math.sin on complex, imaginary part

Percentage Accurate: 54.3% → 99.6%

Time: 19.8s

Alternatives: 20

Speedup: 2.9×

• Unsound rule application detected in e-graph. Results may not be sound.

• 49.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.3% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 10^{-6}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right) + \cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 1e-6)))
     (* (* 0.5 (cos re)) t_0)
     (+
      (* -0.008333333333333333 (* (cos re) (pow im 5.0)))
      (* (cos re) (- (* -0.16666666666666666 (pow im 3.0)) im))))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || !(t_0 <= 1e-6)) {
		tmp = (0.5 * cos(re)) * t_0;
	} else {
		tmp = (-0.008333333333333333 * (cos(re) * pow(im, 5.0))) + (cos(re) * ((-0.16666666666666666 * pow(im, 3.0)) - im));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(-im) - Math.exp(im);
	double tmp;
	if ((t_0 <= -Double.POSITIVE_INFINITY) || !(t_0 <= 1e-6)) {
		tmp = (0.5 * Math.cos(re)) * t_0;
	} else {
		tmp = (-0.008333333333333333 * (Math.cos(re) * Math.pow(im, 5.0))) + (Math.cos(re) * ((-0.16666666666666666 * Math.pow(im, 3.0)) - im));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	tmp = 0
	if (t_0 <= -math.inf) or not (t_0 <= 1e-6):
		tmp = (0.5 * math.cos(re)) * t_0
	else:
		tmp = (-0.008333333333333333 * (math.cos(re) * math.pow(im, 5.0))) + (math.cos(re) * ((-0.16666666666666666 * math.pow(im, 3.0)) - im))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	tmp = 0.0
	if ((t_0 <= Float64(-Inf)) || !(t_0 <= 1e-6))
		tmp = Float64(Float64(0.5 * cos(re)) * t_0);
	else
		tmp = Float64(Float64(-0.008333333333333333 * Float64(cos(re) * (im ^ 5.0))) + Float64(cos(re) * Float64(Float64(-0.16666666666666666 * (im ^ 3.0)) - im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	tmp = 0.0;
	if ((t_0 <= -Inf) || ~((t_0 <= 1e-6)))
		tmp = (0.5 * cos(re)) * t_0;
	else
		tmp = (-0.008333333333333333 * (cos(re) * (im ^ 5.0))) + (cos(re) * ((-0.16666666666666666 * (im ^ 3.0)) - 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, (-Infinity)], N[Not[LessEqual[t$95$0, 1e-6]], $MachinePrecision]], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(-0.008333333333333333 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[re], $MachinePrecision] * N[(N[(-0.16666666666666666 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < -inf.0 or 9.99999999999999955e-7 < (-.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. sub0-neg 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 -inf.0 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 9.99999999999999955e-7

   1. Initial program 8.4%

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

      1. sub0-neg 8.4%

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

   3. Simplified 8.4%

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

   4. Taylor expanded in im around 0 99.8%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + \left(-1 \cdot \left(\cos re \cdot im\right) + -0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right)\right)} \]
   5. Step-by-step derivation

      1. associate-+r+ 99.8%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + -1 \cdot \left(\cos re \cdot im\right)\right) + -0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right)} \]

      2. +-commutative 99.8%

         \[\leadsto \color{blue}{-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right) + \left(-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + -1 \cdot \left(\cos re \cdot im\right)\right)} \]

      3. associate-*r* 99.8%

         \[\leadsto \color{blue}{\left(-0.008333333333333333 \cdot \cos re\right) \cdot {im}^{5}} + \left(-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + -1 \cdot \left(\cos re \cdot im\right)\right) \]

      4. *-commutative 99.8%

         \[\leadsto \color{blue}{{im}^{5} \cdot \left(-0.008333333333333333 \cdot \cos re\right)} + \left(-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + -1 \cdot \left(\cos re \cdot im\right)\right) \]

      5. fma-def 99.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left({im}^{5}, -0.008333333333333333 \cdot \cos re, -0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + -1 \cdot \left(\cos re \cdot im\right)\right)} \]

      6. *-commutative 99.8%

         \[\leadsto \mathsf{fma}\left({im}^{5}, \color{blue}{\cos re \cdot -0.008333333333333333}, -0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + -1 \cdot \left(\cos re \cdot im\right)\right) \]

      7. mul-1-neg 99.8%

         \[\leadsto \mathsf{fma}\left({im}^{5}, \cos re \cdot -0.008333333333333333, -0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + \color{blue}{\left(-\cos re \cdot im\right)}\right) \]

      8. unsub-neg 99.8%

         \[\leadsto \mathsf{fma}\left({im}^{5}, \cos re \cdot -0.008333333333333333, \color{blue}{-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) - \cos re \cdot im}\right) \]

      9. *-commutative 99.8%

         \[\leadsto \mathsf{fma}\left({im}^{5}, \cos re \cdot -0.008333333333333333, \color{blue}{\left(\cos re \cdot {im}^{3}\right) \cdot -0.16666666666666666} - \cos re \cdot im\right) \]

      10. associate-*l* 99.8%

          \[\leadsto \mathsf{fma}\left({im}^{5}, \cos re \cdot -0.008333333333333333, \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)} - \cos re \cdot im\right) \]

      11. distribute-lft-out-- 99.8%

          \[\leadsto \mathsf{fma}\left({im}^{5}, \cos re \cdot -0.008333333333333333, \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)}\right) \]

   6. Simplified 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left({im}^{5}, \cos re \cdot -0.008333333333333333, \cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\right)} \]

   7. Taylor expanded in re around inf 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -\infty \lor \neg \left(e^{-im} - e^{im} \leq 10^{-6}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right) + \cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \end{array} \]

Alternative 2: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 10^{-6}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} + \left(-0.008333333333333333 \cdot {im}^{5} - im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 1e-6)))
     (* (* 0.5 (cos re)) t_0)
     (*
      (cos re)
      (+
       (* -0.16666666666666666 (pow im 3.0))
       (- (* -0.008333333333333333 (pow im 5.0)) im))))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || !(t_0 <= 1e-6)) {
		tmp = (0.5 * cos(re)) * t_0;
	} else {
		tmp = cos(re) * ((-0.16666666666666666 * pow(im, 3.0)) + ((-0.008333333333333333 * pow(im, 5.0)) - im));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(-im) - Math.exp(im);
	double tmp;
	if ((t_0 <= -Double.POSITIVE_INFINITY) || !(t_0 <= 1e-6)) {
		tmp = (0.5 * Math.cos(re)) * t_0;
	} else {
		tmp = Math.cos(re) * ((-0.16666666666666666 * Math.pow(im, 3.0)) + ((-0.008333333333333333 * Math.pow(im, 5.0)) - im));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	tmp = 0
	if (t_0 <= -math.inf) or not (t_0 <= 1e-6):
		tmp = (0.5 * math.cos(re)) * t_0
	else:
		tmp = math.cos(re) * ((-0.16666666666666666 * math.pow(im, 3.0)) + ((-0.008333333333333333 * math.pow(im, 5.0)) - im))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	tmp = 0.0
	if ((t_0 <= Float64(-Inf)) || !(t_0 <= 1e-6))
		tmp = Float64(Float64(0.5 * cos(re)) * t_0);
	else
		tmp = Float64(cos(re) * Float64(Float64(-0.16666666666666666 * (im ^ 3.0)) + Float64(Float64(-0.008333333333333333 * (im ^ 5.0)) - im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	tmp = 0.0;
	if ((t_0 <= -Inf) || ~((t_0 <= 1e-6)))
		tmp = (0.5 * cos(re)) * t_0;
	else
		tmp = cos(re) * ((-0.16666666666666666 * (im ^ 3.0)) + ((-0.008333333333333333 * (im ^ 5.0)) - 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, (-Infinity)], N[Not[LessEqual[t$95$0, 1e-6]], $MachinePrecision]], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(N[(-0.16666666666666666 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.008333333333333333 * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < -inf.0 or 9.99999999999999955e-7 < (-.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. sub0-neg 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 -inf.0 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 9.99999999999999955e-7

   1. Initial program 8.4%

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

      1. sub0-neg 8.4%

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

   3. Simplified 8.4%

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

   4. Taylor expanded in im around 0 99.8%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + \left(-1 \cdot \left(\cos re \cdot im\right) + -0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right)\right)} \]
   5. Step-by-step derivation

      1. *-commutative 99.8%

         \[\leadsto \color{blue}{\left(\cos re \cdot {im}^{3}\right) \cdot -0.16666666666666666} + \left(-1 \cdot \left(\cos re \cdot im\right) + -0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right)\right) \]

      2. associate-*l* 99.8%

         \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)} + \left(-1 \cdot \left(\cos re \cdot im\right) + -0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right)\right) \]

      3. +-commutative 99.8%

         \[\leadsto \cos re \cdot \left({im}^{3} \cdot -0.16666666666666666\right) + \color{blue}{\left(-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right) + -1 \cdot \left(\cos re \cdot im\right)\right)} \]

      4. mul-1-neg 99.8%

         \[\leadsto \cos re \cdot \left({im}^{3} \cdot -0.16666666666666666\right) + \left(-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right) + \color{blue}{\left(-\cos re \cdot im\right)}\right) \]

      5. unsub-neg 99.8%

         \[\leadsto \cos re \cdot \left({im}^{3} \cdot -0.16666666666666666\right) + \color{blue}{\left(-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right) - \cos re \cdot im\right)} \]

      6. *-commutative 99.8%

         \[\leadsto \cos re \cdot \left({im}^{3} \cdot -0.16666666666666666\right) + \left(\color{blue}{\left(\cos re \cdot {im}^{5}\right) \cdot -0.008333333333333333} - \cos re \cdot im\right) \]

      7. associate-*l* 99.8%

         \[\leadsto \cos re \cdot \left({im}^{3} \cdot -0.16666666666666666\right) + \left(\color{blue}{\cos re \cdot \left({im}^{5} \cdot -0.008333333333333333\right)} - \cos re \cdot im\right) \]

      8. distribute-lft-out-- 99.8%

         \[\leadsto \cos re \cdot \left({im}^{3} \cdot -0.16666666666666666\right) + \color{blue}{\cos re \cdot \left({im}^{5} \cdot -0.008333333333333333 - im\right)} \]

      9. distribute-lft-out 99.8%

         \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 + \left({im}^{5} \cdot -0.008333333333333333 - im\right)\right)} \]

   6. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -\infty \lor \neg \left(e^{-im} - e^{im} \leq 10^{-6}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} + \left(-0.008333333333333333 \cdot {im}^{5} - im\right)\right)\\ \end{array} \]

