math.cos on complex, imaginary part

Percentage Accurate: 66.7% → 99.7%

Time: 9.8s

Alternatives: 12

Speedup: 2.8×

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

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 0.05\right):\\ \;\;\;\;t_0 \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(\left({im}^{5} \cdot -0.008333333333333333 - im\right) + \left({im}^{7} \cdot -0.0001984126984126984 + {im}^{3} \cdot -0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 0.05)))
     (* t_0 (* 0.5 (sin re)))
     (*
      (sin re)
      (+
       (- (* (pow im 5.0) -0.008333333333333333) im)
       (+
        (* (pow im 7.0) -0.0001984126984126984)
        (* (pow im 3.0) -0.16666666666666666)))))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || !(t_0 <= 0.05)) {
		tmp = t_0 * (0.5 * sin(re));
	} else {
		tmp = sin(re) * (((pow(im, 5.0) * -0.008333333333333333) - im) + ((pow(im, 7.0) * -0.0001984126984126984) + (pow(im, 3.0) * -0.16666666666666666)));
	}
	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 <= 0.05)) {
		tmp = t_0 * (0.5 * Math.sin(re));
	} else {
		tmp = Math.sin(re) * (((Math.pow(im, 5.0) * -0.008333333333333333) - im) + ((Math.pow(im, 7.0) * -0.0001984126984126984) + (Math.pow(im, 3.0) * -0.16666666666666666)));
	}
	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 <= 0.05):
		tmp = t_0 * (0.5 * math.sin(re))
	else:
		tmp = math.sin(re) * (((math.pow(im, 5.0) * -0.008333333333333333) - im) + ((math.pow(im, 7.0) * -0.0001984126984126984) + (math.pow(im, 3.0) * -0.16666666666666666)))
	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 <= 0.05))
		tmp = Float64(t_0 * Float64(0.5 * sin(re)));
	else
		tmp = Float64(sin(re) * Float64(Float64(Float64((im ^ 5.0) * -0.008333333333333333) - im) + Float64(Float64((im ^ 7.0) * -0.0001984126984126984) + Float64((im ^ 3.0) * -0.16666666666666666))));
	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 <= 0.05)))
		tmp = t_0 * (0.5 * sin(re));
	else
		tmp = sin(re) * ((((im ^ 5.0) * -0.008333333333333333) - im) + (((im ^ 7.0) * -0.0001984126984126984) + ((im ^ 3.0) * -0.16666666666666666)));
	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, 0.05]], $MachinePrecision]], N[(t$95$0 * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(N[(N[(N[Power[im, 5.0], $MachinePrecision] * -0.008333333333333333), $MachinePrecision] - im), $MachinePrecision] + N[(N[(N[Power[im, 7.0], $MachinePrecision] * -0.0001984126984126984), $MachinePrecision] + N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -inf.0 or 0.050000000000000003 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   if -inf.0 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < 0.050000000000000003

   1. Initial program 30.3%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in im around 0 99.8%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + \left(-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right) + \left(-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right) + -1 \cdot \left(\sin re \cdot im\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-+r+ 99.8%

         \[\leadsto \color{blue}{\left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) + \left(-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right) + -1 \cdot \left(\sin re \cdot im\right)\right)} \]

      2. +-commutative 99.8%

         \[\leadsto \color{blue}{\left(-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right) + -1 \cdot \left(\sin re \cdot im\right)\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right)} \]

      3. +-commutative 99.8%

         \[\leadsto \color{blue}{\left(-1 \cdot \left(\sin re \cdot im\right) + -0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right)\right)} + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      4. mul-1-neg 99.8%

         \[\leadsto \left(\color{blue}{\left(-\sin re \cdot im\right)} + -0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right)\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      5. *-commutative 99.8%

         \[\leadsto \left(\left(-\color{blue}{im \cdot \sin re}\right) + -0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right)\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      6. distribute-lft-neg-in 99.8%

         \[\leadsto \left(\color{blue}{\left(-im\right) \cdot \sin re} + -0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right)\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      7. *-commutative 99.8%

         \[\leadsto \left(\left(-im\right) \cdot \sin re + -0.008333333333333333 \cdot \color{blue}{\left({im}^{5} \cdot \sin re\right)}\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      8. associate-*r* 99.8%

         \[\leadsto \left(\left(-im\right) \cdot \sin re + \color{blue}{\left(-0.008333333333333333 \cdot {im}^{5}\right) \cdot \sin re}\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      9. distribute-rgt-out 99.8%

         \[\leadsto \color{blue}{\sin re \cdot \left(\left(-im\right) + -0.008333333333333333 \cdot {im}^{5}\right)} + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      10. *-commutative 99.8%

          \[\leadsto \sin re \cdot \left(\left(-im\right) + -0.008333333333333333 \cdot {im}^{5}\right) + \left(\color{blue}{\left(\sin re \cdot {im}^{7}\right) \cdot -0.0001984126984126984} + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      11. associate-*l* 99.8%

          \[\leadsto \sin re \cdot \left(\left(-im\right) + -0.008333333333333333 \cdot {im}^{5}\right) + \left(\color{blue}{\sin re \cdot \left({im}^{7} \cdot -0.0001984126984126984\right)} + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      12. *-commutative 99.8%

          \[\leadsto \sin re \cdot \left(\left(-im\right) + -0.008333333333333333 \cdot {im}^{5}\right) + \left(\sin re \cdot \left({im}^{7} \cdot -0.0001984126984126984\right) + \color{blue}{\left(\sin re \cdot {im}^{3}\right) \cdot -0.16666666666666666}\right) \]

      13. associate-*l* 99.8%

          \[\leadsto \sin re \cdot \left(\left(-im\right) + -0.008333333333333333 \cdot {im}^{5}\right) + \left(\sin re \cdot \left({im}^{7} \cdot -0.0001984126984126984\right) + \color{blue}{\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)}\right) \]

      14. distribute-lft-out 99.8%

          \[\leadsto \sin re \cdot \left(\left(-im\right) + -0.008333333333333333 \cdot {im}^{5}\right) + \color{blue}{\sin re \cdot \left({im}^{7} \cdot -0.0001984126984126984 + {im}^{3} \cdot -0.16666666666666666\right)} \]

   4. Simplified 99.8%

      \[\leadsto \color{blue}{\sin re \cdot \left(\left({im}^{5} \cdot -0.008333333333333333 - im\right) + \left({im}^{7} \cdot -0.0001984126984126984 + {im}^{3} \cdot -0.16666666666666666\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -\infty \lor \neg \left(e^{-im} - e^{im} \leq 0.05\right):\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(\left({im}^{5} \cdot -0.008333333333333333 - im\right) + \left({im}^{7} \cdot -0.0001984126984126984 + {im}^{3} \cdot -0.16666666666666666\right)\right)\\ \end{array} \]

