math.sin on complex, imaginary part

Percentage Accurate: 53.2% → 99.6%

Time: 13.6s

Alternatives: 21

Speedup: 20.6×

• 48.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{0 - im\_m}\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 - e^{im\_m} \leq -50:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(t\_0 + \frac{-1}{t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(im\_m \cdot \left(-1 + -0.16666666666666666 \cdot \left(im\_m \cdot im\_m\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

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 (- 0.0 im_m))))
   (*
    im_s
    (if (<= (- t_0 (exp im_m)) -50.0)
      (* (* 0.5 (cos re)) (+ t_0 (/ -1.0 t_0)))
      (* (cos re) (* im_m (+ -1.0 (* -0.16666666666666666 (* im_m 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((0.0 - im_m));
	double tmp;
	if ((t_0 - exp(im_m)) <= -50.0) {
		tmp = (0.5 * cos(re)) * (t_0 + (-1.0 / t_0));
	} else {
		tmp = cos(re) * (im_m * (-1.0 + (-0.16666666666666666 * (im_m * 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((0.0d0 - im_m))
    if ((t_0 - exp(im_m)) <= (-50.0d0)) then
        tmp = (0.5d0 * cos(re)) * (t_0 + ((-1.0d0) / t_0))
    else
        tmp = cos(re) * (im_m * ((-1.0d0) + ((-0.16666666666666666d0) * (im_m * 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((0.0 - im_m));
	double tmp;
	if ((t_0 - Math.exp(im_m)) <= -50.0) {
		tmp = (0.5 * Math.cos(re)) * (t_0 + (-1.0 / t_0));
	} else {
		tmp = Math.cos(re) * (im_m * (-1.0 + (-0.16666666666666666 * (im_m * 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((0.0 - im_m))
	tmp = 0
	if (t_0 - math.exp(im_m)) <= -50.0:
		tmp = (0.5 * math.cos(re)) * (t_0 + (-1.0 / t_0))
	else:
		tmp = math.cos(re) * (im_m * (-1.0 + (-0.16666666666666666 * (im_m * 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 = exp(Float64(0.0 - im_m))
	tmp = 0.0
	if (Float64(t_0 - exp(im_m)) <= -50.0)
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(t_0 + Float64(-1.0 / t_0)));
	else
		tmp = Float64(cos(re) * Float64(im_m * Float64(-1.0 + Float64(-0.16666666666666666 * Float64(im_m * 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((0.0 - im_m));
	tmp = 0.0;
	if ((t_0 - exp(im_m)) <= -50.0)
		tmp = (0.5 * cos(re)) * (t_0 + (-1.0 / t_0));
	else
		tmp = cos(re) * (im_m * (-1.0 + (-0.16666666666666666 * (im_m * 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[Exp[N[(0.0 - im$95$m), $MachinePrecision]], $MachinePrecision]}, N[(im$95$s * If[LessEqual[N[(t$95$0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision], -50.0], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(im$95$m * N[(-1.0 + N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)) < -50

   1. Initial program 100.0%

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

      1. /-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, im\right)\right), \left(\frac{e^{im}}{\color{blue}{1}}\right)\right)\right) \]

      2. exp-0 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, im\right)\right), \left(\frac{e^{im}}{e^{0}}\right)\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, im\right)\right), \left(\frac{1}{\color{blue}{\frac{e^{0}}{e^{im}}}}\right)\right)\right) \]

      4. exp-0 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, im\right)\right), \left(\frac{e^{0}}{\frac{\color{blue}{e^{0}}}{e^{im}}}\right)\right)\right) \]

      5. exp-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, im\right)\right), \left(\frac{e^{0}}{e^{0 - im}}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, im\right)\right), \mathsf{/.f64}\left(\left(e^{0}\right), \color{blue}{\left(e^{0 - im}\right)}\right)\right)\right) \]

      7. exp-0 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, im\right)\right), \mathsf{/.f64}\left(1, \left(e^{\color{blue}{0 - im}}\right)\right)\right)\right) \]

      8. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, im\right)\right), \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(0 - im\right)\right)\right)\right)\right) \]

      9. --lowering--.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, im\right)\right), \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, im\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - \color{blue}{\frac{1}{e^{0 - im}}}\right) \]

   if -50 < (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))

   1. Initial program 41.2%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \cos re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \cos re\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \cos re\right) + \color{blue}{-1 \cdot \cos re}\right) \]

      2. mul-1-neg N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \cos re\right) + \left(\mathsf{neg}\left(\cos re\right)\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \cos re\right) - \color{blue}{\cos re}\right) \]

      4. *-commutative N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \left(\cos re \cdot {im}^{2}\right) - \cos re\right) \]

      5. associate-*r* N/A

         \[\leadsto im \cdot \left(\left(\frac{-1}{6} \cdot \cos re\right) \cdot {im}^{2} - \cos \color{blue}{re}\right) \]

      6. distribute-lft-out-- N/A

         \[\leadsto im \cdot \left(\left(\frac{-1}{6} \cdot \cos re\right) \cdot {im}^{2}\right) - \color{blue}{im \cdot \cos re} \]

      7. associate-*r* N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \left(\cos re \cdot {im}^{2}\right)\right) - im \cdot \cos re \]

      8. *-commutative N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \cos re\right)\right) - im \cdot \cos re \]

      9. associate-*r* N/A

         \[\leadsto im \cdot \left(\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \cos re\right) - im \cdot \cos re \]

      10. associate-*r* N/A

          \[\leadsto \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right)\right) \cdot \cos re - \color{blue}{im} \cdot \cos re \]

      11. distribute-rgt-out-- N/A

          \[\leadsto \cos re \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) - im\right)} \]

      12. unsub-neg N/A

          \[\leadsto \cos re \cdot \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(im\right)\right)}\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + \left(\mathsf{neg}\left(im\right)\right)\right)}\right) \]

      14. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right)} + \left(\mathsf{neg}\left(im\right)\right)\right)\right) \]

      15. neg-mul-1 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + -1 \cdot \color{blue}{im}\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + im \cdot \color{blue}{-1}\right)\right) \]

   5. Simplified 92.0%

      \[\leadsto \color{blue}{\cos re \cdot \left(im \cdot \left(-1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{0 - im} - e^{im} \leq -50:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} + \frac{-1}{e^{0 - im}}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(im \cdot \left(-1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{0 - im\_m} - e^{im\_m}\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -50:\\ \;\;\;\;t\_0 \cdot \left(0.5 \cdot \cos re\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(im\_m \cdot \left(-1 + -0.16666666666666666 \cdot \left(im\_m \cdot im\_m\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

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 (- 0.0 im_m)) (exp im_m))))
   (*
    im_s
    (if (<= t_0 -50.0)
      (* t_0 (* 0.5 (cos re)))
      (* (cos re) (* im_m (+ -1.0 (* -0.16666666666666666 (* im_m 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((0.0 - im_m)) - exp(im_m);
	double tmp;
	if (t_0 <= -50.0) {
		tmp = t_0 * (0.5 * cos(re));
	} else {
		tmp = cos(re) * (im_m * (-1.0 + (-0.16666666666666666 * (im_m * 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((0.0d0 - im_m)) - exp(im_m)
    if (t_0 <= (-50.0d0)) then
        tmp = t_0 * (0.5d0 * cos(re))
    else
        tmp = cos(re) * (im_m * ((-1.0d0) + ((-0.16666666666666666d0) * (im_m * 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((0.0 - im_m)) - Math.exp(im_m);
	double tmp;
	if (t_0 <= -50.0) {
		tmp = t_0 * (0.5 * Math.cos(re));
	} else {
		tmp = Math.cos(re) * (im_m * (-1.0 + (-0.16666666666666666 * (im_m * 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((0.0 - im_m)) - math.exp(im_m)
	tmp = 0
	if t_0 <= -50.0:
		tmp = t_0 * (0.5 * math.cos(re))
	else:
		tmp = math.cos(re) * (im_m * (-1.0 + (-0.16666666666666666 * (im_m * 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(0.0 - im_m)) - exp(im_m))
	tmp = 0.0
	if (t_0 <= -50.0)
		tmp = Float64(t_0 * Float64(0.5 * cos(re)));
	else
		tmp = Float64(cos(re) * Float64(im_m * Float64(-1.0 + Float64(-0.16666666666666666 * Float64(im_m * 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((0.0 - im_m)) - exp(im_m);
	tmp = 0.0;
	if (t_0 <= -50.0)
		tmp = t_0 * (0.5 * cos(re));
	else
		tmp = cos(re) * (im_m * (-1.0 + (-0.16666666666666666 * (im_m * 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[N[(0.0 - im$95$m), $MachinePrecision]], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -50.0], N[(t$95$0 * N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(im$95$m * N[(-1.0 + N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)) < -50

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   if -50 < (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))

   1. Initial program 41.2%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \cos re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \cos re\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \cos re\right) + \color{blue}{-1 \cdot \cos re}\right) \]

      2. mul-1-neg N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \cos re\right) + \left(\mathsf{neg}\left(\cos re\right)\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \cos re\right) - \color{blue}{\cos re}\right) \]

      4. *-commutative N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \left(\cos re \cdot {im}^{2}\right) - \cos re\right) \]

      5. associate-*r* N/A

         \[\leadsto im \cdot \left(\left(\frac{-1}{6} \cdot \cos re\right) \cdot {im}^{2} - \cos \color{blue}{re}\right) \]

      6. distribute-lft-out-- N/A

         \[\leadsto im \cdot \left(\left(\frac{-1}{6} \cdot \cos re\right) \cdot {im}^{2}\right) - \color{blue}{im \cdot \cos re} \]

      7. associate-*r* N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \left(\cos re \cdot {im}^{2}\right)\right) - im \cdot \cos re \]

      8. *-commutative N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \cos re\right)\right) - im \cdot \cos re \]

      9. associate-*r* N/A

         \[\leadsto im \cdot \left(\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \cos re\right) - im \cdot \cos re \]

      10. associate-*r* N/A

          \[\leadsto \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right)\right) \cdot \cos re - \color{blue}{im} \cdot \cos re \]

      11. distribute-rgt-out-- N/A

          \[\leadsto \cos re \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) - im\right)} \]

      12. unsub-neg N/A

          \[\leadsto \cos re \cdot \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(im\right)\right)}\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + \left(\mathsf{neg}\left(im\right)\right)\right)}\right) \]

      14. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right)} + \left(\mathsf{neg}\left(im\right)\right)\right)\right) \]

      15. neg-mul-1 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + -1 \cdot \color{blue}{im}\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + im \cdot \color{blue}{-1}\right)\right) \]

   5. Simplified 92.0%

      \[\leadsto \color{blue}{\cos re \cdot \left(im \cdot \left(-1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{0 - im} - e^{im} \leq -50:\\ \;\;\;\;\left(e^{0 - im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(im \cdot \left(-1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 0.023:\\ \;\;\;\;\cos re \cdot \left(im\_m \cdot \left(-1 + -0.16666666666666666 \cdot \left(im\_m \cdot im\_m\right)\right)\right)\\ \mathbf{elif}\;im\_m \leq 3.4 \cdot 10^{+39}:\\ \;\;\;\;\left(\frac{1}{e^{im\_m}} - e^{im\_m}\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(\cos re \cdot \left(-1 + \left(im\_m \cdot im\_m\right) \cdot \left(-0.16666666666666666 + im\_m \cdot \left(im\_m \cdot \left(-0.008333333333333333 + im\_m \cdot \left(im\_m \cdot -0.0001984126984126984\right)\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

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 0.023)
    (* (cos re) (* im_m (+ -1.0 (* -0.16666666666666666 (* im_m im_m)))))
    (if (<= im_m 3.4e+39)
      (* (- (/ 1.0 (exp im_m)) (exp im_m)) (+ 0.5 (* -0.25 (* re re))))
      (*
       im_m
       (*
        (cos re)
        (+
         -1.0
         (*
          (* im_m im_m)
          (+
           -0.16666666666666666
           (*
            im_m
            (*
             im_m
             (+
              -0.008333333333333333
              (* im_m (* im_m -0.0001984126984126984))))))))))))))

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 <= 0.023) {
		tmp = cos(re) * (im_m * (-1.0 + (-0.16666666666666666 * (im_m * im_m))));
	} else if (im_m <= 3.4e+39) {
		tmp = ((1.0 / exp(im_m)) - exp(im_m)) * (0.5 + (-0.25 * (re * re)));
	} else {
		tmp = im_m * (cos(re) * (-1.0 + ((im_m * im_m) * (-0.16666666666666666 + (im_m * (im_m * (-0.008333333333333333 + (im_m * (im_m * -0.0001984126984126984)))))))));
	}
	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 <= 0.023d0) then
        tmp = cos(re) * (im_m * ((-1.0d0) + ((-0.16666666666666666d0) * (im_m * im_m))))
    else if (im_m <= 3.4d+39) then
        tmp = ((1.0d0 / exp(im_m)) - exp(im_m)) * (0.5d0 + ((-0.25d0) * (re * re)))
    else
        tmp = im_m * (cos(re) * ((-1.0d0) + ((im_m * im_m) * ((-0.16666666666666666d0) + (im_m * (im_m * ((-0.008333333333333333d0) + (im_m * (im_m * (-0.0001984126984126984d0))))))))))
    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 <= 0.023) {
		tmp = Math.cos(re) * (im_m * (-1.0 + (-0.16666666666666666 * (im_m * im_m))));
	} else if (im_m <= 3.4e+39) {
		tmp = ((1.0 / Math.exp(im_m)) - Math.exp(im_m)) * (0.5 + (-0.25 * (re * re)));
	} else {
		tmp = im_m * (Math.cos(re) * (-1.0 + ((im_m * im_m) * (-0.16666666666666666 + (im_m * (im_m * (-0.008333333333333333 + (im_m * (im_m * -0.0001984126984126984)))))))));
	}
	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 <= 0.023:
		tmp = math.cos(re) * (im_m * (-1.0 + (-0.16666666666666666 * (im_m * im_m))))
	elif im_m <= 3.4e+39:
		tmp = ((1.0 / math.exp(im_m)) - math.exp(im_m)) * (0.5 + (-0.25 * (re * re)))
	else:
		tmp = im_m * (math.cos(re) * (-1.0 + ((im_m * im_m) * (-0.16666666666666666 + (im_m * (im_m * (-0.008333333333333333 + (im_m * (im_m * -0.0001984126984126984)))))))))
	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 <= 0.023)
		tmp = Float64(cos(re) * Float64(im_m * Float64(-1.0 + Float64(-0.16666666666666666 * Float64(im_m * im_m)))));
	elseif (im_m <= 3.4e+39)
		tmp = Float64(Float64(Float64(1.0 / exp(im_m)) - exp(im_m)) * Float64(0.5 + Float64(-0.25 * Float64(re * re))));
	else
		tmp = Float64(im_m * Float64(cos(re) * Float64(-1.0 + Float64(Float64(im_m * im_m) * Float64(-0.16666666666666666 + Float64(im_m * Float64(im_m * Float64(-0.008333333333333333 + Float64(im_m * Float64(im_m * -0.0001984126984126984))))))))));
	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 <= 0.023)
		tmp = cos(re) * (im_m * (-1.0 + (-0.16666666666666666 * (im_m * im_m))));
	elseif (im_m <= 3.4e+39)
		tmp = ((1.0 / exp(im_m)) - exp(im_m)) * (0.5 + (-0.25 * (re * re)));
	else
		tmp = im_m * (cos(re) * (-1.0 + ((im_m * im_m) * (-0.16666666666666666 + (im_m * (im_m * (-0.008333333333333333 + (im_m * (im_m * -0.0001984126984126984)))))))));
	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, 0.023], N[(N[Cos[re], $MachinePrecision] * N[(im$95$m * N[(-1.0 + N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 3.4e+39], N[(N[(N[(1.0 / N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(N[Cos[re], $MachinePrecision] * N[(-1.0 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(-0.16666666666666666 + N[(im$95$m * N[(im$95$m * N[(-0.008333333333333333 + N[(im$95$m * N[(im$95$m * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 0.023

   1. Initial program 41.2%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \cos re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \cos re\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \cos re\right) + \color{blue}{-1 \cdot \cos re}\right) \]

      2. mul-1-neg N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \cos re\right) + \left(\mathsf{neg}\left(\cos re\right)\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \cos re\right) - \color{blue}{\cos re}\right) \]

      4. *-commutative N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \left(\cos re \cdot {im}^{2}\right) - \cos re\right) \]

      5. associate-*r* N/A

         \[\leadsto im \cdot \left(\left(\frac{-1}{6} \cdot \cos re\right) \cdot {im}^{2} - \cos \color{blue}{re}\right) \]

      6. distribute-lft-out-- N/A

         \[\leadsto im \cdot \left(\left(\frac{-1}{6} \cdot \cos re\right) \cdot {im}^{2}\right) - \color{blue}{im \cdot \cos re} \]

      7. associate-*r* N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \left(\cos re \cdot {im}^{2}\right)\right) - im \cdot \cos re \]

      8. *-commutative N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \cos re\right)\right) - im \cdot \cos re \]

      9. associate-*r* N/A

         \[\leadsto im \cdot \left(\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \cos re\right) - im \cdot \cos re \]

      10. associate-*r* N/A

          \[\leadsto \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right)\right) \cdot \cos re - \color{blue}{im} \cdot \cos re \]

      11. distribute-rgt-out-- N/A

          \[\leadsto \cos re \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) - im\right)} \]

      12. unsub-neg N/A

          \[\leadsto \cos re \cdot \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(im\right)\right)}\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + \left(\mathsf{neg}\left(im\right)\right)\right)}\right) \]

      14. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right)} + \left(\mathsf{neg}\left(im\right)\right)\right)\right) \]

      15. neg-mul-1 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + -1 \cdot \color{blue}{im}\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + im \cdot \color{blue}{-1}\right)\right) \]

   5. Simplified 92.0%

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

   if 0.023 < im < 3.3999999999999999e39

   1. Initial program 99.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) + \color{blue}{\frac{-1}{4} \cdot \left({re}^{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) + \left(\frac{-1}{4} \cdot {re}^{2}\right) \cdot \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)} \]

      3. distribute-rgt-out N/A

         \[\leadsto \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right) \cdot \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right), \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)}\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(e^{\mathsf{neg}\left(im\right)}\right), \left(e^{im}\right)\right), \left(\color{blue}{\frac{1}{2}} + \frac{-1}{4} \cdot {re}^{2}\right)\right) \]

      6. exp-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{1}{e^{im}}\right), \left(e^{im}\right)\right), \left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(e^{im}\right)\right), \left(e^{im}\right)\right), \left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)\right) \]

      8. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(im\right)\right), \left(e^{im}\right)\right), \left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)\right) \]

      9. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(im\right)\right), \mathsf{exp.f64}\left(im\right)\right), \left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(im\right)\right), \mathsf{exp.f64}\left(im\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{-1}{4} \cdot {re}^{2}\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(im\right)\right), \mathsf{exp.f64}\left(im\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{4}, \color{blue}{\left({re}^{2}\right)}\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(im\right)\right), \mathsf{exp.f64}\left(im\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{4}, \left(re \cdot \color{blue}{re}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 87.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(im\right)\right), \mathsf{exp.f64}\left(im\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(re, \color{blue}{re}\right)\right)\right)\right) \]

