math.sin on complex, imaginary part

Percentage Accurate: 54.7% → 99.9%

Time: 12.8s

Alternatives: 19

Speedup: 17.2×

• 50.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.7% 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: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{0 - im\_m} - e^{im\_m}\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -0.2:\\ \;\;\;\;t\_0 \cdot \left(0.5 \cdot \cos re\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(\cos re \cdot \left(-1 + \left(im\_m \cdot im\_m\right) \cdot \left(-0.16666666666666666 + im\_m \cdot \left(im\_m \cdot \left(-0.008333333333333333 + \left(im\_m \cdot im\_m\right) \cdot -0.0001984126984126984\right)\right)\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

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 -0.2)
      (* t_0 (* 0.5 (cos 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 t_0 = exp((0.0 - im_m)) - exp(im_m);
	double tmp;
	if (t_0 <= -0.2) {
		tmp = t_0 * (0.5 * cos(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) :: t_0
    real(8) :: tmp
    t_0 = exp((0.0d0 - im_m)) - exp(im_m)
    if (t_0 <= (-0.2d0)) then
        tmp = t_0 * (0.5d0 * cos(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 t_0 = Math.exp((0.0 - im_m)) - Math.exp(im_m);
	double tmp;
	if (t_0 <= -0.2) {
		tmp = t_0 * (0.5 * Math.cos(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):
	t_0 = math.exp((0.0 - im_m)) - math.exp(im_m)
	tmp = 0
	if t_0 <= -0.2:
		tmp = t_0 * (0.5 * math.cos(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)
	t_0 = Float64(exp(Float64(0.0 - im_m)) - exp(im_m))
	tmp = 0.0
	if (t_0 <= -0.2)
		tmp = Float64(t_0 * Float64(0.5 * cos(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(Float64(im_m * 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)
	t_0 = exp((0.0 - im_m)) - exp(im_m);
	tmp = 0.0;
	if (t_0 <= -0.2)
		tmp = t_0 * (0.5 * cos(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_] := 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, -0.2], N[(t$95$0 * N[(0.5 * N[Cos[re], $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[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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)) < -0.20000000000000001

   1. Initial program 100.0%

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

   if -0.20000000000000001 < (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))

   1. Initial program 40.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 94.4%

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

   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. *-lowering-*.f64 N/A

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

      22. unpow2 N/A

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

      23. *-lowering-*.f64 94.4%

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

   8. Simplified 94.4%

      \[\leadsto im \cdot \left(\cos re \cdot \color{blue}{\left(-1 + \left(im \cdot im\right) \cdot \left(-0.16666666666666666 + im \cdot \left(im \cdot \left(-0.008333333333333333 + \left(im \cdot im\right) \cdot -0.0001984126984126984\right)\right)\right)\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{0 - im} - e^{im} \leq -0.2:\\ \;\;\;\;\left(e^{0 - im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\cos re \cdot \left(-1 + \left(im \cdot im\right) \cdot \left(-0.16666666666666666 + im \cdot \left(im \cdot \left(-0.008333333333333333 + \left(im \cdot im\right) \cdot -0.0001984126984126984\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := im\_m \cdot \left(\cos re \cdot \left(-1 + \left(im\_m \cdot im\_m\right) \cdot \left(-0.16666666666666666 + im\_m \cdot \left(im\_m \cdot \left(-0.008333333333333333 + \left(im\_m \cdot im\_m\right) \cdot -0.0001984126984126984\right)\right)\right)\right)\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 6.2:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im\_m \leq 3.8 \cdot 10^{+44}:\\ \;\;\;\;0.5 \cdot \left(1 - e^{im\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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
          (*
           (cos re)
           (+
            -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 6.2)
      t_0
      (if (<= im_m 3.8e+44) (* 0.5 (- 1.0 (exp 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 * (cos(re) * (-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 <= 6.2) {
		tmp = t_0;
	} else if (im_m <= 3.8e+44) {
		tmp = 0.5 * (1.0 - exp(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 * (cos(re) * ((-1.0d0) + ((im_m * im_m) * ((-0.16666666666666666d0) + (im_m * (im_m * ((-0.008333333333333333d0) + ((im_m * im_m) * (-0.0001984126984126984d0)))))))))
    if (im_m <= 6.2d0) then
        tmp = t_0
    else if (im_m <= 3.8d+44) then
        tmp = 0.5d0 * (1.0d0 - exp(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 * (Math.cos(re) * (-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 <= 6.2) {
		tmp = t_0;
	} else if (im_m <= 3.8e+44) {
		tmp = 0.5 * (1.0 - Math.exp(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 * (math.cos(re) * (-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 <= 6.2:
		tmp = t_0
	elif im_m <= 3.8e+44:
		tmp = 0.5 * (1.0 - math.exp(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(cos(re) * Float64(-1.0 + Float64(Float64(im_m * im_m) * Float64(-0.16666666666666666 + Float64(im_m * Float64(im_m * Float64(-0.008333333333333333 + Float64(Float64(im_m * im_m) * -0.0001984126984126984)))))))))
	tmp = 0.0
	if (im_m <= 6.2)
		tmp = t_0;
	elseif (im_m <= 3.8e+44)
		tmp = Float64(0.5 * Float64(1.0 - exp(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 * (cos(re) * (-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 <= 6.2)
		tmp = t_0;
	elseif (im_m <= 3.8e+44)
		tmp = 0.5 * (1.0 - exp(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[(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[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[im$95$m, 6.2], t$95$0, If[LessEqual[im$95$m, 3.8e+44], N[(0.5 * N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 6.20000000000000018 or 3.8000000000000002e44 < im