Alternative 3: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -0.005 \lor \neg \left(t_0 \leq 10^{-6}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot \left(\cos re \cdot -0.16666666666666666\right) - im \cdot \cos re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 -0.005) (not (<= t_0 1e-6)))
     (* (* 0.5 (cos re)) t_0)
     (- (* (pow im 3.0) (* (cos re) -0.16666666666666666)) (* im (cos re))))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if ((t_0 <= -0.005) || !(t_0 <= 1e-6)) {
		tmp = (0.5 * cos(re)) * t_0;
	} else {
		tmp = (pow(im, 3.0) * (cos(re) * -0.16666666666666666)) - (im * cos(re));
	}
	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 <= (-0.005d0)) .or. (.not. (t_0 <= 1d-6))) then
        tmp = (0.5d0 * cos(re)) * t_0
    else
        tmp = ((im ** 3.0d0) * (cos(re) * (-0.16666666666666666d0))) - (im * cos(re))
    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 <= -0.005) || !(t_0 <= 1e-6)) {
		tmp = (0.5 * Math.cos(re)) * t_0;
	} else {
		tmp = (Math.pow(im, 3.0) * (Math.cos(re) * -0.16666666666666666)) - (im * Math.cos(re));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	tmp = 0
	if (t_0 <= -0.005) or not (t_0 <= 1e-6):
		tmp = (0.5 * math.cos(re)) * t_0
	else:
		tmp = (math.pow(im, 3.0) * (math.cos(re) * -0.16666666666666666)) - (im * math.cos(re))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	tmp = 0.0
	if ((t_0 <= -0.005) || !(t_0 <= 1e-6))
		tmp = Float64(Float64(0.5 * cos(re)) * t_0);
	else
		tmp = Float64(Float64((im ^ 3.0) * Float64(cos(re) * -0.16666666666666666)) - Float64(im * cos(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	tmp = 0.0;
	if ((t_0 <= -0.005) || ~((t_0 <= 1e-6)))
		tmp = (0.5 * cos(re)) * t_0;
	else
		tmp = ((im ^ 3.0) * (cos(re) * -0.16666666666666666)) - (im * cos(re));
	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, -0.005], N[Not[LessEqual[t$95$0, 1e-6]], $MachinePrecision]], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(N[Power[im, 3.0], $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] - N[(im * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < -0.0050000000000000001 or 9.99999999999999955e-7 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 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. sub0-neg 99.9%

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

   3. Simplified 99.9%

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

   if -0.0050000000000000001 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 9.99999999999999955e-7

   1. Initial program 7.7%

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

      1. sub0-neg 7.7%

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

   3. Simplified 7.7%

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

   4. Taylor expanded in im around 0 99.8%

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

      1. mul-1-neg 99.8%

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

      2. unsub-neg 99.8%

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

      3. *-commutative 99.8%

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

      4. *-commutative 99.8%

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

      5. associate-*l* 99.8%

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

      6. *-commutative 99.8%

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

   6. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -0.005 \lor \neg \left(e^{-im} - e^{im} \leq 10^{-6}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot \left(\cos re \cdot -0.16666666666666666\right) - im \cdot \cos re\\ \end{array} \]

Alternative 4: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -0.005 \lor \neg \left(t_0 \leq 10^{-6}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 -0.005) (not (<= t_0 1e-6)))
     (* (* 0.5 (cos re)) t_0)
     (* (cos re) (- (* -0.16666666666666666 (pow im 3.0)) im)))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if ((t_0 <= -0.005) || !(t_0 <= 1e-6)) {
		tmp = (0.5 * cos(re)) * t_0;
	} else {
		tmp = cos(re) * ((-0.16666666666666666 * pow(im, 3.0)) - 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 <= (-0.005d0)) .or. (.not. (t_0 <= 1d-6))) then
        tmp = (0.5d0 * cos(re)) * t_0
    else
        tmp = cos(re) * (((-0.16666666666666666d0) * (im ** 3.0d0)) - 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 <= -0.005) || !(t_0 <= 1e-6)) {
		tmp = (0.5 * Math.cos(re)) * t_0;
	} else {
		tmp = Math.cos(re) * ((-0.16666666666666666 * Math.pow(im, 3.0)) - im);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	tmp = 0
	if (t_0 <= -0.005) or not (t_0 <= 1e-6):
		tmp = (0.5 * math.cos(re)) * t_0
	else:
		tmp = math.cos(re) * ((-0.16666666666666666 * math.pow(im, 3.0)) - im)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	tmp = 0.0
	if ((t_0 <= -0.005) || !(t_0 <= 1e-6))
		tmp = Float64(Float64(0.5 * cos(re)) * t_0);
	else
		tmp = Float64(cos(re) * Float64(Float64(-0.16666666666666666 * (im ^ 3.0)) - im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	tmp = 0.0;
	if ((t_0 <= -0.005) || ~((t_0 <= 1e-6)))
		tmp = (0.5 * cos(re)) * t_0;
	else
		tmp = cos(re) * ((-0.16666666666666666 * (im ^ 3.0)) - 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, -0.005], N[Not[LessEqual[t$95$0, 1e-6]], $MachinePrecision]], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(N[(-0.16666666666666666 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < -0.0050000000000000001 or 9.99999999999999955e-7 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 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. sub0-neg 99.9%

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

   3. Simplified 99.9%

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

   if -0.0050000000000000001 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 9.99999999999999955e-7

   1. Initial program 7.7%

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

      1. sub0-neg 7.7%

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

   3. Simplified 7.7%

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

   4. Taylor expanded in im around 0 99.8%

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

      1. mul-1-neg 99.8%

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

      2. unsub-neg 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-*l* 99.8%

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

      5. distribute-lft-out-- 99.8%

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

   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 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -0.005 \lor \neg \left(e^{-im} - e^{im} \leq 10^{-6}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \end{array} \]

Alternative 5: 97.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ t_1 := -0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right)\\ \mathbf{if}\;im \leq -2.3 \cdot 10^{+64}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -18:\\ \;\;\;\;t_0 \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \mathbf{elif}\;im \leq 0.07:\\ \;\;\;\;\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;0.5 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im)))
        (t_1 (* -0.008333333333333333 (* (cos re) (pow im 5.0)))))
   (if (<= im -2.3e+64)
     t_1
     (if (<= im -18.0)
       (* t_0 (+ 0.5 (* re (* re -0.25))))
       (if (<= im 0.07)
         (* (cos re) (- (* -0.16666666666666666 (pow im 3.0)) im))
         (if (<= im 4.5e+61) (* 0.5 t_0) t_1))))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double t_1 = -0.008333333333333333 * (cos(re) * pow(im, 5.0));
	double tmp;
	if (im <= -2.3e+64) {
		tmp = t_1;
	} else if (im <= -18.0) {
		tmp = t_0 * (0.5 + (re * (re * -0.25)));
	} else if (im <= 0.07) {
		tmp = cos(re) * ((-0.16666666666666666 * pow(im, 3.0)) - im);
	} else if (im <= 4.5e+61) {
		tmp = 0.5 * t_0;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = exp(-im) - exp(im)
    t_1 = (-0.008333333333333333d0) * (cos(re) * (im ** 5.0d0))
    if (im <= (-2.3d+64)) then
        tmp = t_1
    else if (im <= (-18.0d0)) then
        tmp = t_0 * (0.5d0 + (re * (re * (-0.25d0))))
    else if (im <= 0.07d0) then
        tmp = cos(re) * (((-0.16666666666666666d0) * (im ** 3.0d0)) - im)
    else if (im <= 4.5d+61) then
        tmp = 0.5d0 * t_0
    else
        tmp = t_1
    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 t_1 = -0.008333333333333333 * (Math.cos(re) * Math.pow(im, 5.0));
	double tmp;
	if (im <= -2.3e+64) {
		tmp = t_1;
	} else if (im <= -18.0) {
		tmp = t_0 * (0.5 + (re * (re * -0.25)));
	} else if (im <= 0.07) {
		tmp = Math.cos(re) * ((-0.16666666666666666 * Math.pow(im, 3.0)) - im);
	} else if (im <= 4.5e+61) {
		tmp = 0.5 * t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	t_1 = -0.008333333333333333 * (math.cos(re) * math.pow(im, 5.0))
	tmp = 0
	if im <= -2.3e+64:
		tmp = t_1
	elif im <= -18.0:
		tmp = t_0 * (0.5 + (re * (re * -0.25)))
	elif im <= 0.07:
		tmp = math.cos(re) * ((-0.16666666666666666 * math.pow(im, 3.0)) - im)
	elif im <= 4.5e+61:
		tmp = 0.5 * t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	t_1 = Float64(-0.008333333333333333 * Float64(cos(re) * (im ^ 5.0)))
	tmp = 0.0
	if (im <= -2.3e+64)
		tmp = t_1;
	elseif (im <= -18.0)
		tmp = Float64(t_0 * Float64(0.5 + Float64(re * Float64(re * -0.25))));
	elseif (im <= 0.07)
		tmp = Float64(cos(re) * Float64(Float64(-0.16666666666666666 * (im ^ 3.0)) - im));
	elseif (im <= 4.5e+61)
		tmp = Float64(0.5 * t_0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	t_1 = -0.008333333333333333 * (cos(re) * (im ^ 5.0));
	tmp = 0.0;
	if (im <= -2.3e+64)
		tmp = t_1;
	elseif (im <= -18.0)
		tmp = t_0 * (0.5 + (re * (re * -0.25)));
	elseif (im <= 0.07)
		tmp = cos(re) * ((-0.16666666666666666 * (im ^ 3.0)) - im);
	elseif (im <= 4.5e+61)
		tmp = 0.5 * t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-0.008333333333333333 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -2.3e+64], t$95$1, If[LessEqual[im, -18.0], N[(t$95$0 * N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 0.07], N[(N[Cos[re], $MachinePrecision] * N[(N[(-0.16666666666666666 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 4.5e+61], N[(0.5 * t$95$0), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -2.3e64 or 4.5e61 < im

   1. Initial program 100.0%

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

      1. sub0-neg 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%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + \left(-1 \cdot \left(\cos re \cdot im\right) + -0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right)\right)} \]
   5. Step-by-step derivation

      1. associate-+r+ 100.0%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + -1 \cdot \left(\cos re \cdot im\right)\right) + -0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right)} \]

      2. +-commutative 100.0%

         \[\leadsto \color{blue}{-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right) + \left(-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + -1 \cdot \left(\cos re \cdot im\right)\right)} \]

      3. associate-*r* 100.0%

         \[\leadsto \color{blue}{\left(-0.008333333333333333 \cdot \cos re\right) \cdot {im}^{5}} + \left(-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + -1 \cdot \left(\cos re \cdot im\right)\right) \]

      4. *-commutative 100.0%

         \[\leadsto \color{blue}{{im}^{5} \cdot \left(-0.008333333333333333 \cdot \cos re\right)} + \left(-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + -1 \cdot \left(\cos re \cdot im\right)\right) \]

      5. fma-def 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left({im}^{5}, -0.008333333333333333 \cdot \cos re, -0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + -1 \cdot \left(\cos re \cdot im\right)\right)} \]

      6. *-commutative 100.0%

         \[\leadsto \mathsf{fma}\left({im}^{5}, \color{blue}{\cos re \cdot -0.008333333333333333}, -0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + -1 \cdot \left(\cos re \cdot im\right)\right) \]

      7. mul-1-neg 100.0%

         \[\leadsto \mathsf{fma}\left({im}^{5}, \cos re \cdot -0.008333333333333333, -0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + \color{blue}{\left(-\cos re \cdot im\right)}\right) \]

      8. unsub-neg 100.0%

         \[\leadsto \mathsf{fma}\left({im}^{5}, \cos re \cdot -0.008333333333333333, \color{blue}{-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) - \cos re \cdot im}\right) \]

      9. *-commutative 100.0%

         \[\leadsto \mathsf{fma}\left({im}^{5}, \cos re \cdot -0.008333333333333333, \color{blue}{\left(\cos re \cdot {im}^{3}\right) \cdot -0.16666666666666666} - \cos re \cdot im\right) \]

      10. associate-*l* 100.0%

          \[\leadsto \mathsf{fma}\left({im}^{5}, \cos re \cdot -0.008333333333333333, \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)} - \cos re \cdot im\right) \]

