math.cos on complex, imaginary part

Percentage Accurate: 65.9% → 99.9%

Time: 17.7s

Alternatives: 12

Speedup: 2.7×

• 48.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 65.9% 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: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -1 \lor \neg \left(t_0 \leq 0.05\right):\\ \;\;\;\;t_0 \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(\left({im}^{5} \cdot -0.008333333333333333 - im\right) + \left({im}^{3} \cdot -0.16666666666666666 + {im}^{7} \cdot -0.0001984126984126984\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 -1.0) (not (<= t_0 0.05)))
     (* t_0 (* 0.5 (sin re)))
     (*
      (sin re)
      (+
       (- (* (pow im 5.0) -0.008333333333333333) im)
       (+
        (* (pow im 3.0) -0.16666666666666666)
        (* (pow im 7.0) -0.0001984126984126984)))))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if ((t_0 <= -1.0) || !(t_0 <= 0.05)) {
		tmp = t_0 * (0.5 * sin(re));
	} else {
		tmp = sin(re) * (((pow(im, 5.0) * -0.008333333333333333) - im) + ((pow(im, 3.0) * -0.16666666666666666) + (pow(im, 7.0) * -0.0001984126984126984)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-im) - exp(im)
    if ((t_0 <= (-1.0d0)) .or. (.not. (t_0 <= 0.05d0))) then
        tmp = t_0 * (0.5d0 * sin(re))
    else
        tmp = sin(re) * ((((im ** 5.0d0) * (-0.008333333333333333d0)) - im) + (((im ** 3.0d0) * (-0.16666666666666666d0)) + ((im ** 7.0d0) * (-0.0001984126984126984d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(-im) - Math.exp(im);
	double tmp;
	if ((t_0 <= -1.0) || !(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, 3.0) * -0.16666666666666666) + (Math.pow(im, 7.0) * -0.0001984126984126984)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	tmp = 0
	if (t_0 <= -1.0) 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, 3.0) * -0.16666666666666666) + (math.pow(im, 7.0) * -0.0001984126984126984)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	tmp = 0.0
	if ((t_0 <= -1.0) || !(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 ^ 3.0) * -0.16666666666666666) + Float64((im ^ 7.0) * -0.0001984126984126984))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	tmp = 0.0;
	if ((t_0 <= -1.0) || ~((t_0 <= 0.05)))
		tmp = t_0 * (0.5 * sin(re));
	else
		tmp = sin(re) * ((((im ^ 5.0) * -0.008333333333333333) - im) + (((im ^ 3.0) * -0.16666666666666666) + ((im ^ 7.0) * -0.0001984126984126984)));
	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, -1.0], 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, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + N[(N[Power[im, 7.0], $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -1 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 -1 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < 0.050000000000000003

   1. Initial program 31.6%

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

   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. distribute-rgt-neg-in 99.8%

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

      6. *-commutative 99.8%

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

      7. associate-*l* 99.8%

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

      8. distribute-lft-out 99.8%

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

      9. +-commutative 99.8%

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

      10. *-commutative 99.8%

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

      11. associate-*r* 99.8%

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

      12. *-commutative 99.8%

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

      13. associate-*r* 99.8%

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

      14. distribute-rgt-out 99.8%

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

   4. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -1 \lor \neg \left(e^{-im} - e^{im} \leq 0.05\right):\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(\left({im}^{5} \cdot -0.008333333333333333 - im\right) + \left({im}^{3} \cdot -0.16666666666666666 + {im}^{7} \cdot -0.0001984126984126984\right)\right)\\ \end{array} \]

Alternative 2: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -1 \lor \neg \left(t_0 \leq 0.05\right):\\ \;\;\;\;t_0 \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(\left({im}^{5} \cdot -0.008333333333333333 - im\right) + {im}^{3} \cdot -0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 -1.0) (not (<= t_0 0.05)))
     (* t_0 (* 0.5 (sin re)))
     (*
      (sin re)
      (+
       (- (* (pow im 5.0) -0.008333333333333333) im)
       (* (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 <= -1.0) || !(t_0 <= 0.05)) {
		tmp = t_0 * (0.5 * sin(re));
	} else {
		tmp = sin(re) * (((pow(im, 5.0) * -0.008333333333333333) - im) + (pow(im, 3.0) * -0.16666666666666666));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-im) - exp(im)
    if ((t_0 <= (-1.0d0)) .or. (.not. (t_0 <= 0.05d0))) then
        tmp = t_0 * (0.5d0 * sin(re))
    else
        tmp = sin(re) * ((((im ** 5.0d0) * (-0.008333333333333333d0)) - im) + ((im ** 3.0d0) * (-0.16666666666666666d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(-im) - Math.exp(im);
	double tmp;
	if ((t_0 <= -1.0) || !(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, 3.0) * -0.16666666666666666));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	tmp = 0
	if (t_0 <= -1.0) 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, 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 <= -1.0) || !(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((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 <= -1.0) || ~((t_0 <= 0.05)))
		tmp = t_0 * (0.5 * sin(re));
	else
		tmp = sin(re) * ((((im ^ 5.0) * -0.008333333333333333) - im) + ((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, -1.0], 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[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -1 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 -1 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < 0.050000000000000003