Alternative 2: 99.6% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 5e-6)))
     (* t_0 (* 0.5 (sin re)))
     (- (* (pow im 3.0) (* (sin re) -0.16666666666666666)) (* im (sin re))))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || !(t_0 <= 5e-6)) {
		tmp = t_0 * (0.5 * sin(re));
	} else {
		tmp = (pow(im, 3.0) * (sin(re) * -0.16666666666666666)) - (im * sin(re));
	}
	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 <= 5e-6)) {
		tmp = t_0 * (0.5 * Math.sin(re));
	} else {
		tmp = (Math.pow(im, 3.0) * (Math.sin(re) * -0.16666666666666666)) - (im * Math.sin(re));
	}
	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 <= 5e-6):
		tmp = t_0 * (0.5 * math.sin(re))
	else:
		tmp = (math.pow(im, 3.0) * (math.sin(re) * -0.16666666666666666)) - (im * math.sin(re))
	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 <= 5e-6))
		tmp = Float64(t_0 * Float64(0.5 * sin(re)));
	else
		tmp = Float64(Float64((im ^ 3.0) * Float64(sin(re) * -0.16666666666666666)) - Float64(im * sin(re)));
	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 <= 5e-6)))
		tmp = t_0 * (0.5 * sin(re));
	else
		tmp = ((im ^ 3.0) * (sin(re) * -0.16666666666666666)) - (im * sin(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, (-Infinity)], N[Not[LessEqual[t$95$0, 5e-6]], $MachinePrecision]], N[(t$95$0 * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[im, 3.0], $MachinePrecision] * N[(N[Sin[re], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] - N[(im * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -inf.0 or 5.00000000000000041e-6 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   if -inf.0 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < 5.00000000000000041e-6

   1. Initial program 29.8%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in im around 0 99.8%

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

      1. mul-1-neg 99.8%

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

      2. unsub-neg 99.8%

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

      3. *-commutative 99.8%

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

      4. *-commutative 99.8%

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

      5. associate-*l* 99.8%

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

      6. *-commutative 99.8%

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

   4. Simplified 99.8%

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

4. Final simplification 99.9%

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

Alternative 3: 99.6% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 5e-6)))
     (* t_0 (* 0.5 (sin re)))
     (* (sin re) (- (* (pow im 3.0) -0.16666666666666666) im)))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || !(t_0 <= 5e-6)) {
		tmp = t_0 * (0.5 * sin(re));
	} else {
		tmp = sin(re) * ((pow(im, 3.0) * -0.16666666666666666) - 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 <= 5e-6)) {
		tmp = t_0 * (0.5 * Math.sin(re));
	} else {
		tmp = Math.sin(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	tmp = 0
	if (t_0 <= -math.inf) or not (t_0 <= 5e-6):
		tmp = t_0 * (0.5 * math.sin(re))
	else:
		tmp = math.sin(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	tmp = 0.0
	if ((t_0 <= Float64(-Inf)) || !(t_0 <= 5e-6))
		tmp = Float64(t_0 * Float64(0.5 * sin(re)));
	else
		tmp = Float64(sin(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	tmp = 0.0;
	if ((t_0 <= -Inf) || ~((t_0 <= 5e-6)))
		tmp = t_0 * (0.5 * sin(re));
	else
		tmp = sin(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, (-Infinity)], N[Not[LessEqual[t$95$0, 5e-6]], $MachinePrecision]], N[(t$95$0 * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -inf.0 or 5.00000000000000041e-6 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   if -inf.0 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < 5.00000000000000041e-6

   1. Initial program 29.8%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in im around 0 99.8%

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

      1. mul-1-neg 99.8%

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

      2. unsub-neg 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-*l* 99.8%

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

      5. distribute-lft-out-- 99.8%

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

   4. Simplified 99.8%

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

4. Final simplification 99.9%

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

Alternative 4: 95.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -8.5 \cdot 10^{+34} \lor \neg \left(im \leq 1.1 \cdot 10^{+44}\right):\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\left(-im\right) \cdot \sin re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -8.5e+34) (not (<= im 1.1e+44)))
   (* -0.0001984126984126984 (* (sin re) (pow im 7.0)))
   (log1p (expm1 (* (- im) (sin re))))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -8.5e+34) || !(im <= 1.1e+44)) {
		tmp = -0.0001984126984126984 * (sin(re) * pow(im, 7.0));
	} else {
		tmp = log1p(expm1((-im * sin(re))));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -8.5e+34) || !(im <= 1.1e+44)) {
		tmp = -0.0001984126984126984 * (Math.sin(re) * Math.pow(im, 7.0));
	} else {
		tmp = Math.log1p(Math.expm1((-im * Math.sin(re))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -8.5e+34) or not (im <= 1.1e+44):
		tmp = -0.0001984126984126984 * (math.sin(re) * math.pow(im, 7.0))
	else:
		tmp = math.log1p(math.expm1((-im * math.sin(re))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -8.5e+34) || !(im <= 1.1e+44))
		tmp = Float64(-0.0001984126984126984 * Float64(sin(re) * (im ^ 7.0)));
	else
		tmp = log1p(expm1(Float64(Float64(-im) * sin(re))));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -8.5e+34], N[Not[LessEqual[im, 1.1e+44]], $MachinePrecision]], N[(-0.0001984126984126984 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[1 + N[(Exp[N[((-im) * N[Sin[re], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -8.5000000000000003e34 or 1.09999999999999998e44 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in im around 0 99.1%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + \left(-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right) + \left(-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right) + -1 \cdot \left(\sin re \cdot im\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-+r+ 99.1%

         \[\leadsto \color{blue}{\left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) + \left(-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right) + -1 \cdot \left(\sin re \cdot im\right)\right)} \]

      2. +-commutative 99.1%

         \[\leadsto \color{blue}{\left(-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right) + -1 \cdot \left(\sin re \cdot im\right)\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right)} \]

      3. +-commutative 99.1%

         \[\leadsto \color{blue}{\left(-1 \cdot \left(\sin re \cdot im\right) + -0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right)\right)} + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      4. mul-1-neg 99.1%

         \[\leadsto \left(\color{blue}{\left(-\sin re \cdot im\right)} + -0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right)\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      5. *-commutative 99.1%

         \[\leadsto \left(\left(-\color{blue}{im \cdot \sin re}\right) + -0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right)\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      6. distribute-lft-neg-in 99.1%

         \[\leadsto \left(\color{blue}{\left(-im\right) \cdot \sin re} + -0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right)\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      7. *-commutative 99.1%

         \[\leadsto \left(\left(-im\right) \cdot \sin re + -0.008333333333333333 \cdot \color{blue}{\left({im}^{5} \cdot \sin re\right)}\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      8. associate-*r* 99.1%

         \[\leadsto \left(\left(-im\right) \cdot \sin re + \color{blue}{\left(-0.008333333333333333 \cdot {im}^{5}\right) \cdot \sin re}\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      9. distribute-rgt-out 99.1%

         \[\leadsto \color{blue}{\sin re \cdot \left(\left(-im\right) + -0.008333333333333333 \cdot {im}^{5}\right)} + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      10. *-commutative 99.1%

          \[\leadsto \sin re \cdot \left(\left(-im\right) + -0.008333333333333333 \cdot {im}^{5}\right) + \left(\color{blue}{\left(\sin re \cdot {im}^{7}\right) \cdot -0.0001984126984126984} + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      11. associate-*l* 99.1%

          \[\leadsto \sin re \cdot \left(\left(-im\right) + -0.008333333333333333 \cdot {im}^{5}\right) + \left(\color{blue}{\sin re \cdot \left({im}^{7} \cdot -0.0001984126984126984\right)} + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      12. *-commutative 99.1%

          \[\leadsto \sin re \cdot \left(\left(-im\right) + -0.008333333333333333 \cdot {im}^{5}\right) + \left(\sin re \cdot \left({im}^{7} \cdot -0.0001984126984126984\right) + \color{blue}{\left(\sin re \cdot {im}^{3}\right) \cdot -0.16666666666666666}\right) \]