      11. distribute-lft-out-- 100.0%

          \[\leadsto \mathsf{fma}\left({im}^{5}, \cos re \cdot -0.008333333333333333, \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)}\right) \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left({im}^{5}, \cos re \cdot -0.008333333333333333, \cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\right)} \]

   7. Taylor expanded in im around inf 100.0%

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

   if -2.3e64 < im < -18

   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. sub0-neg 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 0.0%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot {re}^{2}\right) + 0.5 \cdot \left(e^{-im} - e^{im}\right)} \]
   5. Step-by-step derivation

      1. +-commutative 0.0%

         \[\leadsto \color{blue}{0.5 \cdot \left(e^{-im} - e^{im}\right) + -0.25 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot {re}^{2}\right)} \]

      2. *-commutative 0.0%

         \[\leadsto 0.5 \cdot \left(e^{-im} - e^{im}\right) + -0.25 \cdot \color{blue}{\left({re}^{2} \cdot \left(e^{-im} - e^{im}\right)\right)} \]

      3. associate-*r* 0.0%

         \[\leadsto 0.5 \cdot \left(e^{-im} - e^{im}\right) + \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot \left(e^{-im} - e^{im}\right)} \]

      4. distribute-rgt-out 89.5%

         \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      5. *-commutative 89.5%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{{re}^{2} \cdot -0.25}\right) \]

      6. unpow2 89.5%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{\left(re \cdot re\right)} \cdot -0.25\right) \]

      7. associate-*l* 89.5%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{re \cdot \left(re \cdot -0.25\right)}\right) \]

   6. Simplified 89.5%

      \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)} \]

   if -18 < im < 0.070000000000000007

   1. Initial program 9.7%

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

      1. sub0-neg 9.7%

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

   3. Simplified 9.7%

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

   4. Taylor expanded in im around 0 98.5%

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

      1. mul-1-neg 98.5%

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

      2. unsub-neg 98.5%

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

      3. *-commutative 98.5%

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

      4. associate-*l* 98.5%

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

      5. distribute-lft-out-- 98.5%

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

   6. Simplified 98.5%

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

   if 0.070000000000000007 < im < 4.5e61

   1. Initial program 100.0%

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

      1. sub0-neg 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 0 100.0%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.3 \cdot 10^{+64}:\\ \;\;\;\;-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right)\\ \mathbf{elif}\;im \leq -18:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \mathbf{elif}\;im \leq 0.07:\\ \;\;\;\;\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right)\\ \end{array} \]

Alternative 6: 95.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(e^{-im} - e^{im}\right)\\ t_1 := -0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right)\\ \mathbf{if}\;im \leq -1.7 \cdot 10^{+93}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -2000000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 3 \cdot 10^{-5}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (- (exp (- im)) (exp im))))
        (t_1 (* -0.008333333333333333 (* (cos re) (pow im 5.0)))))
   (if (<= im -1.7e+93)
     t_1
     (if (<= im -2000000000000.0)
       t_0
       (if (<= im 3e-5) (* (cos re) (- im)) (if (<= im 4.5e+61) t_0 t_1))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (exp(-im) - exp(im));
	double t_1 = -0.008333333333333333 * (cos(re) * pow(im, 5.0));
	double tmp;
	if (im <= -1.7e+93) {
		tmp = t_1;
	} else if (im <= -2000000000000.0) {
		tmp = t_0;
	} else if (im <= 3e-5) {
		tmp = cos(re) * -im;
	} else if (im <= 4.5e+61) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * (exp(-im) - exp(im))
    t_1 = (-0.008333333333333333d0) * (cos(re) * (im ** 5.0d0))
    if (im <= (-1.7d+93)) then
        tmp = t_1
    else if (im <= (-2000000000000.0d0)) then
        tmp = t_0
    else if (im <= 3d-5) then
        tmp = cos(re) * -im
    else if (im <= 4.5d+61) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * (Math.exp(-im) - Math.exp(im));
	double t_1 = -0.008333333333333333 * (Math.cos(re) * Math.pow(im, 5.0));
	double tmp;
	if (im <= -1.7e+93) {
		tmp = t_1;
	} else if (im <= -2000000000000.0) {
		tmp = t_0;
	} else if (im <= 3e-5) {
		tmp = Math.cos(re) * -im;
	} else if (im <= 4.5e+61) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * (math.exp(-im) - math.exp(im))
	t_1 = -0.008333333333333333 * (math.cos(re) * math.pow(im, 5.0))
	tmp = 0
	if im <= -1.7e+93:
		tmp = t_1
	elif im <= -2000000000000.0:
		tmp = t_0
	elif im <= 3e-5:
		tmp = math.cos(re) * -im
	elif im <= 4.5e+61:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(exp(Float64(-im)) - exp(im)))
	t_1 = Float64(-0.008333333333333333 * Float64(cos(re) * (im ^ 5.0)))
	tmp = 0.0
	if (im <= -1.7e+93)
		tmp = t_1;
	elseif (im <= -2000000000000.0)
		tmp = t_0;
	elseif (im <= 3e-5)
		tmp = Float64(cos(re) * Float64(-im));
	elseif (im <= 4.5e+61)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (exp(-im) - exp(im));
	t_1 = -0.008333333333333333 * (cos(re) * (im ^ 5.0));
	tmp = 0.0;
	if (im <= -1.7e+93)
		tmp = t_1;
	elseif (im <= -2000000000000.0)
		tmp = t_0;
	elseif (im <= 3e-5)
		tmp = cos(re) * -im;
	elseif (im <= 4.5e+61)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-0.008333333333333333 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.7e+93], t$95$1, If[LessEqual[im, -2000000000000.0], t$95$0, If[LessEqual[im, 3e-5], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], If[LessEqual[im, 4.5e+61], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.7e93 or 4.5e61 < im

   1. Initial program 100.0%

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

      1. sub0-neg 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%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + \left(-1 \cdot \left(\cos re \cdot im\right) + -0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right)\right)} \]
   5. Step-by-step derivation

      1. associate-+r+ 100.0%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + -1 \cdot \left(\cos re \cdot im\right)\right) + -0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right)} \]

      2. +-commutative 100.0%

         \[\leadsto \color{blue}{-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right) + \left(-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + -1 \cdot \left(\cos re \cdot im\right)\right)} \]

      3. associate-*r* 100.0%

         \[\leadsto \color{blue}{\left(-0.008333333333333333 \cdot \cos re\right) \cdot {im}^{5}} + \left(-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + -1 \cdot \left(\cos re \cdot im\right)\right) \]

      4. *-commutative 100.0%

         \[\leadsto \color{blue}{{im}^{5} \cdot \left(-0.008333333333333333 \cdot \cos re\right)} + \left(-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + -1 \cdot \left(\cos re \cdot im\right)\right) \]

      5. fma-def 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left({im}^{5}, -0.008333333333333333 \cdot \cos re, -0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + -1 \cdot \left(\cos re \cdot im\right)\right)} \]

      6. *-commutative 100.0%

         \[\leadsto \mathsf{fma}\left({im}^{5}, \color{blue}{\cos re \cdot -0.008333333333333333}, -0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + -1 \cdot \left(\cos re \cdot im\right)\right) \]

      7. mul-1-neg 100.0%

         \[\leadsto \mathsf{fma}\left({im}^{5}, \cos re \cdot -0.008333333333333333, -0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + \color{blue}{\left(-\cos re \cdot im\right)}\right) \]

      8. unsub-neg 100.0%

         \[\leadsto \mathsf{fma}\left({im}^{5}, \cos re \cdot -0.008333333333333333, \color{blue}{-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) - \cos re \cdot im}\right) \]

      9. *-commutative 100.0%

         \[\leadsto \mathsf{fma}\left({im}^{5}, \cos re \cdot -0.008333333333333333, \color{blue}{\left(\cos re \cdot {im}^{3}\right) \cdot -0.16666666666666666} - \cos re \cdot im\right) \]

      10. associate-*l* 100.0%

          \[\leadsto \mathsf{fma}\left({im}^{5}, \cos re \cdot -0.008333333333333333, \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)} - \cos re \cdot im\right) \]

      11. distribute-lft-out-- 100.0%

          \[\leadsto \mathsf{fma}\left({im}^{5}, \cos re \cdot -0.008333333333333333, \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)}\right) \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left({im}^{5}, \cos re \cdot -0.008333333333333333, \cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\right)} \]

   7. Taylor expanded in im around inf 100.0%

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

   if -1.7e93 < im < -2e12 or 3.00000000000000008e-5 < im < 4.5e61

   1. Initial program 98.7%

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

      1. sub0-neg 98.7%

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

   3. Simplified 98.7%

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

   4. Taylor expanded in re around 0 85.3%

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

   if -2e12 < im < 3.00000000000000008e-5

   1. Initial program 9.4%

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

      1. sub0-neg 9.4%

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

   3. Simplified 9.4%

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

   4. Taylor expanded in im around 0 97.1%

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

      1. mul-1-neg 97.1%

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

      2. *-commutative 97.1%

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

      3. distribute-lft-neg-in 97.1%

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

   6. Simplified 97.1%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.7 \cdot 10^{+93}:\\ \;\;\;\;-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right)\\ \mathbf{elif}\;im \leq -2000000000000:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 3 \cdot 10^{-5}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right)\\ \end{array} \]

Alternative 7: 95.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(e^{-im} - e^{im}\right)\\ t_1 := -0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right)\\ \mathbf{if}\;im \leq -1.7 \cdot 10^{+93}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -2000000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.28:\\ \;\;\;\;\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (- (exp (- im)) (exp im))))
        (t_1 (* -0.008333333333333333 (* (cos re) (pow im 5.0)))))
   (if (<= im -1.7e+93)
     t_1
     (if (<= im -2000000000000.0)
       t_0
       (if (<= im 0.28)
         (* (cos re) (- (* -0.16666666666666666 (pow im 3.0)) im))
         (if (<= im 4.5e+61) t_0 t_1))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (exp(-im) - exp(im));
	double t_1 = -0.008333333333333333 * (cos(re) * pow(im, 5.0));
	double tmp;
	if (im <= -1.7e+93) {
		tmp = t_1;
	} else if (im <= -2000000000000.0) {
		tmp = t_0;
	} else if (im <= 0.28) {
		tmp = cos(re) * ((-0.16666666666666666 * pow(im, 3.0)) - im);
	} else if (im <= 4.5e+61) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * (exp(-im) - exp(im))
    t_1 = (-0.008333333333333333d0) * (cos(re) * (im ** 5.0d0))
    if (im <= (-1.7d+93)) then
        tmp = t_1
    else if (im <= (-2000000000000.0d0)) then
        tmp = t_0
    else if (im <= 0.28d0) then
        tmp = cos(re) * (((-0.16666666666666666d0) * (im ** 3.0d0)) - im)
    else if (im <= 4.5d+61) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * (Math.exp(-im) - Math.exp(im));
	double t_1 = -0.008333333333333333 * (Math.cos(re) * Math.pow(im, 5.0));
	double tmp;
	if (im <= -1.7e+93) {
		tmp = t_1;
	} else if (im <= -2000000000000.0) {
		tmp = t_0;
	} else if (im <= 0.28) {
		tmp = Math.cos(re) * ((-0.16666666666666666 * Math.pow(im, 3.0)) - im);
	} else if (im <= 4.5e+61) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * (math.exp(-im) - math.exp(im))
	t_1 = -0.008333333333333333 * (math.cos(re) * math.pow(im, 5.0))
	tmp = 0
	if im <= -1.7e+93:
		tmp = t_1
	elif im <= -2000000000000.0:
		tmp = t_0
	elif im <= 0.28:
		tmp = math.cos(re) * ((-0.16666666666666666 * math.pow(im, 3.0)) - im)
	elif im <= 4.5e+61:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(exp(Float64(-im)) - exp(im)))
	t_1 = Float64(-0.008333333333333333 * Float64(cos(re) * (im ^ 5.0)))
	tmp = 0.0
	if (im <= -1.7e+93)
		tmp = t_1;
	elseif (im <= -2000000000000.0)
		tmp = t_0;
	elseif (im <= 0.28)
		tmp = Float64(cos(re) * Float64(Float64(-0.16666666666666666 * (im ^ 3.0)) - im));
	elseif (im <= 4.5e+61)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (exp(-im) - exp(im));
	t_1 = -0.008333333333333333 * (cos(re) * (im ^ 5.0));
	tmp = 0.0;
	if (im <= -1.7e+93)
		tmp = t_1;
	elseif (im <= -2000000000000.0)
		tmp = t_0;
	elseif (im <= 0.28)
		tmp = cos(re) * ((-0.16666666666666666 * (im ^ 3.0)) - im);
	elseif (im <= 4.5e+61)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-0.008333333333333333 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.7e+93], t$95$1, If[LessEqual[im, -2000000000000.0], t$95$0, If[LessEqual[im, 0.28], N[(N[Cos[re], $MachinePrecision] * N[(N[(-0.16666666666666666 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 4.5e+61], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.7e93 or 4.5e61 < im