   5. Simplified 87.6%

      \[\leadsto \color{blue}{\left(\frac{1}{e^{im}} - e^{im}\right) \cdot \left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)} \]

   if 3.3999999999999999e39 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \cos re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \cos re + {im}^{2} \cdot \left(\frac{-1}{120} \cdot \cos re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(-1 \cdot \cos re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \cos re + {im}^{2} \cdot \left(\frac{-1}{120} \cdot \cos re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)\right)}\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left(-1 \cdot \cos re + \left(\left(\frac{-1}{6} \cdot \cos re\right) \cdot {im}^{2} + \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{120} \cdot \cos re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \cos re\right)\right)\right) \cdot {im}^{2}}\right)\right)\right) \]

      3. associate-+r+ N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left(\left(-1 \cdot \cos re + \left(\frac{-1}{6} \cdot \cos re\right) \cdot {im}^{2}\right) + \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{120} \cdot \cos re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \cos re\right)\right)\right) \cdot {im}^{2}}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left(\left(-1 \cdot \cos re + \left(\frac{-1}{6} \cdot \cos re\right) \cdot {im}^{2}\right) + {im}^{2} \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{120} \cdot \cos re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)}\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot \cos re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \cos re\right)\right)\right) + \color{blue}{\left(-1 \cdot \cos re + \left(\frac{-1}{6} \cdot \cos re\right) \cdot {im}^{2}\right)}\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot \cos re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \cos re\right)\right)\right) + \left(\left(\frac{-1}{6} \cdot \cos re\right) \cdot {im}^{2} + \color{blue}{-1 \cdot \cos re}\right)\right)\right) \]

   5. Simplified 100.0%

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

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) - 1\right)}\right)\right) \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) + -1\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(-1 + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right)}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right)\right)}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right)}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{{im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)} - \frac{1}{6}\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{{im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)} - \frac{1}{6}\right)\right)\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) + \frac{-1}{6}\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{6} + \color{blue}{{im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)}\right)\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)\right)}\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{-1}{5040} \cdot {im}^{2}} - \frac{1}{120}\right)\right)\right)\right)\right)\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)\right)}\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)\right)}\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      16. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{5040} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{120}\right)\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{5040} \cdot {im}^{2} + \frac{-1}{120}\right)\right)\right)\right)\right)\right)\right)\right) \]

      18. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{120} + \color{blue}{\frac{-1}{5040} \cdot {im}^{2}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      19. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{120}, \color{blue}{\left(\frac{-1}{5040} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{120}, \left({im}^{2} \cdot \color{blue}{\frac{-1}{5040}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      21. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{120}, \left(\left(im \cdot im\right) \cdot \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      22. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{120}, \left(im \cdot \color{blue}{\left(im \cdot \frac{-1}{5040}\right)}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      23. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \frac{-1}{5040}\right)}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      24. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{5040}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 100.0%

      \[\leadsto im \cdot \left(\cos re \cdot \color{blue}{\left(-1 + \left(im \cdot im\right) \cdot \left(-0.16666666666666666 + im \cdot \left(im \cdot \left(-0.008333333333333333 + im \cdot \left(im \cdot -0.0001984126984126984\right)\right)\right)\right)\right)}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 97.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 0.066:\\ \;\;\;\;\cos re \cdot \left(im\_m \cdot \left(-1 + -0.16666666666666666 \cdot \left(im\_m \cdot im\_m\right)\right)\right)\\ \mathbf{elif}\;im\_m \leq 1.3 \cdot 10^{+62}:\\ \;\;\;\;\frac{0.5}{e^{im\_m}} + e^{im\_m} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(\cos re \cdot \left(-1 + \left(im\_m \cdot im\_m\right) \cdot \left(-0.16666666666666666 + \left(im\_m \cdot im\_m\right) \cdot -0.008333333333333333\right)\right)\right)\\ \end{array} \end{array} \]

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 0.066)
    (* (cos re) (* im_m (+ -1.0 (* -0.16666666666666666 (* im_m im_m)))))
    (if (<= im_m 1.3e+62)
      (+ (/ 0.5 (exp im_m)) (* (exp im_m) -0.5))
      (*
       im_m
       (*
        (cos re)
        (+
         -1.0
         (*
          (* im_m im_m)
          (+
           -0.16666666666666666
           (* (* im_m im_m) -0.008333333333333333))))))))))

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 <= 0.066) {
		tmp = cos(re) * (im_m * (-1.0 + (-0.16666666666666666 * (im_m * im_m))));
	} else if (im_m <= 1.3e+62) {
		tmp = (0.5 / exp(im_m)) + (exp(im_m) * -0.5);
	} else {
		tmp = im_m * (cos(re) * (-1.0 + ((im_m * im_m) * (-0.16666666666666666 + ((im_m * im_m) * -0.008333333333333333)))));
	}
	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 <= 0.066d0) then
        tmp = cos(re) * (im_m * ((-1.0d0) + ((-0.16666666666666666d0) * (im_m * im_m))))
    else if (im_m <= 1.3d+62) then
        tmp = (0.5d0 / exp(im_m)) + (exp(im_m) * (-0.5d0))
    else
        tmp = im_m * (cos(re) * ((-1.0d0) + ((im_m * im_m) * ((-0.16666666666666666d0) + ((im_m * im_m) * (-0.008333333333333333d0))))))
    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 <= 0.066) {
		tmp = Math.cos(re) * (im_m * (-1.0 + (-0.16666666666666666 * (im_m * im_m))));
	} else if (im_m <= 1.3e+62) {
		tmp = (0.5 / Math.exp(im_m)) + (Math.exp(im_m) * -0.5);
	} else {
		tmp = im_m * (Math.cos(re) * (-1.0 + ((im_m * im_m) * (-0.16666666666666666 + ((im_m * im_m) * -0.008333333333333333)))));
	}
	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 <= 0.066:
		tmp = math.cos(re) * (im_m * (-1.0 + (-0.16666666666666666 * (im_m * im_m))))
	elif im_m <= 1.3e+62:
		tmp = (0.5 / math.exp(im_m)) + (math.exp(im_m) * -0.5)
	else:
		tmp = im_m * (math.cos(re) * (-1.0 + ((im_m * im_m) * (-0.16666666666666666 + ((im_m * im_m) * -0.008333333333333333)))))
	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 <= 0.066)
		tmp = Float64(cos(re) * Float64(im_m * Float64(-1.0 + Float64(-0.16666666666666666 * Float64(im_m * im_m)))));
	elseif (im_m <= 1.3e+62)
		tmp = Float64(Float64(0.5 / exp(im_m)) + Float64(exp(im_m) * -0.5));
	else
		tmp = Float64(im_m * Float64(cos(re) * Float64(-1.0 + Float64(Float64(im_m * im_m) * Float64(-0.16666666666666666 + Float64(Float64(im_m * im_m) * -0.008333333333333333))))));
	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 <= 0.066)
		tmp = cos(re) * (im_m * (-1.0 + (-0.16666666666666666 * (im_m * im_m))));
	elseif (im_m <= 1.3e+62)
		tmp = (0.5 / exp(im_m)) + (exp(im_m) * -0.5);
	else
		tmp = im_m * (cos(re) * (-1.0 + ((im_m * im_m) * (-0.16666666666666666 + ((im_m * im_m) * -0.008333333333333333)))));
	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, 0.066], N[(N[Cos[re], $MachinePrecision] * N[(im$95$m * N[(-1.0 + N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 1.3e+62], N[(N[(0.5 / N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] + N[(N[Exp[im$95$m], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(N[Cos[re], $MachinePrecision] * N[(-1.0 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(-0.16666666666666666 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 0.066000000000000003

   1. Initial program 41.2%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \cos re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \cos re\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \cos re\right) + \color{blue}{-1 \cdot \cos re}\right) \]

      2. mul-1-neg N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \cos re\right) + \left(\mathsf{neg}\left(\cos re\right)\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \cos re\right) - \color{blue}{\cos re}\right) \]

      4. *-commutative N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \left(\cos re \cdot {im}^{2}\right) - \cos re\right) \]

      5. associate-*r* N/A

         \[\leadsto im \cdot \left(\left(\frac{-1}{6} \cdot \cos re\right) \cdot {im}^{2} - \cos \color{blue}{re}\right) \]

      6. distribute-lft-out-- N/A

         \[\leadsto im \cdot \left(\left(\frac{-1}{6} \cdot \cos re\right) \cdot {im}^{2}\right) - \color{blue}{im \cdot \cos re} \]

      7. associate-*r* N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \left(\cos re \cdot {im}^{2}\right)\right) - im \cdot \cos re \]

      8. *-commutative N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \cos re\right)\right) - im \cdot \cos re \]

      9. associate-*r* N/A

         \[\leadsto im \cdot \left(\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \cos re\right) - im \cdot \cos re \]

      10. associate-*r* N/A

          \[\leadsto \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right)\right) \cdot \cos re - \color{blue}{im} \cdot \cos re \]

      11. distribute-rgt-out-- N/A

          \[\leadsto \cos re \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) - im\right)} \]

      12. unsub-neg N/A

          \[\leadsto \cos re \cdot \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(im\right)\right)}\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + \left(\mathsf{neg}\left(im\right)\right)\right)}\right) \]

      14. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right)} + \left(\mathsf{neg}\left(im\right)\right)\right)\right) \]

      15. neg-mul-1 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + -1 \cdot \color{blue}{im}\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + im \cdot \color{blue}{-1}\right)\right) \]

   5. Simplified 92.0%

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

   if 0.066000000000000003 < im < 1.29999999999999992e62

   1. Initial program 99.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + \color{blue}{\left(\mathsf{neg}\left(e^{im}\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(e^{im}\right)\right) + \color{blue}{e^{\mathsf{neg}\left(im\right)}}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \left(\mathsf{neg}\left(e^{im}\right)\right) \cdot \frac{1}{2} + \color{blue}{e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\mathsf{neg}\left(e^{im}\right)\right) + \color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2} \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \left(\mathsf{neg}\left(e^{im}\right)\right)\right), \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left(e^{im}\right)\right) \cdot \frac{1}{2}\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      7. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(e^{im} \cdot \frac{1}{2}\right)\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{+.f64}\left(\left(e^{im} \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\left(e^{im} \cdot \frac{-1}{2}\right), \left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(e^{im}\right), \frac{-1}{2}\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      11. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(e^{\color{blue}{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      12. exp-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(\frac{1}{e^{im}} \cdot \frac{1}{2}\right)\right) \]

      13. associate-*l/ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(\frac{1 \cdot \frac{1}{2}}{\color{blue}{e^{im}}}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(\frac{\frac{1}{2}}{e^{\color{blue}{im}}}\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{im}\right)}\right)\right) \]

      16. exp-lowering-exp.f64 52.6%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right)\right) \]

   5. Simplified 52.6%

      \[\leadsto \color{blue}{e^{im} \cdot -0.5 + \frac{0.5}{e^{im}}} \]

   if 1.29999999999999992e62 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \cos re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \cos re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(-1 \cdot \cos re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \cos re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left(\cos re \cdot -1 + \color{blue}{{im}^{2}} \cdot \left(\frac{-1}{6} \cdot \cos re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left(\cos re \cdot -1 + {im}^{2} \cdot \left(\frac{-1}{120} \cdot \left({im}^{2} \cdot \cos re\right) + \color{blue}{\frac{-1}{6} \cdot \cos re}\right)\right)\right) \]

      4. distribute-lft-in N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left(\cos re \cdot -1 + \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot \left({im}^{2} \cdot \cos re\right)\right) + \color{blue}{{im}^{2} \cdot \left(\frac{-1}{6} \cdot \cos re\right)}\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left(\cos re \cdot -1 + \left({im}^{2} \cdot \left(\left(\frac{-1}{120} \cdot {im}^{2}\right) \cdot \cos re\right) + {im}^{\color{blue}{2}} \cdot \left(\frac{-1}{6} \cdot \cos re\right)\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left(\cos re \cdot -1 + \left(\left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2}\right)\right) \cdot \cos re + \color{blue}{{im}^{2}} \cdot \left(\frac{-1}{6} \cdot \cos re\right)\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left(\cos re \cdot -1 + \left(\left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2}\right)\right) \cdot \cos re + \left({im}^{2} \cdot \frac{-1}{6}\right) \cdot \color{blue}{\cos re}\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left(\cos re \cdot -1 + \left(\left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \cos \color{blue}{re}\right)\right)\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left(\cos re \cdot -1 + \cos re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2}\right) + \frac{-1}{6} \cdot {im}^{2}\right)}\right)\right) \]

      10. distribute-lft-out N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left(\cos re \cdot \color{blue}{\left(-1 + \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2}\right) + \frac{-1}{6} \cdot {im}^{2}\right)\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\cos re, \color{blue}{\left(-1 + \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2}\right) + \frac{-1}{6} \cdot {im}^{2}\right)\right)}\right)\right) \]

      12. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{-1} + \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2}\right) + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{im \cdot \left(\cos re \cdot \left(-1 + \left(im \cdot im\right) \cdot \left(-0.16666666666666666 + \left(im \cdot im\right) \cdot -0.008333333333333333\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.066:\\ \;\;\;\;\cos re \cdot \left(im \cdot \left(-1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq 1.3 \cdot 10^{+62}:\\ \;\;\;\;\frac{0.5}{e^{im}} + e^{im} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\cos re \cdot \left(-1 + \left(im \cdot im\right) \cdot \left(-0.16666666666666666 + \left(im \cdot im\right) \cdot -0.008333333333333333\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 2.6:\\ \;\;\;\;\cos re \cdot \left(im\_m \cdot \left(-1 + -0.16666666666666666 \cdot \left(im\_m \cdot im\_m\right)\right)\right)\\ \mathbf{elif}\;im\_m \leq 1.2 \cdot 10^{+62}:\\ \;\;\;\;e^{im\_m} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(\cos re \cdot \left(-1 + \left(im\_m \cdot im\_m\right) \cdot \left(-0.16666666666666666 + \left(im\_m \cdot im\_m\right) \cdot -0.008333333333333333\right)\right)\right)\\ \end{array} \end{array} \]

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 2.6)
    (* (cos re) (* im_m (+ -1.0 (* -0.16666666666666666 (* im_m im_m)))))
    (if (<= im_m 1.2e+62)
      (* (exp im_m) -0.5)
      (*
       im_m
       (*
        (cos re)
        (+
         -1.0
         (*
          (* im_m im_m)
          (+
           -0.16666666666666666
           (* (* im_m im_m) -0.008333333333333333))))))))))

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 <= 2.6) {
		tmp = cos(re) * (im_m * (-1.0 + (-0.16666666666666666 * (im_m * im_m))));
	} else if (im_m <= 1.2e+62) {
		tmp = exp(im_m) * -0.5;
	} else {
		tmp = im_m * (cos(re) * (-1.0 + ((im_m * im_m) * (-0.16666666666666666 + ((im_m * im_m) * -0.008333333333333333)))));
	}
	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 <= 2.6d0) then
        tmp = cos(re) * (im_m * ((-1.0d0) + ((-0.16666666666666666d0) * (im_m * im_m))))
    else if (im_m <= 1.2d+62) then
        tmp = exp(im_m) * (-0.5d0)
    else
        tmp = im_m * (cos(re) * ((-1.0d0) + ((im_m * im_m) * ((-0.16666666666666666d0) + ((im_m * im_m) * (-0.008333333333333333d0))))))
    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 <= 2.6) {
		tmp = Math.cos(re) * (im_m * (-1.0 + (-0.16666666666666666 * (im_m * im_m))));
	} else if (im_m <= 1.2e+62) {
		tmp = Math.exp(im_m) * -0.5;
	} else {
		tmp = im_m * (Math.cos(re) * (-1.0 + ((im_m * im_m) * (-0.16666666666666666 + ((im_m * im_m) * -0.008333333333333333)))));
	}
	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 <= 2.6:
		tmp = math.cos(re) * (im_m * (-1.0 + (-0.16666666666666666 * (im_m * im_m))))
	elif im_m <= 1.2e+62:
		tmp = math.exp(im_m) * -0.5
	else:
		tmp = im_m * (math.cos(re) * (-1.0 + ((im_m * im_m) * (-0.16666666666666666 + ((im_m * im_m) * -0.008333333333333333)))))
	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 <= 2.6)
		tmp = Float64(cos(re) * Float64(im_m * Float64(-1.0 + Float64(-0.16666666666666666 * Float64(im_m * im_m)))));
	elseif (im_m <= 1.2e+62)
		tmp = Float64(exp(im_m) * -0.5);
	else
		tmp = Float64(im_m * Float64(cos(re) * Float64(-1.0 + Float64(Float64(im_m * im_m) * Float64(-0.16666666666666666 + Float64(Float64(im_m * im_m) * -0.008333333333333333))))));
	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 <= 2.6)
		tmp = cos(re) * (im_m * (-1.0 + (-0.16666666666666666 * (im_m * im_m))));
	elseif (im_m <= 1.2e+62)
		tmp = exp(im_m) * -0.5;
	else
		tmp = im_m * (cos(re) * (-1.0 + ((im_m * im_m) * (-0.16666666666666666 + ((im_m * im_m) * -0.008333333333333333)))));
	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, 2.6], N[(N[Cos[re], $MachinePrecision] * N[(im$95$m * N[(-1.0 + N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 1.2e+62], N[(N[Exp[im$95$m], $MachinePrecision] * -0.5), $MachinePrecision], N[(im$95$m * N[(N[Cos[re], $MachinePrecision] * N[(-1.0 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(-0.16666666666666666 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 2.60000000000000009

   1. Initial program 41.2%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \cos re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \cos re\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \cos re\right) + \color{blue}{-1 \cdot \cos re}\right) \]

      2. mul-1-neg N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \cos re\right) + \left(\mathsf{neg}\left(\cos re\right)\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \cos re\right) - \color{blue}{\cos re}\right) \]

      4. *-commutative N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \left(\cos re \cdot {im}^{2}\right) - \cos re\right) \]

      5. associate-*r* N/A

         \[\leadsto im \cdot \left(\left(\frac{-1}{6} \cdot \cos re\right) \cdot {im}^{2} - \cos \color{blue}{re}\right) \]

      6. distribute-lft-out-- N/A

         \[\leadsto im \cdot \left(\left(\frac{-1}{6} \cdot \cos re\right) \cdot {im}^{2}\right) - \color{blue}{im \cdot \cos re} \]

      7. associate-*r* N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \left(\cos re \cdot {im}^{2}\right)\right) - im \cdot \cos re \]

      8. *-commutative N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \cos re\right)\right) - im \cdot \cos re \]

      9. associate-*r* N/A

         \[\leadsto im \cdot \left(\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \cos re\right) - im \cdot \cos re \]

      10. associate-*r* N/A

          \[\leadsto \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right)\right) \cdot \cos re - \color{blue}{im} \cdot \cos re \]

      11. distribute-rgt-out-- N/A

          \[\leadsto \cos re \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) - im\right)} \]

      12. unsub-neg N/A

          \[\leadsto \cos re \cdot \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(im\right)\right)}\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + \left(\mathsf{neg}\left(im\right)\right)\right)}\right) \]

      14. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right)} + \left(\mathsf{neg}\left(im\right)\right)\right)\right) \]

      15. neg-mul-1 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + -1 \cdot \color{blue}{im}\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + im \cdot \color{blue}{-1}\right)\right) \]