   1. Initial program 53.2%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \cos re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \cos re + {im}^{2} \cdot \left(\frac{-1}{120} \cdot \cos re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)\right)} \]
   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 95.6%

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

   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. *-lowering-*.f64 N/A

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

      22. unpow2 N/A

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

      23. *-lowering-*.f64 95.6%

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

   8. Simplified 95.6%

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

   if 6.20000000000000018 < im < 3.8000000000000002e44

   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

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\frac{1}{2}}, \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, im\right)\right), \mathsf{exp.f64}\left(im\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 83.3%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 83.3%

      \[\leadsto 0.5 \cdot \left(\color{blue}{1} - e^{im}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 97.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := im\_m \cdot \left(\cos re \cdot \left(-1 + \left(im\_m \cdot im\_m\right) \cdot \left(-0.16666666666666666 + \left(im\_m \cdot im\_m\right) \cdot -0.008333333333333333\right)\right)\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 6:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im\_m \leq 1.2 \cdot 10^{+62}:\\ \;\;\;\;0.5 \cdot \left(1 - e^{im\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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
          (*
           (cos re)
           (+
            -1.0
            (*
             (* im_m im_m)
             (+
              -0.16666666666666666
              (* (* im_m im_m) -0.008333333333333333))))))))
   (*
    im_s
    (if (<= im_m 6.0)
      t_0
      (if (<= im_m 1.2e+62) (* 0.5 (- 1.0 (exp 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 * (cos(re) * (-1.0 + ((im_m * im_m) * (-0.16666666666666666 + ((im_m * im_m) * -0.008333333333333333)))));
	double tmp;
	if (im_m <= 6.0) {
		tmp = t_0;
	} else if (im_m <= 1.2e+62) {
		tmp = 0.5 * (1.0 - exp(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 * (cos(re) * ((-1.0d0) + ((im_m * im_m) * ((-0.16666666666666666d0) + ((im_m * im_m) * (-0.008333333333333333d0))))))
    if (im_m <= 6.0d0) then
        tmp = t_0
    else if (im_m <= 1.2d+62) then
        tmp = 0.5d0 * (1.0d0 - exp(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 * (Math.cos(re) * (-1.0 + ((im_m * im_m) * (-0.16666666666666666 + ((im_m * im_m) * -0.008333333333333333)))));
	double tmp;
	if (im_m <= 6.0) {
		tmp = t_0;
	} else if (im_m <= 1.2e+62) {
		tmp = 0.5 * (1.0 - Math.exp(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 * (math.cos(re) * (-1.0 + ((im_m * im_m) * (-0.16666666666666666 + ((im_m * im_m) * -0.008333333333333333)))))
	tmp = 0
	if im_m <= 6.0:
		tmp = t_0
	elif im_m <= 1.2e+62:
		tmp = 0.5 * (1.0 - math.exp(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(cos(re) * Float64(-1.0 + Float64(Float64(im_m * im_m) * Float64(-0.16666666666666666 + Float64(Float64(im_m * im_m) * -0.008333333333333333))))))
	tmp = 0.0
	if (im_m <= 6.0)
		tmp = t_0;
	elseif (im_m <= 1.2e+62)
		tmp = Float64(0.5 * Float64(1.0 - exp(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 * (cos(re) * (-1.0 + ((im_m * im_m) * (-0.16666666666666666 + ((im_m * im_m) * -0.008333333333333333)))));
	tmp = 0.0;
	if (im_m <= 6.0)
		tmp = t_0;
	elseif (im_m <= 1.2e+62)
		tmp = 0.5 * (1.0 - exp(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[(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]}, N[(im$95$s * If[LessEqual[im$95$m, 6.0], t$95$0, If[LessEqual[im$95$m, 1.2e+62], N[(0.5 * N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 6 or 1.2e62 < im

