math.cos on complex, imaginary part

Percentage Accurate: 66.4% → 99.6%

Time: 8.3s

Alternatives: 16

Speedup: 2.8×

• 49.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.4% 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: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{-im\_m} - e^{im\_m}\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+30}:\\ \;\;\;\;t\_0 \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left({im\_m}^{2} \cdot \mathsf{fma}\left({im\_m}^{2}, \sin re \cdot \left(-0.008333333333333333 + {im\_m}^{2} \cdot -0.0001984126984126984\right), \sin re \cdot -0.16666666666666666\right) - \sin re\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (- (exp (- im_m)) (exp im_m))))
   (*
    im_s
    (if (<= t_0 -5e+30)
      (* t_0 (* 0.5 (sin re)))
      (*
       im_m
       (-
        (*
         (pow im_m 2.0)
         (fma
          (pow im_m 2.0)
          (*
           (sin re)
           (+ -0.008333333333333333 (* (pow im_m 2.0) -0.0001984126984126984)))
          (* (sin re) -0.16666666666666666)))
        (sin re)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = exp(-im_m) - exp(im_m);
	double tmp;
	if (t_0 <= -5e+30) {
		tmp = t_0 * (0.5 * sin(re));
	} else {
		tmp = im_m * ((pow(im_m, 2.0) * fma(pow(im_m, 2.0), (sin(re) * (-0.008333333333333333 + (pow(im_m, 2.0) * -0.0001984126984126984))), (sin(re) * -0.16666666666666666))) - sin(re));
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(exp(Float64(-im_m)) - exp(im_m))
	tmp = 0.0
	if (t_0 <= -5e+30)
		tmp = Float64(t_0 * Float64(0.5 * sin(re)));
	else
		tmp = Float64(im_m * Float64(Float64((im_m ^ 2.0) * fma((im_m ^ 2.0), Float64(sin(re) * Float64(-0.008333333333333333 + Float64((im_m ^ 2.0) * -0.0001984126984126984))), Float64(sin(re) * -0.16666666666666666))) - sin(re)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -5e+30], N[(t$95$0 * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(N[(N[Power[im$95$m, 2.0], $MachinePrecision] * N[(N[Power[im$95$m, 2.0], $MachinePrecision] * N[(N[Sin[re], $MachinePrecision] * N[(-0.008333333333333333 + N[(N[Power[im$95$m, 2.0], $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[re], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -4.9999999999999998e30

   1. Initial program 100.0%

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

   if -4.9999999999999998e30 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 46.2%

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

   3. Taylor expanded in im around 0 94.5%

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + {im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative 94.5%

         \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + {im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + -1 \cdot \sin re\right)} \]

      2. mul-1-neg 94.5%

         \[\leadsto im \cdot \left({im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + {im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) + \color{blue}{\left(-\sin re\right)}\right) \]

      3. unsub-neg 94.5%

         \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left(-0.16666666666666666 \cdot \sin re + {im}^{2} \cdot \left(-0.008333333333333333 \cdot \sin re + -0.0001984126984126984 \cdot \left({im}^{2} \cdot \sin re\right)\right)\right) - \sin re\right)} \]