   1. Initial program 100.0%

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

      1. sub0-neg 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%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + \left(-1 \cdot \left(\cos re \cdot im\right) + -0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right)\right)} \]
   5. Step-by-step derivation

      1. associate-+r+ 100.0%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + -1 \cdot \left(\cos re \cdot im\right)\right) + -0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right)} \]

      2. +-commutative 100.0%

         \[\leadsto \color{blue}{-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right) + \left(-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + -1 \cdot \left(\cos re \cdot im\right)\right)} \]

      3. associate-*r* 100.0%

         \[\leadsto \color{blue}{\left(-0.008333333333333333 \cdot \cos re\right) \cdot {im}^{5}} + \left(-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + -1 \cdot \left(\cos re \cdot im\right)\right) \]

      4. *-commutative 100.0%

         \[\leadsto \color{blue}{{im}^{5} \cdot \left(-0.008333333333333333 \cdot \cos re\right)} + \left(-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + -1 \cdot \left(\cos re \cdot im\right)\right) \]

      5. fma-def 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left({im}^{5}, -0.008333333333333333 \cdot \cos re, -0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + -1 \cdot \left(\cos re \cdot im\right)\right)} \]

      6. *-commutative 100.0%

         \[\leadsto \mathsf{fma}\left({im}^{5}, \color{blue}{\cos re \cdot -0.008333333333333333}, -0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + -1 \cdot \left(\cos re \cdot im\right)\right) \]

      7. mul-1-neg 100.0%

         \[\leadsto \mathsf{fma}\left({im}^{5}, \cos re \cdot -0.008333333333333333, -0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + \color{blue}{\left(-\cos re \cdot im\right)}\right) \]

      8. unsub-neg 100.0%

         \[\leadsto \mathsf{fma}\left({im}^{5}, \cos re \cdot -0.008333333333333333, \color{blue}{-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) - \cos re \cdot im}\right) \]

      9. *-commutative 100.0%

         \[\leadsto \mathsf{fma}\left({im}^{5}, \cos re \cdot -0.008333333333333333, \color{blue}{\left(\cos re \cdot {im}^{3}\right) \cdot -0.16666666666666666} - \cos re \cdot im\right) \]

      10. associate-*l* 100.0%

          \[\leadsto \mathsf{fma}\left({im}^{5}, \cos re \cdot -0.008333333333333333, \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)} - \cos re \cdot im\right) \]

      11. distribute-lft-out-- 100.0%

          \[\leadsto \mathsf{fma}\left({im}^{5}, \cos re \cdot -0.008333333333333333, \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)}\right) \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left({im}^{5}, \cos re \cdot -0.008333333333333333, \cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\right)} \]

   7. Taylor expanded in im around inf 100.0%

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

   if -1.7e93 < im < -2e12 or 0.28000000000000003 < im < 4.5e61

   1. Initial program 100.0%

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

      1. sub0-neg 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 87.5%

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

   if -2e12 < im < 0.28000000000000003

   1. Initial program 11.1%

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

      1. sub0-neg 11.1%

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

   3. Simplified 11.1%

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

   4. Taylor expanded in im around 0 97.1%

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

      1. mul-1-neg 97.1%

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

      2. unsub-neg 97.1%

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

      3. *-commutative 97.1%

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

      4. associate-*l* 97.1%

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

      5. distribute-lft-out-- 97.1%

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

   6. Simplified 97.1%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.7 \cdot 10^{+93}:\\ \;\;\;\;-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right)\\ \mathbf{elif}\;im \leq -2000000000000:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 0.28:\\ \;\;\;\;\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right)\\ \end{array} \]

Alternative 8: 89.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -3.3 \lor \neg \left(im \leq 3.3\right):\\ \;\;\;\;-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -3.3) (not (<= im 3.3)))
   (* -0.008333333333333333 (* (cos re) (pow im 5.0)))
   (* (cos re) (- im))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -3.3) || !(im <= 3.3)) {
		tmp = -0.008333333333333333 * (cos(re) * pow(im, 5.0));
	} else {
		tmp = cos(re) * -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 <= (-3.3d0)) .or. (.not. (im <= 3.3d0))) then
        tmp = (-0.008333333333333333d0) * (cos(re) * (im ** 5.0d0))
    else
        tmp = cos(re) * -im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -3.3) || !(im <= 3.3)) {
		tmp = -0.008333333333333333 * (Math.cos(re) * Math.pow(im, 5.0));
	} else {
		tmp = Math.cos(re) * -im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -3.3) or not (im <= 3.3):
		tmp = -0.008333333333333333 * (math.cos(re) * math.pow(im, 5.0))
	else:
		tmp = math.cos(re) * -im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -3.3) || !(im <= 3.3))
		tmp = Float64(-0.008333333333333333 * Float64(cos(re) * (im ^ 5.0)));
	else
		tmp = Float64(cos(re) * Float64(-im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -3.3) || ~((im <= 3.3)))
		tmp = -0.008333333333333333 * (cos(re) * (im ^ 5.0));
	else
		tmp = cos(re) * -im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -3.3], N[Not[LessEqual[im, 3.3]], $MachinePrecision]], N[(-0.008333333333333333 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -3.2999999999999998 or 3.2999999999999998 < im

   1. Initial program 100.0%

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

      1. sub0-neg 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 76.2%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + \left(-1 \cdot \left(\cos re \cdot im\right) + -0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right)\right)} \]
   5. Step-by-step derivation

      1. associate-+r+ 76.2%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + -1 \cdot \left(\cos re \cdot im\right)\right) + -0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right)} \]

      2. +-commutative 76.2%

         \[\leadsto \color{blue}{-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right) + \left(-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + -1 \cdot \left(\cos re \cdot im\right)\right)} \]

      3. associate-*r* 76.2%

         \[\leadsto \color{blue}{\left(-0.008333333333333333 \cdot \cos re\right) \cdot {im}^{5}} + \left(-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + -1 \cdot \left(\cos re \cdot im\right)\right) \]

      4. *-commutative 76.2%

         \[\leadsto \color{blue}{{im}^{5} \cdot \left(-0.008333333333333333 \cdot \cos re\right)} + \left(-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + -1 \cdot \left(\cos re \cdot im\right)\right) \]

      5. fma-def 76.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left({im}^{5}, -0.008333333333333333 \cdot \cos re, -0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + -1 \cdot \left(\cos re \cdot im\right)\right)} \]

      6. *-commutative 76.2%

         \[\leadsto \mathsf{fma}\left({im}^{5}, \color{blue}{\cos re \cdot -0.008333333333333333}, -0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + -1 \cdot \left(\cos re \cdot im\right)\right) \]

      7. mul-1-neg 76.2%

         \[\leadsto \mathsf{fma}\left({im}^{5}, \cos re \cdot -0.008333333333333333, -0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + \color{blue}{\left(-\cos re \cdot im\right)}\right) \]

      8. unsub-neg 76.2%

         \[\leadsto \mathsf{fma}\left({im}^{5}, \cos re \cdot -0.008333333333333333, \color{blue}{-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) - \cos re \cdot im}\right) \]

      9. *-commutative 76.2%

         \[\leadsto \mathsf{fma}\left({im}^{5}, \cos re \cdot -0.008333333333333333, \color{blue}{\left(\cos re \cdot {im}^{3}\right) \cdot -0.16666666666666666} - \cos re \cdot im\right) \]

      10. associate-*l* 76.2%

          \[\leadsto \mathsf{fma}\left({im}^{5}, \cos re \cdot -0.008333333333333333, \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)} - \cos re \cdot im\right) \]

      11. distribute-lft-out-- 76.2%

          \[\leadsto \mathsf{fma}\left({im}^{5}, \cos re \cdot -0.008333333333333333, \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)}\right) \]

   6. Simplified 76.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left({im}^{5}, \cos re \cdot -0.008333333333333333, \cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\right)} \]

   7. Taylor expanded in im around inf 76.2%

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

   if -3.2999999999999998 < im < 3.2999999999999998

   1. Initial program 9.0%

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

      1. sub0-neg 9.0%

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

   3. Simplified 9.0%

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

   4. Taylor expanded in im around 0 98.2%

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

      1. mul-1-neg 98.2%

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

      2. *-commutative 98.2%

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

      3. distribute-lft-neg-in 98.2%

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

   6. Simplified 98.2%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3.3 \lor \neg \left(im \leq 3.3\right):\\ \;\;\;\;-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \end{array} \]

Alternative 9: 77.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-0.16666666666666666 \cdot {im}^{3} - im\right) \cdot \left(1 + \left(re \cdot re\right) \cdot -0.5\right)\\ \mathbf{if}\;im \leq -9.6 \cdot 10^{+23}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 600:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 1.65 \cdot 10^{+78}:\\ \;\;\;\;{re}^{4} \cdot \left(im \cdot -0.041666666666666664\right) - im\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (*
          (- (* -0.16666666666666666 (pow im 3.0)) im)
          (+ 1.0 (* (* re re) -0.5)))))
   (if (<= im -9.6e+23)
     t_0
     (if (<= im 600.0)
       (* (cos re) (- im))
       (if (<= im 1.65e+78)
         (- (* (pow re 4.0) (* im -0.041666666666666664)) im)
         t_0)))))