      13. associate-*l* 99.1%

          \[\leadsto \sin re \cdot \left(\left(-im\right) + -0.008333333333333333 \cdot {im}^{5}\right) + \left(\sin re \cdot \left({im}^{7} \cdot -0.0001984126984126984\right) + \color{blue}{\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)}\right) \]

      14. distribute-lft-out 99.1%

          \[\leadsto \sin re \cdot \left(\left(-im\right) + -0.008333333333333333 \cdot {im}^{5}\right) + \color{blue}{\sin re \cdot \left({im}^{7} \cdot -0.0001984126984126984 + {im}^{3} \cdot -0.16666666666666666\right)} \]

   4. Simplified 99.1%

      \[\leadsto \color{blue}{\sin re \cdot \left(\left({im}^{5} \cdot -0.008333333333333333 - im\right) + \left({im}^{7} \cdot -0.0001984126984126984 + {im}^{3} \cdot -0.16666666666666666\right)\right)} \]

   5. Taylor expanded in im around inf 99.1%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)} \]

   if -8.5000000000000003e34 < im < 1.09999999999999998e44

   1. Initial program 41.7%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in im around 0 83.5%

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

      1. mul-1-neg 83.5%

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

      2. *-commutative 83.5%

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

      3. distribute-rgt-neg-in 83.5%

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

   4. Simplified 83.5%

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

      1. log1p-expm1-u 93.0%

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

      2. distribute-rgt-neg-out 93.0%

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

      3. *-commutative 93.0%

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

   6. Applied egg-rr 93.0%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -8.5 \cdot 10^{+34} \lor \neg \left(im \leq 1.1 \cdot 10^{+44}\right):\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\left(-im\right) \cdot \sin re\right)\right)\\ \end{array} \]

Alternative 5: 97.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ t_1 := -0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \mathbf{if}\;im \leq -2.1 \cdot 10^{+55}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.014:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.225:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (- (exp (- im)) (exp im)) (* 0.5 re)))
        (t_1 (* -0.0001984126984126984 (* (sin re) (pow im 7.0)))))
   (if (<= im -2.1e+55)
     t_1
     (if (<= im -0.014)
       t_0
       (if (<= im 0.225)
         (* (sin re) (- (* (pow im 3.0) -0.16666666666666666) im))
         (if (<= im 1.1e+44) t_0 t_1))))))

C

double code(double re, double im) {
	double t_0 = (exp(-im) - exp(im)) * (0.5 * re);
	double t_1 = -0.0001984126984126984 * (sin(re) * pow(im, 7.0));
	double tmp;
	if (im <= -2.1e+55) {
		tmp = t_1;
	} else if (im <= -0.014) {
		tmp = t_0;
	} else if (im <= 0.225) {
		tmp = sin(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	} else if (im <= 1.1e+44) {
		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 = (exp(-im) - exp(im)) * (0.5d0 * re)
    t_1 = (-0.0001984126984126984d0) * (sin(re) * (im ** 7.0d0))
    if (im <= (-2.1d+55)) then
        tmp = t_1
    else if (im <= (-0.014d0)) then
        tmp = t_0
    else if (im <= 0.225d0) then
        tmp = sin(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    else if (im <= 1.1d+44) 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 = (Math.exp(-im) - Math.exp(im)) * (0.5 * re);
	double t_1 = -0.0001984126984126984 * (Math.sin(re) * Math.pow(im, 7.0));
	double tmp;
	if (im <= -2.1e+55) {
		tmp = t_1;
	} else if (im <= -0.014) {
		tmp = t_0;
	} else if (im <= 0.225) {
		tmp = Math.sin(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	} else if (im <= 1.1e+44) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (math.exp(-im) - math.exp(im)) * (0.5 * re)
	t_1 = -0.0001984126984126984 * (math.sin(re) * math.pow(im, 7.0))
	tmp = 0
	if im <= -2.1e+55:
		tmp = t_1
	elif im <= -0.014:
		tmp = t_0
	elif im <= 0.225:
		tmp = math.sin(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	elif im <= 1.1e+44:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(exp(Float64(-im)) - exp(im)) * Float64(0.5 * re))
	t_1 = Float64(-0.0001984126984126984 * Float64(sin(re) * (im ^ 7.0)))
	tmp = 0.0
	if (im <= -2.1e+55)
		tmp = t_1;
	elseif (im <= -0.014)
		tmp = t_0;
	elseif (im <= 0.225)
		tmp = Float64(sin(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	elseif (im <= 1.1e+44)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (exp(-im) - exp(im)) * (0.5 * re);
	t_1 = -0.0001984126984126984 * (sin(re) * (im ^ 7.0));
	tmp = 0.0;
	if (im <= -2.1e+55)
		tmp = t_1;
	elseif (im <= -0.014)
		tmp = t_0;
	elseif (im <= 0.225)
		tmp = sin(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	elseif (im <= 1.1e+44)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-0.0001984126984126984 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -2.1e+55], t$95$1, If[LessEqual[im, -0.014], t$95$0, If[LessEqual[im, 0.225], N[(N[Sin[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.1e+44], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -2.1000000000000001e55 or 1.09999999999999998e44 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + \left(-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right) + \left(-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right) + -1 \cdot \left(\sin re \cdot im\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-+r+ 100.0%

         \[\leadsto \color{blue}{\left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) + \left(-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right) + -1 \cdot \left(\sin re \cdot im\right)\right)} \]

      2. +-commutative 100.0%

         \[\leadsto \color{blue}{\left(-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right) + -1 \cdot \left(\sin re \cdot im\right)\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right)} \]

      3. +-commutative 100.0%

         \[\leadsto \color{blue}{\left(-1 \cdot \left(\sin re \cdot im\right) + -0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right)\right)} + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      4. mul-1-neg 100.0%

         \[\leadsto \left(\color{blue}{\left(-\sin re \cdot im\right)} + -0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right)\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      5. *-commutative 100.0%

         \[\leadsto \left(\left(-\color{blue}{im \cdot \sin re}\right) + -0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right)\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      6. distribute-lft-neg-in 100.0%

         \[\leadsto \left(\color{blue}{\left(-im\right) \cdot \sin re} + -0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right)\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      7. *-commutative 100.0%

         \[\leadsto \left(\left(-im\right) \cdot \sin re + -0.008333333333333333 \cdot \color{blue}{\left({im}^{5} \cdot \sin re\right)}\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      8. associate-*r* 100.0%

         \[\leadsto \left(\left(-im\right) \cdot \sin re + \color{blue}{\left(-0.008333333333333333 \cdot {im}^{5}\right) \cdot \sin re}\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      9. distribute-rgt-out 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left(\left(-im\right) + -0.008333333333333333 \cdot {im}^{5}\right)} + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      10. *-commutative 100.0%

          \[\leadsto \sin re \cdot \left(\left(-im\right) + -0.008333333333333333 \cdot {im}^{5}\right) + \left(\color{blue}{\left(\sin re \cdot {im}^{7}\right) \cdot -0.0001984126984126984} + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      11. associate-*l* 100.0%

          \[\leadsto \sin re \cdot \left(\left(-im\right) + -0.008333333333333333 \cdot {im}^{5}\right) + \left(\color{blue}{\sin re \cdot \left({im}^{7} \cdot -0.0001984126984126984\right)} + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      12. *-commutative 100.0%

          \[\leadsto \sin re \cdot \left(\left(-im\right) + -0.008333333333333333 \cdot {im}^{5}\right) + \left(\sin re \cdot \left({im}^{7} \cdot -0.0001984126984126984\right) + \color{blue}{\left(\sin re \cdot {im}^{3}\right) \cdot -0.16666666666666666}\right) \]