   1. Initial program 52.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 93.7%

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

   if 6 < im < 1.2e62

   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

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\frac{1}{2}}, \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, im\right)\right), \mathsf{exp.f64}\left(im\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 91.7%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 91.7%

      \[\leadsto 0.5 \cdot \left(\color{blue}{1} - e^{im}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 95.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 4.5:\\ \;\;\;\;\cos re \cdot \left(im\_m \cdot \left(\left(im\_m \cdot im\_m\right) \cdot -0.16666666666666666\right) - im\_m\right)\\ \mathbf{elif}\;im\_m \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \left(1 - e^{im\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im\_m \cdot \left(im\_m \cdot im\_m\right)\right) \cdot \left(\cos re \cdot \left(-0.16666666666666666 + \frac{-1}{im\_m \cdot im\_m}\right)\right)\\ \end{array} \end{array} \]

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 4.5)
    (* (cos re) (- (* im_m (* (* im_m im_m) -0.16666666666666666)) im_m))
    (if (<= im_m 5.6e+102)
      (* 0.5 (- 1.0 (exp im_m)))
      (*
       (* im_m (* im_m im_m))
       (* (cos re) (+ -0.16666666666666666 (/ -1.0 (* 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 tmp;
	if (im_m <= 4.5) {
		tmp = cos(re) * ((im_m * ((im_m * im_m) * -0.16666666666666666)) - im_m);
	} else if (im_m <= 5.6e+102) {
		tmp = 0.5 * (1.0 - exp(im_m));
	} else {
		tmp = (im_m * (im_m * im_m)) * (cos(re) * (-0.16666666666666666 + (-1.0 / (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) :: tmp
    if (im_m <= 4.5d0) then
        tmp = cos(re) * ((im_m * ((im_m * im_m) * (-0.16666666666666666d0))) - im_m)
    else if (im_m <= 5.6d+102) then
        tmp = 0.5d0 * (1.0d0 - exp(im_m))
    else
        tmp = (im_m * (im_m * im_m)) * (cos(re) * ((-0.16666666666666666d0) + ((-1.0d0) / (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 tmp;
	if (im_m <= 4.5) {
		tmp = Math.cos(re) * ((im_m * ((im_m * im_m) * -0.16666666666666666)) - im_m);
	} else if (im_m <= 5.6e+102) {
		tmp = 0.5 * (1.0 - Math.exp(im_m));
	} else {
		tmp = (im_m * (im_m * im_m)) * (Math.cos(re) * (-0.16666666666666666 + (-1.0 / (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):
	tmp = 0
	if im_m <= 4.5:
		tmp = math.cos(re) * ((im_m * ((im_m * im_m) * -0.16666666666666666)) - im_m)
	elif im_m <= 5.6e+102:
		tmp = 0.5 * (1.0 - math.exp(im_m))
	else:
		tmp = (im_m * (im_m * im_m)) * (math.cos(re) * (-0.16666666666666666 + (-1.0 / (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)
	tmp = 0.0
	if (im_m <= 4.5)
		tmp = Float64(cos(re) * Float64(Float64(im_m * Float64(Float64(im_m * im_m) * -0.16666666666666666)) - im_m));
	elseif (im_m <= 5.6e+102)
		tmp = Float64(0.5 * Float64(1.0 - exp(im_m)));
	else
		tmp = Float64(Float64(im_m * Float64(im_m * im_m)) * Float64(cos(re) * Float64(-0.16666666666666666 + Float64(-1.0 / 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)
	tmp = 0.0;
	if (im_m <= 4.5)
		tmp = cos(re) * ((im_m * ((im_m * im_m) * -0.16666666666666666)) - im_m);
	elseif (im_m <= 5.6e+102)
		tmp = 0.5 * (1.0 - exp(im_m));
	else
		tmp = (im_m * (im_m * im_m)) * (cos(re) * (-0.16666666666666666 + (-1.0 / (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_] := N[(im$95$s * If[LessEqual[im$95$m, 4.5], N[(N[Cos[re], $MachinePrecision] * N[(N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] - im$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 5.6e+102], N[(0.5 * N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im$95$m * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * N[(-0.16666666666666666 + N[(-1.0 / N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 4.5