   5. Simplified 94.5%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -5 \cdot 10^{+30}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left({im}^{2} \cdot \mathsf{fma}\left({im}^{2}, \sin re \cdot \left(-0.008333333333333333 + {im}^{2} \cdot -0.0001984126984126984\right), \sin re \cdot -0.16666666666666666\right) - \sin re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{-im\_m} - e^{im\_m}\\ t_1 := 0.5 \cdot \sin re\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+30}:\\ \;\;\;\;t\_0 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(im\_m \cdot \left({im\_m}^{2} \cdot \left({im\_m}^{2} \cdot \left({im\_m}^{2} \cdot -0.0003968253968253968 - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (- (exp (- im_m)) (exp im_m))) (t_1 (* 0.5 (sin re))))
   (*
    im_s
    (if (<= t_0 -5e+30)
      (* t_0 t_1)
      (*
       t_1
       (*
        im_m
        (-
         (*
          (pow im_m 2.0)
          (-
           (*
            (pow im_m 2.0)
            (- (* (pow im_m 2.0) -0.0003968253968253968) 0.016666666666666666))
           0.3333333333333333))
         2.0)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = exp(-im_m) - exp(im_m);
	double t_1 = 0.5 * sin(re);
	double tmp;
	if (t_0 <= -5e+30) {
		tmp = t_0 * t_1;
	} else {
		tmp = t_1 * (im_m * ((pow(im_m, 2.0) * ((pow(im_m, 2.0) * ((pow(im_m, 2.0) * -0.0003968253968253968) - 0.016666666666666666)) - 0.3333333333333333)) - 2.0));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(-im_m) - exp(im_m)
    t_1 = 0.5d0 * sin(re)
    if (t_0 <= (-5d+30)) then
        tmp = t_0 * t_1
    else
        tmp = t_1 * (im_m * (((im_m ** 2.0d0) * (((im_m ** 2.0d0) * (((im_m ** 2.0d0) * (-0.0003968253968253968d0)) - 0.016666666666666666d0)) - 0.3333333333333333d0)) - 2.0d0))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double t_0 = Math.exp(-im_m) - Math.exp(im_m);
	double t_1 = 0.5 * Math.sin(re);
	double tmp;
	if (t_0 <= -5e+30) {
		tmp = t_0 * t_1;
	} else {
		tmp = t_1 * (im_m * ((Math.pow(im_m, 2.0) * ((Math.pow(im_m, 2.0) * ((Math.pow(im_m, 2.0) * -0.0003968253968253968) - 0.016666666666666666)) - 0.3333333333333333)) - 2.0));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	t_0 = math.exp(-im_m) - math.exp(im_m)
	t_1 = 0.5 * math.sin(re)
	tmp = 0
	if t_0 <= -5e+30:
		tmp = t_0 * t_1
	else:
		tmp = t_1 * (im_m * ((math.pow(im_m, 2.0) * ((math.pow(im_m, 2.0) * ((math.pow(im_m, 2.0) * -0.0003968253968253968) - 0.016666666666666666)) - 0.3333333333333333)) - 2.0))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(exp(Float64(-im_m)) - exp(im_m))
	t_1 = Float64(0.5 * sin(re))
	tmp = 0.0
	if (t_0 <= -5e+30)
		tmp = Float64(t_0 * t_1);
	else
		tmp = Float64(t_1 * Float64(im_m * Float64(Float64((im_m ^ 2.0) * Float64(Float64((im_m ^ 2.0) * Float64(Float64((im_m ^ 2.0) * -0.0003968253968253968) - 0.016666666666666666)) - 0.3333333333333333)) - 2.0)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	t_0 = exp(-im_m) - exp(im_m);
	t_1 = 0.5 * sin(re);
	tmp = 0.0;
	if (t_0 <= -5e+30)
		tmp = t_0 * t_1;
	else
		tmp = t_1 * (im_m * (((im_m ^ 2.0) * (((im_m ^ 2.0) * (((im_m ^ 2.0) * -0.0003968253968253968) - 0.016666666666666666)) - 0.3333333333333333)) - 2.0));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -5e+30], N[(t$95$0 * t$95$1), $MachinePrecision], N[(t$95$1 * N[(im$95$m * N[(N[(N[Power[im$95$m, 2.0], $MachinePrecision] * N[(N[(N[Power[im$95$m, 2.0], $MachinePrecision] * N[(N[(N[Power[im$95$m, 2.0], $MachinePrecision] * -0.0003968253968253968), $MachinePrecision] - 0.016666666666666666), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -4.9999999999999998e30

   1. Initial program 100.0%

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

   if -4.9999999999999998e30 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 46.2%

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

   3. Taylor expanded in im around 0 96.4%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.0003968253968253968 \cdot {im}^{2} - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -5 \cdot 10^{+30}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot -0.0003968253968253968 - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{-im\_m} - e^{im\_m}\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -0.04:\\ \;\;\;\;t\_0 \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(-0.16666666666666666 \cdot {im\_m}^{3} - im\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (- (exp (- im_m)) (exp im_m))))
   (*
    im_s
    (if (<= t_0 -0.04)
      (* t_0 (* 0.5 (sin re)))
      (* (sin re) (- (* -0.16666666666666666 (pow im_m 3.0)) im_m))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = exp(-im_m) - exp(im_m);
	double tmp;
	if (t_0 <= -0.04) {
		tmp = t_0 * (0.5 * sin(re));
	} else {
		tmp = sin(re) * ((-0.16666666666666666 * pow(im_m, 3.0)) - im_m);
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-im_m) - exp(im_m)
    if (t_0 <= (-0.04d0)) then
        tmp = t_0 * (0.5d0 * sin(re))
    else
        tmp = sin(re) * (((-0.16666666666666666d0) * (im_m ** 3.0d0)) - im_m)
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double t_0 = Math.exp(-im_m) - Math.exp(im_m);
	double tmp;
	if (t_0 <= -0.04) {
		tmp = t_0 * (0.5 * Math.sin(re));
	} else {
		tmp = Math.sin(re) * ((-0.16666666666666666 * Math.pow(im_m, 3.0)) - im_m);
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	t_0 = math.exp(-im_m) - math.exp(im_m)
	tmp = 0
	if t_0 <= -0.04:
		tmp = t_0 * (0.5 * math.sin(re))
	else:
		tmp = math.sin(re) * ((-0.16666666666666666 * math.pow(im_m, 3.0)) - im_m)
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(exp(Float64(-im_m)) - exp(im_m))
	tmp = 0.0
	if (t_0 <= -0.04)
		tmp = Float64(t_0 * Float64(0.5 * sin(re)));
	else
		tmp = Float64(sin(re) * Float64(Float64(-0.16666666666666666 * (im_m ^ 3.0)) - im_m));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	t_0 = exp(-im_m) - exp(im_m);
	tmp = 0.0;
	if (t_0 <= -0.04)
		tmp = t_0 * (0.5 * sin(re));
	else
		tmp = sin(re) * ((-0.16666666666666666 * (im_m ^ 3.0)) - im_m);
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -0.04], N[(t$95$0 * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(N[(-0.16666666666666666 * N[Power[im$95$m, 3.0], $MachinePrecision]), $MachinePrecision] - im$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -0.0400000000000000008