      13. associate-*l* 100.0%

          \[\leadsto \sin re \cdot \left(\left(-im\right) + -0.008333333333333333 \cdot {im}^{5}\right) + \left(\sin re \cdot \left({im}^{7} \cdot -0.0001984126984126984\right) + \color{blue}{\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)}\right) \]

      14. distribute-lft-out 100.0%

          \[\leadsto \sin re \cdot \left(\left(-im\right) + -0.008333333333333333 \cdot {im}^{5}\right) + \color{blue}{\sin re \cdot \left({im}^{7} \cdot -0.0001984126984126984 + {im}^{3} \cdot -0.16666666666666666\right)} \]

   4. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(\left({im}^{5} \cdot -0.008333333333333333 - im\right) + \left({im}^{7} \cdot -0.0001984126984126984 + {im}^{3} \cdot -0.16666666666666666\right)\right)} \]

   5. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)} \]

   if -2.1000000000000001e55 < im < -0.0140000000000000003 or 0.225000000000000006 < im < 1.09999999999999998e44

   1. Initial program 99.8%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in re around 0 79.8%

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

   if -0.0140000000000000003 < im < 0.225000000000000006

   1. Initial program 29.8%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in im around 0 99.8%

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

      1. mul-1-neg 99.8%

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

      2. unsub-neg 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-*l* 99.8%

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

      5. distribute-lft-out-- 99.8%

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

   4. Simplified 99.8%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.1 \cdot 10^{+55}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \mathbf{elif}\;im \leq -0.014:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;im \leq 0.225:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \end{array} \]

Alternative 6: 94.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \mathbf{if}\;im \leq -9 \cdot 10^{+26}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -10500000000:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(0.5 \cdot \left(re \cdot -3\right)\right)\right)\\ \mathbf{elif}\;im \leq 600:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot \left(-re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* -0.0001984126984126984 (* (sin re) (pow im 7.0)))))
   (if (<= im -9e+26)
     t_0
     (if (<= im -10500000000.0)
       (log1p (expm1 (* 0.5 (* re -3.0))))
       (if (<= im 600.0)
         (* (- im) (sin re))
         (if (<= im 1.1e+44) (log1p (expm1 (* im (- re)))) t_0))))))

C

double code(double re, double im) {
	double t_0 = -0.0001984126984126984 * (sin(re) * pow(im, 7.0));
	double tmp;
	if (im <= -9e+26) {
		tmp = t_0;
	} else if (im <= -10500000000.0) {
		tmp = log1p(expm1((0.5 * (re * -3.0))));
	} else if (im <= 600.0) {
		tmp = -im * sin(re);
	} else if (im <= 1.1e+44) {
		tmp = log1p(expm1((im * -re)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = -0.0001984126984126984 * (Math.sin(re) * Math.pow(im, 7.0));
	double tmp;
	if (im <= -9e+26) {
		tmp = t_0;
	} else if (im <= -10500000000.0) {
		tmp = Math.log1p(Math.expm1((0.5 * (re * -3.0))));
	} else if (im <= 600.0) {
		tmp = -im * Math.sin(re);
	} else if (im <= 1.1e+44) {
		tmp = Math.log1p(Math.expm1((im * -re)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = -0.0001984126984126984 * (math.sin(re) * math.pow(im, 7.0))
	tmp = 0
	if im <= -9e+26:
		tmp = t_0
	elif im <= -10500000000.0:
		tmp = math.log1p(math.expm1((0.5 * (re * -3.0))))
	elif im <= 600.0:
		tmp = -im * math.sin(re)
	elif im <= 1.1e+44:
		tmp = math.log1p(math.expm1((im * -re)))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(-0.0001984126984126984 * Float64(sin(re) * (im ^ 7.0)))
	tmp = 0.0
	if (im <= -9e+26)
		tmp = t_0;
	elseif (im <= -10500000000.0)
		tmp = log1p(expm1(Float64(0.5 * Float64(re * -3.0))));
	elseif (im <= 600.0)
		tmp = Float64(Float64(-im) * sin(re));
	elseif (im <= 1.1e+44)
		tmp = log1p(expm1(Float64(im * Float64(-re))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(-0.0001984126984126984 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -9e+26], t$95$0, If[LessEqual[im, -10500000000.0], N[Log[1 + N[(Exp[N[(0.5 * N[(re * -3.0), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision], If[LessEqual[im, 600.0], N[((-im) * N[Sin[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.1e+44], N[Log[1 + N[(Exp[N[(im * (-re)), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -8.99999999999999957e26 or 1.09999999999999998e44 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in im around 0 96.4%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + \left(-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right) + \left(-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right) + -1 \cdot \left(\sin re \cdot im\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-+r+ 96.4%

         \[\leadsto \color{blue}{\left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) + \left(-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right) + -1 \cdot \left(\sin re \cdot im\right)\right)} \]

      2. +-commutative 96.4%

         \[\leadsto \color{blue}{\left(-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right) + -1 \cdot \left(\sin re \cdot im\right)\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right)} \]

      3. +-commutative 96.4%

         \[\leadsto \color{blue}{\left(-1 \cdot \left(\sin re \cdot im\right) + -0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right)\right)} + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      4. mul-1-neg 96.4%

         \[\leadsto \left(\color{blue}{\left(-\sin re \cdot im\right)} + -0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right)\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      5. *-commutative 96.4%

         \[\leadsto \left(\left(-\color{blue}{im \cdot \sin re}\right) + -0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right)\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      6. distribute-lft-neg-in 96.4%

         \[\leadsto \left(\color{blue}{\left(-im\right) \cdot \sin re} + -0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right)\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      7. *-commutative 96.4%

         \[\leadsto \left(\left(-im\right) \cdot \sin re + -0.008333333333333333 \cdot \color{blue}{\left({im}^{5} \cdot \sin re\right)}\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      8. associate-*r* 96.4%

         \[\leadsto \left(\left(-im\right) \cdot \sin re + \color{blue}{\left(-0.008333333333333333 \cdot {im}^{5}\right) \cdot \sin re}\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      9. distribute-rgt-out 96.4%

         \[\leadsto \color{blue}{\sin re \cdot \left(\left(-im\right) + -0.008333333333333333 \cdot {im}^{5}\right)} + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      10. *-commutative 96.4%

          \[\leadsto \sin re \cdot \left(\left(-im\right) + -0.008333333333333333 \cdot {im}^{5}\right) + \left(\color{blue}{\left(\sin re \cdot {im}^{7}\right) \cdot -0.0001984126984126984} + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      11. associate-*l* 96.4%

          \[\leadsto \sin re \cdot \left(\left(-im\right) + -0.008333333333333333 \cdot {im}^{5}\right) + \left(\color{blue}{\sin re \cdot \left({im}^{7} \cdot -0.0001984126984126984\right)} + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      12. *-commutative 96.4%

          \[\leadsto \sin re \cdot \left(\left(-im\right) + -0.008333333333333333 \cdot {im}^{5}\right) + \left(\sin re \cdot \left({im}^{7} \cdot -0.0001984126984126984\right) + \color{blue}{\left(\sin re \cdot {im}^{3}\right) \cdot -0.16666666666666666}\right) \]