   1. Initial program 40.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \cos re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \cos re\right)\right)} \]
   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 85.0%

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. neg-mul-1 N/A

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

      4. unsub-neg N/A

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

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

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

      6. *-commutative N/A

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

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

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

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

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

      9. *-lowering-*.f64 85.0%

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

   7. Applied egg-rr 85.0%

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

   if 4.5 < im < 5.60000000000000037e102

   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

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\frac{1}{2}}, \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, im\right)\right), \mathsf{exp.f64}\left(im\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 81.3%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 81.3%

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

   if 5.60000000000000037e102 < 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

      \[\leadsto \color{blue}{{im}^{3} \cdot \left(-1 \cdot \frac{\cos re}{{im}^{2}} + \frac{-1}{6} \cdot \cos re\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\left({im}^{3}\right), \color{blue}{\left(-1 \cdot \frac{\cos re}{{im}^{2}} + \frac{-1}{6} \cdot \cos re\right)}\right) \]

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. associate-*r/ N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

      12. neg-mul-1 N/A

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

      13. *-commutative N/A

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

      14. distribute-lft-out N/A

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

      15. metadata-eval N/A

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

      16. distribute-lft-in N/A

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

      17. +-commutative N/A

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

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

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

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

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

      20. distribute-lft-in N/A

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

      21. metadata-eval N/A

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

   8. Simplified 100.0%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.5:\\ \;\;\;\;\cos re \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot -0.16666666666666666\right) - im\right)\\ \mathbf{elif}\;im \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \left(1 - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \left(im \cdot im\right)\right) \cdot \left(\cos re \cdot \left(-0.16666666666666666 + \frac{-1}{im \cdot im}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(im\_m \cdot im\_m\right) \cdot -0.16666666666666666\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 3.7:\\ \;\;\;\;\cos re \cdot \left(im\_m \cdot t\_0 - im\_m\right)\\ \mathbf{elif}\;im\_m \leq 5.8 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \left(1 - e^{im\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(im\_m \cdot \left(-1 + t\_0\right)\right)\\ \end{array} \end{array} \end{array} \]