   1. Initial program 99.8%

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

   if -0.0400000000000000008 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 46.0%

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

   3. Taylor expanded in im around 0 88.4%

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

      1. +-commutative 88.4%

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

      2. mul-1-neg 88.4%

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

      3. unsub-neg 88.4%

         \[\leadsto im \cdot \color{blue}{\left(-0.16666666666666666 \cdot \left({im}^{2} \cdot \sin re\right) - \sin re\right)} \]

      4. *-commutative 88.4%

         \[\leadsto im \cdot \left(-0.16666666666666666 \cdot \color{blue}{\left(\sin re \cdot {im}^{2}\right)} - \sin re\right) \]

      5. associate-*r* 88.4%

         \[\leadsto im \cdot \left(\color{blue}{\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}} - \sin re\right) \]

      6. distribute-lft-out-- 88.4%

         \[\leadsto \color{blue}{im \cdot \left(\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right) - im \cdot \sin re} \]

      7. associate-*r* 88.4%

         \[\leadsto im \cdot \color{blue}{\left(-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{2}\right)\right)} - im \cdot \sin re \]

      8. *-commutative 88.4%

         \[\leadsto im \cdot \left(-0.16666666666666666 \cdot \color{blue}{\left({im}^{2} \cdot \sin re\right)}\right) - im \cdot \sin re \]

      9. associate-*r* 88.4%

         \[\leadsto im \cdot \color{blue}{\left(\left(-0.16666666666666666 \cdot {im}^{2}\right) \cdot \sin re\right)} - im \cdot \sin re \]

      10. associate-*r* 89.9%

          \[\leadsto \color{blue}{\left(im \cdot \left(-0.16666666666666666 \cdot {im}^{2}\right)\right) \cdot \sin re} - im \cdot \sin re \]

      11. distribute-rgt-out-- 89.9%

          \[\leadsto \color{blue}{\sin re \cdot \left(im \cdot \left(-0.16666666666666666 \cdot {im}^{2}\right) - im\right)} \]

      12. unsub-neg 89.9%

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

      13. unsub-neg 89.9%

          \[\leadsto \sin re \cdot \color{blue}{\left(im \cdot \left(-0.16666666666666666 \cdot {im}^{2}\right) - im\right)} \]

      14. *-commutative 89.9%

          \[\leadsto \sin re \cdot \left(im \cdot \color{blue}{\left({im}^{2} \cdot -0.16666666666666666\right)} - im\right) \]

      15. associate-*r* 89.9%

          \[\leadsto \sin re \cdot \left(\color{blue}{\left(im \cdot {im}^{2}\right) \cdot -0.16666666666666666} - im\right) \]