   1. Initial program 31.6%

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

   2. Taylor expanded in im around 0 99.7%

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

      1. associate-*r* 99.7%

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

      2. *-commutative 99.7%

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

      3. associate-*l* 99.7%

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

      4. +-commutative 99.7%

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

      5. mul-1-neg 99.7%

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

      6. distribute-rgt-neg-in 99.7%

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

      7. *-commutative 99.7%

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

      8. associate-*l* 99.7%

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

      9. distribute-lft-out 99.7%

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

      10. distribute-lft-out 99.8%

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

      11. *-commutative 99.8%

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

      12. +-commutative 99.8%

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

      13. sub-neg 99.8%

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

   4. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -1 \lor \neg \left(e^{-im} - e^{im} \leq 0.05\right):\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(\left({im}^{5} \cdot -0.008333333333333333 - im\right) + {im}^{3} \cdot -0.16666666666666666\right)\\ \end{array} \]

Alternative 3: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -1 \lor \neg \left(t_0 \leq 0.0005\right):\\ \;\;\;\;t_0 \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 -1.0) (not (<= t_0 0.0005)))
     (* 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 <= -1.0) || !(t_0 <= 0.0005)) {
		tmp = t_0 * (0.5 * sin(re));
	} else {
		tmp = sin(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-im) - exp(im)
    if ((t_0 <= (-1.0d0)) .or. (.not. (t_0 <= 0.0005d0))) then
        tmp = t_0 * (0.5d0 * sin(re))
    else
        tmp = sin(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(-im) - Math.exp(im);
	double tmp;
	if ((t_0 <= -1.0) || !(t_0 <= 0.0005)) {
		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 <= -1.0) or not (t_0 <= 0.0005):
		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 <= -1.0) || !(t_0 <= 0.0005))
		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 <= -1.0) || ~((t_0 <= 0.0005)))
		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, -1.0], N[Not[LessEqual[t$95$0, 0.0005]], $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)) < -1 or 5.0000000000000001e-4 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 99.9%

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

   if -1 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < 5.0000000000000001e-4

   1. Initial program 31.1%

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

   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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -1 \lor \neg \left(e^{-im} - e^{im} \leq 0.0005\right):\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternative 4: 96.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ t_1 := -0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \mathbf{if}\;im \leq -4.4 \cdot 10^{+123}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.6:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.66:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

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 -4.4e+123)
     t_1
     (if (<= im -0.6)
       t_0
       (if (<= im 0.66)
         (* (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 <= -4.4e+123) {
		tmp = t_1;
	} else if (im <= -0.6) {
		tmp = t_0;
	} else if (im <= 0.66) {
		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 <= (-4.4d+123)) then
        tmp = t_1
    else if (im <= (-0.6d0)) then
        tmp = t_0
    else if (im <= 0.66d0) 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 <= -4.4e+123) {
		tmp = t_1;
	} else if (im <= -0.6) {
		tmp = t_0;
	} else if (im <= 0.66) {
		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 <= -4.4e+123:
		tmp = t_1
	elif im <= -0.6:
		tmp = t_0
	elif im <= 0.66:
		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 <= -4.4e+123)
		tmp = t_1;
	elseif (im <= -0.6)
		tmp = t_0;
	elseif (im <= 0.66)
		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 <= -4.4e+123)
		tmp = t_1;
	elseif (im <= -0.6)
		tmp = t_0;
	elseif (im <= 0.66)
		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, -4.4e+123], t$95$1, If[LessEqual[im, -0.6], t$95$0, If[LessEqual[im, 0.66], 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 < -4.39999999999999984e123 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. distribute-rgt-neg-in 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*l* 100.0%