   5. Simplified 92.0%

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

   if 2.60000000000000009 < im < 1.2e62

   1. Initial program 99.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + \color{blue}{\left(\mathsf{neg}\left(e^{im}\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(e^{im}\right)\right) + \color{blue}{e^{\mathsf{neg}\left(im\right)}}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \left(\mathsf{neg}\left(e^{im}\right)\right) \cdot \frac{1}{2} + \color{blue}{e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\mathsf{neg}\left(e^{im}\right)\right) + \color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2} \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \left(\mathsf{neg}\left(e^{im}\right)\right)\right), \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left(e^{im}\right)\right) \cdot \frac{1}{2}\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      7. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(e^{im} \cdot \frac{1}{2}\right)\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{+.f64}\left(\left(e^{im} \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\left(e^{im} \cdot \frac{-1}{2}\right), \left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(e^{im}\right), \frac{-1}{2}\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      11. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(e^{\color{blue}{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      12. exp-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(\frac{1}{e^{im}} \cdot \frac{1}{2}\right)\right) \]

      13. associate-*l/ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(\frac{1 \cdot \frac{1}{2}}{\color{blue}{e^{im}}}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(\frac{\frac{1}{2}}{e^{\color{blue}{im}}}\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{im}\right)}\right)\right) \]

      16. exp-lowering-exp.f64 52.6%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right)\right) \]

   5. Simplified 52.6%

      \[\leadsto \color{blue}{e^{im} \cdot -0.5 + \frac{0.5}{e^{im}}} \]

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(1 + im \cdot \left(1 + \frac{1}{2} \cdot im\right)\right)}\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \color{blue}{\left(im \cdot \left(1 + \frac{1}{2} \cdot im\right)\right)}\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(1 + \frac{1}{2} \cdot im\right)}\right)\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot im\right)}\right)\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 49.2%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 49.2%

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

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot e^{im}} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left(e^{im}\right)}\right) \]

       2. exp-lowering-exp.f64 49.3%

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{exp.f64}\left(im\right)\right) \]

   11. Simplified 49.3%

       \[\leadsto \color{blue}{-0.5 \cdot e^{im}} \]

   if 1.2e62 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \cos re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \cos re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(-1 \cdot \cos re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \cos re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left(\cos re \cdot -1 + \color{blue}{{im}^{2}} \cdot \left(\frac{-1}{6} \cdot \cos re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left(\cos re \cdot -1 + {im}^{2} \cdot \left(\frac{-1}{120} \cdot \left({im}^{2} \cdot \cos re\right) + \color{blue}{\frac{-1}{6} \cdot \cos re}\right)\right)\right) \]

      4. distribute-lft-in N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left(\cos re \cdot -1 + \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot \left({im}^{2} \cdot \cos re\right)\right) + \color{blue}{{im}^{2} \cdot \left(\frac{-1}{6} \cdot \cos re\right)}\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left(\cos re \cdot -1 + \left({im}^{2} \cdot \left(\left(\frac{-1}{120} \cdot {im}^{2}\right) \cdot \cos re\right) + {im}^{\color{blue}{2}} \cdot \left(\frac{-1}{6} \cdot \cos re\right)\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left(\cos re \cdot -1 + \left(\left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2}\right)\right) \cdot \cos re + \color{blue}{{im}^{2}} \cdot \left(\frac{-1}{6} \cdot \cos re\right)\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left(\cos re \cdot -1 + \left(\left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2}\right)\right) \cdot \cos re + \left({im}^{2} \cdot \frac{-1}{6}\right) \cdot \color{blue}{\cos re}\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left(\cos re \cdot -1 + \left(\left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \cos \color{blue}{re}\right)\right)\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left(\cos re \cdot -1 + \cos re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2}\right) + \frac{-1}{6} \cdot {im}^{2}\right)}\right)\right) \]

      10. distribute-lft-out N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left(\cos re \cdot \color{blue}{\left(-1 + \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2}\right) + \frac{-1}{6} \cdot {im}^{2}\right)\right)}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\cos re, \color{blue}{\left(-1 + \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2}\right) + \frac{-1}{6} \cdot {im}^{2}\right)\right)}\right)\right) \]

      12. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{-1} + \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2}\right) + \frac{-1}{6} \cdot {im}^{2}\right)\right)\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{im \cdot \left(\cos re \cdot \left(-1 + \left(im \cdot im\right) \cdot \left(-0.16666666666666666 + \left(im \cdot im\right) \cdot -0.008333333333333333\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.6:\\ \;\;\;\;\cos re \cdot \left(im \cdot \left(-1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq 1.2 \cdot 10^{+62}:\\ \;\;\;\;e^{im} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\cos re \cdot \left(-1 + \left(im \cdot im\right) \cdot \left(-0.16666666666666666 + \left(im \cdot im\right) \cdot -0.008333333333333333\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 2.1:\\ \;\;\;\;\cos re \cdot \left(im\_m \cdot \left(-1 + -0.16666666666666666 \cdot \left(im\_m \cdot im\_m\right)\right)\right)\\ \mathbf{elif}\;im\_m \leq 1.15 \cdot 10^{+103}:\\ \;\;\;\;e^{im\_m} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot \left(\cos re \cdot -0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

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 2.1)
    (* (cos re) (* im_m (+ -1.0 (* -0.16666666666666666 (* im_m im_m)))))
    (if (<= im_m 1.15e+103)
      (* (exp im_m) -0.5)
      (* im_m (* im_m (* im_m (* (cos re) -0.16666666666666666))))))))

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 <= 2.1) {
		tmp = cos(re) * (im_m * (-1.0 + (-0.16666666666666666 * (im_m * im_m))));
	} else if (im_m <= 1.15e+103) {
		tmp = exp(im_m) * -0.5;
	} else {
		tmp = im_m * (im_m * (im_m * (cos(re) * -0.16666666666666666)));
	}
	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 <= 2.1d0) then
        tmp = cos(re) * (im_m * ((-1.0d0) + ((-0.16666666666666666d0) * (im_m * im_m))))
    else if (im_m <= 1.15d+103) then
        tmp = exp(im_m) * (-0.5d0)
    else
        tmp = im_m * (im_m * (im_m * (cos(re) * (-0.16666666666666666d0))))
    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 <= 2.1) {
		tmp = Math.cos(re) * (im_m * (-1.0 + (-0.16666666666666666 * (im_m * im_m))));
	} else if (im_m <= 1.15e+103) {
		tmp = Math.exp(im_m) * -0.5;
	} else {
		tmp = im_m * (im_m * (im_m * (Math.cos(re) * -0.16666666666666666)));
	}
	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 <= 2.1:
		tmp = math.cos(re) * (im_m * (-1.0 + (-0.16666666666666666 * (im_m * im_m))))
	elif im_m <= 1.15e+103:
		tmp = math.exp(im_m) * -0.5
	else:
		tmp = im_m * (im_m * (im_m * (math.cos(re) * -0.16666666666666666)))
	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 <= 2.1)
		tmp = Float64(cos(re) * Float64(im_m * Float64(-1.0 + Float64(-0.16666666666666666 * Float64(im_m * im_m)))));
	elseif (im_m <= 1.15e+103)
		tmp = Float64(exp(im_m) * -0.5);
	else
		tmp = Float64(im_m * Float64(im_m * Float64(im_m * Float64(cos(re) * -0.16666666666666666))));
	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 <= 2.1)
		tmp = cos(re) * (im_m * (-1.0 + (-0.16666666666666666 * (im_m * im_m))));
	elseif (im_m <= 1.15e+103)
		tmp = exp(im_m) * -0.5;
	else
		tmp = im_m * (im_m * (im_m * (cos(re) * -0.16666666666666666)));
	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, 2.1], N[(N[Cos[re], $MachinePrecision] * N[(im$95$m * N[(-1.0 + N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 1.15e+103], N[(N[Exp[im$95$m], $MachinePrecision] * -0.5), $MachinePrecision], N[(im$95$m * N[(im$95$m * N[(im$95$m * N[(N[Cos[re], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 2.10000000000000009

   1. Initial program 41.2%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \cos re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \cos re\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \cos re\right) + \color{blue}{-1 \cdot \cos re}\right) \]

      2. mul-1-neg N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \cos re\right) + \left(\mathsf{neg}\left(\cos re\right)\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \cos re\right) - \color{blue}{\cos re}\right) \]

      4. *-commutative N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \left(\cos re \cdot {im}^{2}\right) - \cos re\right) \]

      5. associate-*r* N/A

         \[\leadsto im \cdot \left(\left(\frac{-1}{6} \cdot \cos re\right) \cdot {im}^{2} - \cos \color{blue}{re}\right) \]

      6. distribute-lft-out-- N/A

         \[\leadsto im \cdot \left(\left(\frac{-1}{6} \cdot \cos re\right) \cdot {im}^{2}\right) - \color{blue}{im \cdot \cos re} \]

      7. associate-*r* N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \left(\cos re \cdot {im}^{2}\right)\right) - im \cdot \cos re \]

      8. *-commutative N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \cos re\right)\right) - im \cdot \cos re \]

      9. associate-*r* N/A

         \[\leadsto im \cdot \left(\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \cos re\right) - im \cdot \cos re \]

      10. associate-*r* N/A

          \[\leadsto \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right)\right) \cdot \cos re - \color{blue}{im} \cdot \cos re \]

      11. distribute-rgt-out-- N/A

          \[\leadsto \cos re \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) - im\right)} \]

      12. unsub-neg N/A

          \[\leadsto \cos re \cdot \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(im\right)\right)}\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + \left(\mathsf{neg}\left(im\right)\right)\right)}\right) \]

      14. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right)} + \left(\mathsf{neg}\left(im\right)\right)\right)\right) \]

      15. neg-mul-1 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + -1 \cdot \color{blue}{im}\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right) + im \cdot \color{blue}{-1}\right)\right) \]

   5. Simplified 92.0%

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

   if 2.10000000000000009 < im < 1.15000000000000004e103

   1. Initial program 99.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + \color{blue}{\left(\mathsf{neg}\left(e^{im}\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(e^{im}\right)\right) + \color{blue}{e^{\mathsf{neg}\left(im\right)}}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \left(\mathsf{neg}\left(e^{im}\right)\right) \cdot \frac{1}{2} + \color{blue}{e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\mathsf{neg}\left(e^{im}\right)\right) + \color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2} \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \left(\mathsf{neg}\left(e^{im}\right)\right)\right), \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left(e^{im}\right)\right) \cdot \frac{1}{2}\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      7. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(e^{im} \cdot \frac{1}{2}\right)\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{+.f64}\left(\left(e^{im} \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\left(e^{im} \cdot \frac{-1}{2}\right), \left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(e^{im}\right), \frac{-1}{2}\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      11. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(e^{\color{blue}{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      12. exp-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(\frac{1}{e^{im}} \cdot \frac{1}{2}\right)\right) \]

      13. associate-*l/ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(\frac{1 \cdot \frac{1}{2}}{\color{blue}{e^{im}}}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(\frac{\frac{1}{2}}{e^{\color{blue}{im}}}\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{im}\right)}\right)\right) \]

      16. exp-lowering-exp.f64 58.1%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right)\right) \]

   5. Simplified 58.1%

      \[\leadsto \color{blue}{e^{im} \cdot -0.5 + \frac{0.5}{e^{im}}} \]

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(1 + im \cdot \left(1 + \frac{1}{2} \cdot im\right)\right)}\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \color{blue}{\left(im \cdot \left(1 + \frac{1}{2} \cdot im\right)\right)}\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(1 + \frac{1}{2} \cdot im\right)}\right)\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot im\right)}\right)\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 56.0%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 56.0%

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

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot e^{im}} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left(e^{im}\right)}\right) \]

       2. exp-lowering-exp.f64 56.0%

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{exp.f64}\left(im\right)\right) \]

   11. Simplified 56.0%

       \[\leadsto \color{blue}{-0.5 \cdot e^{im}} \]

   if 1.15000000000000004e103 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \color{blue}{\left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{3} \cdot {im}^{2} - 2\right)}\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left(\frac{-1}{3} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left(\frac{-1}{3} \cdot {im}^{2} + -2\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left(-2 + \color{blue}{\frac{-1}{3} \cdot {im}^{2}}\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \color{blue}{\left(\frac{-1}{3} \cdot {im}^{2}\right)}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \left({im}^{2} \cdot \color{blue}{\frac{-1}{3}}\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \left(\left(im \cdot im\right) \cdot \frac{-1}{3}\right)\right)\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \left(im \cdot \color{blue}{\left(im \cdot \frac{-1}{3}\right)}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \frac{-1}{3}\right)}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{3}}\right)\right)\right)\right)\right) \]

   5. Simplified 100.0%

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

   6. Taylor expanded in im around inf

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

      1. *-commutative N/A

         \[\leadsto \frac{-1}{6} \cdot \left(\cos re \cdot \color{blue}{{im}^{3}}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{6} \cdot \cos re\right) \cdot \color{blue}{{im}^{3}} \]

      3. unpow3 N/A

         \[\leadsto \left(\frac{-1}{6} \cdot \cos re\right) \cdot \left(\left(im \cdot im\right) \cdot \color{blue}{im}\right) \]

      4. unpow2 N/A

         \[\leadsto \left(\frac{-1}{6} \cdot \cos re\right) \cdot \left({im}^{2} \cdot im\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-1}{6} \cdot \cos re\right) \cdot {im}^{2}\right) \cdot \color{blue}{im} \]

      6. *-commutative N/A

         \[\leadsto im \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot \cos re\right) \cdot {im}^{2}\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(\left(\frac{-1}{6} \cdot \cos re\right) \cdot {im}^{2}\right)}\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left(\left(\frac{-1}{6} \cdot \cos re\right) \cdot \left(im \cdot \color{blue}{im}\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left(\left(\left(\frac{-1}{6} \cdot \cos re\right) \cdot im\right) \cdot \color{blue}{im}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(\left(\frac{-1}{6} \cdot \cos re\right) \cdot im\right), \color{blue}{im}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{6} \cdot \cos re\right), im\right), im\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\cos re \cdot \frac{-1}{6}\right), im\right), im\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\cos re, \frac{-1}{6}\right), im\right), im\right)\right) \]

      14. cos-lowering-cos.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \frac{-1}{6}\right), im\right), im\right)\right) \]

   8. Simplified 100.0%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.1:\\ \;\;\;\;\cos re \cdot \left(im \cdot \left(-1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+103}:\\ \;\;\;\;e^{im} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(im \cdot \left(im \cdot \left(\cos re \cdot -0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 95.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 2.7:\\ \;\;\;\;0 - im\_m \cdot \cos re\\ \mathbf{elif}\;im\_m \leq 1.1 \cdot 10^{+103}:\\ \;\;\;\;e^{im\_m} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot \left(\cos re \cdot -0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

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 2.7)
    (- 0.0 (* im_m (cos re)))
    (if (<= im_m 1.1e+103)
      (* (exp im_m) -0.5)
      (* im_m (* im_m (* im_m (* (cos re) -0.16666666666666666))))))))

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 <= 2.7) {
		tmp = 0.0 - (im_m * cos(re));
	} else if (im_m <= 1.1e+103) {
		tmp = exp(im_m) * -0.5;
	} else {
		tmp = im_m * (im_m * (im_m * (cos(re) * -0.16666666666666666)));
	}
	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 <= 2.7d0) then
        tmp = 0.0d0 - (im_m * cos(re))
    else if (im_m <= 1.1d+103) then
        tmp = exp(im_m) * (-0.5d0)
    else
        tmp = im_m * (im_m * (im_m * (cos(re) * (-0.16666666666666666d0))))
    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 <= 2.7) {
		tmp = 0.0 - (im_m * Math.cos(re));
	} else if (im_m <= 1.1e+103) {
		tmp = Math.exp(im_m) * -0.5;
	} else {
		tmp = im_m * (im_m * (im_m * (Math.cos(re) * -0.16666666666666666)));
	}
	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 <= 2.7:
		tmp = 0.0 - (im_m * math.cos(re))
	elif im_m <= 1.1e+103:
		tmp = math.exp(im_m) * -0.5
	else:
		tmp = im_m * (im_m * (im_m * (math.cos(re) * -0.16666666666666666)))
	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 <= 2.7)
		tmp = Float64(0.0 - Float64(im_m * cos(re)));
	elseif (im_m <= 1.1e+103)
		tmp = Float64(exp(im_m) * -0.5);
	else
		tmp = Float64(im_m * Float64(im_m * Float64(im_m * Float64(cos(re) * -0.16666666666666666))));
	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 <= 2.7)
		tmp = 0.0 - (im_m * cos(re));
	elseif (im_m <= 1.1e+103)
		tmp = exp(im_m) * -0.5;
	else
		tmp = im_m * (im_m * (im_m * (cos(re) * -0.16666666666666666)));
	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, 2.7], N[(0.0 - N[(im$95$m * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 1.1e+103], N[(N[Exp[im$95$m], $MachinePrecision] * -0.5), $MachinePrecision], N[(im$95$m * N[(im$95$m * N[(im$95$m * N[(N[Cos[re], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 2.7000000000000002

   1. Initial program 41.2%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(im \cdot \cos re\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{im \cdot \cos re} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(im \cdot \cos re\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \color{blue}{\cos re}\right)\right) \]

      5. cos-lowering-cos.f64 65.0%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \mathsf{cos.f64}\left(re\right)\right)\right) \]

   5. Simplified 65.0%

      \[\leadsto \color{blue}{0 - im \cdot \cos re} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(im \cdot \cos re\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\cos re \cdot im\right) \]

      3. distribute-lft-neg-in N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{neg}\left(\cos re\right)\right), \color{blue}{im}\right) \]

      5. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{neg.f64}\left(\cos re\right), im\right) \]

      6. cos-lowering-cos.f64 65.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{cos.f64}\left(re\right)\right), im\right) \]

   7. Applied egg-rr 65.0%

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

   if 2.7000000000000002 < im < 1.09999999999999996e103

   1. Initial program 99.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + \color{blue}{\left(\mathsf{neg}\left(e^{im}\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(e^{im}\right)\right) + \color{blue}{e^{\mathsf{neg}\left(im\right)}}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \left(\mathsf{neg}\left(e^{im}\right)\right) \cdot \frac{1}{2} + \color{blue}{e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\mathsf{neg}\left(e^{im}\right)\right) + \color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2} \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \left(\mathsf{neg}\left(e^{im}\right)\right)\right), \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left(e^{im}\right)\right) \cdot \frac{1}{2}\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      7. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(e^{im} \cdot \frac{1}{2}\right)\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{+.f64}\left(\left(e^{im} \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\left(e^{im} \cdot \frac{-1}{2}\right), \left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(e^{im}\right), \frac{-1}{2}\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      11. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(e^{\color{blue}{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      12. exp-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(\frac{1}{e^{im}} \cdot \frac{1}{2}\right)\right) \]

      13. associate-*l/ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(\frac{1 \cdot \frac{1}{2}}{\color{blue}{e^{im}}}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(\frac{\frac{1}{2}}{e^{\color{blue}{im}}}\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{im}\right)}\right)\right) \]

      16. exp-lowering-exp.f64 58.1%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right)\right) \]

   5. Simplified 58.1%

      \[\leadsto \color{blue}{e^{im} \cdot -0.5 + \frac{0.5}{e^{im}}} \]

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(1 + im \cdot \left(1 + \frac{1}{2} \cdot im\right)\right)}\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \color{blue}{\left(im \cdot \left(1 + \frac{1}{2} \cdot im\right)\right)}\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(1 + \frac{1}{2} \cdot im\right)}\right)\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot im\right)}\right)\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 56.0%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 56.0%

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

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot e^{im}} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left(e^{im}\right)}\right) \]

       2. exp-lowering-exp.f64 56.0%

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{exp.f64}\left(im\right)\right) \]