   5. Simplified 89.9%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -0.04:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 0.009:\\ \;\;\;\;\sin re \cdot \left(-0.16666666666666666 \cdot {im\_m}^{3} - im\_m\right)\\ \mathbf{elif}\;im\_m \leq 2.45 \cdot 10^{+38}:\\ \;\;\;\;\left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im\_m}^{7}\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 0.009)
    (* (sin re) (- (* -0.16666666666666666 (pow im_m 3.0)) im_m))
    (if (<= im_m 2.45e+38)
      (* (- (exp (- im_m)) (exp im_m)) (* 0.5 re))
      (* -0.0001984126984126984 (* (sin re) (pow im_m 7.0)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 0.009) {
		tmp = sin(re) * ((-0.16666666666666666 * pow(im_m, 3.0)) - im_m);
	} else if (im_m <= 2.45e+38) {
		tmp = (exp(-im_m) - exp(im_m)) * (0.5 * re);
	} else {
		tmp = -0.0001984126984126984 * (sin(re) * pow(im_m, 7.0));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 0.009d0) then
        tmp = sin(re) * (((-0.16666666666666666d0) * (im_m ** 3.0d0)) - im_m)
    else if (im_m <= 2.45d+38) then
        tmp = (exp(-im_m) - exp(im_m)) * (0.5d0 * re)
    else
        tmp = (-0.0001984126984126984d0) * (sin(re) * (im_m ** 7.0d0))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 0.009) {
		tmp = Math.sin(re) * ((-0.16666666666666666 * Math.pow(im_m, 3.0)) - im_m);
	} else if (im_m <= 2.45e+38) {
		tmp = (Math.exp(-im_m) - Math.exp(im_m)) * (0.5 * re);
	} else {
		tmp = -0.0001984126984126984 * (Math.sin(re) * Math.pow(im_m, 7.0));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 0.009:
		tmp = math.sin(re) * ((-0.16666666666666666 * math.pow(im_m, 3.0)) - im_m)
	elif im_m <= 2.45e+38:
		tmp = (math.exp(-im_m) - math.exp(im_m)) * (0.5 * re)
	else:
		tmp = -0.0001984126984126984 * (math.sin(re) * math.pow(im_m, 7.0))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 0.009)
		tmp = Float64(sin(re) * Float64(Float64(-0.16666666666666666 * (im_m ^ 3.0)) - im_m));
	elseif (im_m <= 2.45e+38)
		tmp = Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * re));
	else
		tmp = Float64(-0.0001984126984126984 * Float64(sin(re) * (im_m ^ 7.0)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 0.009)
		tmp = sin(re) * ((-0.16666666666666666 * (im_m ^ 3.0)) - im_m);
	elseif (im_m <= 2.45e+38)
		tmp = (exp(-im_m) - exp(im_m)) * (0.5 * re);
	else
		tmp = -0.0001984126984126984 * (sin(re) * (im_m ^ 7.0));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 0.009], N[(N[Sin[re], $MachinePrecision] * N[(N[(-0.16666666666666666 * N[Power[im$95$m, 3.0], $MachinePrecision]), $MachinePrecision] - im$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 2.45e+38], N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], N[(-0.0001984126984126984 * N[(N[Sin[re], $MachinePrecision] * N[Power[im$95$m, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 0.00899999999999999932

   1. Initial program 46.0%

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

   3. Taylor expanded in im around 0 88.4%

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

      1. +-commutative 88.4%

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

      2. mul-1-neg 88.4%

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

      3. unsub-neg 88.4%

         \[\leadsto im \cdot \color{blue}{\left(-0.16666666666666666 \cdot \left({im}^{2} \cdot \sin re\right) - \sin re\right)} \]

      4. *-commutative 88.4%

         \[\leadsto im \cdot \left(-0.16666666666666666 \cdot \color{blue}{\left(\sin re \cdot {im}^{2}\right)} - \sin re\right) \]

      5. associate-*r* 88.4%

         \[\leadsto im \cdot \left(\color{blue}{\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}} - \sin re\right) \]

      6. distribute-lft-out-- 88.4%

         \[\leadsto \color{blue}{im \cdot \left(\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right) - im \cdot \sin re} \]

      7. associate-*r* 88.4%

         \[\leadsto im \cdot \color{blue}{\left(-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{2}\right)\right)} - im \cdot \sin re \]

      8. *-commutative 88.4%

         \[\leadsto im \cdot \left(-0.16666666666666666 \cdot \color{blue}{\left({im}^{2} \cdot \sin re\right)}\right) - im \cdot \sin re \]

      9. associate-*r* 88.4%

         \[\leadsto im \cdot \color{blue}{\left(\left(-0.16666666666666666 \cdot {im}^{2}\right) \cdot \sin re\right)} - im \cdot \sin re \]

      10. associate-*r* 89.9%

          \[\leadsto \color{blue}{\left(im \cdot \left(-0.16666666666666666 \cdot {im}^{2}\right)\right) \cdot \sin re} - im \cdot \sin re \]

      11. distribute-rgt-out-- 89.9%

          \[\leadsto \color{blue}{\sin re \cdot \left(im \cdot \left(-0.16666666666666666 \cdot {im}^{2}\right) - im\right)} \]

      12. unsub-neg 89.9%

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

      13. unsub-neg 89.9%

          \[\leadsto \sin re \cdot \color{blue}{\left(im \cdot \left(-0.16666666666666666 \cdot {im}^{2}\right) - im\right)} \]

      14. *-commutative 89.9%

          \[\leadsto \sin re \cdot \left(im \cdot \color{blue}{\left({im}^{2} \cdot -0.16666666666666666\right)} - im\right) \]

      15. associate-*r* 89.9%

          \[\leadsto \sin re \cdot \left(\color{blue}{\left(im \cdot {im}^{2}\right) \cdot -0.16666666666666666} - im\right) \]

   5. Simplified 89.9%

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

   if 0.00899999999999999932 < im < 2.45000000000000001e38

   1. Initial program 98.7%

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

   3. Taylor expanded in re around 0 74.0%

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

      1. associate-*r* 74.0%

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

      2. *-commutative 74.0%

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

   5. Simplified 74.0%

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

   if 2.45000000000000001e38 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 98.5%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.0003968253968253968 \cdot {im}^{2} - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)} \]