      13. associate-*l* 96.4%

          \[\leadsto \sin re \cdot \left(\left(-im\right) + -0.008333333333333333 \cdot {im}^{5}\right) + \left(\sin re \cdot \left({im}^{7} \cdot -0.0001984126984126984\right) + \color{blue}{\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)}\right) \]

      14. distribute-lft-out 96.4%

          \[\leadsto \sin re \cdot \left(\left(-im\right) + -0.008333333333333333 \cdot {im}^{5}\right) + \color{blue}{\sin re \cdot \left({im}^{7} \cdot -0.0001984126984126984 + {im}^{3} \cdot -0.16666666666666666\right)} \]

   4. Simplified 96.4%

      \[\leadsto \color{blue}{\sin re \cdot \left(\left({im}^{5} \cdot -0.008333333333333333 - im\right) + \left({im}^{7} \cdot -0.0001984126984126984 + {im}^{3} \cdot -0.16666666666666666\right)\right)} \]

   5. Taylor expanded in im around inf 96.4%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)} \]

   if -8.99999999999999957e26 < im < -1.05e10

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in re around 0 25.0%

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

   4. Applied egg-rr 4.3%

      \[\leadsto \left(0.5 \cdot re\right) \cdot \color{blue}{-3} \]
   5. Step-by-step derivation

      1. log1p-expm1-u 75.2%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(0.5 \cdot re\right) \cdot -3\right)\right)} \]

      2. associate-*l* 75.2%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{0.5 \cdot \left(re \cdot -3\right)}\right)\right) \]

   6. Applied egg-rr 75.2%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(0.5 \cdot \left(re \cdot -3\right)\right)\right)} \]

   if -1.05e10 < im < 600

   1. Initial program 32.9%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in im around 0 95.6%

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

      1. mul-1-neg 95.6%

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

      2. *-commutative 95.6%

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

      3. distribute-rgt-neg-in 95.6%

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

   4. Simplified 95.6%

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

   if 600 < im < 1.09999999999999998e44

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in im around 0 3.0%

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

      1. mul-1-neg 3.0%

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

      2. *-commutative 3.0%

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

      3. distribute-rgt-neg-in 3.0%

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

   4. Simplified 3.0%

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

      1. log1p-expm1-u 56.7%

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

      2. distribute-rgt-neg-out 56.7%

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

      3. *-commutative 56.7%

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

   6. Applied egg-rr 56.7%

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

   7. Taylor expanded in re around 0 56.7%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(-\color{blue}{re \cdot im}\right)\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -9 \cdot 10^{+26}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \mathbf{elif}\;im \leq -10500000000:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(0.5 \cdot \left(re \cdot -3\right)\right)\right)\\ \mathbf{elif}\;im \leq 600:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot \left(-re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \end{array} \]

Alternative 7: 94.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \mathbf{if}\;im \leq -9 \cdot 10^{+26}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -10500000000:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(0.5 \cdot \left(re \cdot -3\right)\right)\right)\\ \mathbf{elif}\;im \leq 92000:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot \left(-re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* -0.0001984126984126984 (* (sin re) (pow im 7.0)))))
   (if (<= im -9e+26)
     t_0
     (if (<= im -10500000000.0)
       (log1p (expm1 (* 0.5 (* re -3.0))))
       (if (<= im 92000.0)
         (* (sin re) (- (* (pow im 3.0) -0.16666666666666666) im))
         (if (<= im 1.1e+44) (log1p (expm1 (* im (- re)))) t_0))))))

C

double code(double re, double im) {
	double t_0 = -0.0001984126984126984 * (sin(re) * pow(im, 7.0));
	double tmp;
	if (im <= -9e+26) {
		tmp = t_0;
	} else if (im <= -10500000000.0) {
		tmp = log1p(expm1((0.5 * (re * -3.0))));
	} else if (im <= 92000.0) {
		tmp = sin(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	} else if (im <= 1.1e+44) {
		tmp = log1p(expm1((im * -re)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = -0.0001984126984126984 * (Math.sin(re) * Math.pow(im, 7.0));
	double tmp;
	if (im <= -9e+26) {
		tmp = t_0;
	} else if (im <= -10500000000.0) {
		tmp = Math.log1p(Math.expm1((0.5 * (re * -3.0))));
	} else if (im <= 92000.0) {
		tmp = Math.sin(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	} else if (im <= 1.1e+44) {
		tmp = Math.log1p(Math.expm1((im * -re)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = -0.0001984126984126984 * (math.sin(re) * math.pow(im, 7.0))
	tmp = 0
	if im <= -9e+26:
		tmp = t_0
	elif im <= -10500000000.0:
		tmp = math.log1p(math.expm1((0.5 * (re * -3.0))))
	elif im <= 92000.0:
		tmp = math.sin(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	elif im <= 1.1e+44:
		tmp = math.log1p(math.expm1((im * -re)))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(-0.0001984126984126984 * Float64(sin(re) * (im ^ 7.0)))
	tmp = 0.0
	if (im <= -9e+26)
		tmp = t_0;
	elseif (im <= -10500000000.0)
		tmp = log1p(expm1(Float64(0.5 * Float64(re * -3.0))));
	elseif (im <= 92000.0)
		tmp = Float64(sin(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	elseif (im <= 1.1e+44)
		tmp = log1p(expm1(Float64(im * Float64(-re))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(-0.0001984126984126984 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -9e+26], t$95$0, If[LessEqual[im, -10500000000.0], N[Log[1 + N[(Exp[N[(0.5 * N[(re * -3.0), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision], If[LessEqual[im, 92000.0], N[(N[Sin[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.1e+44], N[Log[1 + N[(Exp[N[(im * (-re)), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -8.99999999999999957e26 or 1.09999999999999998e44 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in im around 0 96.4%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + \left(-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right) + \left(-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right) + -1 \cdot \left(\sin re \cdot im\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-+r+ 96.4%

         \[\leadsto \color{blue}{\left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) + \left(-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right) + -1 \cdot \left(\sin re \cdot im\right)\right)} \]

      2. +-commutative 96.4%

         \[\leadsto \color{blue}{\left(-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right) + -1 \cdot \left(\sin re \cdot im\right)\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right)} \]

      3. +-commutative 96.4%

         \[\leadsto \color{blue}{\left(-1 \cdot \left(\sin re \cdot im\right) + -0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right)\right)} + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      4. mul-1-neg 96.4%

         \[\leadsto \left(\color{blue}{\left(-\sin re \cdot im\right)} + -0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right)\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      5. *-commutative 96.4%

         \[\leadsto \left(\left(-\color{blue}{im \cdot \sin re}\right) + -0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right)\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      6. distribute-lft-neg-in 96.4%

         \[\leadsto \left(\color{blue}{\left(-im\right) \cdot \sin re} + -0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right)\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      7. *-commutative 96.4%

         \[\leadsto \left(\left(-im\right) \cdot \sin re + -0.008333333333333333 \cdot \color{blue}{\left({im}^{5} \cdot \sin re\right)}\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      8. associate-*r* 96.4%

         \[\leadsto \left(\left(-im\right) \cdot \sin re + \color{blue}{\left(-0.008333333333333333 \cdot {im}^{5}\right) \cdot \sin re}\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      9. distribute-rgt-out 96.4%

         \[\leadsto \color{blue}{\sin re \cdot \left(\left(-im\right) + -0.008333333333333333 \cdot {im}^{5}\right)} + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      10. *-commutative 96.4%

          \[\leadsto \sin re \cdot \left(\left(-im\right) + -0.008333333333333333 \cdot {im}^{5}\right) + \left(\color{blue}{\left(\sin re \cdot {im}^{7}\right) \cdot -0.0001984126984126984} + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      11. associate-*l* 96.4%

          \[\leadsto \sin re \cdot \left(\left(-im\right) + -0.008333333333333333 \cdot {im}^{5}\right) + \left(\color{blue}{\sin re \cdot \left({im}^{7} \cdot -0.0001984126984126984\right)} + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      12. *-commutative 96.4%

          \[\leadsto \sin re \cdot \left(\left(-im\right) + -0.008333333333333333 \cdot {im}^{5}\right) + \left(\sin re \cdot \left({im}^{7} \cdot -0.0001984126984126984\right) + \color{blue}{\left(\sin re \cdot {im}^{3}\right) \cdot -0.16666666666666666}\right) \]