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 (* (* im_m im_m) -0.16666666666666666)))
   (*
    im_s
    (if (<= im_m 3.7)
      (* (cos re) (- (* im_m t_0) im_m))
      (if (<= im_m 5.8e+102)
        (* 0.5 (- 1.0 (exp im_m)))
        (* (cos re) (* im_m (+ -1.0 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 * im_m) * -0.16666666666666666;
	double tmp;
	if (im_m <= 3.7) {
		tmp = cos(re) * ((im_m * t_0) - im_m);
	} else if (im_m <= 5.8e+102) {
		tmp = 0.5 * (1.0 - exp(im_m));
	} else {
		tmp = cos(re) * (im_m * (-1.0 + 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 * im_m) * (-0.16666666666666666d0)
    if (im_m <= 3.7d0) then
        tmp = cos(re) * ((im_m * t_0) - im_m)
    else if (im_m <= 5.8d+102) then
        tmp = 0.5d0 * (1.0d0 - exp(im_m))
    else
        tmp = cos(re) * (im_m * ((-1.0d0) + 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 * im_m) * -0.16666666666666666;
	double tmp;
	if (im_m <= 3.7) {
		tmp = Math.cos(re) * ((im_m * t_0) - im_m);
	} else if (im_m <= 5.8e+102) {
		tmp = 0.5 * (1.0 - Math.exp(im_m));
	} else {
		tmp = Math.cos(re) * (im_m * (-1.0 + 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 * im_m) * -0.16666666666666666
	tmp = 0
	if im_m <= 3.7:
		tmp = math.cos(re) * ((im_m * t_0) - im_m)
	elif im_m <= 5.8e+102:
		tmp = 0.5 * (1.0 - math.exp(im_m))
	else:
		tmp = math.cos(re) * (im_m * (-1.0 + 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(Float64(im_m * im_m) * -0.16666666666666666)
	tmp = 0.0
	if (im_m <= 3.7)
		tmp = Float64(cos(re) * Float64(Float64(im_m * t_0) - im_m));
	elseif (im_m <= 5.8e+102)
		tmp = Float64(0.5 * Float64(1.0 - exp(im_m)));
	else
		tmp = Float64(cos(re) * Float64(im_m * Float64(-1.0 + 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 * im_m) * -0.16666666666666666;
	tmp = 0.0;
	if (im_m <= 3.7)
		tmp = cos(re) * ((im_m * t_0) - im_m);
	elseif (im_m <= 5.8e+102)
		tmp = 0.5 * (1.0 - exp(im_m));
	else
		tmp = cos(re) * (im_m * (-1.0 + 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[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]}, N[(im$95$s * If[LessEqual[im$95$m, 3.7], N[(N[Cos[re], $MachinePrecision] * N[(N[(im$95$m * t$95$0), $MachinePrecision] - im$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 5.8e+102], N[(0.5 * N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(im$95$m * N[(-1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if im < 3.7000000000000002

   1. Initial program 40.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \cos re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \cos re\right)\right)} \]
   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 85.0%

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. neg-mul-1 N/A

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

      4. unsub-neg N/A

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

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

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

      6. *-commutative N/A

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

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

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

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

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

      9. *-lowering-*.f64 85.0%

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

   7. Applied egg-rr 85.0%

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

   if 3.7000000000000002 < im < 5.8000000000000005e102

   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

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\frac{1}{2}}, \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, im\right)\right), \mathsf{exp.f64}\left(im\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 81.3%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 81.3%

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

   if 5.8000000000000005e102 < 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

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \cos re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \cos re\right)\right)} \]
   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 100.0%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.7:\\ \;\;\;\;\cos re \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot -0.16666666666666666\right) - im\right)\\ \mathbf{elif}\;im \leq 5.8 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \left(1 - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(im \cdot \left(-1 + \left(im \cdot im\right) \cdot -0.16666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \cos re \cdot \left(im\_m \cdot \left(-1 + \left(im\_m \cdot im\_m\right) \cdot -0.16666666666666666\right)\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 3.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im\_m \leq 5.8 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \left(1 - e^{im\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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
         (*
          (cos re)
          (* im_m (+ -1.0 (* (* im_m im_m) -0.16666666666666666))))))
   (*
    im_s
    (if (<= im_m 3.5)
      t_0
      (if (<= im_m 5.8e+102) (* 0.5 (- 1.0 (exp 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 = cos(re) * (im_m * (-1.0 + ((im_m * im_m) * -0.16666666666666666)));
	double tmp;
	if (im_m <= 3.5) {
		tmp = t_0;
	} else if (im_m <= 5.8e+102) {
		tmp = 0.5 * (1.0 - exp(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 = cos(re) * (im_m * ((-1.0d0) + ((im_m * im_m) * (-0.16666666666666666d0))))
    if (im_m <= 3.5d0) then
        tmp = t_0
    else if (im_m <= 5.8d+102) then
        tmp = 0.5d0 * (1.0d0 - exp(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 = Math.cos(re) * (im_m * (-1.0 + ((im_m * im_m) * -0.16666666666666666)));
	double tmp;
	if (im_m <= 3.5) {
		tmp = t_0;
	} else if (im_m <= 5.8e+102) {
		tmp = 0.5 * (1.0 - Math.exp(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 = math.cos(re) * (im_m * (-1.0 + ((im_m * im_m) * -0.16666666666666666)))
	tmp = 0
	if im_m <= 3.5:
		tmp = t_0
	elif im_m <= 5.8e+102:
		tmp = 0.5 * (1.0 - math.exp(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(cos(re) * Float64(im_m * Float64(-1.0 + Float64(Float64(im_m * im_m) * -0.16666666666666666))))
	tmp = 0.0
	if (im_m <= 3.5)
		tmp = t_0;
	elseif (im_m <= 5.8e+102)
		tmp = Float64(0.5 * Float64(1.0 - exp(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 = cos(re) * (im_m * (-1.0 + ((im_m * im_m) * -0.16666666666666666)));
	tmp = 0.0;
	if (im_m <= 3.5)
		tmp = t_0;
	elseif (im_m <= 5.8e+102)
		tmp = 0.5 * (1.0 - exp(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[(N[Cos[re], $MachinePrecision] * N[(im$95$m * N[(-1.0 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[im$95$m, 3.5], t$95$0, If[LessEqual[im$95$m, 5.8e+102], N[(0.5 * N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 3.5 or 5.8000000000000005e102 < im