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

      8. distribute-lft-out 100.0%

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

      9. +-commutative 100.0%

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

      10. *-commutative 100.0%

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

      11. associate-*r* 100.0%

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

      12. *-commutative 100.0%

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

      13. associate-*r* 100.0%

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

      14. distribute-rgt-out 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   if -4.39999999999999984e123 < im < -0.599999999999999978 or 0.660000000000000031 < im < 1.09999999999999998e44

   1. Initial program 99.9%

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

   2. Taylor expanded in re around 0 85.1%

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

   if -0.599999999999999978 < im < 0.660000000000000031

   1. Initial program 32.1%

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

   2. Taylor expanded in im around 0 99.0%

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

      1. mul-1-neg 99.0%

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

      2. unsub-neg 99.0%

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

      3. *-commutative 99.0%

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

      4. associate-*l* 99.0%

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

      5. distribute-lft-out-- 99.0%

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

   4. Simplified 99.0%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -4.4 \cdot 10^{+123}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \mathbf{elif}\;im \leq -0.6:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;im \leq 0.66:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \end{array} \]

Alternative 5: 93.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -5.5 \lor \neg \left(im \leq 5.5\right):\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -5.5) (not (<= im 5.5)))
   (* -0.0001984126984126984 (* (sin re) (pow im 7.0)))
   (* (sin re) (- (* (pow im 3.0) -0.16666666666666666) im))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -5.5) || !(im <= 5.5)) {
		tmp = -0.0001984126984126984 * (sin(re) * pow(im, 7.0));
	} else {
		tmp = sin(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-5.5d0)) .or. (.not. (im <= 5.5d0))) then
        tmp = (-0.0001984126984126984d0) * (sin(re) * (im ** 7.0d0))
    else
        tmp = sin(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -5.5) || !(im <= 5.5)) {
		tmp = -0.0001984126984126984 * (Math.sin(re) * Math.pow(im, 7.0));
	} else {
		tmp = Math.sin(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -5.5) or not (im <= 5.5):
		tmp = -0.0001984126984126984 * (math.sin(re) * math.pow(im, 7.0))
	else:
		tmp = math.sin(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -5.5) || !(im <= 5.5))
		tmp = Float64(-0.0001984126984126984 * Float64(sin(re) * (im ^ 7.0)));
	else
		tmp = Float64(sin(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -5.5) || ~((im <= 5.5)))
		tmp = -0.0001984126984126984 * (sin(re) * (im ^ 7.0));
	else
		tmp = sin(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -5.5], N[Not[LessEqual[im, 5.5]], $MachinePrecision]], N[(-0.0001984126984126984 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 7.0], $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 im < -5.5 or 5.5 < 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 87.5%