C

double code(double re, double im) {
	double t_0 = ((-0.16666666666666666 * pow(im, 3.0)) - im) * (1.0 + ((re * re) * -0.5));
	double tmp;
	if (im <= -9.6e+23) {
		tmp = t_0;
	} else if (im <= 600.0) {
		tmp = cos(re) * -im;
	} else if (im <= 1.65e+78) {
		tmp = (pow(re, 4.0) * (im * -0.041666666666666664)) - 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 = (((-0.16666666666666666d0) * (im ** 3.0d0)) - im) * (1.0d0 + ((re * re) * (-0.5d0)))
    if (im <= (-9.6d+23)) then
        tmp = t_0
    else if (im <= 600.0d0) then
        tmp = cos(re) * -im
    else if (im <= 1.65d+78) then
        tmp = ((re ** 4.0d0) * (im * (-0.041666666666666664d0))) - im
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = ((-0.16666666666666666 * Math.pow(im, 3.0)) - im) * (1.0 + ((re * re) * -0.5));
	double tmp;
	if (im <= -9.6e+23) {
		tmp = t_0;
	} else if (im <= 600.0) {
		tmp = Math.cos(re) * -im;
	} else if (im <= 1.65e+78) {
		tmp = (Math.pow(re, 4.0) * (im * -0.041666666666666664)) - im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = ((-0.16666666666666666 * math.pow(im, 3.0)) - im) * (1.0 + ((re * re) * -0.5))
	tmp = 0
	if im <= -9.6e+23:
		tmp = t_0
	elif im <= 600.0:
		tmp = math.cos(re) * -im
	elif im <= 1.65e+78:
		tmp = (math.pow(re, 4.0) * (im * -0.041666666666666664)) - im
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(Float64(-0.16666666666666666 * (im ^ 3.0)) - im) * Float64(1.0 + Float64(Float64(re * re) * -0.5)))
	tmp = 0.0
	if (im <= -9.6e+23)
		tmp = t_0;
	elseif (im <= 600.0)
		tmp = Float64(cos(re) * Float64(-im));
	elseif (im <= 1.65e+78)
		tmp = Float64(Float64((re ^ 4.0) * Float64(im * -0.041666666666666664)) - im);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = ((-0.16666666666666666 * (im ^ 3.0)) - im) * (1.0 + ((re * re) * -0.5));
	tmp = 0.0;
	if (im <= -9.6e+23)
		tmp = t_0;
	elseif (im <= 600.0)
		tmp = cos(re) * -im;
	elseif (im <= 1.65e+78)
		tmp = ((re ^ 4.0) * (im * -0.041666666666666664)) - im;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[(-0.16666666666666666 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision] * N[(1.0 + N[(N[(re * re), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -9.6e+23], t$95$0, If[LessEqual[im, 600.0], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], If[LessEqual[im, 1.65e+78], N[(N[(N[Power[re, 4.0], $MachinePrecision] * N[(im * -0.041666666666666664), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -9.6e23 or 1.65e78 < 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. sub0-neg 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 79.9%

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

      1. mul-1-neg 79.9%

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

      2. unsub-neg 79.9%

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

      3. *-commutative 79.9%

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

      4. associate-*l* 79.9%

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

      5. distribute-lft-out-- 79.9%

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

   6. Simplified 79.9%

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

   7. Taylor expanded in re around 0 63.0%

      \[\leadsto \color{blue}{\left(1 + -0.5 \cdot {re}^{2}\right)} \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right) \]
   8. Step-by-step derivation

      1. *-commutative 63.0%

         \[\leadsto \left(1 + \color{blue}{{re}^{2} \cdot -0.5}\right) \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right) \]

      2. unpow2 63.0%

         \[\leadsto \left(1 + \color{blue}{\left(re \cdot re\right)} \cdot -0.5\right) \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right) \]

   9. Simplified 63.0%

      \[\leadsto \color{blue}{\left(1 + \left(re \cdot re\right) \cdot -0.5\right)} \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right) \]

   if -9.6e23 < im < 600

   1. Initial program 13.0%

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

      1. sub0-neg 13.0%

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

   3. Simplified 13.0%

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

   4. Taylor expanded in im around 0 94.2%

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

      1. mul-1-neg 94.2%

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

      2. *-commutative 94.2%

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

      3. distribute-lft-neg-in 94.2%

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

   6. Simplified 94.2%

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

   if 600 < im < 1.65e78

   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. sub0-neg 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%

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

      1. mul-1-neg 3.4%

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

      2. *-commutative 3.4%

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

      3. distribute-lft-neg-in 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 16.5%

      \[\leadsto \color{blue}{-0.041666666666666664 \cdot \left({re}^{4} \cdot im\right) + \left(-1 \cdot im + 0.5 \cdot \left({re}^{2} \cdot im\right)\right)} \]
   8. Step-by-step derivation

      1. neg-mul-1 16.5%

         \[\leadsto -0.041666666666666664 \cdot \left({re}^{4} \cdot im\right) + \left(\color{blue}{\left(-im\right)} + 0.5 \cdot \left({re}^{2} \cdot im\right)\right) \]

      2. associate-+r+ 16.5%

         \[\leadsto \color{blue}{\left(-0.041666666666666664 \cdot \left({re}^{4} \cdot im\right) + \left(-im\right)\right) + 0.5 \cdot \left({re}^{2} \cdot im\right)} \]

      3. +-commutative 16.5%

         \[\leadsto \color{blue}{0.5 \cdot \left({re}^{2} \cdot im\right) + \left(-0.041666666666666664 \cdot \left({re}^{4} \cdot im\right) + \left(-im\right)\right)} \]

      4. associate-+r+ 16.5%

         \[\leadsto \color{blue}{\left(0.5 \cdot \left({re}^{2} \cdot im\right) + -0.041666666666666664 \cdot \left({re}^{4} \cdot im\right)\right) + \left(-im\right)} \]

      5. sub-neg 16.5%

         \[\leadsto \color{blue}{\left(0.5 \cdot \left({re}^{2} \cdot im\right) + -0.041666666666666664 \cdot \left({re}^{4} \cdot im\right)\right) - im} \]

      6. associate-*r* 16.5%

         \[\leadsto \left(\color{blue}{\left(0.5 \cdot {re}^{2}\right) \cdot im} + -0.041666666666666664 \cdot \left({re}^{4} \cdot im\right)\right) - im \]

      7. associate-*r* 16.5%

         \[\leadsto \left(\left(0.5 \cdot {re}^{2}\right) \cdot im + \color{blue}{\left(-0.041666666666666664 \cdot {re}^{4}\right) \cdot im}\right) - im \]

      8. distribute-rgt-out 16.5%

         \[\leadsto \color{blue}{im \cdot \left(0.5 \cdot {re}^{2} + -0.041666666666666664 \cdot {re}^{4}\right)} - im \]

      9. *-commutative 16.5%

         \[\leadsto im \cdot \left(\color{blue}{{re}^{2} \cdot 0.5} + -0.041666666666666664 \cdot {re}^{4}\right) - im \]

      10. unpow2 16.5%

          \[\leadsto im \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot 0.5 + -0.041666666666666664 \cdot {re}^{4}\right) - im \]

      11. associate-*l* 16.5%

          \[\leadsto im \cdot \left(\color{blue}{re \cdot \left(re \cdot 0.5\right)} + -0.041666666666666664 \cdot {re}^{4}\right) - im \]

      12. *-commutative 16.5%

          \[\leadsto im \cdot \left(re \cdot \left(re \cdot 0.5\right) + \color{blue}{{re}^{4} \cdot -0.041666666666666664}\right) - im \]

   9. Simplified 16.5%

      \[\leadsto \color{blue}{im \cdot \left(re \cdot \left(re \cdot 0.5\right) + {re}^{4} \cdot -0.041666666666666664\right) - im} \]

   10. Taylor expanded in re around inf 37.9%

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

       1. *-commutative 37.9%

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

       2. associate-*l* 37.9%

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

   12. Simplified 37.9%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -9.6 \cdot 10^{+23}:\\ \;\;\;\;\left(-0.16666666666666666 \cdot {im}^{3} - im\right) \cdot \left(1 + \left(re \cdot re\right) \cdot -0.5\right)\\ \mathbf{elif}\;im \leq 600:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 1.65 \cdot 10^{+78}:\\ \;\;\;\;{re}^{4} \cdot \left(im \cdot -0.041666666666666664\right) - im\\ \mathbf{else}:\\ \;\;\;\;\left(-0.16666666666666666 \cdot {im}^{3} - im\right) \cdot \left(1 + \left(re \cdot re\right) \cdot -0.5\right)\\ \end{array} \]

Alternative 10: 76.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.16666666666666666 \cdot {im}^{3}\\ \mathbf{if}\;im \leq -3.4 \cdot 10^{+52}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 510:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 6.8 \cdot 10^{+85}:\\ \;\;\;\;{re}^{4} \cdot \left(im \cdot -0.041666666666666664\right) - im\\ \mathbf{else}:\\ \;\;\;\;t_0 - im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* -0.16666666666666666 (pow im 3.0))))
   (if (<= im -3.4e+52)
     t_0
     (if (<= im 510.0)
       (* (cos re) (- im))
       (if (<= im 6.8e+85)
         (- (* (pow re 4.0) (* im -0.041666666666666664)) im)
         (- t_0 im))))))

C

double code(double re, double im) {
	double t_0 = -0.16666666666666666 * pow(im, 3.0);
	double tmp;
	if (im <= -3.4e+52) {
		tmp = t_0;
	} else if (im <= 510.0) {
		tmp = cos(re) * -im;
	} else if (im <= 6.8e+85) {
		tmp = (pow(re, 4.0) * (im * -0.041666666666666664)) - im;
	} else {
		tmp = t_0 - 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 = (-0.16666666666666666d0) * (im ** 3.0d0)
    if (im <= (-3.4d+52)) then
        tmp = t_0
    else if (im <= 510.0d0) then
        tmp = cos(re) * -im
    else if (im <= 6.8d+85) then
        tmp = ((re ** 4.0d0) * (im * (-0.041666666666666664d0))) - im
    else
        tmp = t_0 - im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = -0.16666666666666666 * Math.pow(im, 3.0);
	double tmp;
	if (im <= -3.4e+52) {
		tmp = t_0;
	} else if (im <= 510.0) {
		tmp = Math.cos(re) * -im;
	} else if (im <= 6.8e+85) {
		tmp = (Math.pow(re, 4.0) * (im * -0.041666666666666664)) - im;
	} else {
		tmp = t_0 - im;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = -0.16666666666666666 * math.pow(im, 3.0)
	tmp = 0
	if im <= -3.4e+52:
		tmp = t_0
	elif im <= 510.0:
		tmp = math.cos(re) * -im
	elif im <= 6.8e+85:
		tmp = (math.pow(re, 4.0) * (im * -0.041666666666666664)) - im
	else:
		tmp = t_0 - im
	return tmp

Julia

function code(re, im)
	t_0 = Float64(-0.16666666666666666 * (im ^ 3.0))
	tmp = 0.0
	if (im <= -3.4e+52)
		tmp = t_0;
	elseif (im <= 510.0)
		tmp = Float64(cos(re) * Float64(-im));
	elseif (im <= 6.8e+85)
		tmp = Float64(Float64((re ^ 4.0) * Float64(im * -0.041666666666666664)) - im);
	else
		tmp = Float64(t_0 - im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = -0.16666666666666666 * (im ^ 3.0);
	tmp = 0.0;
	if (im <= -3.4e+52)
		tmp = t_0;
	elseif (im <= 510.0)
		tmp = cos(re) * -im;
	elseif (im <= 6.8e+85)
		tmp = ((re ^ 4.0) * (im * -0.041666666666666664)) - im;
	else
		tmp = t_0 - im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(-0.16666666666666666 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -3.4e+52], t$95$0, If[LessEqual[im, 510.0], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], If[LessEqual[im, 6.8e+85], N[(N[(N[Power[re, 4.0], $MachinePrecision] * N[(im * -0.041666666666666664), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], N[(t$95$0 - im), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -3.4e52

   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. sub0-neg 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 88.1%