      13. associate-*l* 96.4%

          \[\leadsto \sin re \cdot \left(\left(-im\right) + -0.008333333333333333 \cdot {im}^{5}\right) + \left(\sin re \cdot \left({im}^{7} \cdot -0.0001984126984126984\right) + \color{blue}{\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)}\right) \]

      14. distribute-lft-out 96.4%

          \[\leadsto \sin re \cdot \left(\left(-im\right) + -0.008333333333333333 \cdot {im}^{5}\right) + \color{blue}{\sin re \cdot \left({im}^{7} \cdot -0.0001984126984126984 + {im}^{3} \cdot -0.16666666666666666\right)} \]

   4. Simplified 96.4%

      \[\leadsto \color{blue}{\sin re \cdot \left(\left({im}^{5} \cdot -0.008333333333333333 - im\right) + \left({im}^{7} \cdot -0.0001984126984126984 + {im}^{3} \cdot -0.16666666666666666\right)\right)} \]

   5. Taylor expanded in im around inf 96.4%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)} \]

   if -8.99999999999999957e26 < im < -1.05e10

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in re around 0 25.0%

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

   4. Applied egg-rr 4.3%

      \[\leadsto \left(0.5 \cdot re\right) \cdot \color{blue}{-3} \]
   5. Step-by-step derivation

      1. log1p-expm1-u 75.2%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(0.5 \cdot re\right) \cdot -3\right)\right)} \]

      2. associate-*l* 75.2%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{0.5 \cdot \left(re \cdot -3\right)}\right)\right) \]

   6. Applied egg-rr 75.2%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(0.5 \cdot \left(re \cdot -3\right)\right)\right)} \]

   if -1.05e10 < im < 92000

   1. Initial program 33.4%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in im around 0 95.2%

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

      1. mul-1-neg 95.2%

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

      2. unsub-neg 95.2%

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

      3. *-commutative 95.2%

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

      4. associate-*l* 95.2%

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

      5. distribute-lft-out-- 95.2%

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

   4. Simplified 95.2%

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

   if 92000 < im < 1.09999999999999998e44

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in im around 0 3.1%

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

      1. mul-1-neg 3.1%

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

      2. *-commutative 3.1%

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

      3. distribute-rgt-neg-in 3.1%

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

   4. Simplified 3.1%

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

      1. log1p-expm1-u 63.5%

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

      2. distribute-rgt-neg-out 63.5%

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

      3. *-commutative 63.5%

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

   6. Applied egg-rr 63.5%

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

   7. Taylor expanded in re around 0 63.5%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(-\color{blue}{re \cdot im}\right)\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -9 \cdot 10^{+26}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \mathbf{elif}\;im \leq -10500000000:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(0.5 \cdot \left(re \cdot -3\right)\right)\right)\\ \mathbf{elif}\;im \leq 92000:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot \left(-re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \end{array} \]

Alternative 8: 83.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.0001984126984126984 \cdot \left(re \cdot {im}^{7}\right)\\ \mathbf{if}\;im \leq -9 \cdot 10^{+26}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -10500000000:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(0.5 \cdot \left(re \cdot -3\right)\right)\right)\\ \mathbf{elif}\;im \leq 700:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* -0.0001984126984126984 (* re (pow im 7.0)))))
   (if (<= im -9e+26)
     t_0
     (if (<= im -10500000000.0)
       (log1p (expm1 (* 0.5 (* re -3.0))))
       (if (<= im 700.0) (* (- im) (sin re)) t_0)))))

C

double code(double re, double im) {
	double t_0 = -0.0001984126984126984 * (re * pow(im, 7.0));
	double tmp;
	if (im <= -9e+26) {
		tmp = t_0;
	} else if (im <= -10500000000.0) {
		tmp = log1p(expm1((0.5 * (re * -3.0))));
	} else if (im <= 700.0) {
		tmp = -im * sin(re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = -0.0001984126984126984 * (re * Math.pow(im, 7.0));
	double tmp;
	if (im <= -9e+26) {
		tmp = t_0;
	} else if (im <= -10500000000.0) {
		tmp = Math.log1p(Math.expm1((0.5 * (re * -3.0))));
	} else if (im <= 700.0) {
		tmp = -im * Math.sin(re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = -0.0001984126984126984 * (re * math.pow(im, 7.0))
	tmp = 0
	if im <= -9e+26:
		tmp = t_0
	elif im <= -10500000000.0:
		tmp = math.log1p(math.expm1((0.5 * (re * -3.0))))
	elif im <= 700.0:
		tmp = -im * math.sin(re)
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(-0.0001984126984126984 * Float64(re * (im ^ 7.0)))
	tmp = 0.0
	if (im <= -9e+26)
		tmp = t_0;
	elseif (im <= -10500000000.0)
		tmp = log1p(expm1(Float64(0.5 * Float64(re * -3.0))));
	elseif (im <= 700.0)
		tmp = Float64(Float64(-im) * sin(re));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(-0.0001984126984126984 * N[(re * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -9e+26], t$95$0, If[LessEqual[im, -10500000000.0], N[Log[1 + N[(Exp[N[(0.5 * N[(re * -3.0), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision], If[LessEqual[im, 700.0], N[((-im) * N[Sin[re], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -8.99999999999999957e26 or 700 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in im around 0 89.3%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + \left(-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right) + \left(-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right) + -1 \cdot \left(\sin re \cdot im\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-+r+ 89.3%

         \[\leadsto \color{blue}{\left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) + \left(-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right) + -1 \cdot \left(\sin re \cdot im\right)\right)} \]

      2. +-commutative 89.3%

         \[\leadsto \color{blue}{\left(-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right) + -1 \cdot \left(\sin re \cdot im\right)\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right)} \]