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

      1. associate-+r+ 87.5%

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

      2. +-commutative 87.5%

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

      3. +-commutative 87.5%

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

      4. mul-1-neg 87.5%

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

      5. distribute-rgt-neg-in 87.5%

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

      6. *-commutative 87.5%

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

      7. associate-*l* 87.5%

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

      8. distribute-lft-out 87.5%

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

      9. +-commutative 87.5%

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

      10. *-commutative 87.5%

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

      11. associate-*r* 87.5%

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

      12. *-commutative 87.5%

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

      13. associate-*r* 87.5%

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

      14. distribute-rgt-out 87.5%

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

   4. Simplified 87.5%

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

   5. Taylor expanded in im around inf 87.5%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 87.5%

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

   7. Simplified 87.5%

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

   if -5.5 < im < 5.5

   1. Initial program 32.1%

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

   2. Taylor expanded in im around 0 99.0%

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

      1. mul-1-neg 99.0%

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

      2. unsub-neg 99.0%

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

      3. *-commutative 99.0%

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

      4. associate-*l* 99.0%

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

      5. distribute-lft-out-- 99.0%

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

   4. Simplified 99.0%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -5.5 \lor \neg \left(im \leq 5.5\right):\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternative 6: 92.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -4 \lor \neg \left(im \leq 4.2\right):\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -4.0) (not (<= im 4.2)))
   (* -0.0001984126984126984 (* (sin re) (pow im 7.0)))
   (* im (- (sin re)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -4.0) || !(im <= 4.2)) {
		tmp = -0.0001984126984126984 * (sin(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 <= (-4.0d0)) .or. (.not. (im <= 4.2d0))) then
        tmp = (-0.0001984126984126984d0) * (sin(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 <= -4.0) || !(im <= 4.2)) {
		tmp = -0.0001984126984126984 * (Math.sin(re) * Math.pow(im, 7.0));
	} else {
		tmp = im * -Math.sin(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -4.0) or not (im <= 4.2):
		tmp = -0.0001984126984126984 * (math.sin(re) * math.pow(im, 7.0))
	else:
		tmp = im * -math.sin(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -4.0) || !(im <= 4.2))
		tmp = Float64(-0.0001984126984126984 * Float64(sin(re) * (im ^ 7.0)));
	else
		tmp = Float64(im * Float64(-sin(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -4.0) || ~((im <= 4.2)))
		tmp = -0.0001984126984126984 * (sin(re) * (im ^ 7.0));
	else
		tmp = im * -sin(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -4.0], N[Not[LessEqual[im, 4.2]], $MachinePrecision]], N[(-0.0001984126984126984 * N[(N[Sin[re], $MachinePrecision] * 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 < -4 or 4.20000000000000018 < 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 87.5%

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

      1. associate-+r+ 87.5%

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

      2. +-commutative 87.5%

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

      3. +-commutative 87.5%

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

      4. mul-1-neg 87.5%

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

      5. distribute-rgt-neg-in 87.5%

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

      6. *-commutative 87.5%

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

      7. associate-*l* 87.5%

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

      8. distribute-lft-out 87.5%

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

      9. +-commutative 87.5%

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

      10. *-commutative 87.5%

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

      11. associate-*r* 87.5%

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

      12. *-commutative 87.5%

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

      13. associate-*r* 87.5%

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

      14. distribute-rgt-out 87.5%

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

   4. Simplified 87.5%

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

   5. Taylor expanded in im around inf 87.5%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 87.5%

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

   7. Simplified 87.5%

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

   if -4 < im < 4.20000000000000018

   1. Initial program 32.1%

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

   2. Taylor expanded in im around 0 98.2%

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

      1. mul-1-neg 98.2%

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

      2. *-commutative 98.2%

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

      3. distribute-lft-neg-in 98.2%

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

   4. Simplified 98.2%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -4 \lor \neg \left(im \leq 4.2\right):\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \]

Alternative 7: 80.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left({im}^{7} \cdot -0.0001984126984126984\right)\\ \mathbf{if}\;im \leq -60000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 7.6 \cdot 10^{+32}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 10^{+208} \lor \neg \left(im \leq 4.2 \cdot 10^{+275}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left({re}^{3} \cdot 0.16666666666666666 - re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (* (pow im 7.0) -0.0001984126984126984))))
   (if (<= im -60000.0)
     t_0
     (if (<= im 7.6e+32)
       (* im (- (sin re)))
       (if (or (<= im 1e+208) (not (<= im 4.2e+275)))
         t_0
         (* im (- (* (pow re 3.0) 0.16666666666666666) re)))))))