   11. Simplified 56.0%

       \[\leadsto \color{blue}{-0.5 \cdot e^{im}} \]

   if 1.09999999999999996e103 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \color{blue}{\left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{3} \cdot {im}^{2} - 2\right)}\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left(\frac{-1}{3} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left(\frac{-1}{3} \cdot {im}^{2} + -2\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left(-2 + \color{blue}{\frac{-1}{3} \cdot {im}^{2}}\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \color{blue}{\left(\frac{-1}{3} \cdot {im}^{2}\right)}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \left({im}^{2} \cdot \color{blue}{\frac{-1}{3}}\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \left(\left(im \cdot im\right) \cdot \frac{-1}{3}\right)\right)\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \left(im \cdot \color{blue}{\left(im \cdot \frac{-1}{3}\right)}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \frac{-1}{3}\right)}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{3}}\right)\right)\right)\right)\right) \]

   5. Simplified 100.0%

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

   6. Taylor expanded in im around inf

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

      1. *-commutative N/A

         \[\leadsto \frac{-1}{6} \cdot \left(\cos re \cdot \color{blue}{{im}^{3}}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{6} \cdot \cos re\right) \cdot \color{blue}{{im}^{3}} \]

      3. unpow3 N/A

         \[\leadsto \left(\frac{-1}{6} \cdot \cos re\right) \cdot \left(\left(im \cdot im\right) \cdot \color{blue}{im}\right) \]

      4. unpow2 N/A

         \[\leadsto \left(\frac{-1}{6} \cdot \cos re\right) \cdot \left({im}^{2} \cdot im\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-1}{6} \cdot \cos re\right) \cdot {im}^{2}\right) \cdot \color{blue}{im} \]

      6. *-commutative N/A

         \[\leadsto im \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot \cos re\right) \cdot {im}^{2}\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(\left(\frac{-1}{6} \cdot \cos re\right) \cdot {im}^{2}\right)}\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left(\left(\frac{-1}{6} \cdot \cos re\right) \cdot \left(im \cdot \color{blue}{im}\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left(\left(\left(\frac{-1}{6} \cdot \cos re\right) \cdot im\right) \cdot \color{blue}{im}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(\left(\frac{-1}{6} \cdot \cos re\right) \cdot im\right), \color{blue}{im}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{6} \cdot \cos re\right), im\right), im\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\cos re \cdot \frac{-1}{6}\right), im\right), im\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\cos re, \frac{-1}{6}\right), im\right), im\right)\right) \]

      14. cos-lowering-cos.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \frac{-1}{6}\right), im\right), im\right)\right) \]

   8. Simplified 100.0%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.7:\\ \;\;\;\;0 - im \cdot \cos re\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+103}:\\ \;\;\;\;e^{im} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(im \cdot \left(im \cdot \left(\cos re \cdot -0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 86.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 3:\\ \;\;\;\;0 - im\_m \cdot \cos re\\ \mathbf{else}:\\ \;\;\;\;e^{im\_m} \cdot -0.5\\ \end{array} \end{array} \]

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 3.0) (- 0.0 (* im_m (cos re))) (* (exp im_m) -0.5))))

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 <= 3.0) {
		tmp = 0.0 - (im_m * cos(re));
	} else {
		tmp = exp(im_m) * -0.5;
	}
	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 <= 3.0d0) then
        tmp = 0.0d0 - (im_m * cos(re))
    else
        tmp = exp(im_m) * (-0.5d0)
    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 <= 3.0) {
		tmp = 0.0 - (im_m * Math.cos(re));
	} else {
		tmp = Math.exp(im_m) * -0.5;
	}
	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 <= 3.0:
		tmp = 0.0 - (im_m * math.cos(re))
	else:
		tmp = math.exp(im_m) * -0.5
	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 <= 3.0)
		tmp = Float64(0.0 - Float64(im_m * cos(re)));
	else
		tmp = Float64(exp(im_m) * -0.5);
	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 <= 3.0)
		tmp = 0.0 - (im_m * cos(re));
	else
		tmp = exp(im_m) * -0.5;
	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, 3.0], N[(0.0 - N[(im$95$m * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[im$95$m], $MachinePrecision] * -0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 3

   1. Initial program 41.2%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(im \cdot \cos re\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{im \cdot \cos re} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(im \cdot \cos re\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \color{blue}{\cos re}\right)\right) \]

      5. cos-lowering-cos.f64 65.0%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \mathsf{cos.f64}\left(re\right)\right)\right) \]

   5. Simplified 65.0%

      \[\leadsto \color{blue}{0 - im \cdot \cos re} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(im \cdot \cos re\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\cos re \cdot im\right) \]

      3. distribute-lft-neg-in N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{neg}\left(\cos re\right)\right), \color{blue}{im}\right) \]

      5. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{neg.f64}\left(\cos re\right), im\right) \]

      6. cos-lowering-cos.f64 65.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{cos.f64}\left(re\right)\right), im\right) \]

   7. Applied egg-rr 65.0%

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

   if 3 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + \color{blue}{\left(\mathsf{neg}\left(e^{im}\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(e^{im}\right)\right) + \color{blue}{e^{\mathsf{neg}\left(im\right)}}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \left(\mathsf{neg}\left(e^{im}\right)\right) \cdot \frac{1}{2} + \color{blue}{e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\mathsf{neg}\left(e^{im}\right)\right) + \color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2} \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \left(\mathsf{neg}\left(e^{im}\right)\right)\right), \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left(e^{im}\right)\right) \cdot \frac{1}{2}\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      7. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(e^{im} \cdot \frac{1}{2}\right)\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{+.f64}\left(\left(e^{im} \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\left(e^{im} \cdot \frac{-1}{2}\right), \left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(e^{im}\right), \frac{-1}{2}\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      11. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(e^{\color{blue}{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      12. exp-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(\frac{1}{e^{im}} \cdot \frac{1}{2}\right)\right) \]

      13. associate-*l/ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(\frac{1 \cdot \frac{1}{2}}{\color{blue}{e^{im}}}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(\frac{\frac{1}{2}}{e^{\color{blue}{im}}}\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{im}\right)}\right)\right) \]

      16. exp-lowering-exp.f64 69.8%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right)\right) \]

   5. Simplified 69.8%

      \[\leadsto \color{blue}{e^{im} \cdot -0.5 + \frac{0.5}{e^{im}}} \]

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(1 + im \cdot \left(1 + \frac{1}{2} \cdot im\right)\right)}\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \color{blue}{\left(im \cdot \left(1 + \frac{1}{2} \cdot im\right)\right)}\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(1 + \frac{1}{2} \cdot im\right)}\right)\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot im\right)}\right)\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 68.8%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 68.8%

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

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot e^{im}} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left(e^{im}\right)}\right) \]

       2. exp-lowering-exp.f64 68.8%

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{exp.f64}\left(im\right)\right) \]

   11. Simplified 68.8%

       \[\leadsto \color{blue}{-0.5 \cdot e^{im}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3:\\ \;\;\;\;0 - im \cdot \cos re\\ \mathbf{else}:\\ \;\;\;\;e^{im} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 1.65:\\ \;\;\;\;im\_m \cdot \left(-1 + -0.16666666666666666 \cdot \left(im\_m \cdot im\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{im\_m} \cdot -0.5\\ \end{array} \end{array} \]

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 1.65)
    (* im_m (+ -1.0 (* -0.16666666666666666 (* im_m im_m))))
    (* (exp im_m) -0.5))))

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 <= 1.65) {
		tmp = im_m * (-1.0 + (-0.16666666666666666 * (im_m * im_m)));
	} else {
		tmp = exp(im_m) * -0.5;
	}
	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 <= 1.65d0) then
        tmp = im_m * ((-1.0d0) + ((-0.16666666666666666d0) * (im_m * im_m)))
    else
        tmp = exp(im_m) * (-0.5d0)
    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 <= 1.65) {
		tmp = im_m * (-1.0 + (-0.16666666666666666 * (im_m * im_m)));
	} else {
		tmp = Math.exp(im_m) * -0.5;
	}
	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 <= 1.65:
		tmp = im_m * (-1.0 + (-0.16666666666666666 * (im_m * im_m)))
	else:
		tmp = math.exp(im_m) * -0.5
	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 <= 1.65)
		tmp = Float64(im_m * Float64(-1.0 + Float64(-0.16666666666666666 * Float64(im_m * im_m))));
	else
		tmp = Float64(exp(im_m) * -0.5);
	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 <= 1.65)
		tmp = im_m * (-1.0 + (-0.16666666666666666 * (im_m * im_m)));
	else
		tmp = exp(im_m) * -0.5;
	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, 1.65], N[(im$95$m * N[(-1.0 + N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[im$95$m], $MachinePrecision] * -0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 1.6499999999999999

   1. Initial program 41.2%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + \color{blue}{\left(\mathsf{neg}\left(e^{im}\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(e^{im}\right)\right) + \color{blue}{e^{\mathsf{neg}\left(im\right)}}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \left(\mathsf{neg}\left(e^{im}\right)\right) \cdot \frac{1}{2} + \color{blue}{e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\mathsf{neg}\left(e^{im}\right)\right) + \color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2} \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \left(\mathsf{neg}\left(e^{im}\right)\right)\right), \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left(e^{im}\right)\right) \cdot \frac{1}{2}\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      7. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(e^{im} \cdot \frac{1}{2}\right)\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{+.f64}\left(\left(e^{im} \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\left(e^{im} \cdot \frac{-1}{2}\right), \left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(e^{im}\right), \frac{-1}{2}\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      11. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(e^{\color{blue}{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      12. exp-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(\frac{1}{e^{im}} \cdot \frac{1}{2}\right)\right) \]

      13. associate-*l/ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(\frac{1 \cdot \frac{1}{2}}{\color{blue}{e^{im}}}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(\frac{\frac{1}{2}}{e^{\color{blue}{im}}}\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{im}\right)}\right)\right) \]

      16. exp-lowering-exp.f64 30.6%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right)\right) \]

   5. Simplified 30.6%

      \[\leadsto \color{blue}{e^{im} \cdot -0.5 + \frac{0.5}{e^{im}}} \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(\frac{-1}{6} \cdot {im}^{2} - 1\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2} - 1\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left(\frac{-1}{6} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left(\frac{-1}{6} \cdot {im}^{2} + -1\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left(-1 + \color{blue}{\frac{-1}{6} \cdot {im}^{2}}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{-1}{6}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{-1}{6}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 56.0%

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   8. Simplified 56.0%

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

   if 1.6499999999999999 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + \color{blue}{\left(\mathsf{neg}\left(e^{im}\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(e^{im}\right)\right) + \color{blue}{e^{\mathsf{neg}\left(im\right)}}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \left(\mathsf{neg}\left(e^{im}\right)\right) \cdot \frac{1}{2} + \color{blue}{e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\mathsf{neg}\left(e^{im}\right)\right) + \color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2} \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \left(\mathsf{neg}\left(e^{im}\right)\right)\right), \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left(e^{im}\right)\right) \cdot \frac{1}{2}\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      7. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(e^{im} \cdot \frac{1}{2}\right)\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{+.f64}\left(\left(e^{im} \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\left(e^{im} \cdot \frac{-1}{2}\right), \left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(e^{im}\right), \frac{-1}{2}\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      11. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(e^{\color{blue}{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      12. exp-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(\frac{1}{e^{im}} \cdot \frac{1}{2}\right)\right) \]

      13. associate-*l/ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(\frac{1 \cdot \frac{1}{2}}{\color{blue}{e^{im}}}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(\frac{\frac{1}{2}}{e^{\color{blue}{im}}}\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{im}\right)}\right)\right) \]

      16. exp-lowering-exp.f64 69.8%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right)\right) \]

   5. Simplified 69.8%

      \[\leadsto \color{blue}{e^{im} \cdot -0.5 + \frac{0.5}{e^{im}}} \]

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(1 + im \cdot \left(1 + \frac{1}{2} \cdot im\right)\right)}\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \color{blue}{\left(im \cdot \left(1 + \frac{1}{2} \cdot im\right)\right)}\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(1 + \frac{1}{2} \cdot im\right)}\right)\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot im\right)}\right)\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 68.8%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 68.8%

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

   9. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot e^{im}} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left(e^{im}\right)}\right) \]

       2. exp-lowering-exp.f64 68.8%

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{exp.f64}\left(im\right)\right) \]

   11. Simplified 68.8%

       \[\leadsto \color{blue}{-0.5 \cdot e^{im}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.65:\\ \;\;\;\;im \cdot \left(-1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{im} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 60.5% accurate, 6.1× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := -1 + \left(im\_m \cdot im\_m\right) \cdot \left(-0.16666666666666666 + im\_m \cdot \left(im\_m \cdot \left(-0.008333333333333333 + im\_m \cdot \left(im\_m \cdot -0.0001984126984126984\right)\right)\right)\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 3.05 \cdot 10^{-61}:\\ \;\;\;\;0 - im\_m\\ \mathbf{elif}\;im\_m \leq 4.8 \cdot 10^{+64}:\\ \;\;\;\;im\_m \cdot \left(t\_0 \cdot \left(1 + \left(re \cdot re\right) \cdot \left(-0.5 + \left(re \cdot re\right) \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.001388888888888889\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot t\_0\\ \end{array} \end{array} \end{array} \]

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
         (+
          -1.0
          (*
           (* im_m im_m)
           (+
            -0.16666666666666666
            (*
             im_m
             (*
              im_m
              (+
               -0.008333333333333333
               (* im_m (* im_m -0.0001984126984126984))))))))))
   (*
    im_s
    (if (<= im_m 3.05e-61)
      (- 0.0 im_m)
      (if (<= im_m 4.8e+64)
        (*
         im_m
         (*
          t_0
          (+
           1.0
           (*
            (* re re)
            (+
             -0.5
             (*
              (* re re)
              (+
               0.041666666666666664
               (* (* re re) -0.001388888888888889))))))))
        (* im_m t_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 = -1.0 + ((im_m * im_m) * (-0.16666666666666666 + (im_m * (im_m * (-0.008333333333333333 + (im_m * (im_m * -0.0001984126984126984)))))));
	double tmp;
	if (im_m <= 3.05e-61) {
		tmp = 0.0 - im_m;
	} else if (im_m <= 4.8e+64) {
		tmp = im_m * (t_0 * (1.0 + ((re * re) * (-0.5 + ((re * re) * (0.041666666666666664 + ((re * re) * -0.001388888888888889)))))));
	} else {
		tmp = im_m * t_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) :: tmp
    t_0 = (-1.0d0) + ((im_m * im_m) * ((-0.16666666666666666d0) + (im_m * (im_m * ((-0.008333333333333333d0) + (im_m * (im_m * (-0.0001984126984126984d0))))))))
    if (im_m <= 3.05d-61) then
        tmp = 0.0d0 - im_m
    else if (im_m <= 4.8d+64) then
        tmp = im_m * (t_0 * (1.0d0 + ((re * re) * ((-0.5d0) + ((re * re) * (0.041666666666666664d0 + ((re * re) * (-0.001388888888888889d0))))))))
    else
        tmp = im_m * t_0
    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 = -1.0 + ((im_m * im_m) * (-0.16666666666666666 + (im_m * (im_m * (-0.008333333333333333 + (im_m * (im_m * -0.0001984126984126984)))))));
	double tmp;
	if (im_m <= 3.05e-61) {
		tmp = 0.0 - im_m;
	} else if (im_m <= 4.8e+64) {
		tmp = im_m * (t_0 * (1.0 + ((re * re) * (-0.5 + ((re * re) * (0.041666666666666664 + ((re * re) * -0.001388888888888889)))))));
	} else {
		tmp = im_m * t_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 = -1.0 + ((im_m * im_m) * (-0.16666666666666666 + (im_m * (im_m * (-0.008333333333333333 + (im_m * (im_m * -0.0001984126984126984)))))))
	tmp = 0
	if im_m <= 3.05e-61:
		tmp = 0.0 - im_m
	elif im_m <= 4.8e+64:
		tmp = im_m * (t_0 * (1.0 + ((re * re) * (-0.5 + ((re * re) * (0.041666666666666664 + ((re * re) * -0.001388888888888889)))))))
	else:
		tmp = im_m * t_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(-1.0 + Float64(Float64(im_m * im_m) * Float64(-0.16666666666666666 + Float64(im_m * Float64(im_m * Float64(-0.008333333333333333 + Float64(im_m * Float64(im_m * -0.0001984126984126984))))))))
	tmp = 0.0
	if (im_m <= 3.05e-61)
		tmp = Float64(0.0 - im_m);
	elseif (im_m <= 4.8e+64)
		tmp = Float64(im_m * Float64(t_0 * Float64(1.0 + Float64(Float64(re * re) * Float64(-0.5 + Float64(Float64(re * re) * Float64(0.041666666666666664 + Float64(Float64(re * re) * -0.001388888888888889))))))));
	else
		tmp = Float64(im_m * t_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 = -1.0 + ((im_m * im_m) * (-0.16666666666666666 + (im_m * (im_m * (-0.008333333333333333 + (im_m * (im_m * -0.0001984126984126984)))))));
	tmp = 0.0;
	if (im_m <= 3.05e-61)
		tmp = 0.0 - im_m;
	elseif (im_m <= 4.8e+64)
		tmp = im_m * (t_0 * (1.0 + ((re * re) * (-0.5 + ((re * re) * (0.041666666666666664 + ((re * re) * -0.001388888888888889)))))));
	else
		tmp = im_m * t_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[(-1.0 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(-0.16666666666666666 + N[(im$95$m * N[(im$95$m * N[(-0.008333333333333333 + N[(im$95$m * N[(im$95$m * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[im$95$m, 3.05e-61], N[(0.0 - im$95$m), $MachinePrecision], If[LessEqual[im$95$m, 4.8e+64], N[(im$95$m * N[(t$95$0 * N[(1.0 + N[(N[(re * re), $MachinePrecision] * N[(-0.5 + N[(N[(re * re), $MachinePrecision] * N[(0.041666666666666664 + N[(N[(re * re), $MachinePrecision] * -0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im$95$m * t$95$0), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if im < 3.05e-61

   1. Initial program 42.4%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(im \cdot \cos re\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{im \cdot \cos re} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(im \cdot \cos re\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \color{blue}{\cos re}\right)\right) \]

      5. cos-lowering-cos.f64 63.6%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \mathsf{cos.f64}\left(re\right)\right)\right) \]

   5. Simplified 63.6%

      \[\leadsto \color{blue}{0 - im \cdot \cos re} \]

   6. Taylor expanded in re around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(im\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 35.8%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{im}\right) \]

   8. Simplified 35.8%

      \[\leadsto \color{blue}{0 - im} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(im\right) \]

      2. neg-lowering-neg.f64 35.8%

         \[\leadsto \mathsf{neg.f64}\left(im\right) \]

   10. Applied egg-rr 35.8%

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

   if 3.05e-61 < im < 4.79999999999999999e64

   1. Initial program 76.7%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \cos re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \cos re + {im}^{2} \cdot \left(\frac{-1}{120} \cdot \cos re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(-1 \cdot \cos re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \cos re + {im}^{2} \cdot \left(\frac{-1}{120} \cdot \cos re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)\right)}\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left(-1 \cdot \cos re + \left(\left(\frac{-1}{6} \cdot \cos re\right) \cdot {im}^{2} + \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{120} \cdot \cos re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \cos re\right)\right)\right) \cdot {im}^{2}}\right)\right)\right) \]