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

      1. mul-1-neg 88.1%

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

      2. unsub-neg 88.1%

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

      3. *-commutative 88.1%

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

      4. associate-*l* 88.1%

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

      5. distribute-lft-out-- 88.1%

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

   6. Simplified 88.1%

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

   7. Taylor expanded in re around 0 69.8%

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

   8. Taylor expanded in im around inf 69.8%

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

   if -3.4e52 < im < 510

   1. Initial program 19.4%

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

      1. sub0-neg 19.4%

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

   3. Simplified 19.4%

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

   4. Taylor expanded in im around 0 87.5%

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

      1. mul-1-neg 87.5%

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

      2. *-commutative 87.5%

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

      3. distribute-lft-neg-in 87.5%

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

   6. Simplified 87.5%

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

   if 510 < im < 6.8000000000000007e85

   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. sub0-neg 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%

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

      1. mul-1-neg 3.4%

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

      2. *-commutative 3.4%

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

      3. distribute-lft-neg-in 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 16.5%

      \[\leadsto \color{blue}{-0.041666666666666664 \cdot \left({re}^{4} \cdot im\right) + \left(-1 \cdot im + 0.5 \cdot \left({re}^{2} \cdot im\right)\right)} \]
   8. Step-by-step derivation

      1. neg-mul-1 16.5%

         \[\leadsto -0.041666666666666664 \cdot \left({re}^{4} \cdot im\right) + \left(\color{blue}{\left(-im\right)} + 0.5 \cdot \left({re}^{2} \cdot im\right)\right) \]

      2. associate-+r+ 16.5%

         \[\leadsto \color{blue}{\left(-0.041666666666666664 \cdot \left({re}^{4} \cdot im\right) + \left(-im\right)\right) + 0.5 \cdot \left({re}^{2} \cdot im\right)} \]

      3. +-commutative 16.5%

         \[\leadsto \color{blue}{0.5 \cdot \left({re}^{2} \cdot im\right) + \left(-0.041666666666666664 \cdot \left({re}^{4} \cdot im\right) + \left(-im\right)\right)} \]

      4. associate-+r+ 16.5%

         \[\leadsto \color{blue}{\left(0.5 \cdot \left({re}^{2} \cdot im\right) + -0.041666666666666664 \cdot \left({re}^{4} \cdot im\right)\right) + \left(-im\right)} \]

      5. sub-neg 16.5%

         \[\leadsto \color{blue}{\left(0.5 \cdot \left({re}^{2} \cdot im\right) + -0.041666666666666664 \cdot \left({re}^{4} \cdot im\right)\right) - im} \]

      6. associate-*r* 16.5%

         \[\leadsto \left(\color{blue}{\left(0.5 \cdot {re}^{2}\right) \cdot im} + -0.041666666666666664 \cdot \left({re}^{4} \cdot im\right)\right) - im \]

      7. associate-*r* 16.5%

         \[\leadsto \left(\left(0.5 \cdot {re}^{2}\right) \cdot im + \color{blue}{\left(-0.041666666666666664 \cdot {re}^{4}\right) \cdot im}\right) - im \]

      8. distribute-rgt-out 16.5%

         \[\leadsto \color{blue}{im \cdot \left(0.5 \cdot {re}^{2} + -0.041666666666666664 \cdot {re}^{4}\right)} - im \]

      9. *-commutative 16.5%

         \[\leadsto im \cdot \left(\color{blue}{{re}^{2} \cdot 0.5} + -0.041666666666666664 \cdot {re}^{4}\right) - im \]

      10. unpow2 16.5%

          \[\leadsto im \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot 0.5 + -0.041666666666666664 \cdot {re}^{4}\right) - im \]

      11. associate-*l* 16.5%

          \[\leadsto im \cdot \left(\color{blue}{re \cdot \left(re \cdot 0.5\right)} + -0.041666666666666664 \cdot {re}^{4}\right) - im \]

      12. *-commutative 16.5%

          \[\leadsto im \cdot \left(re \cdot \left(re \cdot 0.5\right) + \color{blue}{{re}^{4} \cdot -0.041666666666666664}\right) - im \]

   9. Simplified 16.5%

      \[\leadsto \color{blue}{im \cdot \left(re \cdot \left(re \cdot 0.5\right) + {re}^{4} \cdot -0.041666666666666664\right) - im} \]

   10. Taylor expanded in re around inf 37.9%

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

       1. *-commutative 37.9%

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

       2. associate-*l* 37.9%

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

   12. Simplified 37.9%

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

   if 6.8000000000000007e85 < 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. sub0-neg 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 90.1%

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

      1. mul-1-neg 90.1%

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

      2. unsub-neg 90.1%

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

      3. *-commutative 90.1%

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

      4. associate-*l* 90.1%

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

      5. distribute-lft-out-- 90.1%

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

   6. Simplified 90.1%

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

   7. Taylor expanded in re around 0 51.8%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3} - im} \]
3. Recombined 4 regimes into one program.

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3.4 \cdot 10^{+52}:\\ \;\;\;\;-0.16666666666666666 \cdot {im}^{3}\\ \mathbf{elif}\;im \leq 510:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{elif}\;im \leq 6.8 \cdot 10^{+85}:\\ \;\;\;\;{re}^{4} \cdot \left(im \cdot -0.041666666666666664\right) - im\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot {im}^{3} - im\\ \end{array} \]

Alternative 11: 74.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.16666666666666666 \cdot {im}^{3}\\ \mathbf{if}\;im \leq -8.2 \cdot 10^{+52}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 6.9 \cdot 10^{-18}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 - im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* -0.16666666666666666 (pow im 3.0))))
   (if (<= im -8.2e+52)
     t_0
     (if (<= im 6.9e-18) (* (cos re) (- im)) (- t_0 im)))))

C

double code(double re, double im) {
	double t_0 = -0.16666666666666666 * pow(im, 3.0);
	double tmp;
	if (im <= -8.2e+52) {
		tmp = t_0;
	} else if (im <= 6.9e-18) {
		tmp = cos(re) * -im;
	} else {
		tmp = t_0 - 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 = (-0.16666666666666666d0) * (im ** 3.0d0)
    if (im <= (-8.2d+52)) then
        tmp = t_0
    else if (im <= 6.9d-18) then
        tmp = cos(re) * -im
    else
        tmp = t_0 - im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = -0.16666666666666666 * Math.pow(im, 3.0);
	double tmp;
	if (im <= -8.2e+52) {
		tmp = t_0;
	} else if (im <= 6.9e-18) {
		tmp = Math.cos(re) * -im;
	} else {
		tmp = t_0 - im;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = -0.16666666666666666 * math.pow(im, 3.0)
	tmp = 0
	if im <= -8.2e+52:
		tmp = t_0
	elif im <= 6.9e-18:
		tmp = math.cos(re) * -im
	else:
		tmp = t_0 - im
	return tmp

Julia

function code(re, im)
	t_0 = Float64(-0.16666666666666666 * (im ^ 3.0))
	tmp = 0.0
	if (im <= -8.2e+52)
		tmp = t_0;
	elseif (im <= 6.9e-18)
		tmp = Float64(cos(re) * Float64(-im));
	else
		tmp = Float64(t_0 - im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = -0.16666666666666666 * (im ^ 3.0);
	tmp = 0.0;
	if (im <= -8.2e+52)
		tmp = t_0;
	elseif (im <= 6.9e-18)
		tmp = cos(re) * -im;
	else
		tmp = t_0 - im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(-0.16666666666666666 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -8.2e+52], t$95$0, If[LessEqual[im, 6.9e-18], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision], N[(t$95$0 - im), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < -8.1999999999999999e52

   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. sub0-neg 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 88.1%

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

      1. mul-1-neg 88.1%

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

      2. unsub-neg 88.1%

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

      3. *-commutative 88.1%

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

      4. associate-*l* 88.1%

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

      5. distribute-lft-out-- 88.1%

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

   6. Simplified 88.1%

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

   7. Taylor expanded in re around 0 69.8%

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

   8. Taylor expanded in im around inf 69.8%

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

   if -8.1999999999999999e52 < im < 6.9000000000000003e-18

   1. Initial program 18.1%

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

      1. sub0-neg 18.1%

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

   3. Simplified 18.1%

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

   4. Taylor expanded in im around 0 88.1%

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

      1. mul-1-neg 88.1%

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

      2. *-commutative 88.1%

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

      3. distribute-lft-neg-in 88.1%

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

   6. Simplified 88.1%

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

   if 6.9000000000000003e-18 < im

   1. Initial program 97.4%

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

      1. sub0-neg 97.4%

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

   3. Simplified 97.4%

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

   4. Taylor expanded in im around 0 68.5%

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

      1. mul-1-neg 68.5%

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

      2. unsub-neg 68.5%

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

      3. *-commutative 68.5%

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

      4. associate-*l* 68.5%

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

      5. distribute-lft-out-- 68.5%

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

   6. Simplified 68.5%

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

   7. Taylor expanded in re around 0 41.7%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3} - im} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -8.2 \cdot 10^{+52}:\\ \;\;\;\;-0.16666666666666666 \cdot {im}^{3}\\ \mathbf{elif}\;im \leq 6.9 \cdot 10^{-18}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot {im}^{3} - im\\ \end{array} \]

Alternative 12: 52.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -2000000000000 \lor \neg \left(im \leq 3.4\right):\\ \;\;\;\;-0.16666666666666666 \cdot {im}^{3}\\ \mathbf{else}:\\ \;\;\;\;-im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -2000000000000.0) (not (<= im 3.4)))
   (* -0.16666666666666666 (pow im 3.0))
   (- im)))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -2000000000000.0) || !(im <= 3.4)) {
		tmp = -0.16666666666666666 * pow(im, 3.0);
	} else {
		tmp = -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 <= (-2000000000000.0d0)) .or. (.not. (im <= 3.4d0))) then
        tmp = (-0.16666666666666666d0) * (im ** 3.0d0)
    else
        tmp = -im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -2000000000000.0) || !(im <= 3.4)) {
		tmp = -0.16666666666666666 * Math.pow(im, 3.0);
	} else {
		tmp = -im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -2000000000000.0) or not (im <= 3.4):
		tmp = -0.16666666666666666 * math.pow(im, 3.0)
	else:
		tmp = -im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -2000000000000.0) || !(im <= 3.4))
		tmp = Float64(-0.16666666666666666 * (im ^ 3.0));
	else
		tmp = Float64(-im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -2000000000000.0) || ~((im <= 3.4)))
		tmp = -0.16666666666666666 * (im ^ 3.0);
	else
		tmp = -im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -2000000000000.0], N[Not[LessEqual[im, 3.4]], $MachinePrecision]], N[(-0.16666666666666666 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision], (-im)]

Derivation

1. Split input into 2 regimes

2. if im < -2e12 or 3.39999999999999991 < 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. sub0-neg 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 69.1%

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

      1. mul-1-neg 69.1%

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

      2. unsub-neg 69.1%

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

      3. *-commutative 69.1%

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

      4. associate-*l* 69.1%

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

      5. distribute-lft-out-- 69.1%

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

   6. Simplified 69.1%

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

   7. Taylor expanded in re around 0 48.8%

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

   8. Taylor expanded in im around inf 48.8%

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

   if -2e12 < im < 3.39999999999999991

   1. Initial program 11.1%

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

      1. sub0-neg 11.1%

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

   3. Simplified 11.1%

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

   4. Taylor expanded in im around 0 96.2%

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

      1. mul-1-neg 96.2%

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

      2. *-commutative 96.2%

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

      3. distribute-lft-neg-in 96.2%

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

   6. Simplified 96.2%

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

   7. Taylor expanded in re around 0 50.5%

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

      1. neg-mul-1 50.5%

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

   9. Simplified 50.5%

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

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2000000000000 \lor \neg \left(im \leq 3.4\right):\\ \;\;\;\;-0.16666666666666666 \cdot {im}^{3}\\ \mathbf{else}:\\ \;\;\;\;-im\\ \end{array} \]