C

double code(double re, double im) {
	double t_0 = re * (pow(im, 7.0) * -0.0001984126984126984);
	double tmp;
	if (im <= -60000.0) {
		tmp = t_0;
	} else if (im <= 7.6e+32) {
		tmp = im * -sin(re);
	} else if ((im <= 1e+208) || !(im <= 4.2e+275)) {
		tmp = t_0;
	} else {
		tmp = im * ((pow(re, 3.0) * 0.16666666666666666) - re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = re * ((im ** 7.0d0) * (-0.0001984126984126984d0))
    if (im <= (-60000.0d0)) then
        tmp = t_0
    else if (im <= 7.6d+32) then
        tmp = im * -sin(re)
    else if ((im <= 1d+208) .or. (.not. (im <= 4.2d+275))) then
        tmp = t_0
    else
        tmp = im * (((re ** 3.0d0) * 0.16666666666666666d0) - re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re * (Math.pow(im, 7.0) * -0.0001984126984126984);
	double tmp;
	if (im <= -60000.0) {
		tmp = t_0;
	} else if (im <= 7.6e+32) {
		tmp = im * -Math.sin(re);
	} else if ((im <= 1e+208) || !(im <= 4.2e+275)) {
		tmp = t_0;
	} else {
		tmp = im * ((Math.pow(re, 3.0) * 0.16666666666666666) - re);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * (math.pow(im, 7.0) * -0.0001984126984126984)
	tmp = 0
	if im <= -60000.0:
		tmp = t_0
	elif im <= 7.6e+32:
		tmp = im * -math.sin(re)
	elif (im <= 1e+208) or not (im <= 4.2e+275):
		tmp = t_0
	else:
		tmp = im * ((math.pow(re, 3.0) * 0.16666666666666666) - re)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64((im ^ 7.0) * -0.0001984126984126984))
	tmp = 0.0
	if (im <= -60000.0)
		tmp = t_0;
	elseif (im <= 7.6e+32)
		tmp = Float64(im * Float64(-sin(re)));
	elseif ((im <= 1e+208) || !(im <= 4.2e+275))
		tmp = t_0;
	else
		tmp = Float64(im * Float64(Float64((re ^ 3.0) * 0.16666666666666666) - re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * ((im ^ 7.0) * -0.0001984126984126984);
	tmp = 0.0;
	if (im <= -60000.0)
		tmp = t_0;
	elseif (im <= 7.6e+32)
		tmp = im * -sin(re);
	elseif ((im <= 1e+208) || ~((im <= 4.2e+275)))
		tmp = t_0;
	else
		tmp = im * (((re ^ 3.0) * 0.16666666666666666) - re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(N[Power[im, 7.0], $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -60000.0], t$95$0, If[LessEqual[im, 7.6e+32], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], If[Or[LessEqual[im, 1e+208], N[Not[LessEqual[im, 4.2e+275]], $MachinePrecision]], t$95$0, N[(im * N[(N[(N[Power[re, 3.0], $MachinePrecision] * 0.16666666666666666), $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -6e4 or 7.6000000000000006e32 < im < 9.9999999999999998e207 or 4.2000000000000002e275 < 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 90.9%

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

      1. associate-+r+ 90.9%

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

      2. +-commutative 90.9%

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

      3. +-commutative 90.9%

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

      4. mul-1-neg 90.9%

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

      5. distribute-rgt-neg-in 90.9%

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

      6. *-commutative 90.9%

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

      7. associate-*l* 90.9%

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

      8. distribute-lft-out 90.9%

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

      9. +-commutative 90.9%

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

      10. *-commutative 90.9%

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

      11. associate-*r* 90.9%

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

      12. *-commutative 90.9%

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

      13. associate-*r* 90.9%

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

      14. distribute-rgt-out 90.9%

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

   4. Simplified 90.9%

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

   5. Taylor expanded in im around inf 90.9%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 90.9%

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

   7. Simplified 90.9%

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

   8. Taylor expanded in re around 0 76.5%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left(re \cdot {im}^{7}\right)} \]
   9. Step-by-step derivation

      1. *-commutative 76.5%

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

      2. associate-*l* 76.5%

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

      3. *-commutative 76.5%

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

   10. Simplified 76.5%

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

   if -6e4 < im < 7.6000000000000006e32

   1. Initial program 35.7%

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

   2. Taylor expanded in im around 0 93.3%

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

      1. mul-1-neg 93.3%

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

      2. *-commutative 93.3%

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

      3. distribute-lft-neg-in 93.3%

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

   4. Simplified 93.3%

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

   if 9.9999999999999998e207 < im < 4.2000000000000002e275

   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 6.4%

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

      1. mul-1-neg 6.4%

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

      2. *-commutative 6.4%

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

      3. distribute-lft-neg-in 6.4%

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

   4. Simplified 6.4%

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

   5. Taylor expanded in re around 0 1.0%

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

      1. +-commutative 1.0%

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

      2. mul-1-neg 1.0%

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

      3. unsub-neg 1.0%

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

      4. associate-*r* 1.0%

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

      5. distribute-rgt-out-- 59.8%

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

      6. *-commutative 59.8%

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

   7. Simplified 59.8%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -60000:\\ \;\;\;\;re \cdot \left({im}^{7} \cdot -0.0001984126984126984\right)\\ \mathbf{elif}\;im \leq 7.6 \cdot 10^{+32}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 10^{+208} \lor \neg \left(im \leq 4.2 \cdot 10^{+275}\right):\\ \;\;\;\;re \cdot \left({im}^{7} \cdot -0.0001984126984126984\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left({re}^{3} \cdot 0.16666666666666666 - re\right)\\ \end{array} \]