      3. associate-+r+ N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left(\left(-1 \cdot \cos re + \left(\frac{-1}{6} \cdot \cos re\right) \cdot {im}^{2}\right) + \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{120} \cdot \cos re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \cos re\right)\right)\right) \cdot {im}^{2}}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left(\left(-1 \cdot \cos re + \left(\frac{-1}{6} \cdot \cos re\right) \cdot {im}^{2}\right) + {im}^{2} \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{120} \cdot \cos re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)}\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot \cos re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \cos re\right)\right)\right) + \color{blue}{\left(-1 \cdot \cos re + \left(\frac{-1}{6} \cdot \cos re\right) \cdot {im}^{2}\right)}\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot \cos re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \cos re\right)\right)\right) + \left(\left(\frac{-1}{6} \cdot \cos re\right) \cdot {im}^{2} + \color{blue}{-1 \cdot \cos re}\right)\right)\right) \]

   5. Simplified 50.2%

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

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) - 1\right)}\right)\right) \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) + -1\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(-1 + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right)}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right)\right)}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right)}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{{im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)} - \frac{1}{6}\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{{im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)} - \frac{1}{6}\right)\right)\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) + \frac{-1}{6}\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{6} + \color{blue}{{im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)}\right)\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)\right)}\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{-1}{5040} \cdot {im}^{2}} - \frac{1}{120}\right)\right)\right)\right)\right)\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)\right)}\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)\right)}\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      16. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{5040} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{120}\right)\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{5040} \cdot {im}^{2} + \frac{-1}{120}\right)\right)\right)\right)\right)\right)\right)\right) \]

      18. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{120} + \color{blue}{\frac{-1}{5040} \cdot {im}^{2}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      19. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{120}, \color{blue}{\left(\frac{-1}{5040} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{120}, \left({im}^{2} \cdot \color{blue}{\frac{-1}{5040}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      21. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{120}, \left(\left(im \cdot im\right) \cdot \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      22. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{120}, \left(im \cdot \color{blue}{\left(im \cdot \frac{-1}{5040}\right)}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      23. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \frac{-1}{5040}\right)}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      24. *-lowering-*.f64 50.2%

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{5040}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 50.2%

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

   9. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\color{blue}{\left(1 + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)}, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{-1}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({re}^{2}\right), \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(re \cdot re\right), \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       5. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right) + \frac{-1}{2}\right)\right)\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       7. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{-1}{2} + {re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left({re}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({re}^{2}\right), \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(re \cdot re\right), \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{1}{24} + \frac{-1}{720} \cdot {re}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       12. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(\frac{-1}{720} \cdot {re}^{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({re}^{2} \cdot \frac{-1}{720}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({re}^{2}\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       15. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       16. *-lowering-*.f64 55.6%

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{720}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   11. Simplified 55.6%

       \[\leadsto im \cdot \left(\color{blue}{\left(1 + \left(re \cdot re\right) \cdot \left(-0.5 + \left(re \cdot re\right) \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.001388888888888889\right)\right)\right)} \cdot \left(-1 + \left(im \cdot im\right) \cdot \left(-0.16666666666666666 + im \cdot \left(im \cdot \left(-0.008333333333333333 + im \cdot \left(im \cdot -0.0001984126984126984\right)\right)\right)\right)\right)\right) \]

   if 4.79999999999999999e64 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + \color{blue}{\left(\mathsf{neg}\left(e^{im}\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(e^{im}\right)\right) + \color{blue}{e^{\mathsf{neg}\left(im\right)}}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \left(\mathsf{neg}\left(e^{im}\right)\right) \cdot \frac{1}{2} + \color{blue}{e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\mathsf{neg}\left(e^{im}\right)\right) + \color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2} \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \left(\mathsf{neg}\left(e^{im}\right)\right)\right), \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left(e^{im}\right)\right) \cdot \frac{1}{2}\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      7. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(e^{im} \cdot \frac{1}{2}\right)\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{+.f64}\left(\left(e^{im} \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\left(e^{im} \cdot \frac{-1}{2}\right), \left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(e^{im}\right), \frac{-1}{2}\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      11. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(e^{\color{blue}{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      12. exp-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(\frac{1}{e^{im}} \cdot \frac{1}{2}\right)\right) \]

      13. associate-*l/ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(\frac{1 \cdot \frac{1}{2}}{\color{blue}{e^{im}}}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(\frac{\frac{1}{2}}{e^{\color{blue}{im}}}\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{im}\right)}\right)\right) \]

      16. exp-lowering-exp.f64 82.9%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right)\right) \]

   5. Simplified 82.9%

      \[\leadsto \color{blue}{e^{im} \cdot -0.5 + \frac{0.5}{e^{im}}} \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) - 1\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) - 1\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) + -1\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left(-1 + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right)}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{{im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)} - \frac{1}{6}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{{im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)} - \frac{1}{6}\right)\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) + \frac{-1}{6}\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{6} + \color{blue}{{im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)}\right)\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)\right)}\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{-1}{5040} \cdot {im}^{2}} - \frac{1}{120}\right)\right)\right)\right)\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)\right)}\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)\right)}\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)}\right)\right)\right)\right)\right)\right) \]

      17. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{5040} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{120}\right)\right)}\right)\right)\right)\right)\right)\right)\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{5040} \cdot {im}^{2} + \frac{-1}{120}\right)\right)\right)\right)\right)\right)\right) \]

      19. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{120} + \color{blue}{\frac{-1}{5040} \cdot {im}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

      20. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{120}, \color{blue}{\left(\frac{-1}{5040} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 82.9%

      \[\leadsto \color{blue}{im \cdot \left(-1 + \left(im \cdot im\right) \cdot \left(-0.16666666666666666 + im \cdot \left(im \cdot \left(-0.008333333333333333 + im \cdot \left(im \cdot -0.0001984126984126984\right)\right)\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.05 \cdot 10^{-61}:\\ \;\;\;\;0 - im\\ \mathbf{elif}\;im \leq 4.8 \cdot 10^{+64}:\\ \;\;\;\;im \cdot \left(\left(-1 + \left(im \cdot im\right) \cdot \left(-0.16666666666666666 + im \cdot \left(im \cdot \left(-0.008333333333333333 + im \cdot \left(im \cdot -0.0001984126984126984\right)\right)\right)\right)\right) \cdot \left(1 + \left(re \cdot re\right) \cdot \left(-0.5 + \left(re \cdot re\right) \cdot \left(0.041666666666666664 + \left(re \cdot re\right) \cdot -0.001388888888888889\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-1 + \left(im \cdot im\right) \cdot \left(-0.16666666666666666 + im \cdot \left(im \cdot \left(-0.008333333333333333 + im \cdot \left(im \cdot -0.0001984126984126984\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 60.2% accurate, 7.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := -1 + \left(im\_m \cdot im\_m\right) \cdot \left(-0.16666666666666666 + im\_m \cdot \left(im\_m \cdot \left(-0.008333333333333333 + im\_m \cdot \left(im\_m \cdot -0.0001984126984126984\right)\right)\right)\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 3.05 \cdot 10^{-61}:\\ \;\;\;\;0 - im\_m\\ \mathbf{elif}\;im\_m \leq 3 \cdot 10^{+65}:\\ \;\;\;\;im\_m \cdot \left(t\_0 \cdot \left(1 + re \cdot \left(re \cdot -0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot t\_0\\ \end{array} \end{array} \end{array} \]

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
         (+
          -1.0
          (*
           (* im_m im_m)
           (+
            -0.16666666666666666
            (*
             im_m
             (*
              im_m
              (+
               -0.008333333333333333
               (* im_m (* im_m -0.0001984126984126984))))))))))
   (*
    im_s
    (if (<= im_m 3.05e-61)
      (- 0.0 im_m)
      (if (<= im_m 3e+65)
        (* im_m (* t_0 (+ 1.0 (* re (* re -0.5)))))
        (* im_m t_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 = -1.0 + ((im_m * im_m) * (-0.16666666666666666 + (im_m * (im_m * (-0.008333333333333333 + (im_m * (im_m * -0.0001984126984126984)))))));
	double tmp;
	if (im_m <= 3.05e-61) {
		tmp = 0.0 - im_m;
	} else if (im_m <= 3e+65) {
		tmp = im_m * (t_0 * (1.0 + (re * (re * -0.5))));
	} else {
		tmp = im_m * t_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) :: tmp
    t_0 = (-1.0d0) + ((im_m * im_m) * ((-0.16666666666666666d0) + (im_m * (im_m * ((-0.008333333333333333d0) + (im_m * (im_m * (-0.0001984126984126984d0))))))))
    if (im_m <= 3.05d-61) then
        tmp = 0.0d0 - im_m
    else if (im_m <= 3d+65) then
        tmp = im_m * (t_0 * (1.0d0 + (re * (re * (-0.5d0)))))
    else
        tmp = im_m * t_0
    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 = -1.0 + ((im_m * im_m) * (-0.16666666666666666 + (im_m * (im_m * (-0.008333333333333333 + (im_m * (im_m * -0.0001984126984126984)))))));
	double tmp;
	if (im_m <= 3.05e-61) {
		tmp = 0.0 - im_m;
	} else if (im_m <= 3e+65) {
		tmp = im_m * (t_0 * (1.0 + (re * (re * -0.5))));
	} else {
		tmp = im_m * t_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 = -1.0 + ((im_m * im_m) * (-0.16666666666666666 + (im_m * (im_m * (-0.008333333333333333 + (im_m * (im_m * -0.0001984126984126984)))))))
	tmp = 0
	if im_m <= 3.05e-61:
		tmp = 0.0 - im_m
	elif im_m <= 3e+65:
		tmp = im_m * (t_0 * (1.0 + (re * (re * -0.5))))
	else:
		tmp = im_m * t_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(-1.0 + Float64(Float64(im_m * im_m) * Float64(-0.16666666666666666 + Float64(im_m * Float64(im_m * Float64(-0.008333333333333333 + Float64(im_m * Float64(im_m * -0.0001984126984126984))))))))
	tmp = 0.0
	if (im_m <= 3.05e-61)
		tmp = Float64(0.0 - im_m);
	elseif (im_m <= 3e+65)
		tmp = Float64(im_m * Float64(t_0 * Float64(1.0 + Float64(re * Float64(re * -0.5)))));
	else
		tmp = Float64(im_m * t_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 = -1.0 + ((im_m * im_m) * (-0.16666666666666666 + (im_m * (im_m * (-0.008333333333333333 + (im_m * (im_m * -0.0001984126984126984)))))));
	tmp = 0.0;
	if (im_m <= 3.05e-61)
		tmp = 0.0 - im_m;
	elseif (im_m <= 3e+65)
		tmp = im_m * (t_0 * (1.0 + (re * (re * -0.5))));
	else
		tmp = im_m * t_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[(-1.0 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(-0.16666666666666666 + N[(im$95$m * N[(im$95$m * N[(-0.008333333333333333 + N[(im$95$m * N[(im$95$m * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[im$95$m, 3.05e-61], N[(0.0 - im$95$m), $MachinePrecision], If[LessEqual[im$95$m, 3e+65], N[(im$95$m * N[(t$95$0 * N[(1.0 + N[(re * N[(re * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im$95$m * t$95$0), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if im < 3.05e-61

   1. Initial program 42.4%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(im \cdot \cos re\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{im \cdot \cos re} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(im \cdot \cos re\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \color{blue}{\cos re}\right)\right) \]

      5. cos-lowering-cos.f64 63.6%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \mathsf{cos.f64}\left(re\right)\right)\right) \]

   5. Simplified 63.6%

      \[\leadsto \color{blue}{0 - im \cdot \cos re} \]

   6. Taylor expanded in re around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(im\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 35.8%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{im}\right) \]

   8. Simplified 35.8%

      \[\leadsto \color{blue}{0 - im} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(im\right) \]

      2. neg-lowering-neg.f64 35.8%

         \[\leadsto \mathsf{neg.f64}\left(im\right) \]

   10. Applied egg-rr 35.8%

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

   if 3.05e-61 < im < 3.0000000000000002e65

   1. Initial program 76.7%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \cos re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \cos re + {im}^{2} \cdot \left(\frac{-1}{120} \cdot \cos re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(-1 \cdot \cos re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \cos re + {im}^{2} \cdot \left(\frac{-1}{120} \cdot \cos re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)\right)}\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left(-1 \cdot \cos re + \left(\left(\frac{-1}{6} \cdot \cos re\right) \cdot {im}^{2} + \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{120} \cdot \cos re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \cos re\right)\right)\right) \cdot {im}^{2}}\right)\right)\right) \]

      3. associate-+r+ N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left(\left(-1 \cdot \cos re + \left(\frac{-1}{6} \cdot \cos re\right) \cdot {im}^{2}\right) + \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{120} \cdot \cos re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \cos re\right)\right)\right) \cdot {im}^{2}}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left(\left(-1 \cdot \cos re + \left(\frac{-1}{6} \cdot \cos re\right) \cdot {im}^{2}\right) + {im}^{2} \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{120} \cdot \cos re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)}\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot \cos re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \cos re\right)\right)\right) + \color{blue}{\left(-1 \cdot \cos re + \left(\frac{-1}{6} \cdot \cos re\right) \cdot {im}^{2}\right)}\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot \cos re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \cos re\right)\right)\right) + \left(\left(\frac{-1}{6} \cdot \cos re\right) \cdot {im}^{2} + \color{blue}{-1 \cdot \cos re}\right)\right)\right) \]

   5. Simplified 50.2%

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

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) - 1\right)}\right)\right) \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) + -1\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(-1 + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right)}\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right)\right)}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right)}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{{im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)} - \frac{1}{6}\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{{im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)} - \frac{1}{6}\right)\right)\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) + \frac{-1}{6}\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{6} + \color{blue}{{im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)}\right)\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)\right)}\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{-1}{5040} \cdot {im}^{2}} - \frac{1}{120}\right)\right)\right)\right)\right)\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)\right)}\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)\right)}\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      16. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{5040} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{120}\right)\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{5040} \cdot {im}^{2} + \frac{-1}{120}\right)\right)\right)\right)\right)\right)\right)\right) \]

      18. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{120} + \color{blue}{\frac{-1}{5040} \cdot {im}^{2}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      19. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{120}, \color{blue}{\left(\frac{-1}{5040} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{120}, \left({im}^{2} \cdot \color{blue}{\frac{-1}{5040}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      21. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{120}, \left(\left(im \cdot im\right) \cdot \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      22. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{120}, \left(im \cdot \color{blue}{\left(im \cdot \frac{-1}{5040}\right)}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      23. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \frac{-1}{5040}\right)}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      24. *-lowering-*.f64 50.2%

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{5040}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 50.2%

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

   9. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)}, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{2} \cdot {re}^{2}\right)\right), \mathsf{+.f64}\left(\color{blue}{-1}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({re}^{2} \cdot \frac{-1}{2}\right)\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\left(re \cdot re\right) \cdot \frac{-1}{2}\right)\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(re \cdot \left(re \cdot \frac{-1}{2}\right)\right)\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \left(re \cdot \frac{-1}{2}\right)\right)\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       6. *-lowering-*.f64 49.4%

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \frac{-1}{2}\right)\right)\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   11. Simplified 49.4%

       \[\leadsto im \cdot \left(\color{blue}{\left(1 + re \cdot \left(re \cdot -0.5\right)\right)} \cdot \left(-1 + \left(im \cdot im\right) \cdot \left(-0.16666666666666666 + im \cdot \left(im \cdot \left(-0.008333333333333333 + im \cdot \left(im \cdot -0.0001984126984126984\right)\right)\right)\right)\right)\right) \]

   if 3.0000000000000002e65 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + \color{blue}{\left(\mathsf{neg}\left(e^{im}\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(e^{im}\right)\right) + \color{blue}{e^{\mathsf{neg}\left(im\right)}}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \left(\mathsf{neg}\left(e^{im}\right)\right) \cdot \frac{1}{2} + \color{blue}{e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\mathsf{neg}\left(e^{im}\right)\right) + \color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2} \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \left(\mathsf{neg}\left(e^{im}\right)\right)\right), \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left(e^{im}\right)\right) \cdot \frac{1}{2}\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      7. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(e^{im} \cdot \frac{1}{2}\right)\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{+.f64}\left(\left(e^{im} \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\left(e^{im} \cdot \frac{-1}{2}\right), \left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(e^{im}\right), \frac{-1}{2}\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      11. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(e^{\color{blue}{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      12. exp-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(\frac{1}{e^{im}} \cdot \frac{1}{2}\right)\right) \]

      13. associate-*l/ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(\frac{1 \cdot \frac{1}{2}}{\color{blue}{e^{im}}}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(\frac{\frac{1}{2}}{e^{\color{blue}{im}}}\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{im}\right)}\right)\right) \]

      16. exp-lowering-exp.f64 82.9%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right)\right) \]

   5. Simplified 82.9%

      \[\leadsto \color{blue}{e^{im} \cdot -0.5 + \frac{0.5}{e^{im}}} \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) - 1\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) - 1\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) + -1\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left(-1 + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right)}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{{im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)} - \frac{1}{6}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{{im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)} - \frac{1}{6}\right)\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) + \frac{-1}{6}\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{6} + \color{blue}{{im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)}\right)\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)\right)}\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{-1}{5040} \cdot {im}^{2}} - \frac{1}{120}\right)\right)\right)\right)\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)\right)}\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)\right)}\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)}\right)\right)\right)\right)\right)\right) \]

      17. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{5040} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{120}\right)\right)}\right)\right)\right)\right)\right)\right)\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{5040} \cdot {im}^{2} + \frac{-1}{120}\right)\right)\right)\right)\right)\right)\right) \]

      19. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{120} + \color{blue}{\frac{-1}{5040} \cdot {im}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