   4. Taylor expanded in im around inf 98.5%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.009:\\ \;\;\;\;\sin re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \mathbf{elif}\;im \leq 2.45 \cdot 10^{+38}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 5.5:\\ \;\;\;\;\sin re \cdot \left(-0.16666666666666666 \cdot {im\_m}^{3} - im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im\_m}^{7}\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 5.5)
    (* (sin re) (- (* -0.16666666666666666 (pow im_m 3.0)) im_m))
    (* -0.0001984126984126984 (* (sin re) (pow im_m 7.0))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 5.5) {
		tmp = sin(re) * ((-0.16666666666666666 * pow(im_m, 3.0)) - im_m);
	} else {
		tmp = -0.0001984126984126984 * (sin(re) * pow(im_m, 7.0));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 5.5d0) then
        tmp = sin(re) * (((-0.16666666666666666d0) * (im_m ** 3.0d0)) - im_m)
    else
        tmp = (-0.0001984126984126984d0) * (sin(re) * (im_m ** 7.0d0))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 5.5) {
		tmp = Math.sin(re) * ((-0.16666666666666666 * Math.pow(im_m, 3.0)) - im_m);
	} else {
		tmp = -0.0001984126984126984 * (Math.sin(re) * Math.pow(im_m, 7.0));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 5.5:
		tmp = math.sin(re) * ((-0.16666666666666666 * math.pow(im_m, 3.0)) - im_m)
	else:
		tmp = -0.0001984126984126984 * (math.sin(re) * math.pow(im_m, 7.0))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 5.5)
		tmp = Float64(sin(re) * Float64(Float64(-0.16666666666666666 * (im_m ^ 3.0)) - im_m));
	else
		tmp = Float64(-0.0001984126984126984 * Float64(sin(re) * (im_m ^ 7.0)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 5.5)
		tmp = sin(re) * ((-0.16666666666666666 * (im_m ^ 3.0)) - im_m);
	else
		tmp = -0.0001984126984126984 * (sin(re) * (im_m ^ 7.0));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 5.5], N[(N[Sin[re], $MachinePrecision] * N[(N[(-0.16666666666666666 * N[Power[im$95$m, 3.0], $MachinePrecision]), $MachinePrecision] - im$95$m), $MachinePrecision]), $MachinePrecision], N[(-0.0001984126984126984 * N[(N[Sin[re], $MachinePrecision] * N[Power[im$95$m, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 5.5

   1. Initial program 46.2%

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

   3. Taylor expanded in im around 0 88.2%

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

      1. +-commutative 88.2%

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

      2. mul-1-neg 88.2%

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

      3. unsub-neg 88.2%

         \[\leadsto im \cdot \color{blue}{\left(-0.16666666666666666 \cdot \left({im}^{2} \cdot \sin re\right) - \sin re\right)} \]

      4. *-commutative 88.2%

         \[\leadsto im \cdot \left(-0.16666666666666666 \cdot \color{blue}{\left(\sin re \cdot {im}^{2}\right)} - \sin re\right) \]

      5. associate-*r* 88.2%

         \[\leadsto im \cdot \left(\color{blue}{\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}} - \sin re\right) \]

      6. distribute-lft-out-- 88.2%

         \[\leadsto \color{blue}{im \cdot \left(\left(-0.16666666666666666 \cdot \sin re\right) \cdot {im}^{2}\right) - im \cdot \sin re} \]

      7. associate-*r* 88.2%

         \[\leadsto im \cdot \color{blue}{\left(-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{2}\right)\right)} - im \cdot \sin re \]

      8. *-commutative 88.2%

         \[\leadsto im \cdot \left(-0.16666666666666666 \cdot \color{blue}{\left({im}^{2} \cdot \sin re\right)}\right) - im \cdot \sin re \]

      9. associate-*r* 88.2%

         \[\leadsto im \cdot \color{blue}{\left(\left(-0.16666666666666666 \cdot {im}^{2}\right) \cdot \sin re\right)} - im \cdot \sin re \]

      10. associate-*r* 89.7%

          \[\leadsto \color{blue}{\left(im \cdot \left(-0.16666666666666666 \cdot {im}^{2}\right)\right) \cdot \sin re} - im \cdot \sin re \]

      11. distribute-rgt-out-- 89.7%

          \[\leadsto \color{blue}{\sin re \cdot \left(im \cdot \left(-0.16666666666666666 \cdot {im}^{2}\right) - im\right)} \]

      12. unsub-neg 89.7%

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

      13. unsub-neg 89.7%

          \[\leadsto \sin re \cdot \color{blue}{\left(im \cdot \left(-0.16666666666666666 \cdot {im}^{2}\right) - im\right)} \]

      14. *-commutative 89.7%

          \[\leadsto \sin re \cdot \left(im \cdot \color{blue}{\left({im}^{2} \cdot -0.16666666666666666\right)} - im\right) \]