Alternative 8: 79.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left({im}^{7} \cdot -0.0001984126984126984\right)\\ \mathbf{if}\;im \leq -130000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 7.6 \cdot 10^{+32}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 10^{+208} \lor \neg \left(im \leq 4.2 \cdot 10^{+275}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(im \cdot {re}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (* (pow im 7.0) -0.0001984126984126984))))
   (if (<= im -130000.0)
     t_0
     (if (<= im 7.6e+32)
       (* im (- (sin re)))
       (if (or (<= im 1e+208) (not (<= im 4.2e+275)))
         t_0
         (* 0.16666666666666666 (* im (pow re 3.0))))))))

C

double code(double re, double im) {
	double t_0 = re * (pow(im, 7.0) * -0.0001984126984126984);
	double tmp;
	if (im <= -130000.0) {
		tmp = t_0;
	} else if (im <= 7.6e+32) {
		tmp = im * -sin(re);
	} else if ((im <= 1e+208) || !(im <= 4.2e+275)) {
		tmp = t_0;
	} else {
		tmp = 0.16666666666666666 * (im * pow(re, 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = re * ((im ** 7.0d0) * (-0.0001984126984126984d0))
    if (im <= (-130000.0d0)) then
        tmp = t_0
    else if (im <= 7.6d+32) then
        tmp = im * -sin(re)
    else if ((im <= 1d+208) .or. (.not. (im <= 4.2d+275))) then
        tmp = t_0
    else
        tmp = 0.16666666666666666d0 * (im * (re ** 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re * (Math.pow(im, 7.0) * -0.0001984126984126984);
	double tmp;
	if (im <= -130000.0) {
		tmp = t_0;
	} else if (im <= 7.6e+32) {
		tmp = im * -Math.sin(re);
	} else if ((im <= 1e+208) || !(im <= 4.2e+275)) {
		tmp = t_0;
	} else {
		tmp = 0.16666666666666666 * (im * Math.pow(re, 3.0));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * (math.pow(im, 7.0) * -0.0001984126984126984)
	tmp = 0
	if im <= -130000.0:
		tmp = t_0
	elif im <= 7.6e+32:
		tmp = im * -math.sin(re)
	elif (im <= 1e+208) or not (im <= 4.2e+275):
		tmp = t_0
	else:
		tmp = 0.16666666666666666 * (im * math.pow(re, 3.0))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64((im ^ 7.0) * -0.0001984126984126984))
	tmp = 0.0
	if (im <= -130000.0)
		tmp = t_0;
	elseif (im <= 7.6e+32)
		tmp = Float64(im * Float64(-sin(re)));
	elseif ((im <= 1e+208) || !(im <= 4.2e+275))
		tmp = t_0;
	else
		tmp = Float64(0.16666666666666666 * Float64(im * (re ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * ((im ^ 7.0) * -0.0001984126984126984);
	tmp = 0.0;
	if (im <= -130000.0)
		tmp = t_0;
	elseif (im <= 7.6e+32)
		tmp = im * -sin(re);
	elseif ((im <= 1e+208) || ~((im <= 4.2e+275)))
		tmp = t_0;
	else
		tmp = 0.16666666666666666 * (im * (re ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(N[Power[im, 7.0], $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -130000.0], t$95$0, If[LessEqual[im, 7.6e+32], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], If[Or[LessEqual[im, 1e+208], N[Not[LessEqual[im, 4.2e+275]], $MachinePrecision]], t$95$0, N[(0.16666666666666666 * N[(im * N[Power[re, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.3e5 or 7.6000000000000006e32 < im < 9.9999999999999998e207 or 4.2000000000000002e275 < 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 90.9%

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

      1. associate-+r+ 90.9%

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

      2. +-commutative 90.9%

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

      3. +-commutative 90.9%

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

      4. mul-1-neg 90.9%

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

      5. distribute-rgt-neg-in 90.9%

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

      6. *-commutative 90.9%

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

      7. associate-*l* 90.9%

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

      8. distribute-lft-out 90.9%

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

      9. +-commutative 90.9%

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

      10. *-commutative 90.9%

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

      11. associate-*r* 90.9%

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

      12. *-commutative 90.9%

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

      13. associate-*r* 90.9%

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

      14. distribute-rgt-out 90.9%

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

   4. Simplified 90.9%

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

   5. Taylor expanded in im around inf 90.9%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 90.9%

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

   7. Simplified 90.9%

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

   8. Taylor expanded in re around 0 76.5%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left(re \cdot {im}^{7}\right)} \]
   9. Step-by-step derivation