      20. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{120}, \color{blue}{\left(\frac{-1}{5040} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 82.9%

      \[\leadsto \color{blue}{im \cdot \left(-1 + \left(im \cdot im\right) \cdot \left(-0.16666666666666666 + im \cdot \left(im \cdot \left(-0.008333333333333333 + im \cdot \left(im \cdot -0.0001984126984126984\right)\right)\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.05 \cdot 10^{-61}:\\ \;\;\;\;0 - im\\ \mathbf{elif}\;im \leq 3 \cdot 10^{+65}:\\ \;\;\;\;im \cdot \left(\left(-1 + \left(im \cdot im\right) \cdot \left(-0.16666666666666666 + im \cdot \left(im \cdot \left(-0.008333333333333333 + im \cdot \left(im \cdot -0.0001984126984126984\right)\right)\right)\right)\right) \cdot \left(1 + re \cdot \left(re \cdot -0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-1 + \left(im \cdot im\right) \cdot \left(-0.16666666666666666 + im \cdot \left(im \cdot \left(-0.008333333333333333 + im \cdot \left(im \cdot -0.0001984126984126984\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 60.6% accurate, 9.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := im\_m \cdot \left(-1 + \left(im\_m \cdot im\_m\right) \cdot \left(-0.16666666666666666 + im\_m \cdot \left(im\_m \cdot \left(-0.008333333333333333 + im\_m \cdot \left(im\_m \cdot -0.0001984126984126984\right)\right)\right)\right)\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 26:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im\_m \leq 3.4 \cdot 10^{+39}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(\left(re \cdot re\right) \cdot \left(im\_m \cdot \left(\left(re \cdot re\right) \cdot 0.001388888888888889 + -0.041666666666666664\right)\right) + im\_m \cdot 0.5\right) - im\_m\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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
         (*
          im_m
          (+
           -1.0
           (*
            (* im_m im_m)
            (+
             -0.16666666666666666
             (*
              im_m
              (*
               im_m
               (+
                -0.008333333333333333
                (* im_m (* im_m -0.0001984126984126984)))))))))))
   (*
    im_s
    (if (<= im_m 26.0)
      t_0
      (if (<= im_m 3.4e+39)
        (-
         (*
          (* re re)
          (+
           (*
            (* re re)
            (*
             im_m
             (+ (* (* re re) 0.001388888888888889) -0.041666666666666664)))
           (* im_m 0.5)))
         im_m)
        t_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 = im_m * (-1.0 + ((im_m * im_m) * (-0.16666666666666666 + (im_m * (im_m * (-0.008333333333333333 + (im_m * (im_m * -0.0001984126984126984))))))));
	double tmp;
	if (im_m <= 26.0) {
		tmp = t_0;
	} else if (im_m <= 3.4e+39) {
		tmp = ((re * re) * (((re * re) * (im_m * (((re * re) * 0.001388888888888889) + -0.041666666666666664))) + (im_m * 0.5))) - im_m;
	} else {
		tmp = t_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) :: tmp
    t_0 = im_m * ((-1.0d0) + ((im_m * im_m) * ((-0.16666666666666666d0) + (im_m * (im_m * ((-0.008333333333333333d0) + (im_m * (im_m * (-0.0001984126984126984d0)))))))))
    if (im_m <= 26.0d0) then
        tmp = t_0
    else if (im_m <= 3.4d+39) then
        tmp = ((re * re) * (((re * re) * (im_m * (((re * re) * 0.001388888888888889d0) + (-0.041666666666666664d0)))) + (im_m * 0.5d0))) - im_m
    else
        tmp = t_0
    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 = im_m * (-1.0 + ((im_m * im_m) * (-0.16666666666666666 + (im_m * (im_m * (-0.008333333333333333 + (im_m * (im_m * -0.0001984126984126984))))))));
	double tmp;
	if (im_m <= 26.0) {
		tmp = t_0;
	} else if (im_m <= 3.4e+39) {
		tmp = ((re * re) * (((re * re) * (im_m * (((re * re) * 0.001388888888888889) + -0.041666666666666664))) + (im_m * 0.5))) - im_m;
	} else {
		tmp = t_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 = im_m * (-1.0 + ((im_m * im_m) * (-0.16666666666666666 + (im_m * (im_m * (-0.008333333333333333 + (im_m * (im_m * -0.0001984126984126984))))))))
	tmp = 0
	if im_m <= 26.0:
		tmp = t_0
	elif im_m <= 3.4e+39:
		tmp = ((re * re) * (((re * re) * (im_m * (((re * re) * 0.001388888888888889) + -0.041666666666666664))) + (im_m * 0.5))) - im_m
	else:
		tmp = t_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(im_m * Float64(-1.0 + Float64(Float64(im_m * im_m) * Float64(-0.16666666666666666 + Float64(im_m * Float64(im_m * Float64(-0.008333333333333333 + Float64(im_m * Float64(im_m * -0.0001984126984126984)))))))))
	tmp = 0.0
	if (im_m <= 26.0)
		tmp = t_0;
	elseif (im_m <= 3.4e+39)
		tmp = Float64(Float64(Float64(re * re) * Float64(Float64(Float64(re * re) * Float64(im_m * Float64(Float64(Float64(re * re) * 0.001388888888888889) + -0.041666666666666664))) + Float64(im_m * 0.5))) - im_m);
	else
		tmp = t_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 = im_m * (-1.0 + ((im_m * im_m) * (-0.16666666666666666 + (im_m * (im_m * (-0.008333333333333333 + (im_m * (im_m * -0.0001984126984126984))))))));
	tmp = 0.0;
	if (im_m <= 26.0)
		tmp = t_0;
	elseif (im_m <= 3.4e+39)
		tmp = ((re * re) * (((re * re) * (im_m * (((re * re) * 0.001388888888888889) + -0.041666666666666664))) + (im_m * 0.5))) - im_m;
	else
		tmp = t_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[(im$95$m * N[(-1.0 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(-0.16666666666666666 + N[(im$95$m * N[(im$95$m * N[(-0.008333333333333333 + N[(im$95$m * N[(im$95$m * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[im$95$m, 26.0], t$95$0, If[LessEqual[im$95$m, 3.4e+39], N[(N[(N[(re * re), $MachinePrecision] * N[(N[(N[(re * re), $MachinePrecision] * N[(im$95$m * N[(N[(N[(re * re), $MachinePrecision] * 0.001388888888888889), $MachinePrecision] + -0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(im$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im$95$m), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 26 or 3.3999999999999999e39 < im

   1. Initial program 52.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + \color{blue}{\left(\mathsf{neg}\left(e^{im}\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(e^{im}\right)\right) + \color{blue}{e^{\mathsf{neg}\left(im\right)}}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \left(\mathsf{neg}\left(e^{im}\right)\right) \cdot \frac{1}{2} + \color{blue}{e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\mathsf{neg}\left(e^{im}\right)\right) + \color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2} \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \left(\mathsf{neg}\left(e^{im}\right)\right)\right), \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left(e^{im}\right)\right) \cdot \frac{1}{2}\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      7. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(e^{im} \cdot \frac{1}{2}\right)\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{+.f64}\left(\left(e^{im} \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\left(e^{im} \cdot \frac{-1}{2}\right), \left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(e^{im}\right), \frac{-1}{2}\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      11. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(e^{\color{blue}{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      12. exp-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(\frac{1}{e^{im}} \cdot \frac{1}{2}\right)\right) \]

      13. associate-*l/ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(\frac{1 \cdot \frac{1}{2}}{\color{blue}{e^{im}}}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(\frac{\frac{1}{2}}{e^{\color{blue}{im}}}\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{im}\right)}\right)\right) \]

      16. exp-lowering-exp.f64 40.3%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right)\right) \]

   5. Simplified 40.3%

      \[\leadsto \color{blue}{e^{im} \cdot -0.5 + \frac{0.5}{e^{im}}} \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) - 1\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) - 1\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) + -1\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left(-1 + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right)\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right)}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{{im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)} - \frac{1}{6}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{{im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)} - \frac{1}{6}\right)\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) + \frac{-1}{6}\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{6} + \color{blue}{{im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)}\right)\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)\right)}\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{-1}{5040} \cdot {im}^{2}} - \frac{1}{120}\right)\right)\right)\right)\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)\right)}\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)\right)}\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)}\right)\right)\right)\right)\right)\right) \]

      17. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{5040} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{120}\right)\right)}\right)\right)\right)\right)\right)\right)\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{5040} \cdot {im}^{2} + \frac{-1}{120}\right)\right)\right)\right)\right)\right)\right) \]

      19. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{120} + \color{blue}{\frac{-1}{5040} \cdot {im}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

      20. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{120}, \color{blue}{\left(\frac{-1}{5040} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 63.4%

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

   if 26 < im < 3.3999999999999999e39

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(im \cdot \cos re\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{im \cdot \cos re} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(im \cdot \cos re\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \color{blue}{\cos re}\right)\right) \]

      5. cos-lowering-cos.f64 3.8%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \mathsf{cos.f64}\left(re\right)\right)\right) \]

   5. Simplified 3.8%

      \[\leadsto \color{blue}{0 - im \cdot \cos re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{{re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{720} \cdot \left(im \cdot {re}^{2}\right) - \frac{1}{24} \cdot im\right) - \frac{-1}{2} \cdot im\right) - im} \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{720} \cdot \left(im \cdot {re}^{2}\right) - \frac{1}{24} \cdot im\right) - \frac{-1}{2} \cdot im\right)\right), \color{blue}{im}\right) \]

   8. Simplified 41.9%

      \[\leadsto \color{blue}{\left(re \cdot re\right) \cdot \left(\left(re \cdot re\right) \cdot \left(im \cdot \left(\left(re \cdot re\right) \cdot 0.001388888888888889 + -0.041666666666666664\right)\right) + im \cdot 0.5\right) - im} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 57.1% accurate, 13.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 0.00012:\\ \;\;\;\;im\_m \cdot \left(-1 + -0.16666666666666666 \cdot \left(im\_m \cdot im\_m\right)\right)\\ \mathbf{elif}\;im\_m \leq 4.2 \cdot 10^{+65}:\\ \;\;\;\;im\_m \cdot \left(-1 + 0.5 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(-1 + \left(im\_m \cdot im\_m\right) \cdot \left(-0.08333333333333333 + im\_m \cdot -0.14583333333333334\right)\right)\\ \end{array} \end{array} \]

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 0.00012)
    (* im_m (+ -1.0 (* -0.16666666666666666 (* im_m im_m))))
    (if (<= im_m 4.2e+65)
      (* im_m (+ -1.0 (* 0.5 (* re re))))
      (*
       im_m
       (+
        -1.0
        (*
         (* im_m im_m)
         (+ -0.08333333333333333 (* im_m -0.14583333333333334)))))))))

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 <= 0.00012) {
		tmp = im_m * (-1.0 + (-0.16666666666666666 * (im_m * im_m)));
	} else if (im_m <= 4.2e+65) {
		tmp = im_m * (-1.0 + (0.5 * (re * re)));
	} else {
		tmp = im_m * (-1.0 + ((im_m * im_m) * (-0.08333333333333333 + (im_m * -0.14583333333333334))));
	}
	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 <= 0.00012d0) then
        tmp = im_m * ((-1.0d0) + ((-0.16666666666666666d0) * (im_m * im_m)))
    else if (im_m <= 4.2d+65) then
        tmp = im_m * ((-1.0d0) + (0.5d0 * (re * re)))
    else
        tmp = im_m * ((-1.0d0) + ((im_m * im_m) * ((-0.08333333333333333d0) + (im_m * (-0.14583333333333334d0)))))
    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 <= 0.00012) {
		tmp = im_m * (-1.0 + (-0.16666666666666666 * (im_m * im_m)));
	} else if (im_m <= 4.2e+65) {
		tmp = im_m * (-1.0 + (0.5 * (re * re)));
	} else {
		tmp = im_m * (-1.0 + ((im_m * im_m) * (-0.08333333333333333 + (im_m * -0.14583333333333334))));
	}
	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 <= 0.00012:
		tmp = im_m * (-1.0 + (-0.16666666666666666 * (im_m * im_m)))
	elif im_m <= 4.2e+65:
		tmp = im_m * (-1.0 + (0.5 * (re * re)))
	else:
		tmp = im_m * (-1.0 + ((im_m * im_m) * (-0.08333333333333333 + (im_m * -0.14583333333333334))))
	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 <= 0.00012)
		tmp = Float64(im_m * Float64(-1.0 + Float64(-0.16666666666666666 * Float64(im_m * im_m))));
	elseif (im_m <= 4.2e+65)
		tmp = Float64(im_m * Float64(-1.0 + Float64(0.5 * Float64(re * re))));
	else
		tmp = Float64(im_m * Float64(-1.0 + Float64(Float64(im_m * im_m) * Float64(-0.08333333333333333 + Float64(im_m * -0.14583333333333334)))));
	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 <= 0.00012)
		tmp = im_m * (-1.0 + (-0.16666666666666666 * (im_m * im_m)));
	elseif (im_m <= 4.2e+65)
		tmp = im_m * (-1.0 + (0.5 * (re * re)));
	else
		tmp = im_m * (-1.0 + ((im_m * im_m) * (-0.08333333333333333 + (im_m * -0.14583333333333334))));
	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, 0.00012], N[(im$95$m * N[(-1.0 + N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 4.2e+65], N[(im$95$m * N[(-1.0 + N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(-1.0 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(-0.08333333333333333 + N[(im$95$m * -0.14583333333333334), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 1.20000000000000003e-4

   1. Initial program 41.2%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + \color{blue}{\left(\mathsf{neg}\left(e^{im}\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(e^{im}\right)\right) + \color{blue}{e^{\mathsf{neg}\left(im\right)}}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \left(\mathsf{neg}\left(e^{im}\right)\right) \cdot \frac{1}{2} + \color{blue}{e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\mathsf{neg}\left(e^{im}\right)\right) + \color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2} \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \left(\mathsf{neg}\left(e^{im}\right)\right)\right), \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left(e^{im}\right)\right) \cdot \frac{1}{2}\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      7. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(e^{im} \cdot \frac{1}{2}\right)\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{+.f64}\left(\left(e^{im} \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\left(e^{im} \cdot \frac{-1}{2}\right), \left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(e^{im}\right), \frac{-1}{2}\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      11. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(e^{\color{blue}{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      12. exp-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(\frac{1}{e^{im}} \cdot \frac{1}{2}\right)\right) \]

      13. associate-*l/ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(\frac{1 \cdot \frac{1}{2}}{\color{blue}{e^{im}}}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(\frac{\frac{1}{2}}{e^{\color{blue}{im}}}\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{im}\right)}\right)\right) \]

      16. exp-lowering-exp.f64 30.6%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right)\right) \]

   5. Simplified 30.6%

      \[\leadsto \color{blue}{e^{im} \cdot -0.5 + \frac{0.5}{e^{im}}} \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(\frac{-1}{6} \cdot {im}^{2} - 1\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2} - 1\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left(\frac{-1}{6} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left(\frac{-1}{6} \cdot {im}^{2} + -1\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left(-1 + \color{blue}{\frac{-1}{6} \cdot {im}^{2}}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{-1}{6}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{-1}{6}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 56.0%

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   8. Simplified 56.0%

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

   if 1.20000000000000003e-4 < im < 4.19999999999999983e65

   1. Initial program 99.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(im \cdot \cos re\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{im \cdot \cos re} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(im \cdot \cos re\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \color{blue}{\cos re}\right)\right) \]

      5. cos-lowering-cos.f64 4.3%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \mathsf{cos.f64}\left(re\right)\right)\right) \]

   5. Simplified 4.3%

      \[\leadsto \color{blue}{0 - im \cdot \cos re} \]

   6. Taylor expanded in re around 0

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

      1. sub-neg N/A

         \[\leadsto \frac{1}{2} \cdot \left(im \cdot {re}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(im\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left({re}^{2} \cdot im\right) + \left(\mathsf{neg}\left(im\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\frac{1}{2} \cdot {re}^{2}\right) \cdot im + \left(\mathsf{neg}\left(\color{blue}{im}\right)\right) \]

      4. mul-1-neg N/A

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

      5. distribute-rgt-out N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{2} \cdot {re}^{2} + -1\right)}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot {re}^{2}\right), \color{blue}{-1}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({re}^{2}\right)\right), -1\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(re \cdot re\right)\right), -1\right)\right) \]

      10. *-lowering-*.f64 26.4%

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, re\right)\right), -1\right)\right) \]

   8. Simplified 26.4%

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

   if 4.19999999999999983e65 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + \color{blue}{\left(\mathsf{neg}\left(e^{im}\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(e^{im}\right)\right) + \color{blue}{e^{\mathsf{neg}\left(im\right)}}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \left(\mathsf{neg}\left(e^{im}\right)\right) \cdot \frac{1}{2} + \color{blue}{e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\mathsf{neg}\left(e^{im}\right)\right) + \color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2} \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \left(\mathsf{neg}\left(e^{im}\right)\right)\right), \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left(e^{im}\right)\right) \cdot \frac{1}{2}\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      7. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(e^{im} \cdot \frac{1}{2}\right)\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{+.f64}\left(\left(e^{im} \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\left(e^{im} \cdot \frac{-1}{2}\right), \left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(e^{im}\right), \frac{-1}{2}\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      11. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(e^{\color{blue}{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      12. exp-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(\frac{1}{e^{im}} \cdot \frac{1}{2}\right)\right) \]

      13. associate-*l/ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(\frac{1 \cdot \frac{1}{2}}{\color{blue}{e^{im}}}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(\frac{\frac{1}{2}}{e^{\color{blue}{im}}}\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{im}\right)}\right)\right) \]

      16. exp-lowering-exp.f64 82.9%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right)\right) \]

   5. Simplified 82.9%

      \[\leadsto \color{blue}{e^{im} \cdot -0.5 + \frac{0.5}{e^{im}}} \]

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(1 + im \cdot \left(1 + \frac{1}{2} \cdot im\right)\right)}\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \color{blue}{\left(im \cdot \left(1 + \frac{1}{2} \cdot im\right)\right)}\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(1 + \frac{1}{2} \cdot im\right)}\right)\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot im\right)}\right)\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 82.9%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 82.9%

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

   9. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left({im}^{2} \cdot \left(\frac{-7}{48} \cdot im - \frac{1}{12}\right) - 1\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left({im}^{2} \cdot \left(\frac{-7}{48} \cdot im - \frac{1}{12}\right) - 1\right)}\right) \]

       2. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left(\frac{-7}{48} \cdot im - \frac{1}{12}\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

       3. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left(\frac{-7}{48} \cdot im - \frac{1}{12}\right) + -1\right)\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left(-1 + \color{blue}{{im}^{2} \cdot \left(\frac{-7}{48} \cdot im - \frac{1}{12}\right)}\right)\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \color{blue}{\left({im}^{2} \cdot \left(\frac{-7}{48} \cdot im - \frac{1}{12}\right)\right)}\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{-7}{48} \cdot im - \frac{1}{12}\right)}\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{-7}{48} \cdot im} - \frac{1}{12}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{-7}{48} \cdot im} - \frac{1}{12}\right)\right)\right)\right) \]

       9. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-7}{48} \cdot im + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{12}\right)\right)}\right)\right)\right)\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-7}{48} \cdot im + \frac{-1}{12}\right)\right)\right)\right) \]

       11. +-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{12} + \color{blue}{\frac{-7}{48} \cdot im}\right)\right)\right)\right) \]

       12. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{12}, \color{blue}{\left(\frac{-7}{48} \cdot im\right)}\right)\right)\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{12}, \left(im \cdot \color{blue}{\frac{-7}{48}}\right)\right)\right)\right)\right) \]

       14. *-lowering-*.f64 80.7%

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(im, \color{blue}{\frac{-7}{48}}\right)\right)\right)\right)\right) \]

   11. Simplified 80.7%

       \[\leadsto \color{blue}{im \cdot \left(-1 + \left(im \cdot im\right) \cdot \left(-0.08333333333333333 + im \cdot -0.14583333333333334\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.00012:\\ \;\;\;\;im \cdot \left(-1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{+65}:\\ \;\;\;\;im \cdot \left(-1 + 0.5 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-1 + \left(im \cdot im\right) \cdot \left(-0.08333333333333333 + im \cdot -0.14583333333333334\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 59.5% accurate, 14.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(im\_m \cdot \left(-1 + \left(im\_m \cdot im\_m\right) \cdot \left(-0.16666666666666666 + im\_m \cdot \left(im\_m \cdot \left(-0.008333333333333333 + im\_m \cdot \left(im\_m \cdot -0.0001984126984126984\right)\right)\right)\right)\right)\right) \end{array} \]

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
   (+
    -1.0
    (*
     (* im_m im_m)
     (+
      -0.16666666666666666
      (*
       im_m
       (*
        im_m
        (+
         -0.008333333333333333
         (* im_m (* im_m -0.0001984126984126984)))))))))))

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 * (-1.0 + ((im_m * im_m) * (-0.16666666666666666 + (im_m * (im_m * (-0.008333333333333333 + (im_m * (im_m * -0.0001984126984126984)))))))));
}

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 * ((-1.0d0) + ((im_m * im_m) * ((-0.16666666666666666d0) + (im_m * (im_m * ((-0.008333333333333333d0) + (im_m * (im_m * (-0.0001984126984126984d0))))))))))
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 * (-1.0 + ((im_m * im_m) * (-0.16666666666666666 + (im_m * (im_m * (-0.008333333333333333 + (im_m * (im_m * -0.0001984126984126984)))))))));
}

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 * (-1.0 + ((im_m * im_m) * (-0.16666666666666666 + (im_m * (im_m * (-0.008333333333333333 + (im_m * (im_m * -0.0001984126984126984)))))))))