Alternative 13: 75.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -2.5 \cdot 10^{+54} \lor \neg \left(im \leq 1600000000000\right):\\ \;\;\;\;-0.16666666666666666 \cdot {im}^{3}\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -2.5e+54) (not (<= im 1600000000000.0)))
   (* -0.16666666666666666 (pow im 3.0))
   (* (cos re) (- im))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -2.5e+54) || !(im <= 1600000000000.0)) {
		tmp = -0.16666666666666666 * pow(im, 3.0);
	} else {
		tmp = cos(re) * -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 <= (-2.5d+54)) .or. (.not. (im <= 1600000000000.0d0))) then
        tmp = (-0.16666666666666666d0) * (im ** 3.0d0)
    else
        tmp = cos(re) * -im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -2.5e+54) || !(im <= 1600000000000.0)) {
		tmp = -0.16666666666666666 * Math.pow(im, 3.0);
	} else {
		tmp = Math.cos(re) * -im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -2.5e+54) or not (im <= 1600000000000.0):
		tmp = -0.16666666666666666 * math.pow(im, 3.0)
	else:
		tmp = math.cos(re) * -im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -2.5e+54) || !(im <= 1600000000000.0))
		tmp = Float64(-0.16666666666666666 * (im ^ 3.0));
	else
		tmp = Float64(cos(re) * Float64(-im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -2.5e+54) || ~((im <= 1600000000000.0)))
		tmp = -0.16666666666666666 * (im ^ 3.0);
	else
		tmp = cos(re) * -im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -2.5e+54], N[Not[LessEqual[im, 1600000000000.0]], $MachinePrecision]], N[(-0.16666666666666666 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * (-im)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -2.50000000000000003e54 or 1.6e12 < 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. sub0-neg 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 77.7%

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

      1. mul-1-neg 77.7%

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

      2. unsub-neg 77.7%

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

      3. *-commutative 77.7%

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

      4. associate-*l* 77.7%

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

      5. distribute-lft-out-- 77.7%

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

   6. Simplified 77.7%

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

   7. Taylor expanded in re around 0 54.8%

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

   8. Taylor expanded in im around inf 54.8%

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

   if -2.50000000000000003e54 < im < 1.6e12

   1. Initial program 19.4%

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

      1. sub0-neg 19.4%

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

   3. Simplified 19.4%

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

   4. Taylor expanded in im around 0 87.5%

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

      1. mul-1-neg 87.5%

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

      2. *-commutative 87.5%

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

      3. distribute-lft-neg-in 87.5%

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

   6. Simplified 87.5%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.5 \cdot 10^{+54} \lor \neg \left(im \leq 1600000000000\right):\\ \;\;\;\;-0.16666666666666666 \cdot {im}^{3}\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(-im\right)\\ \end{array} \]

Alternative 14: 38.4% accurate, 20.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -5.5 \cdot 10^{+155}:\\ \;\;\;\;\frac{im \cdot im}{im \cdot \left(re \cdot \left(re \cdot -0.5\right)\right) - im}\\ \mathbf{elif}\;im \leq 9 \cdot 10^{+76}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right) + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -5.5e+155)
   (/ (* im im) (- (* im (* re (* re -0.5))) im))
   (if (<= im 9e+76) (- im) (* im (+ (* 0.5 (* re re)) -1.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -5.5e+155) {
		tmp = (im * im) / ((im * (re * (re * -0.5))) - im);
	} else if (im <= 9e+76) {
		tmp = -im;
	} else {
		tmp = im * ((0.5 * (re * re)) + -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= (-5.5d+155)) then
        tmp = (im * im) / ((im * (re * (re * (-0.5d0)))) - im)
    else if (im <= 9d+76) then
        tmp = -im
    else
        tmp = im * ((0.5d0 * (re * re)) + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -5.5e+155) {
		tmp = (im * im) / ((im * (re * (re * -0.5))) - im);
	} else if (im <= 9e+76) {
		tmp = -im;
	} else {
		tmp = im * ((0.5 * (re * re)) + -1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -5.5e+155:
		tmp = (im * im) / ((im * (re * (re * -0.5))) - im)
	elif im <= 9e+76:
		tmp = -im
	else:
		tmp = im * ((0.5 * (re * re)) + -1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -5.5e+155)
		tmp = Float64(Float64(im * im) / Float64(Float64(im * Float64(re * Float64(re * -0.5))) - im));
	elseif (im <= 9e+76)
		tmp = Float64(-im);
	else
		tmp = Float64(im * Float64(Float64(0.5 * Float64(re * re)) + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -5.5e+155)
		tmp = (im * im) / ((im * (re * (re * -0.5))) - im);
	elseif (im <= 9e+76)
		tmp = -im;
	else
		tmp = im * ((0.5 * (re * re)) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -5.5e+155], N[(N[(im * im), $MachinePrecision] / N[(N[(im * N[(re * N[(re * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 9e+76], (-im), N[(im * N[(N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < -5.5000000000000001e155

   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. sub0-neg 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 7.5%

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

      1. mul-1-neg 7.5%

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

      2. *-commutative 7.5%

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

      3. distribute-lft-neg-in 7.5%

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

   6. Simplified 7.5%

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

   7. Taylor expanded in re around 0 26.8%

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

      1. neg-mul-1 26.8%

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

      2. +-commutative 26.8%

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

      3. unsub-neg 26.8%

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

      4. *-commutative 26.8%

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

      5. associate-*l* 26.8%

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

      6. unpow2 26.8%

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

      7. *-commutative 26.8%

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

      8. associate-*l* 26.8%

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

      9. *-commutative 26.8%

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

   9. Simplified 26.8%

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

       1. flip-- 47.2%

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

       2. frac-2neg 47.2%

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

   11. Applied egg-rr 0.0%

       \[\leadsto \color{blue}{\frac{-\left({re}^{4} \cdot \left(0.25 \cdot \left(im \cdot im\right)\right) - im \cdot im\right)}{-\mathsf{fma}\left(re, \left(re \cdot 0.5\right) \cdot im, im\right)}} \]
   12. Step-by-step derivation

   13. Simplified 0.0%

       \[\leadsto \color{blue}{\frac{\left({re}^{4} \cdot -0.25\right) \cdot \left(im \cdot im\right) + im \cdot im}{im \cdot \left(\left(-0.5 \cdot re\right) \cdot re\right) - im}} \]

   14. Taylor expanded in re around 0 61.1%

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

       1. unpow2 61.1%

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

   16. Simplified 61.1%

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

   if -5.5000000000000001e155 < im < 8.9999999999999995e76

   1. Initial program 33.9%

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

      1. sub0-neg 33.9%

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

   3. Simplified 33.9%

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

   4. Taylor expanded in im around 0 72.4%

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

      1. mul-1-neg 72.4%

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

      2. *-commutative 72.4%

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

      3. distribute-lft-neg-in 72.4%

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

   6. Simplified 72.4%

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

   7. Taylor expanded in re around 0 38.3%

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

      1. neg-mul-1 38.3%

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

   9. Simplified 38.3%

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

   if 8.9999999999999995e76 < 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. sub0-neg 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 6.1%

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

      1. mul-1-neg 6.1%

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

      2. *-commutative 6.1%

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

      3. distribute-lft-neg-in 6.1%

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

   6. Simplified 6.1%

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

   7. Taylor expanded in re around 0 35.0%

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

      1. neg-mul-1 35.0%

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

      2. +-commutative 35.0%

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

      3. unsub-neg 35.0%

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

      4. *-commutative 35.0%

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

      5. associate-*l* 35.0%

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

      6. unpow2 35.0%

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

      7. *-commutative 35.0%

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

      8. associate-*l* 35.0%

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

      9. *-commutative 35.0%

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

   9. Simplified 35.0%

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

       1. flip-- 27.2%

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

       2. frac-2neg 27.2%

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

   11. Applied egg-rr 5.6%

       \[\leadsto \color{blue}{\frac{-\left({re}^{4} \cdot \left(0.25 \cdot \left(im \cdot im\right)\right) - im \cdot im\right)}{-\mathsf{fma}\left(re, \left(re \cdot 0.5\right) \cdot im, im\right)}} \]
   12. Step-by-step derivation

   13. Simplified 5.6%

       \[\leadsto \color{blue}{\frac{\left({re}^{4} \cdot -0.25\right) \cdot \left(im \cdot im\right) + im \cdot im}{im \cdot \left(\left(-0.5 \cdot re\right) \cdot re\right) - im}} \]

   14. Taylor expanded in re around 0 35.0%

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

       1. neg-mul-1 35.0%

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

       2. +-commutative 35.0%

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

       3. associate-*r* 35.0%

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

       4. neg-mul-1 35.0%

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

       5. distribute-rgt-out 35.0%

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

       6. *-commutative 35.0%

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

       7. unpow2 35.0%

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

   16. Simplified 35.0%

       \[\leadsto \color{blue}{im \cdot \left(\left(re \cdot re\right) \cdot 0.5 + -1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 41.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -5.5 \cdot 10^{+155}:\\ \;\;\;\;\frac{im \cdot im}{im \cdot \left(re \cdot \left(re \cdot -0.5\right)\right) - im}\\ \mathbf{elif}\;im \leq 9 \cdot 10^{+76}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right) + -1\right)\\ \end{array} \]

Alternative 15: 36.3% accurate, 27.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -17 \lor \neg \left(im \leq 4.1 \cdot 10^{+89}\right):\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -17.0) (not (<= im 4.1e+89))) (* im (* 0.5 (* re re))) (- im)))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -17.0) || !(im <= 4.1e+89)) {
		tmp = im * (0.5 * (re * re));
	} else {
		tmp = -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 <= (-17.0d0)) .or. (.not. (im <= 4.1d+89))) then
        tmp = im * (0.5d0 * (re * re))
    else
        tmp = -im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -17.0) || !(im <= 4.1e+89)) {
		tmp = im * (0.5 * (re * re));
	} else {
		tmp = -im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -17.0) or not (im <= 4.1e+89):
		tmp = im * (0.5 * (re * re))
	else:
		tmp = -im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -17.0) || !(im <= 4.1e+89))
		tmp = Float64(im * Float64(0.5 * Float64(re * re)));
	else
		tmp = Float64(-im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -17.0) || ~((im <= 4.1e+89)))
		tmp = im * (0.5 * (re * re));
	else
		tmp = -im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -17.0], N[Not[LessEqual[im, 4.1e+89]], $MachinePrecision]], N[(im * N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-im)]

Derivation

1. Split input into 2 regimes

2. if im < -17 or 4.09999999999999985e89 < 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. sub0-neg 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 5.9%

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

      1. mul-1-neg 5.9%

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

      2. *-commutative 5.9%

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

      3. distribute-lft-neg-in 5.9%

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

   6. Simplified 5.9%

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

   7. Taylor expanded in re around 0 24.9%

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

      1. neg-mul-1 24.9%

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

      2. +-commutative 24.9%

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

      3. unsub-neg 24.9%

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

      4. *-commutative 24.9%

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

      5. associate-*l* 24.9%

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

      6. unpow2 24.9%

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

      7. *-commutative 24.9%

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

      8. associate-*l* 24.9%

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

      9. *-commutative 24.9%

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

   9. Simplified 24.9%

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

       1. flip-- 27.9%

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

       2. frac-2neg 27.9%

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

   11. Applied egg-rr 4.6%

       \[\leadsto \color{blue}{\frac{-\left({re}^{4} \cdot \left(0.25 \cdot \left(im \cdot im\right)\right) - im \cdot im\right)}{-\mathsf{fma}\left(re, \left(re \cdot 0.5\right) \cdot im, im\right)}} \]
   12. Step-by-step derivation