      1. *-commutative 76.5%

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

      2. associate-*l* 76.5%

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

      3. *-commutative 76.5%

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

   10. Simplified 76.5%

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

   if -1.3e5 < im < 7.6000000000000006e32

   1. Initial program 35.7%

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

   2. Taylor expanded in im around 0 93.3%

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

      1. mul-1-neg 93.3%

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

      2. *-commutative 93.3%

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

      3. distribute-lft-neg-in 93.3%

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

   4. Simplified 93.3%

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

   if 9.9999999999999998e207 < im < 4.2000000000000002e275

   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 6.4%

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

      1. mul-1-neg 6.4%

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

      2. *-commutative 6.4%

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

      3. distribute-lft-neg-in 6.4%

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

   4. Simplified 6.4%

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

   5. Taylor expanded in re around 0 1.0%

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

   6. Taylor expanded in re around inf 59.4%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -130000:\\ \;\;\;\;re \cdot \left({im}^{7} \cdot -0.0001984126984126984\right)\\ \mathbf{elif}\;im \leq 7.6 \cdot 10^{+32}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 10^{+208} \lor \neg \left(im \leq 4.2 \cdot 10^{+275}\right):\\ \;\;\;\;re \cdot \left({im}^{7} \cdot -0.0001984126984126984\right)\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(im \cdot {re}^{3}\right)\\ \end{array} \]

Alternative 9: 59.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -9.8 \cdot 10^{+123} \lor \neg \left(im \leq 120000000000\right):\\ \;\;\;\;0.16666666666666666 \cdot \left(im \cdot {re}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -9.8e+123) (not (<= im 120000000000.0)))
   (* 0.16666666666666666 (* im (pow re 3.0)))
   (* im (- (sin re)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -9.8e+123) || !(im <= 120000000000.0)) {
		tmp = 0.16666666666666666 * (im * pow(re, 3.0));
	} 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 <= (-9.8d+123)) .or. (.not. (im <= 120000000000.0d0))) then
        tmp = 0.16666666666666666d0 * (im * (re ** 3.0d0))
    else
        tmp = im * -sin(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -9.8e+123) || !(im <= 120000000000.0)) {
		tmp = 0.16666666666666666 * (im * Math.pow(re, 3.0));
	} else {
		tmp = im * -Math.sin(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -9.8e+123) or not (im <= 120000000000.0):
		tmp = 0.16666666666666666 * (im * math.pow(re, 3.0))
	else:
		tmp = im * -math.sin(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -9.8e+123) || !(im <= 120000000000.0))
		tmp = Float64(0.16666666666666666 * Float64(im * (re ^ 3.0)));
	else
		tmp = Float64(im * Float64(-sin(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -9.8e+123) || ~((im <= 120000000000.0)))
		tmp = 0.16666666666666666 * (im * (re ^ 3.0));
	else
		tmp = im * -sin(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -9.8e+123], N[Not[LessEqual[im, 120000000000.0]], $MachinePrecision]], N[(0.16666666666666666 * N[(im * N[Power[re, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -9.79999999999999952e123 or 1.2e11 < 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-lft-neg-in 4.6%

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

   4. Simplified 4.6%

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

   5. Taylor expanded in re around 0 7.4%

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

   6. Taylor expanded in re around inf 25.8%

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

   if -9.79999999999999952e123 < im < 1.2e11

   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 84.7%

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

      1. mul-1-neg 84.7%

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

      2. *-commutative 84.7%

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

      3. distribute-lft-neg-in 84.7%

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

   4. Simplified 84.7%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -9.8 \cdot 10^{+123} \lor \neg \left(im \leq 120000000000\right):\\ \;\;\;\;0.16666666666666666 \cdot \left(im \cdot {re}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \]

Alternative 10: 56.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -740000000000 \lor \neg \left(im \leq 1.5 \cdot 10^{+83}\right):\\ \;\;\;\;\left(-im\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -740000000000.0) (not (<= im 1.5e+83)))
   (* (- im) re)
   (* im (- (sin re)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -740000000000.0) || !(im <= 1.5e+83)) {
		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 <= (-740000000000.0d0)) .or. (.not. (im <= 1.5d+83))) 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 <= -740000000000.0) || !(im <= 1.5e+83)) {
		tmp = -im * re;
	} else {
		tmp = im * -Math.sin(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -740000000000.0) or not (im <= 1.5e+83):
		tmp = -im * re
	else:
		tmp = im * -math.sin(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -740000000000.0) || !(im <= 1.5e+83))
		tmp = Float64(Float64(-im) * re);
	else
		tmp = Float64(im * Float64(-sin(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -740000000000.0) || ~((im <= 1.5e+83)))
		tmp = -im * re;
	else
		tmp = im * -sin(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -740000000000.0], N[Not[LessEqual[im, 1.5e+83]], $MachinePrecision]], N[((-im) * re), $MachinePrecision], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -7.4e11 or 1.5e83 < 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 4.7%