      15. associate-*r* 89.7%

          \[\leadsto \sin re \cdot \left(\color{blue}{\left(im \cdot {im}^{2}\right) \cdot -0.16666666666666666} - im\right) \]

   5. Simplified 89.7%

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

   if 5.5 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 88.2%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.0003968253968253968 \cdot {im}^{2} - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)} \]

   4. Taylor expanded in im around inf 88.2%

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

4. Final simplification 89.3%

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

Alternative 6: 92.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 4:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im\_m}^{7}\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 4.0)
    (* (- im_m) (sin re))
    (* -0.0001984126984126984 (* (sin re) (pow im_m 7.0))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 4.0) {
		tmp = -im_m * sin(re);
	} else {
		tmp = -0.0001984126984126984 * (sin(re) * pow(im_m, 7.0));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 4.0d0) then
        tmp = -im_m * sin(re)
    else
        tmp = (-0.0001984126984126984d0) * (sin(re) * (im_m ** 7.0d0))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 4.0) {
		tmp = -im_m * Math.sin(re);
	} else {
		tmp = -0.0001984126984126984 * (Math.sin(re) * Math.pow(im_m, 7.0));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 4.0:
		tmp = -im_m * math.sin(re)
	else:
		tmp = -0.0001984126984126984 * (math.sin(re) * math.pow(im_m, 7.0))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 4.0)
		tmp = Float64(Float64(-im_m) * sin(re));
	else
		tmp = Float64(-0.0001984126984126984 * Float64(sin(re) * (im_m ^ 7.0)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 4.0)
		tmp = -im_m * sin(re);
	else
		tmp = -0.0001984126984126984 * (sin(re) * (im_m ^ 7.0));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 4.0], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], N[(-0.0001984126984126984 * N[(N[Sin[re], $MachinePrecision] * N[Power[im$95$m, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 4

   1. Initial program 46.2%

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

   3. Taylor expanded in im around 0 71.6%

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

      1. associate-*r* 71.6%

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

      2. neg-mul-1 71.6%

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

   5. Simplified 71.6%

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

   if 4 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 88.2%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.0003968253968253968 \cdot {im}^{2} - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)} \]

   4. Taylor expanded in im around inf 88.2%

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

4. Final simplification 75.8%

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

Alternative 7: 82.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 5000:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(re \cdot {im\_m}^{7}\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 5000.0)
    (* (- im_m) (sin re))
    (* -0.0001984126984126984 (* re (pow im_m 7.0))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 5000.0) {
		tmp = -im_m * sin(re);
	} else {
		tmp = -0.0001984126984126984 * (re * pow(im_m, 7.0));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 5000.0d0) then
        tmp = -im_m * sin(re)
    else
        tmp = (-0.0001984126984126984d0) * (re * (im_m ** 7.0d0))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 5000.0) {
		tmp = -im_m * Math.sin(re);
	} else {
		tmp = -0.0001984126984126984 * (re * Math.pow(im_m, 7.0));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 5000.0:
		tmp = -im_m * math.sin(re)
	else:
		tmp = -0.0001984126984126984 * (re * math.pow(im_m, 7.0))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 5000.0)
		tmp = Float64(Float64(-im_m) * sin(re));
	else
		tmp = Float64(-0.0001984126984126984 * Float64(re * (im_m ^ 7.0)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 5000.0)
		tmp = -im_m * sin(re);
	else
		tmp = -0.0001984126984126984 * (re * (im_m ^ 7.0));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 5000.0], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], N[(-0.0001984126984126984 * N[(re * N[Power[im$95$m, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 5e3

   1. Initial program 46.7%

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

   3. Taylor expanded in im around 0 71.0%

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

      1. associate-*r* 71.0%

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

      2. neg-mul-1 71.0%

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

   5. Simplified 71.0%

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

   if 5e3 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 90.8%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.0003968253968253968 \cdot {im}^{2} - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)} \]

   4. Taylor expanded in im around inf 90.8%

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

   5. Taylor expanded in re around 0 64.8%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 5000:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(re \cdot {im}^{7}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 0.00073:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im\_m \cdot \left(-0.3333333333333333 \cdot \left(im\_m \cdot im\_m\right) - 2\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 0.00073)
    (* (- im_m) (sin re))
    (* (* 0.5 re) (* im_m (- (* -0.3333333333333333 (* im_m im_m)) 2.0))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 0.00073) {
		tmp = -im_m * sin(re);
	} else {
		tmp = (0.5 * re) * (im_m * ((-0.3333333333333333 * (im_m * im_m)) - 2.0));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 0.00073d0) then
        tmp = -im_m * sin(re)
    else
        tmp = (0.5d0 * re) * (im_m * (((-0.3333333333333333d0) * (im_m * im_m)) - 2.0d0))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 0.00073) {
		tmp = -im_m * Math.sin(re);
	} else {
		tmp = (0.5 * re) * (im_m * ((-0.3333333333333333 * (im_m * im_m)) - 2.0));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 0.00073:
		tmp = -im_m * math.sin(re)
	else:
		tmp = (0.5 * re) * (im_m * ((-0.3333333333333333 * (im_m * im_m)) - 2.0))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 0.00073)
		tmp = Float64(Float64(-im_m) * sin(re));
	else
		tmp = Float64(Float64(0.5 * re) * Float64(im_m * Float64(Float64(-0.3333333333333333 * Float64(im_m * im_m)) - 2.0)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 0.00073)
		tmp = -im_m * sin(re);
	else
		tmp = (0.5 * re) * (im_m * ((-0.3333333333333333 * (im_m * im_m)) - 2.0));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 0.00073], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(im$95$m * N[(N[(-0.3333333333333333 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 7.2999999999999996e-4