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(-1.0 + Float64(Float64(im_m * im_m) * Float64(-0.16666666666666666 + Float64(im_m * Float64(im_m * Float64(-0.008333333333333333 + Float64(im_m * Float64(im_m * -0.0001984126984126984))))))))))
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 * (-1.0 + ((im_m * im_m) * (-0.16666666666666666 + (im_m * (im_m * (-0.008333333333333333 + (im_m * (im_m * -0.0001984126984126984)))))))));
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[(-1.0 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(-0.16666666666666666 + N[(im$95$m * N[(im$95$m * N[(-0.008333333333333333 + N[(im$95$m * N[(im$95$m * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.6%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in re around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)} \]
4. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + \color{blue}{\left(\mathsf{neg}\left(e^{im}\right)\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(e^{im}\right)\right) + \color{blue}{e^{\mathsf{neg}\left(im\right)}}\right) \]

   3. distribute-rgt-in N/A

      \[\leadsto \left(\mathsf{neg}\left(e^{im}\right)\right) \cdot \frac{1}{2} + \color{blue}{e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}} \]

   4. *-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{neg}\left(e^{im}\right)\right) + \color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2} \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \left(\mathsf{neg}\left(e^{im}\right)\right)\right), \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)}\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left(e^{im}\right)\right) \cdot \frac{1}{2}\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

   7. distribute-lft-neg-out N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(e^{im} \cdot \frac{1}{2}\right)\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

   8. distribute-rgt-neg-in N/A

      \[\leadsto \mathsf{+.f64}\left(\left(e^{im} \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(\left(e^{im} \cdot \frac{-1}{2}\right), \left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(e^{im}\right), \frac{-1}{2}\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

   11. exp-lowering-exp.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(e^{\color{blue}{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

   12. exp-neg N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(\frac{1}{e^{im}} \cdot \frac{1}{2}\right)\right) \]

   13. associate-*l/ N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(\frac{1 \cdot \frac{1}{2}}{\color{blue}{e^{im}}}\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(\frac{\frac{1}{2}}{e^{\color{blue}{im}}}\right)\right) \]

   15. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{im}\right)}\right)\right) \]

   16. exp-lowering-exp.f64 40.3%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right)\right) \]

5. Simplified 40.3%

   \[\leadsto \color{blue}{e^{im} \cdot -0.5 + \frac{0.5}{e^{im}}} \]

6. Taylor expanded in im around 0

   \[\leadsto \color{blue}{im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) - 1\right)} \]
7. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) - 1\right)}\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) + -1\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(im, \left(-1 + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right)}\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right)\right)}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right)}\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{{im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)} - \frac{1}{6}\right)\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{{im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)} - \frac{1}{6}\right)\right)\right)\right) \]

   9. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) + \frac{-1}{6}\right)\right)\right)\right) \]

   11. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{6} + \color{blue}{{im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)}\right)\right)\right)\right) \]

   12. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)\right)}\right)\right)\right)\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{-1}{5040} \cdot {im}^{2}} - \frac{1}{120}\right)\right)\right)\right)\right)\right) \]

   14. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)\right)}\right)\right)\right)\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)\right)}\right)\right)\right)\right)\right) \]

   16. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)}\right)\right)\right)\right)\right)\right) \]

   17. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{5040} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{120}\right)\right)}\right)\right)\right)\right)\right)\right)\right) \]

   18. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{5040} \cdot {im}^{2} + \frac{-1}{120}\right)\right)\right)\right)\right)\right)\right) \]

   19. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{120} + \color{blue}{\frac{-1}{5040} \cdot {im}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

   20. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{120}, \color{blue}{\left(\frac{-1}{5040} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

8. Simplified 59.8%

   \[\leadsto \color{blue}{im \cdot \left(-1 + \left(im \cdot im\right) \cdot \left(-0.16666666666666666 + im \cdot \left(im \cdot \left(-0.008333333333333333 + im \cdot \left(im \cdot -0.0001984126984126984\right)\right)\right)\right)\right)} \]
9. Add Preprocessing

Alternative 15: 54.8% accurate, 16.2× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := im\_m \cdot \left(-1 + -0.16666666666666666 \cdot \left(im\_m \cdot im\_m\right)\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 0.00012:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im\_m \leq 2.5 \cdot 10^{+83}:\\ \;\;\;\;im\_m \cdot \left(-1 + 0.5 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 (* im_m (+ -1.0 (* -0.16666666666666666 (* im_m im_m))))))
   (*
    im_s
    (if (<= im_m 0.00012)
      t_0
      (if (<= im_m 2.5e+83) (* im_m (+ -1.0 (* 0.5 (* re re)))) t_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 = im_m * (-1.0 + (-0.16666666666666666 * (im_m * im_m)));
	double tmp;
	if (im_m <= 0.00012) {
		tmp = t_0;
	} else if (im_m <= 2.5e+83) {
		tmp = im_m * (-1.0 + (0.5 * (re * re)));
	} else {
		tmp = t_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) :: tmp
    t_0 = im_m * ((-1.0d0) + ((-0.16666666666666666d0) * (im_m * im_m)))
    if (im_m <= 0.00012d0) then
        tmp = t_0
    else if (im_m <= 2.5d+83) then
        tmp = im_m * ((-1.0d0) + (0.5d0 * (re * re)))
    else
        tmp = t_0
    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 = im_m * (-1.0 + (-0.16666666666666666 * (im_m * im_m)));
	double tmp;
	if (im_m <= 0.00012) {
		tmp = t_0;
	} else if (im_m <= 2.5e+83) {
		tmp = im_m * (-1.0 + (0.5 * (re * re)));
	} else {
		tmp = t_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 = im_m * (-1.0 + (-0.16666666666666666 * (im_m * im_m)))
	tmp = 0
	if im_m <= 0.00012:
		tmp = t_0
	elif im_m <= 2.5e+83:
		tmp = im_m * (-1.0 + (0.5 * (re * re)))
	else:
		tmp = t_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(im_m * Float64(-1.0 + Float64(-0.16666666666666666 * Float64(im_m * im_m))))
	tmp = 0.0
	if (im_m <= 0.00012)
		tmp = t_0;
	elseif (im_m <= 2.5e+83)
		tmp = Float64(im_m * Float64(-1.0 + Float64(0.5 * Float64(re * re))));
	else
		tmp = t_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 = im_m * (-1.0 + (-0.16666666666666666 * (im_m * im_m)));
	tmp = 0.0;
	if (im_m <= 0.00012)
		tmp = t_0;
	elseif (im_m <= 2.5e+83)
		tmp = im_m * (-1.0 + (0.5 * (re * re)));
	else
		tmp = t_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[(im$95$m * N[(-1.0 + N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[im$95$m, 0.00012], t$95$0, If[LessEqual[im$95$m, 2.5e+83], N[(im$95$m * N[(-1.0 + N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.20000000000000003e-4 or 2.50000000000000014e83 < im

   1. Initial program 50.2%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + \color{blue}{\left(\mathsf{neg}\left(e^{im}\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(e^{im}\right)\right) + \color{blue}{e^{\mathsf{neg}\left(im\right)}}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \left(\mathsf{neg}\left(e^{im}\right)\right) \cdot \frac{1}{2} + \color{blue}{e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\mathsf{neg}\left(e^{im}\right)\right) + \color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2} \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \left(\mathsf{neg}\left(e^{im}\right)\right)\right), \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left(e^{im}\right)\right) \cdot \frac{1}{2}\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      7. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(e^{im} \cdot \frac{1}{2}\right)\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{+.f64}\left(\left(e^{im} \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\left(e^{im} \cdot \frac{-1}{2}\right), \left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(e^{im}\right), \frac{-1}{2}\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      11. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(e^{\color{blue}{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

      12. exp-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(\frac{1}{e^{im}} \cdot \frac{1}{2}\right)\right) \]

      13. associate-*l/ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(\frac{1 \cdot \frac{1}{2}}{\color{blue}{e^{im}}}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(\frac{\frac{1}{2}}{e^{\color{blue}{im}}}\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{im}\right)}\right)\right) \]

      16. exp-lowering-exp.f64 38.6%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right)\right) \]

   5. Simplified 38.6%

      \[\leadsto \color{blue}{e^{im} \cdot -0.5 + \frac{0.5}{e^{im}}} \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(\frac{-1}{6} \cdot {im}^{2} - 1\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2} - 1\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left(\frac{-1}{6} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left(\frac{-1}{6} \cdot {im}^{2} + -1\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left(-1 + \color{blue}{\frac{-1}{6} \cdot {im}^{2}}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{-1}{6}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{-1}{6}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 58.9%

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   8. Simplified 58.9%

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

   if 1.20000000000000003e-4 < im < 2.50000000000000014e83

   1. Initial program 99.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(im \cdot \cos re\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{im \cdot \cos re} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(im \cdot \cos re\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \color{blue}{\cos re}\right)\right) \]

      5. cos-lowering-cos.f64 4.2%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \mathsf{cos.f64}\left(re\right)\right)\right) \]

   5. Simplified 4.2%

      \[\leadsto \color{blue}{0 - im \cdot \cos re} \]

   6. Taylor expanded in re around 0

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

      1. sub-neg N/A

         \[\leadsto \frac{1}{2} \cdot \left(im \cdot {re}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(im\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left({re}^{2} \cdot im\right) + \left(\mathsf{neg}\left(im\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\frac{1}{2} \cdot {re}^{2}\right) \cdot im + \left(\mathsf{neg}\left(\color{blue}{im}\right)\right) \]

      4. mul-1-neg N/A

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

      5. distribute-rgt-out N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{2} \cdot {re}^{2} + -1\right)}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot {re}^{2}\right), \color{blue}{-1}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({re}^{2}\right)\right), -1\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(re \cdot re\right)\right), -1\right)\right) \]

      10. *-lowering-*.f64 24.8%

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, re\right)\right), -1\right)\right) \]

   8. Simplified 24.8%

      \[\leadsto \color{blue}{im \cdot \left(0.5 \cdot \left(re \cdot re\right) + -1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.00012:\\ \;\;\;\;im \cdot \left(-1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{+83}:\\ \;\;\;\;im \cdot \left(-1 + 0.5 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 59.4% accurate, 16.3× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(im\_m \cdot \left(-1 + \left(im\_m \cdot im\_m\right) \cdot \left(-0.16666666666666666 + -0.0001984126984126984 \cdot \left(im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot im\_m\right)\right)\right)\right)\right)\right) \end{array} \]

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
   (+
    -1.0
    (*
     (* im_m im_m)
     (+
      -0.16666666666666666
      (* -0.0001984126984126984 (* im_m (* im_m (* im_m im_m))))))))))

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 * (-1.0 + ((im_m * im_m) * (-0.16666666666666666 + (-0.0001984126984126984 * (im_m * (im_m * (im_m * im_m))))))));
}

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 * ((-1.0d0) + ((im_m * im_m) * ((-0.16666666666666666d0) + ((-0.0001984126984126984d0) * (im_m * (im_m * (im_m * im_m))))))))
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 * (-1.0 + ((im_m * im_m) * (-0.16666666666666666 + (-0.0001984126984126984 * (im_m * (im_m * (im_m * im_m))))))));
}

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 * (-1.0 + ((im_m * im_m) * (-0.16666666666666666 + (-0.0001984126984126984 * (im_m * (im_m * (im_m * im_m))))))))

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(-1.0 + Float64(Float64(im_m * im_m) * Float64(-0.16666666666666666 + Float64(-0.0001984126984126984 * Float64(im_m * Float64(im_m * Float64(im_m * im_m)))))))))
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 * (-1.0 + ((im_m * im_m) * (-0.16666666666666666 + (-0.0001984126984126984 * (im_m * (im_m * (im_m * im_m))))))));
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[(-1.0 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(-0.16666666666666666 + N[(-0.0001984126984126984 * N[(im$95$m * N[(im$95$m * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.6%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in re around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)} \]
4. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + \color{blue}{\left(\mathsf{neg}\left(e^{im}\right)\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(e^{im}\right)\right) + \color{blue}{e^{\mathsf{neg}\left(im\right)}}\right) \]

   3. distribute-rgt-in N/A

      \[\leadsto \left(\mathsf{neg}\left(e^{im}\right)\right) \cdot \frac{1}{2} + \color{blue}{e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}} \]

   4. *-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{neg}\left(e^{im}\right)\right) + \color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2} \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \left(\mathsf{neg}\left(e^{im}\right)\right)\right), \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)}\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left(e^{im}\right)\right) \cdot \frac{1}{2}\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

   7. distribute-lft-neg-out N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(e^{im} \cdot \frac{1}{2}\right)\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

   8. distribute-rgt-neg-in N/A

      \[\leadsto \mathsf{+.f64}\left(\left(e^{im} \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(\left(e^{im} \cdot \frac{-1}{2}\right), \left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(e^{im}\right), \frac{-1}{2}\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

   11. exp-lowering-exp.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(e^{\color{blue}{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

   12. exp-neg N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(\frac{1}{e^{im}} \cdot \frac{1}{2}\right)\right) \]

   13. associate-*l/ N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(\frac{1 \cdot \frac{1}{2}}{\color{blue}{e^{im}}}\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(\frac{\frac{1}{2}}{e^{\color{blue}{im}}}\right)\right) \]

   15. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{im}\right)}\right)\right) \]

   16. exp-lowering-exp.f64 40.3%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right)\right) \]

5. Simplified 40.3%

   \[\leadsto \color{blue}{e^{im} \cdot -0.5 + \frac{0.5}{e^{im}}} \]

6. Taylor expanded in im around 0

   \[\leadsto \color{blue}{im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) - 1\right)} \]
7. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) - 1\right)}\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) + -1\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(im, \left(-1 + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right)}\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right)\right)}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right)}\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{{im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)} - \frac{1}{6}\right)\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{{im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)} - \frac{1}{6}\right)\right)\right)\right) \]

   9. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) + \frac{-1}{6}\right)\right)\right)\right) \]

   11. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{6} + \color{blue}{{im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)}\right)\right)\right)\right) \]

   12. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)\right)}\right)\right)\right)\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{-1}{5040} \cdot {im}^{2}} - \frac{1}{120}\right)\right)\right)\right)\right)\right) \]

   14. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)\right)}\right)\right)\right)\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)\right)}\right)\right)\right)\right)\right) \]

   16. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)}\right)\right)\right)\right)\right)\right) \]

   17. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{5040} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{120}\right)\right)}\right)\right)\right)\right)\right)\right)\right) \]

   18. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{5040} \cdot {im}^{2} + \frac{-1}{120}\right)\right)\right)\right)\right)\right)\right) \]

   19. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{120} + \color{blue}{\frac{-1}{5040} \cdot {im}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

   20. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{120}, \color{blue}{\left(\frac{-1}{5040} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

8. Simplified 59.8%

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

9. Taylor expanded in im around inf

   \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left(\frac{-1}{5040} \cdot {im}^{4}\right)}\right)\right)\right)\right) \]
10. Step-by-step derivation

    1. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\frac{-1}{5040}, \color{blue}{\left({im}^{4}\right)}\right)\right)\right)\right)\right) \]

    2. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\frac{-1}{5040}, \left({im}^{\left(3 + \color{blue}{1}\right)}\right)\right)\right)\right)\right)\right) \]

    3. pow-plus N/A

       \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\frac{-1}{5040}, \left({im}^{3} \cdot \color{blue}{im}\right)\right)\right)\right)\right)\right) \]

    4. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\frac{-1}{5040}, \mathsf{*.f64}\left(\left({im}^{3}\right), \color{blue}{im}\right)\right)\right)\right)\right)\right) \]

    5. cube-mult N/A

       \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\frac{-1}{5040}, \mathsf{*.f64}\left(\left(im \cdot \left(im \cdot im\right)\right), im\right)\right)\right)\right)\right)\right) \]

    6. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\frac{-1}{5040}, \mathsf{*.f64}\left(\left(im \cdot {im}^{2}\right), im\right)\right)\right)\right)\right)\right) \]

    7. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\frac{-1}{5040}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, \left({im}^{2}\right)\right), im\right)\right)\right)\right)\right)\right) \]

    8. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\frac{-1}{5040}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, \left(im \cdot im\right)\right), im\right)\right)\right)\right)\right)\right) \]

    9. *-lowering-*.f64 59.8%

       \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\frac{-1}{5040}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, im\right)\right), im\right)\right)\right)\right)\right)\right) \]

11. Simplified 59.8%

    \[\leadsto im \cdot \left(-1 + \left(im \cdot im\right) \cdot \left(-0.16666666666666666 + \color{blue}{-0.0001984126984126984 \cdot \left(\left(im \cdot \left(im \cdot im\right)\right) \cdot im\right)}\right)\right) \]

12. Final simplification 59.8%

    \[\leadsto im \cdot \left(-1 + \left(im \cdot im\right) \cdot \left(-0.16666666666666666 + -0.0001984126984126984 \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\right)\right) \]
13. Add Preprocessing

Alternative 17: 59.2% accurate, 18.2× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(im\_m \cdot \left(-1 + \left(im\_m \cdot \left(im\_m \cdot -0.0001984126984126984\right)\right) \cdot \left(im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot im\_m\right)\right)\right)\right)\right) \end{array} \]

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
   (+
    -1.0
    (*
     (* im_m (* im_m -0.0001984126984126984))
     (* im_m (* im_m (* im_m im_m))))))))

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 * (-1.0 + ((im_m * (im_m * -0.0001984126984126984)) * (im_m * (im_m * (im_m * im_m))))));
}

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 * ((-1.0d0) + ((im_m * (im_m * (-0.0001984126984126984d0))) * (im_m * (im_m * (im_m * im_m))))))
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 * (-1.0 + ((im_m * (im_m * -0.0001984126984126984)) * (im_m * (im_m * (im_m * im_m))))));
}

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 * (-1.0 + ((im_m * (im_m * -0.0001984126984126984)) * (im_m * (im_m * (im_m * im_m))))))

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(-1.0 + Float64(Float64(im_m * Float64(im_m * -0.0001984126984126984)) * Float64(im_m * Float64(im_m * Float64(im_m * im_m)))))))
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 * (-1.0 + ((im_m * (im_m * -0.0001984126984126984)) * (im_m * (im_m * (im_m * im_m))))));
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[(-1.0 + N[(N[(im$95$m * N[(im$95$m * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision] * N[(im$95$m * N[(im$95$m * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.6%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in re around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)} \]
4. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + \color{blue}{\left(\mathsf{neg}\left(e^{im}\right)\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(e^{im}\right)\right) + \color{blue}{e^{\mathsf{neg}\left(im\right)}}\right) \]

   3. distribute-rgt-in N/A

      \[\leadsto \left(\mathsf{neg}\left(e^{im}\right)\right) \cdot \frac{1}{2} + \color{blue}{e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}} \]

   4. *-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{neg}\left(e^{im}\right)\right) + \color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2} \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \left(\mathsf{neg}\left(e^{im}\right)\right)\right), \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)}\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left(e^{im}\right)\right) \cdot \frac{1}{2}\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

   7. distribute-lft-neg-out N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(e^{im} \cdot \frac{1}{2}\right)\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

   8. distribute-rgt-neg-in N/A

      \[\leadsto \mathsf{+.f64}\left(\left(e^{im} \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(\left(e^{im} \cdot \frac{-1}{2}\right), \left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(e^{im}\right), \frac{-1}{2}\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

   11. exp-lowering-exp.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(e^{\color{blue}{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

   12. exp-neg N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(\frac{1}{e^{im}} \cdot \frac{1}{2}\right)\right) \]