      3. +-commutative 89.3%

         \[\leadsto \color{blue}{\left(-1 \cdot \left(\sin re \cdot im\right) + -0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right)\right)} + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      4. mul-1-neg 89.3%

         \[\leadsto \left(\color{blue}{\left(-\sin re \cdot im\right)} + -0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right)\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      5. *-commutative 89.3%

         \[\leadsto \left(\left(-\color{blue}{im \cdot \sin re}\right) + -0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right)\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      6. distribute-lft-neg-in 89.3%

         \[\leadsto \left(\color{blue}{\left(-im\right) \cdot \sin re} + -0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right)\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      7. *-commutative 89.3%

         \[\leadsto \left(\left(-im\right) \cdot \sin re + -0.008333333333333333 \cdot \color{blue}{\left({im}^{5} \cdot \sin re\right)}\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      8. associate-*r* 89.3%

         \[\leadsto \left(\left(-im\right) \cdot \sin re + \color{blue}{\left(-0.008333333333333333 \cdot {im}^{5}\right) \cdot \sin re}\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      9. distribute-rgt-out 89.3%

         \[\leadsto \color{blue}{\sin re \cdot \left(\left(-im\right) + -0.008333333333333333 \cdot {im}^{5}\right)} + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      10. *-commutative 89.3%

          \[\leadsto \sin re \cdot \left(\left(-im\right) + -0.008333333333333333 \cdot {im}^{5}\right) + \left(\color{blue}{\left(\sin re \cdot {im}^{7}\right) \cdot -0.0001984126984126984} + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      11. associate-*l* 89.3%

          \[\leadsto \sin re \cdot \left(\left(-im\right) + -0.008333333333333333 \cdot {im}^{5}\right) + \left(\color{blue}{\sin re \cdot \left({im}^{7} \cdot -0.0001984126984126984\right)} + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      12. *-commutative 89.3%

          \[\leadsto \sin re \cdot \left(\left(-im\right) + -0.008333333333333333 \cdot {im}^{5}\right) + \left(\sin re \cdot \left({im}^{7} \cdot -0.0001984126984126984\right) + \color{blue}{\left(\sin re \cdot {im}^{3}\right) \cdot -0.16666666666666666}\right) \]

      13. associate-*l* 89.3%

          \[\leadsto \sin re \cdot \left(\left(-im\right) + -0.008333333333333333 \cdot {im}^{5}\right) + \left(\sin re \cdot \left({im}^{7} \cdot -0.0001984126984126984\right) + \color{blue}{\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)}\right) \]

      14. distribute-lft-out 89.3%

          \[\leadsto \sin re \cdot \left(\left(-im\right) + -0.008333333333333333 \cdot {im}^{5}\right) + \color{blue}{\sin re \cdot \left({im}^{7} \cdot -0.0001984126984126984 + {im}^{3} \cdot -0.16666666666666666\right)} \]

   4. Simplified 89.3%

      \[\leadsto \color{blue}{\sin re \cdot \left(\left({im}^{5} \cdot -0.008333333333333333 - im\right) + \left({im}^{7} \cdot -0.0001984126984126984 + {im}^{3} \cdot -0.16666666666666666\right)\right)} \]

   5. Taylor expanded in im around inf 89.3%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)} \]

   6. Taylor expanded in re around 0 66.5%

      \[\leadsto -0.0001984126984126984 \cdot \color{blue}{\left(re \cdot {im}^{7}\right)} \]

   if -8.99999999999999957e26 < im < -1.05e10

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in re around 0 25.0%

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

   4. Applied egg-rr 4.3%

      \[\leadsto \left(0.5 \cdot re\right) \cdot \color{blue}{-3} \]
   5. Step-by-step derivation

      1. log1p-expm1-u 75.2%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(0.5 \cdot re\right) \cdot -3\right)\right)} \]

      2. associate-*l* 75.2%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{0.5 \cdot \left(re \cdot -3\right)}\right)\right) \]

   6. Applied egg-rr 75.2%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(0.5 \cdot \left(re \cdot -3\right)\right)\right)} \]

   if -1.05e10 < im < 700

   1. Initial program 32.9%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in im around 0 95.6%

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

      1. mul-1-neg 95.6%

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

      2. *-commutative 95.6%

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

      3. distribute-rgt-neg-in 95.6%

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

   4. Simplified 95.6%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -9 \cdot 10^{+26}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(re \cdot {im}^{7}\right)\\ \mathbf{elif}\;im \leq -10500000000:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(0.5 \cdot \left(re \cdot -3\right)\right)\right)\\ \mathbf{elif}\;im \leq 700:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(re \cdot {im}^{7}\right)\\ \end{array} \]

Alternative 9: 83.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -8 \cdot 10^{+25} \lor \neg \left(im \leq 360\right):\\ \;\;\;\;-0.0001984126984126984 \cdot \left(re \cdot {im}^{7}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -8e+25) (not (<= im 360.0)))
   (* -0.0001984126984126984 (* re (pow im 7.0)))
   (* (- im) (sin re))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -8e+25) || !(im <= 360.0)) {
		tmp = -0.0001984126984126984 * (re * pow(im, 7.0));
	} else {
		tmp = -im * sin(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-8d+25)) .or. (.not. (im <= 360.0d0))) then
        tmp = (-0.0001984126984126984d0) * (re * (im ** 7.0d0))
    else
        tmp = -im * sin(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -8e+25) || !(im <= 360.0)) {
		tmp = -0.0001984126984126984 * (re * Math.pow(im, 7.0));
	} else {
		tmp = -im * Math.sin(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -8e+25) or not (im <= 360.0):
		tmp = -0.0001984126984126984 * (re * math.pow(im, 7.0))
	else:
		tmp = -im * math.sin(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -8e+25) || !(im <= 360.0))
		tmp = Float64(-0.0001984126984126984 * Float64(re * (im ^ 7.0)));
	else
		tmp = Float64(Float64(-im) * sin(re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -8e+25) || ~((im <= 360.0)))
		tmp = -0.0001984126984126984 * (re * (im ^ 7.0));
	else
		tmp = -im * sin(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -8e+25], N[Not[LessEqual[im, 360.0]], $MachinePrecision]], N[(-0.0001984126984126984 * N[(re * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-im) * N[Sin[re], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -8.00000000000000072e25 or 360 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in im around 0 89.3%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + \left(-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right) + \left(-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right) + -1 \cdot \left(\sin re \cdot im\right)\right)\right)} \]
   3. Step-by-step derivation

      1. associate-+r+ 89.3%

         \[\leadsto \color{blue}{\left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) + \left(-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right) + -1 \cdot \left(\sin re \cdot im\right)\right)} \]

      2. +-commutative 89.3%

         \[\leadsto \color{blue}{\left(-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right) + -1 \cdot \left(\sin re \cdot im\right)\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right)} \]

      3. +-commutative 89.3%

         \[\leadsto \color{blue}{\left(-1 \cdot \left(\sin re \cdot im\right) + -0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right)\right)} + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      4. mul-1-neg 89.3%

         \[\leadsto \left(\color{blue}{\left(-\sin re \cdot im\right)} + -0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right)\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      5. *-commutative 89.3%

         \[\leadsto \left(\left(-\color{blue}{im \cdot \sin re}\right) + -0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right)\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      6. distribute-lft-neg-in 89.3%

         \[\leadsto \left(\color{blue}{\left(-im\right) \cdot \sin re} + -0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right)\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      7. *-commutative 89.3%

         \[\leadsto \left(\left(-im\right) \cdot \sin re + -0.008333333333333333 \cdot \color{blue}{\left({im}^{5} \cdot \sin re\right)}\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      8. associate-*r* 89.3%

         \[\leadsto \left(\left(-im\right) \cdot \sin re + \color{blue}{\left(-0.008333333333333333 \cdot {im}^{5}\right) \cdot \sin re}\right) + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      9. distribute-rgt-out 89.3%

         \[\leadsto \color{blue}{\sin re \cdot \left(\left(-im\right) + -0.008333333333333333 \cdot {im}^{5}\right)} + \left(-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right) + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      10. *-commutative 89.3%

          \[\leadsto \sin re \cdot \left(\left(-im\right) + -0.008333333333333333 \cdot {im}^{5}\right) + \left(\color{blue}{\left(\sin re \cdot {im}^{7}\right) \cdot -0.0001984126984126984} + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      11. associate-*l* 89.3%

          \[\leadsto \sin re \cdot \left(\left(-im\right) + -0.008333333333333333 \cdot {im}^{5}\right) + \left(\color{blue}{\sin re \cdot \left({im}^{7} \cdot -0.0001984126984126984\right)} + -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\right) \]