   1. Initial program 46.0%

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

   3. Taylor expanded in im around 0 71.8%

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

      1. associate-*r* 71.8%

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

      2. neg-mul-1 71.8%

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

   5. Simplified 71.8%

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

   if 7.2999999999999996e-4 < im

   1. Initial program 99.8%

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

   3. Taylor expanded in re around 0 70.6%

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

      1. associate-*r* 70.6%

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

      2. *-commutative 70.6%

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

   5. Simplified 70.6%

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

   6. Taylor expanded in im around 0 55.5%

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

      1. unpow2 55.5%

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

   8. Applied egg-rr 55.5%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.00073:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im \cdot \left(-0.3333333333333333 \cdot \left(im \cdot im\right) - 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 53.4% accurate, 23.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(\left(0.5 \cdot re\right) \cdot \left(im\_m \cdot \left(-0.3333333333333333 \cdot \left(im\_m \cdot im\_m\right) - 2\right)\right)\right) \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (* (* 0.5 re) (* im_m (- (* -0.3333333333333333 (* im_m im_m)) 2.0)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * ((0.5 * re) * (im_m * ((-0.3333333333333333 * (im_m * im_m)) - 2.0)));
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = im_s * ((0.5d0 * re) * (im_m * (((-0.3333333333333333d0) * (im_m * im_m)) - 2.0d0)))
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	return im_s * ((0.5 * re) * (im_m * ((-0.3333333333333333 * (im_m * im_m)) - 2.0)));
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * ((0.5 * re) * (im_m * ((-0.3333333333333333 * (im_m * im_m)) - 2.0)))

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * Float64(Float64(0.5 * re) * Float64(im_m * Float64(Float64(-0.3333333333333333 * Float64(im_m * im_m)) - 2.0))))
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * ((0.5 * re) * (im_m * ((-0.3333333333333333 * (im_m * im_m)) - 2.0)));
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * N[(N[(0.5 * re), $MachinePrecision] * N[(im$95$m * N[(N[(-0.3333333333333333 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.6%

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

3. Taylor expanded in re around 0 44.7%

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

   1. associate-*r* 44.7%

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

   2. *-commutative 44.7%

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

5. Simplified 44.7%

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

6. Taylor expanded in im around 0 49.0%

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

   1. unpow2 49.0%

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

8. Applied egg-rr 49.0%

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

9. Final simplification 49.0%

   \[\leadsto \left(0.5 \cdot re\right) \cdot \left(im \cdot \left(-0.3333333333333333 \cdot \left(im \cdot im\right) - 2\right)\right) \]
10. Add Preprocessing

Alternative 10: 50.4% accurate, 28.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(im\_m \cdot \left(re \cdot \left(-0.16666666666666666 \cdot \left(im\_m \cdot im\_m\right) + -1\right)\right)\right) \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (* im_s (* im_m (* re (+ (* -0.16666666666666666 (* im_m im_m)) -1.0)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * (im_m * (re * ((-0.16666666666666666 * (im_m * im_m)) + -1.0)));
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = im_s * (im_m * (re * (((-0.16666666666666666d0) * (im_m * im_m)) + (-1.0d0))))
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	return im_s * (im_m * (re * ((-0.16666666666666666 * (im_m * im_m)) + -1.0)));
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * (im_m * (re * ((-0.16666666666666666 * (im_m * im_m)) + -1.0)))

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * Float64(im_m * Float64(re * Float64(Float64(-0.16666666666666666 * Float64(im_m * im_m)) + -1.0))))
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * (im_m * (re * ((-0.16666666666666666 * (im_m * im_m)) + -1.0)));
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * N[(im$95$m * N[(re * N[(N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.6%

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

3. Taylor expanded in im around 0 83.0%

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

   1. associate-*r* 83.0%

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

   2. distribute-rgt-out 83.0%

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

   3. *-commutative 83.0%

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

5. Simplified 83.0%

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

6. Taylor expanded in re around 0 46.8%

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

   1. unpow2 49.0%

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

8. Applied egg-rr 46.8%

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

9. Final simplification 46.8%

   \[\leadsto im \cdot \left(re \cdot \left(-0.16666666666666666 \cdot \left(im \cdot im\right) + -1\right)\right) \]
10. Add Preprocessing

