval 93.1%

         \[\leadsto \cos re \cdot \left(\color{blue}{-1} \cdot im + 0.5 \cdot \left(-0.016666666666666666 \cdot {im}^{5}\right)\right) \]

      7. associate-*r* 93.1%

         \[\leadsto \cos re \cdot \left(-1 \cdot im + \color{blue}{\left(0.5 \cdot -0.016666666666666666\right) \cdot {im}^{5}}\right) \]

      8. metadata-eval 93.1%

         \[\leadsto \cos re \cdot \left(-1 \cdot im + \color{blue}{-0.008333333333333333} \cdot {im}^{5}\right) \]

      9. mul-1-neg 93.1%

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

      10. *-commutative 93.1%

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

   8. Simplified 93.1%

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

   9. Taylor expanded in re around 0 71.0%

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

   if -1.4799999999999999e23 < im < 1.70000000000000005e97

   1. Initial program 22.3%

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

      1. neg-sub0 22.3%

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

   3. Simplified 22.3%

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

   4. Taylor expanded in im around 0 84.3%

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

      1. associate-*r* 84.3%

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

      2. neg-mul-1 84.3%

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

   6. Simplified 84.3%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.48 \cdot 10^{+23} \lor \neg \left(im \leq 1.7 \cdot 10^{+97}\right):\\ \;\;\;\;{im}^{5} \cdot -0.008333333333333333 - im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \]

Alternative 10: 51.9% 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 51.1%

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

   1. neg-sub0 51.1%

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

3. Simplified 51.1%

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

4. Taylor expanded in im around 0 55.2%

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

   1. associate-*r* 55.2%

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

   2. neg-mul-1 55.2%

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

6. Simplified 55.2%

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

7. Final simplification 55.2%

   \[\leadsto im \cdot \left(-\cos re\right) \]

Alternative 11: 29.9% 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 51.1%

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

   1. neg-sub0 51.1%

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

3. Simplified 51.1%

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

4. Taylor expanded in im around 0 55.2%

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

   1. associate-*r* 55.2%

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

   2. neg-mul-1 55.2%

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

6. Simplified 55.2%

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

7. Taylor expanded in re around 0 30.8%

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

   1. mul-1-neg 30.8%

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

9. Simplified 30.8%

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

10. Final simplification 30.8%

    \[\leadsto -im \]

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 2023318 
(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))))