      12. *-commutative 89.3%

          \[\leadsto \sin re \cdot \left(\left(-im\right) + -0.008333333333333333 \cdot {im}^{5}\right) + \left(\sin re \cdot \left({im}^{7} \cdot -0.0001984126984126984\right) + \color{blue}{\left(\sin re \cdot {im}^{3}\right) \cdot -0.16666666666666666}\right) \]

      13. associate-*l* 89.3%

          \[\leadsto \sin re \cdot \left(\left(-im\right) + -0.008333333333333333 \cdot {im}^{5}\right) + \left(\sin re \cdot \left({im}^{7} \cdot -0.0001984126984126984\right) + \color{blue}{\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)}\right) \]

      14. distribute-lft-out 89.3%

          \[\leadsto \sin re \cdot \left(\left(-im\right) + -0.008333333333333333 \cdot {im}^{5}\right) + \color{blue}{\sin re \cdot \left({im}^{7} \cdot -0.0001984126984126984 + {im}^{3} \cdot -0.16666666666666666\right)} \]

   4. Simplified 89.3%

      \[\leadsto \color{blue}{\sin re \cdot \left(\left({im}^{5} \cdot -0.008333333333333333 - im\right) + \left({im}^{7} \cdot -0.0001984126984126984 + {im}^{3} \cdot -0.16666666666666666\right)\right)} \]

   5. Taylor expanded in im around inf 89.3%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)} \]

   6. Taylor expanded in re around 0 66.5%

      \[\leadsto -0.0001984126984126984 \cdot \color{blue}{\left(re \cdot {im}^{7}\right)} \]

   if -8.00000000000000072e25 < im < 360

   1. Initial program 36.7%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in im around 0 90.3%

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

      1. mul-1-neg 90.3%

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

      2. *-commutative 90.3%

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

      3. distribute-rgt-neg-in 90.3%

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

   4. Simplified 90.3%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -8 \cdot 10^{+25} \lor \neg \left(im \leq 360\right):\\ \;\;\;\;-0.0001984126984126984 \cdot \left(re \cdot {im}^{7}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \end{array} \]

Alternative 10: 56.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -5.4 \cdot 10^{+31} \lor \neg \left(im \leq 160000\right):\\ \;\;\;\;im \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -5.4e+31) (not (<= im 160000.0)))
   (* im (- re))
   (* (- im) (sin re))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -5.4e+31) || !(im <= 160000.0)) {
		tmp = im * -re;
	} else {
		tmp = -im * sin(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-5.4d+31)) .or. (.not. (im <= 160000.0d0))) then
        tmp = im * -re
    else
        tmp = -im * sin(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -5.4e+31) || !(im <= 160000.0)) {
		tmp = im * -re;
	} else {
		tmp = -im * Math.sin(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -5.4e+31) or not (im <= 160000.0):
		tmp = im * -re
	else:
		tmp = -im * math.sin(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -5.4e+31) || !(im <= 160000.0))
		tmp = Float64(im * Float64(-re));
	else
		tmp = Float64(Float64(-im) * sin(re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -5.4e+31) || ~((im <= 160000.0)))
		tmp = im * -re;
	else
		tmp = -im * sin(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -5.4e+31], N[Not[LessEqual[im, 160000.0]], $MachinePrecision]], N[(im * (-re)), $MachinePrecision], N[((-im) * N[Sin[re], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -5.39999999999999971e31 or 1.6e5 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in im around 0 4.6%

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

      1. mul-1-neg 4.6%

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

      2. *-commutative 4.6%

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

      3. distribute-rgt-neg-in 4.6%

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

   4. Simplified 4.6%

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

   5. Taylor expanded in re around 0 15.1%

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

      1. mul-1-neg 15.1%

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

      2. *-commutative 15.1%

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

      3. distribute-rgt-neg-in 15.1%

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

   7. Simplified 15.1%

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

   if -5.39999999999999971e31 < im < 1.6e5

   1. Initial program 37.6%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

   2. Taylor expanded in im around 0 89.1%

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

      1. mul-1-neg 89.1%

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

      2. *-commutative 89.1%

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

      3. distribute-rgt-neg-in 89.1%

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

   4. Simplified 89.1%

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

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -5.4 \cdot 10^{+31} \lor \neg \left(im \leq 160000\right):\\ \;\;\;\;im \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \end{array} \]

Alternative 11: 33.3% accurate, 77.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.1%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

2. Taylor expanded in im around 0 51.8%

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

   1. mul-1-neg 51.8%

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

   2. *-commutative 51.8%

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

   3. distribute-rgt-neg-in 51.8%

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

4. Simplified 51.8%

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

5. Taylor expanded in re around 0 32.6%

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

   1. mul-1-neg 32.6%

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

   2. *-commutative 32.6%

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

   3. distribute-rgt-neg-in 32.6%

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

7. Simplified 32.6%

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

8. Final simplification 32.6%

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

Alternative 12: 3.2% accurate, 102.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return re * -1.5

Julia

function code(re, im)
	return Float64(re * -1.5)
end

MATLAB

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

Wolfram

code[re_, im_] := N[(re * -1.5), $MachinePrecision]

Derivation

1. Initial program 65.1%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]

2. Taylor expanded in re around 0 51.0%

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

4. Applied egg-rr 3.2%

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

5. Taylor expanded in re around 0 3.2%

   \[\leadsto \color{blue}{-1.5 \cdot re} \]
6. Step-by-step derivation

   1. *-commutative 3.2%

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

7. Simplified 3.2%

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

8. Final simplification 3.2%

   \[\leadsto re \cdot -1.5 \]

Developer target: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	double tmp;
	if (fabs(im) < 1.0) {
		tmp = -(sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * sin(re)) * (exp(-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 = -(sin(re) * ((im + (((0.16666666666666666d0 * im) * im) * im)) + (((((0.008333333333333333d0 * im) * im) * im) * im) * im)))
    else
        tmp = (0.5d0 * sin(re)) * (exp(-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.sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * Math.sin(re)) * (Math.exp(-im) - Math.exp(im));
	}
	return tmp;
}

Python

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

Julia

function code(re, im)
	tmp = 0.0
	if (abs(im) < 1.0)
		tmp = Float64(-Float64(sin(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 * sin(re)) * Float64(exp(Float64(-im)) - exp(im)));
	end
	return tmp
end

MATLAB

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

Wolfram

code[re_, im_] := If[Less[N[Abs[im], $MachinePrecision], 1.0], (-N[(N[Sin[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[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2023171 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"
  :precision binary64

  :herbie-target
  (if (< (fabs im) 1.0) (- (* (sin re) (+ (+ im (* (* (* 0.16666666666666666 im) im) im)) (* (* (* (* (* 0.008333333333333333 im) im) im) im) im)))) (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))

  (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))