   13. associate-*l/ N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(\frac{1 \cdot \frac{1}{2}}{\color{blue}{e^{im}}}\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(\frac{\frac{1}{2}}{e^{\color{blue}{im}}}\right)\right) \]

   15. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{im}\right)}\right)\right) \]

   16. exp-lowering-exp.f64 40.3%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right)\right) \]

5. Simplified 40.3%

   \[\leadsto \color{blue}{e^{im} \cdot -0.5 + \frac{0.5}{e^{im}}} \]

6. Taylor expanded in im around 0

   \[\leadsto \color{blue}{im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) - 1\right)} \]
7. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) - 1\right)}\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) + -1\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(im, \left(-1 + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right)}\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right)\right)}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right)}\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{{im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)} - \frac{1}{6}\right)\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{{im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)} - \frac{1}{6}\right)\right)\right)\right) \]

   9. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) + \frac{-1}{6}\right)\right)\right)\right) \]

   11. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{6} + \color{blue}{{im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)}\right)\right)\right)\right) \]

   12. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)\right)}\right)\right)\right)\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{-1}{5040} \cdot {im}^{2}} - \frac{1}{120}\right)\right)\right)\right)\right)\right) \]

   14. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)\right)}\right)\right)\right)\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)\right)}\right)\right)\right)\right)\right) \]

   16. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)}\right)\right)\right)\right)\right)\right) \]

   17. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{5040} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{120}\right)\right)}\right)\right)\right)\right)\right)\right)\right) \]

   18. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{5040} \cdot {im}^{2} + \frac{-1}{120}\right)\right)\right)\right)\right)\right)\right) \]

   19. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{120} + \color{blue}{\frac{-1}{5040} \cdot {im}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

   20. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{120}, \color{blue}{\left(\frac{-1}{5040} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

8. Simplified 59.8%

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

   1. distribute-rgt-in N/A

      \[\leadsto \mathsf{*.f64}\left(im, \left(-1 + \left(\frac{-1}{6} \cdot \left(im \cdot im\right) + \color{blue}{\left(im \cdot \left(im \cdot \left(\frac{-1}{120} + im \cdot \left(im \cdot \frac{-1}{5040}\right)\right)\right)\right) \cdot \left(im \cdot im\right)}\right)\right)\right) \]

   2. associate-+r+ N/A

      \[\leadsto \mathsf{*.f64}\left(im, \left(\left(-1 + \frac{-1}{6} \cdot \left(im \cdot im\right)\right) + \color{blue}{\left(im \cdot \left(im \cdot \left(\frac{-1}{120} + im \cdot \left(im \cdot \frac{-1}{5040}\right)\right)\right)\right) \cdot \left(im \cdot im\right)}\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(im, \left(\left(-1 + \frac{-1}{6} \cdot \left(im \cdot im\right)\right) + \left(im \cdot im\right) \cdot \color{blue}{\left(im \cdot \left(im \cdot \left(\frac{-1}{120} + im \cdot \left(im \cdot \frac{-1}{5040}\right)\right)\right)\right)}\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(im, \left(\left(-1 + \frac{-1}{6} \cdot \left(im \cdot im\right)\right) + \left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot \color{blue}{\left(\frac{-1}{120} + im \cdot \left(im \cdot \frac{-1}{5040}\right)\right)}\right)\right)\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(im, \left(\left(-1 + \frac{-1}{6} \cdot \left(im \cdot im\right)\right) + \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{\left(\frac{-1}{120} + im \cdot \left(im \cdot \frac{-1}{5040}\right)\right)}\right)\right) \]

   6. distribute-lft-in N/A

      \[\leadsto \mathsf{*.f64}\left(im, \left(\left(-1 + \frac{-1}{6} \cdot \left(im \cdot im\right)\right) + \left(\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{120} + \color{blue}{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(im \cdot \left(im \cdot \frac{-1}{5040}\right)\right)}\right)\right)\right) \]

   7. associate-+r+ N/A

      \[\leadsto \mathsf{*.f64}\left(im, \left(\left(\left(-1 + \frac{-1}{6} \cdot \left(im \cdot im\right)\right) + \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{120}\right) + \color{blue}{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(im \cdot \left(im \cdot \frac{-1}{5040}\right)\right)}\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(im, \left(\left(\left(-1 + \frac{-1}{6} \cdot \left(im \cdot im\right)\right) + \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{120}\right) + \left(im \cdot \left(im \cdot \frac{-1}{5040}\right)\right) \cdot \color{blue}{\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)}\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\left(\left(-1 + \frac{-1}{6} \cdot \left(im \cdot im\right)\right) + \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{120}\right), \color{blue}{\left(\left(im \cdot \left(im \cdot \frac{-1}{5040}\right)\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)}\right)\right) \]

10. Applied egg-rr 59.8%

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

11. Taylor expanded in im around 0

    \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\color{blue}{-1}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{5040}\right)\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right)\right) \]
12. Step-by-step derivation

13. Simplified 59.7%

    \[\leadsto im \cdot \left(\color{blue}{-1} + \left(im \cdot \left(im \cdot -0.0001984126984126984\right)\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\right) \]
14. Add Preprocessing

Alternative 18: 57.7% accurate, 20.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(im\_m \cdot \left(-1 + im\_m \cdot \left(im\_m \cdot \left(-0.16666666666666666 + \left(im\_m \cdot im\_m\right) \cdot -0.008333333333333333\right)\right)\right)\right) \end{array} \]

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
   (+
    -1.0
    (*
     im_m
     (*
      im_m
      (+ -0.16666666666666666 (* (* im_m im_m) -0.008333333333333333))))))))

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 * (-1.0 + (im_m * (im_m * (-0.16666666666666666 + ((im_m * im_m) * -0.008333333333333333))))));
}

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 * ((-1.0d0) + (im_m * (im_m * ((-0.16666666666666666d0) + ((im_m * im_m) * (-0.008333333333333333d0)))))))
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 * (-1.0 + (im_m * (im_m * (-0.16666666666666666 + ((im_m * im_m) * -0.008333333333333333))))));
}

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 * (-1.0 + (im_m * (im_m * (-0.16666666666666666 + ((im_m * im_m) * -0.008333333333333333))))))

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(-1.0 + Float64(im_m * Float64(im_m * Float64(-0.16666666666666666 + Float64(Float64(im_m * im_m) * -0.008333333333333333)))))))
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 * (-1.0 + (im_m * (im_m * (-0.16666666666666666 + ((im_m * im_m) * -0.008333333333333333))))));
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[(-1.0 + N[(im$95$m * N[(im$95$m * N[(-0.16666666666666666 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.6%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in re around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)} \]
4. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + \color{blue}{\left(\mathsf{neg}\left(e^{im}\right)\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(e^{im}\right)\right) + \color{blue}{e^{\mathsf{neg}\left(im\right)}}\right) \]

   3. distribute-rgt-in N/A

      \[\leadsto \left(\mathsf{neg}\left(e^{im}\right)\right) \cdot \frac{1}{2} + \color{blue}{e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}} \]

   4. *-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{neg}\left(e^{im}\right)\right) + \color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2} \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \left(\mathsf{neg}\left(e^{im}\right)\right)\right), \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)}\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left(e^{im}\right)\right) \cdot \frac{1}{2}\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

   7. distribute-lft-neg-out N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(e^{im} \cdot \frac{1}{2}\right)\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

   8. distribute-rgt-neg-in N/A

      \[\leadsto \mathsf{+.f64}\left(\left(e^{im} \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(\left(e^{im} \cdot \frac{-1}{2}\right), \left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(e^{im}\right), \frac{-1}{2}\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

   11. exp-lowering-exp.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(e^{\color{blue}{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

   12. exp-neg N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(\frac{1}{e^{im}} \cdot \frac{1}{2}\right)\right) \]

   13. associate-*l/ N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(\frac{1 \cdot \frac{1}{2}}{\color{blue}{e^{im}}}\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(\frac{\frac{1}{2}}{e^{\color{blue}{im}}}\right)\right) \]

   15. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{im}\right)}\right)\right) \]

   16. exp-lowering-exp.f64 40.3%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right)\right) \]

5. Simplified 40.3%

   \[\leadsto \color{blue}{e^{im} \cdot -0.5 + \frac{0.5}{e^{im}}} \]

6. Taylor expanded in im around 0

   \[\leadsto \color{blue}{im \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)} \]
7. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)}\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) + -1\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(im, \left(-1 + \color{blue}{{im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right)}\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)}\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{-1}{120} \cdot {im}^{2}} - \frac{1}{6}\right)\right)\right)\right) \]

   7. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)}\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)}\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right)}\right)\right)\right)\right) \]

   10. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{120} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{120} \cdot {im}^{2} + \frac{-1}{6}\right)\right)\right)\right)\right) \]

   12. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{6} + \color{blue}{\frac{-1}{120} \cdot {im}^{2}}\right)\right)\right)\right)\right) \]

   13. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left(\frac{-1}{120} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \left({im}^{2} \cdot \color{blue}{\frac{-1}{120}}\right)\right)\right)\right)\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{-1}{120}}\right)\right)\right)\right)\right)\right) \]

   16. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{-1}{120}\right)\right)\right)\right)\right)\right) \]

   17. *-lowering-*.f64 58.3%

       \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{120}\right)\right)\right)\right)\right)\right) \]

8. Simplified 58.3%

   \[\leadsto \color{blue}{im \cdot \left(-1 + im \cdot \left(im \cdot \left(-0.16666666666666666 + \left(im \cdot im\right) \cdot -0.008333333333333333\right)\right)\right)} \]
9. Add Preprocessing

Alternative 19: 47.2% accurate, 25.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 5 \cdot 10^{-105}:\\ \;\;\;\;0 - im\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{im\_m \cdot im\_m}{0 - im\_m}\\ \end{array} \end{array} \]

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 5e-105) (- 0.0 im_m) (/ (* im_m im_m) (- 0.0 im_m)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 5e-105) {
		tmp = 0.0 - im_m;
	} else {
		tmp = (im_m * im_m) / (0.0 - 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) :: tmp
    if (im_m <= 5d-105) then
        tmp = 0.0d0 - im_m
    else
        tmp = (im_m * im_m) / (0.0d0 - 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 tmp;
	if (im_m <= 5e-105) {
		tmp = 0.0 - im_m;
	} else {
		tmp = (im_m * im_m) / (0.0 - 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):
	tmp = 0
	if im_m <= 5e-105:
		tmp = 0.0 - im_m
	else:
		tmp = (im_m * im_m) / (0.0 - im_m)
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 5e-105)
		tmp = Float64(0.0 - im_m);
	else
		tmp = Float64(Float64(im_m * im_m) / Float64(0.0 - 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)
	tmp = 0.0;
	if (im_m <= 5e-105)
		tmp = 0.0 - im_m;
	else
		tmp = (im_m * im_m) / (0.0 - 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_] := N[(im$95$s * If[LessEqual[im$95$m, 5e-105], N[(0.0 - im$95$m), $MachinePrecision], N[(N[(im$95$m * im$95$m), $MachinePrecision] / N[(0.0 - im$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 4.99999999999999963e-105

   1. Initial program 45.8%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(im \cdot \cos re\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{im \cdot \cos re} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(im \cdot \cos re\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \color{blue}{\cos re}\right)\right) \]

      5. cos-lowering-cos.f64 60.4%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \mathsf{cos.f64}\left(re\right)\right)\right) \]

   5. Simplified 60.4%

      \[\leadsto \color{blue}{0 - im \cdot \cos re} \]

   6. Taylor expanded in re around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(im\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 34.9%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{im}\right) \]

   8. Simplified 34.9%

      \[\leadsto \color{blue}{0 - im} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(im\right) \]

      2. neg-lowering-neg.f64 34.9%

         \[\leadsto \mathsf{neg.f64}\left(im\right) \]

   10. Applied egg-rr 34.9%

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

   if 4.99999999999999963e-105 < im

   1. Initial program 75.1%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(im \cdot \cos re\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{im \cdot \cos re} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(im \cdot \cos re\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \color{blue}{\cos re}\right)\right) \]

      5. cos-lowering-cos.f64 30.3%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \mathsf{cos.f64}\left(re\right)\right)\right) \]

   5. Simplified 30.3%

      \[\leadsto \color{blue}{0 - im \cdot \cos re} \]

   6. Taylor expanded in re around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(im\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 16.7%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{im}\right) \]

   8. Simplified 16.7%

      \[\leadsto \color{blue}{0 - im} \]
   9. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \frac{0 \cdot 0 - im \cdot im}{\color{blue}{0 + im}} \]

      2. +-lft-identity N/A

         \[\leadsto \frac{0 \cdot 0 - im \cdot im}{im} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(0 \cdot 0 - im \cdot im\right), \color{blue}{im}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(0 - im \cdot im\right), im\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(im \cdot im\right)\right), im\right) \]

      6. *-lowering-*.f64 37.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, im\right)\right), im\right) \]

   10. Applied egg-rr 37.2%

       \[\leadsto \color{blue}{\frac{0 - im \cdot im}{im}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 5 \cdot 10^{-105}:\\ \;\;\;\;0 - im\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot im}{0 - im}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 53.2% accurate, 34.3× speedup

Math

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

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 (+ -1.0 (* -0.16666666666666666 (* im_m im_m))))))

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 * (-1.0 + (-0.16666666666666666 * (im_m * im_m))));
}

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 * ((-1.0d0) + ((-0.16666666666666666d0) * (im_m * im_m))))
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 * (-1.0 + (-0.16666666666666666 * (im_m * im_m))));
}

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 * (-1.0 + (-0.16666666666666666 * (im_m * im_m))))

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(-1.0 + Float64(-0.16666666666666666 * Float64(im_m * im_m)))))
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 * (-1.0 + (-0.16666666666666666 * (im_m * im_m))));
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[(-1.0 + N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.6%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in re around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)} \]
4. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + \color{blue}{\left(\mathsf{neg}\left(e^{im}\right)\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(\mathsf{neg}\left(e^{im}\right)\right) + \color{blue}{e^{\mathsf{neg}\left(im\right)}}\right) \]

   3. distribute-rgt-in N/A

      \[\leadsto \left(\mathsf{neg}\left(e^{im}\right)\right) \cdot \frac{1}{2} + \color{blue}{e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}} \]

   4. *-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \left(\mathsf{neg}\left(e^{im}\right)\right) + \color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2} \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \left(\mathsf{neg}\left(e^{im}\right)\right)\right), \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)}\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left(e^{im}\right)\right) \cdot \frac{1}{2}\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

   7. distribute-lft-neg-out N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{neg}\left(e^{im} \cdot \frac{1}{2}\right)\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

   8. distribute-rgt-neg-in N/A

      \[\leadsto \mathsf{+.f64}\left(\left(e^{im} \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(\left(e^{im} \cdot \frac{-1}{2}\right), \left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(e^{im}\right), \frac{-1}{2}\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

   11. exp-lowering-exp.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(e^{\color{blue}{\mathsf{neg}\left(im\right)}} \cdot \frac{1}{2}\right)\right) \]

   12. exp-neg N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(\frac{1}{e^{im}} \cdot \frac{1}{2}\right)\right) \]

   13. associate-*l/ N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(\frac{1 \cdot \frac{1}{2}}{\color{blue}{e^{im}}}\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \left(\frac{\frac{1}{2}}{e^{\color{blue}{im}}}\right)\right) \]

   15. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{im}\right)}\right)\right) \]

   16. exp-lowering-exp.f64 40.3%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right)\right) \]

5. Simplified 40.3%

   \[\leadsto \color{blue}{e^{im} \cdot -0.5 + \frac{0.5}{e^{im}}} \]

6. Taylor expanded in im around 0

   \[\leadsto \color{blue}{im \cdot \left(\frac{-1}{6} \cdot {im}^{2} - 1\right)} \]
7. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2} - 1\right)}\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(im, \left(\frac{-1}{6} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(im, \left(\frac{-1}{6} \cdot {im}^{2} + -1\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(im, \left(-1 + \color{blue}{\frac{-1}{6} \cdot {im}^{2}}\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{-1}{6}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{-1}{6}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right) \]

   8. *-lowering-*.f64 52.8%

      \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

8. Simplified 52.8%

   \[\leadsto \color{blue}{im \cdot \left(-1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)} \]
9. Add Preprocessing

Alternative 21: 30.1% accurate, 103.0× speedup

Math

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

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 (- 0.0 im_m)))

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 - im_m);
}

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 - im_m)
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 - im_m);
}

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 - im_m)

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * Float64(0.0 - im_m))
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 - im_m);
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[(0.0 - im$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.6%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in im around 0

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

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(im \cdot \cos re\right) \]

   2. neg-sub0 N/A

      \[\leadsto 0 - \color{blue}{im \cdot \cos re} \]

   3. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(im \cdot \cos re\right)}\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \color{blue}{\cos re}\right)\right) \]

   5. cos-lowering-cos.f64 50.3%

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \mathsf{cos.f64}\left(re\right)\right)\right) \]

5. Simplified 50.3%

   \[\leadsto \color{blue}{0 - im \cdot \cos re} \]

6. Taylor expanded in re around 0

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

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(im\right) \]

   2. neg-sub0 N/A

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

   3. --lowering--.f64 28.7%

      \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{im}\right) \]

8. Simplified 28.7%

   \[\leadsto \color{blue}{0 - im} \]
9. Step-by-step derivation

   1. sub0-neg N/A

      \[\leadsto \mathsf{neg}\left(im\right) \]

   2. neg-lowering-neg.f64 28.7%

      \[\leadsto \mathsf{neg.f64}\left(im\right) \]

10. Applied egg-rr 28.7%

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

11. Final simplification 28.7%

    \[\leadsto 0 - im \]
12. Add Preprocessing

Developer Target 1: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|im\right| < 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(0.16666666666666666 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (< (fabs im) 1.0)
   (-
    (*
     (cos re)
     (+
      (+ im (* (* (* 0.16666666666666666 im) im) im))
      (* (* (* (* (* 0.008333333333333333 im) im) im) im) im))))
   (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if (fabs(im) < 1.0) {
		tmp = -(cos(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (abs(im) < 1.0d0) then
        tmp = -(cos(re) * ((im + (((0.16666666666666666d0 * im) * im) * im)) + (((((0.008333333333333333d0 * im) * im) * im) * im) * im)))
    else
        tmp = (0.5d0 * cos(re)) * (exp((0.0d0 - im)) - exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.abs(im) < 1.0) {
		tmp = -(Math.cos(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * Math.cos(re)) * (Math.exp((0.0 - im)) - Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.fabs(im) < 1.0:
		tmp = -(math.cos(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)))
	else:
		tmp = (0.5 * math.cos(re)) * (math.exp((0.0 - im)) - math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (abs(im) < 1.0)
		tmp = Float64(-Float64(cos(re) * Float64(Float64(im + Float64(Float64(Float64(0.16666666666666666 * im) * im) * im)) + Float64(Float64(Float64(Float64(Float64(0.008333333333333333 * im) * im) * im) * im) * im))));
	else
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - im)) - exp(im)));
	end
	return tmp
end

MATLAB

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

Wolfram

code[re_, im_] := If[Less[N[Abs[im], $MachinePrecision], 1.0], (-N[(N[Cos[re], $MachinePrecision] * N[(N[(im + N[(N[(N[(0.16666666666666666 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[(0.008333333333333333 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024154 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (fabs im) 1) (- (* (cos re) (+ im (* 1/6 im im im) (* 1/120 im im im im im)))) (* (* 1/2 (cos re)) (- (exp (- 0 im)) (exp im)))))

  (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))