Alternative 11: 33.3% accurate, 77.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(im\_m \cdot \left(-re\right)\right) \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m) :precision binary64 (* im_s (* im_m (- re))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * (im_m * -re);
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = im_s * (im_m * -re)
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	return im_s * (im_m * -re);
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * (im_m * -re)

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * Float64(im_m * Float64(-re)))
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * (im_m * -re);
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * N[(im$95$m * (-re)), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.6%

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

3. Taylor expanded in im around 0 54.9%

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

   1. associate-*r* 54.9%

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

   2. neg-mul-1 54.9%

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

5. Simplified 54.9%

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

6. Taylor expanded in re around 0 33.1%

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

   1. associate-*r* 33.1%

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

   2. neg-mul-1 33.1%

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

8. Simplified 33.1%

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

9. Final simplification 33.1%

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

Alternative 12: 15.2% accurate, 308.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot 0 \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m) :precision binary64 (* im_s 0.0))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * 0.0;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = im_s * 0.0d0
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	return im_s * 0.0;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * 0.0

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * 0.0)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * 0.0;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * 0.0), $MachinePrecision]

Derivation

1. Initial program 59.6%

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

3. Taylor expanded in im around 0 94.3%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.0003968253968253968 \cdot {im}^{2} - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)} \]

5. Applied egg-rr 12.4%

   \[\leadsto \color{blue}{0} \]
6. Add Preprocessing

Alternative 13: 2.8% accurate, 308.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot -9.922903012752122 \cdot 10^{-17} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m) :precision binary64 (* im_s -9.922903012752122e-17))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * -9.922903012752122e-17;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = im_s * (-9.922903012752122d-17)
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	return im_s * -9.922903012752122e-17;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * -9.922903012752122e-17

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * -9.922903012752122e-17)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * -9.922903012752122e-17;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * -9.922903012752122e-17), $MachinePrecision]

Derivation

1. Initial program 59.6%

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

3. Taylor expanded in im around 0 94.3%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.0003968253968253968 \cdot {im}^{2} - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)} \]

5. Applied egg-rr 3.0%

   \[\leadsto \color{blue}{-9.922903012752122 \cdot 10^{-17}} \]
6. Add Preprocessing

Alternative 14: 2.7% accurate, 308.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot -2.1433470507544583 \cdot 10^{-11} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m) :precision binary64 (* im_s -2.1433470507544583e-11))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * -2.1433470507544583e-11;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = im_s * (-2.1433470507544583d-11)
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	return im_s * -2.1433470507544583e-11;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * -2.1433470507544583e-11

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * -2.1433470507544583e-11)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * -2.1433470507544583e-11;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * -2.1433470507544583e-11), $MachinePrecision]

Derivation

1. Initial program 59.6%

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

3. Taylor expanded in im around 0 94.3%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.0003968253968253968 \cdot {im}^{2} - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)} \]

5. Applied egg-rr 3.0%

   \[\leadsto \color{blue}{-2.1433470507544583 \cdot 10^{-11}} \]
6. Add Preprocessing

Alternative 15: 2.7% accurate, 308.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot -7.71604938271605 \cdot 10^{-8} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m) :precision binary64 (* im_s -7.71604938271605e-8))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * -7.71604938271605e-8;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = im_s * (-7.71604938271605d-8)
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	return im_s * -7.71604938271605e-8;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * -7.71604938271605e-8

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * -7.71604938271605e-8)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * -7.71604938271605e-8;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * -7.71604938271605e-8), $MachinePrecision]

Derivation

1. Initial program 59.6%

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

3. Taylor expanded in im around 0 94.3%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.0003968253968253968 \cdot {im}^{2} - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)} \]

5. Applied egg-rr 2.9%

   \[\leadsto \color{blue}{-7.71604938271605 \cdot 10^{-8}} \]
6. Add Preprocessing

Alternative 16: 2.7% accurate, 308.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot -512 \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m) :precision binary64 (* im_s -512.0))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * -512.0;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = im_s * (-512.0d0)
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	return im_s * -512.0;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * -512.0

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * -512.0)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * -512.0;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * -512.0), $MachinePrecision]

Derivation

1. Initial program 59.6%

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

3. Taylor expanded in im around 0 94.3%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(-0.0003968253968253968 \cdot {im}^{2} - 0.016666666666666666\right) - 0.3333333333333333\right) - 2\right)\right)} \]

5. Applied egg-rr 2.9%

   \[\leadsto \color{blue}{-512} \]
6. Add Preprocessing

Developer target: 99.8% accurate, 0.7× 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 2024106 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"
  :precision binary64

  :alt
  (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))))