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

      1. mul-1-neg 4.7%

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

      2. *-commutative 4.7%

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

      3. distribute-lft-neg-in 4.7%

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

   4. Simplified 4.7%

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

   5. Taylor expanded in re around 0 17.9%

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

   if -7.4e11 < im < 1.5e83

   1. Initial program 42.5%

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

   2. Taylor expanded in im around 0 83.7%

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

      1. mul-1-neg 83.7%

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

      2. *-commutative 83.7%

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

      3. distribute-lft-neg-in 83.7%

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

   4. Simplified 83.7%

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

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -740000000000 \lor \neg \left(im \leq 1.5 \cdot 10^{+83}\right):\\ \;\;\;\;\left(-im\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \]

Alternative 11: 33.6% accurate, 77.0× speedup

Math

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

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(Float64(-im) * re)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.3%

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

2. Taylor expanded in im around 0 51.0%

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

   1. mul-1-neg 51.0%

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

   2. *-commutative 51.0%

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

   3. distribute-lft-neg-in 51.0%

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

4. Simplified 51.0%

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

5. Taylor expanded in re around 0 31.4%

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

6. Final simplification 31.4%

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

Alternative 12: 15.7% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.0)

C

double code(double re, double im) {
	return 0.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.0d0
end function

Java

public static double code(double re, double im) {
	return 0.0;
}

Python

def code(re, im):
	return 0.0

Julia

function code(re, im)
	return 0.0
end

MATLAB

function tmp = code(re, im)
	tmp = 0.0;
end

Wolfram

code[re_, im_] := 0.0

Derivation

1. Initial program 66.3%

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

2. Taylor expanded in re around 0 52.1%

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

   1. expm1-log1p-u 34.5%

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

   2. expm1-udef 33.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(0.5 \cdot re\right) \cdot \left(e^{-im} - e^{im}\right)\right)} - 1} \]

   3. *-commutative 33.2%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(re \cdot 0.5\right)} \cdot \left(e^{-im} - e^{im}\right)\right)} - 1 \]

   4. add-sqr-sqrt 16.4%

      \[\leadsto e^{\mathsf{log1p}\left(\left(re \cdot 0.5\right) \cdot \left(e^{\color{blue}{\sqrt{-im} \cdot \sqrt{-im}}} - e^{im}\right)\right)} - 1 \]

   5. sqrt-unprod 23.9%

      \[\leadsto e^{\mathsf{log1p}\left(\left(re \cdot 0.5\right) \cdot \left(e^{\color{blue}{\sqrt{\left(-im\right) \cdot \left(-im\right)}}} - e^{im}\right)\right)} - 1 \]

   6. sqr-neg 23.9%

      \[\leadsto e^{\mathsf{log1p}\left(\left(re \cdot 0.5\right) \cdot \left(e^{\sqrt{\color{blue}{im \cdot im}}} - e^{im}\right)\right)} - 1 \]

   7. sqrt-unprod 7.4%

      \[\leadsto e^{\mathsf{log1p}\left(\left(re \cdot 0.5\right) \cdot \left(e^{\color{blue}{\sqrt{im} \cdot \sqrt{im}}} - e^{im}\right)\right)} - 1 \]

   8. add-sqr-sqrt 13.7%

      \[\leadsto e^{\mathsf{log1p}\left(\left(re \cdot 0.5\right) \cdot \left(e^{\color{blue}{im}} - e^{im}\right)\right)} - 1 \]

4. Applied egg-rr 13.7%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(re \cdot 0.5\right) \cdot \left(e^{im} - e^{im}\right)\right)} - 1} \]
5. Step-by-step derivation

   1. expm1-def 13.7%

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

   2. expm1-log1p 13.7%

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

   3. +-inverses 14.1%

      \[\leadsto \left(re \cdot 0.5\right) \cdot \color{blue}{0} \]

   4. mul0-rgt 14.1%

      \[\leadsto \color{blue}{0} \]

6. Simplified 14.1%

   \[\leadsto \color{blue}{0} \]

7. Final simplification 14.1%

   \[\leadsto 0 \]

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 2023274 
(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))))