   13. Simplified 4.6%

       \[\leadsto \color{blue}{\frac{\left({re}^{4} \cdot -0.25\right) \cdot \left(im \cdot im\right) + im \cdot im}{im \cdot \left(\left(-0.5 \cdot re\right) \cdot re\right) - im}} \]

   14. Taylor expanded in re around inf 22.6%

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

       1. *-commutative 22.6%

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

       2. *-commutative 22.6%

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

       3. associate-*l* 22.6%

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

       4. unpow2 22.6%

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

   16. Simplified 22.6%

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

   if -17 < im < 4.09999999999999985e89

   1. Initial program 18.8%

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

      1. sub0-neg 18.8%

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

   3. Simplified 18.8%

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

   4. Taylor expanded in im around 0 88.0%

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

      1. mul-1-neg 88.0%

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

      2. *-commutative 88.0%

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

      3. distribute-lft-neg-in 88.0%

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

   6. Simplified 88.0%

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

   7. Taylor expanded in re around 0 46.4%

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

      1. neg-mul-1 46.4%

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

   9. Simplified 46.4%

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

4. Final simplification 36.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -17 \lor \neg \left(im \leq 4.1 \cdot 10^{+89}\right):\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-im\\ \end{array} \]

Alternative 16: 35.8% accurate, 34.3× speedup

Math

\[\begin{array}{l} \\ im \cdot \left(0.5 \cdot \left(re \cdot re\right) + -1\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* im (+ (* 0.5 (* re re)) -1.0)))

C

double code(double re, double im) {
	return im * ((0.5 * (re * re)) + -1.0);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = im * ((0.5d0 * (re * re)) + (-1.0d0))
end function

Java

public static double code(double re, double im) {
	return im * ((0.5 * (re * re)) + -1.0);
}

Python

def code(re, im):
	return im * ((0.5 * (re * re)) + -1.0)

Julia

function code(re, im)
	return Float64(im * Float64(Float64(0.5 * Float64(re * re)) + -1.0))
end

MATLAB

function tmp = code(re, im)
	tmp = im * ((0.5 * (re * re)) + -1.0);
end

Wolfram

code[re_, im_] := N[(im * N[(N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.7%

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

   1. sub0-neg 52.7%

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

3. Simplified 52.7%

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

4. Taylor expanded in im around 0 53.7%

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

   1. mul-1-neg 53.7%

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

   2. *-commutative 53.7%

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

   3. distribute-lft-neg-in 53.7%

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

6. Simplified 53.7%

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

7. Taylor expanded in re around 0 34.5%

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

   1. neg-mul-1 34.5%

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

   2. +-commutative 34.5%

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

   3. unsub-neg 34.5%

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

   4. *-commutative 34.5%

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

   5. associate-*l* 34.5%

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

   6. unpow2 34.5%

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

   7. *-commutative 34.5%

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

   8. associate-*l* 34.5%

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

   9. *-commutative 34.5%

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

9. Simplified 34.5%

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

    1. flip-- 22.9%

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

    2. frac-2neg 22.9%

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

11. Applied egg-rr 13.0%

    \[\leadsto \color{blue}{\frac{-\left({re}^{4} \cdot \left(0.25 \cdot \left(im \cdot im\right)\right) - im \cdot im\right)}{-\mathsf{fma}\left(re, \left(re \cdot 0.5\right) \cdot im, im\right)}} \]
12. Step-by-step derivation

13. Simplified 12.9%

    \[\leadsto \color{blue}{\frac{\left({re}^{4} \cdot -0.25\right) \cdot \left(im \cdot im\right) + im \cdot im}{im \cdot \left(\left(-0.5 \cdot re\right) \cdot re\right) - im}} \]

14. Taylor expanded in re around 0 34.5%

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

    1. neg-mul-1 34.5%

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

    2. +-commutative 34.5%

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

    3. associate-*r* 34.5%

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

    4. neg-mul-1 34.5%

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

    5. distribute-rgt-out 34.5%

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

    6. *-commutative 34.5%

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

    7. unpow2 34.5%

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

16. Simplified 34.5%

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

17. Final simplification 34.5%

    \[\leadsto im \cdot \left(0.5 \cdot \left(re \cdot re\right) + -1\right) \]

Alternative 17: 35.9% accurate, 34.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.7%

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

   1. sub0-neg 52.7%

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

3. Simplified 52.7%

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

4. Taylor expanded in im around 0 53.7%

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

   1. mul-1-neg 53.7%

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

   2. *-commutative 53.7%

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

   3. distribute-lft-neg-in 53.7%

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

6. Simplified 53.7%

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

7. Taylor expanded in re around 0 34.5%

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

   1. neg-mul-1 34.5%

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

   2. +-commutative 34.5%

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

   3. unsub-neg 34.5%

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

   4. *-commutative 34.5%

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

   5. associate-*l* 34.5%

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

   6. unpow2 34.5%

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

   7. *-commutative 34.5%

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

   8. associate-*l* 34.5%

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

   9. *-commutative 34.5%

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

9. Simplified 34.5%

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

10. Final simplification 34.5%

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

Alternative 18: 30.7% accurate, 43.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.6 \cdot 10^{+94}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.75\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.6e+94) (- im) (* re (* re 0.75))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.6e+94) {
		tmp = -im;
	} else {
		tmp = re * (re * 0.75);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 2.6d+94) then
        tmp = -im
    else
        tmp = re * (re * 0.75d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.6e+94) {
		tmp = -im;
	} else {
		tmp = re * (re * 0.75);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.6e+94:
		tmp = -im
	else:
		tmp = re * (re * 0.75)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.6e+94)
		tmp = Float64(-im);
	else
		tmp = Float64(re * Float64(re * 0.75));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.6e+94)
		tmp = -im;
	else
		tmp = re * (re * 0.75);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.6e+94], (-im), N[(re * N[(re * 0.75), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.5999999999999999e94

   1. Initial program 45.3%

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

      1. sub0-neg 45.3%

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

   3. Simplified 45.3%

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

   4. Taylor expanded in im around 0 61.2%

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

      1. mul-1-neg 61.2%

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

      2. *-commutative 61.2%

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

      3. distribute-lft-neg-in 61.2%

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

   6. Simplified 61.2%

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

   7. Taylor expanded in re around 0 32.8%

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

      1. neg-mul-1 32.8%

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

   9. Simplified 32.8%

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

   if 2.5999999999999999e94 < 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. sub0-neg 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 0.0%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot {re}^{2}\right) + 0.5 \cdot \left(e^{-im} - e^{im}\right)} \]
   5. Step-by-step derivation

      1. +-commutative 0.0%

         \[\leadsto \color{blue}{0.5 \cdot \left(e^{-im} - e^{im}\right) + -0.25 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot {re}^{2}\right)} \]

      2. *-commutative 0.0%

         \[\leadsto 0.5 \cdot \left(e^{-im} - e^{im}\right) + -0.25 \cdot \color{blue}{\left({re}^{2} \cdot \left(e^{-im} - e^{im}\right)\right)} \]

      3. associate-*r* 0.0%

         \[\leadsto 0.5 \cdot \left(e^{-im} - e^{im}\right) + \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot \left(e^{-im} - e^{im}\right)} \]

      4. distribute-rgt-out 71.4%

         \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      5. *-commutative 71.4%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{{re}^{2} \cdot -0.25}\right) \]

      6. unpow2 71.4%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{\left(re \cdot re\right)} \cdot -0.25\right) \]

      7. associate-*l* 71.4%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{re \cdot \left(re \cdot -0.25\right)}\right) \]

   6. Simplified 71.4%

      \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)} \]

   8. Applied egg-rr 27.5%

      \[\leadsto \color{blue}{-3} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \]

   9. Taylor expanded in re around inf 27.0%

      \[\leadsto \color{blue}{0.75 \cdot {re}^{2}} \]
   10. Step-by-step derivation

       1. *-commutative 27.0%

          \[\leadsto \color{blue}{{re}^{2} \cdot 0.75} \]

       2. unpow2 27.0%

          \[\leadsto \color{blue}{\left(re \cdot re\right)} \cdot 0.75 \]

       3. associate-*l* 27.0%

          \[\leadsto \color{blue}{re \cdot \left(re \cdot 0.75\right)} \]

   11. Simplified 27.0%

       \[\leadsto \color{blue}{re \cdot \left(re \cdot 0.75\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.6 \cdot 10^{+94}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot 0.75\right)\\ \end{array} \]

Alternative 19: 29.4% 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 52.7%

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

   1. sub0-neg 52.7%

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

3. Simplified 52.7%

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

4. Taylor expanded in im around 0 53.7%

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

   1. mul-1-neg 53.7%

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

   2. *-commutative 53.7%

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

   3. distribute-lft-neg-in 53.7%

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

6. Simplified 53.7%

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

7. Taylor expanded in re around 0 28.8%

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

   1. neg-mul-1 28.8%

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

9. Simplified 28.8%

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

10. Final simplification 28.8%

    \[\leadsto -im \]

Alternative 20: 2.9% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ -1.5 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 -1.5)

C

double code(double re, double im) {
	return -1.5;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = -1.5d0
end function

Java

public static double code(double re, double im) {
	return -1.5;
}

Python

def code(re, im):
	return -1.5

Julia

function code(re, im)
	return -1.5
end

MATLAB

function tmp = code(re, im)
	tmp = -1.5;
end

Wolfram

code[re_, im_] := -1.5

Derivation

1. Initial program 52.7%

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

   1. sub0-neg 52.7%

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

3. Simplified 52.7%

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

4. Taylor expanded in re around 0 2.9%

   \[\leadsto \color{blue}{-0.25 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot {re}^{2}\right) + 0.5 \cdot \left(e^{-im} - e^{im}\right)} \]
5. Step-by-step derivation

   1. +-commutative 2.9%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{-im} - e^{im}\right) + -0.25 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot {re}^{2}\right)} \]

   2. *-commutative 2.9%

      \[\leadsto 0.5 \cdot \left(e^{-im} - e^{im}\right) + -0.25 \cdot \color{blue}{\left({re}^{2} \cdot \left(e^{-im} - e^{im}\right)\right)} \]

   3. associate-*r* 2.9%

      \[\leadsto 0.5 \cdot \left(e^{-im} - e^{im}\right) + \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot \left(e^{-im} - e^{im}\right)} \]

   4. distribute-rgt-out 38.5%

      \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

   5. *-commutative 38.5%

      \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{{re}^{2} \cdot -0.25}\right) \]

   6. unpow2 38.5%

      \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{\left(re \cdot re\right)} \cdot -0.25\right) \]

   7. associate-*l* 38.5%

      \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{re \cdot \left(re \cdot -0.25\right)}\right) \]

6. Simplified 38.5%

   \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)} \]

8. Applied egg-rr 7.3%

   \[\leadsto \color{blue}{-3} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \]

9. Taylor expanded in re around 0 2.8%

   \[\leadsto \color{blue}{-1.5} \]

10. Final simplification 2.8%

    \[\leadsto -1.5 \]

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 2023187 
(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))))