   1. Initial program 51.2%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 87.8%

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

   if 3.5 < im < 5.8000000000000005e102

   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

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\frac{1}{2}}, \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, im\right)\right), \mathsf{exp.f64}\left(im\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 81.3%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 81.3%

      \[\leadsto 0.5 \cdot \left(\color{blue}{1} - e^{im}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.5:\\ \;\;\;\;\cos re \cdot \left(im \cdot \left(-1 + \left(im \cdot im\right) \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;im \leq 5.8 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \left(1 - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(im \cdot \left(-1 + \left(im \cdot im\right) \cdot -0.16666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 2.3:\\ \;\;\;\;im\_m \cdot \left(0 - \cos re\right)\\ \mathbf{elif}\;im\_m \leq 2.45 \cdot 10^{+151}:\\ \;\;\;\;0.5 \cdot \left(1 - e^{im\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos re \cdot \left(im\_m \cdot im\_m\right)}{0 - im\_m}\\ \end{array} \end{array} \]

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 2.3)
    (* im_m (- 0.0 (cos re)))
    (if (<= im_m 2.45e+151)
      (* 0.5 (- 1.0 (exp im_m)))
      (/ (* (cos re) (* 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 <= 2.3) {
		tmp = im_m * (0.0 - cos(re));
	} else if (im_m <= 2.45e+151) {
		tmp = 0.5 * (1.0 - exp(im_m));
	} else {
		tmp = (cos(re) * (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 <= 2.3d0) then
        tmp = im_m * (0.0d0 - cos(re))
    else if (im_m <= 2.45d+151) then
        tmp = 0.5d0 * (1.0d0 - exp(im_m))
    else
        tmp = (cos(re) * (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 <= 2.3) {
		tmp = im_m * (0.0 - Math.cos(re));
	} else if (im_m <= 2.45e+151) {
		tmp = 0.5 * (1.0 - Math.exp(im_m));
	} else {
		tmp = (Math.cos(re) * (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 <= 2.3:
		tmp = im_m * (0.0 - math.cos(re))
	elif im_m <= 2.45e+151:
		tmp = 0.5 * (1.0 - math.exp(im_m))
	else:
		tmp = (math.cos(re) * (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 <= 2.3)
		tmp = Float64(im_m * Float64(0.0 - cos(re)));
	elseif (im_m <= 2.45e+151)
		tmp = Float64(0.5 * Float64(1.0 - exp(im_m)));
	else
		tmp = Float64(Float64(cos(re) * 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 <= 2.3)
		tmp = im_m * (0.0 - cos(re));
	elseif (im_m <= 2.45e+151)
		tmp = 0.5 * (1.0 - exp(im_m));
	else
		tmp = (cos(re) * (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, 2.3], N[(im$95$m * N[(0.0 - N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 2.45e+151], N[(0.5 * N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[re], $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] / N[(0.0 - im$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 2.2999999999999998

   1. Initial program 40.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 66.7%

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

   5. Simplified 66.7%

      \[\leadsto \color{blue}{0 - im \cdot \cos re} \]
   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 66.7%

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

   7. Applied egg-rr 66.7%

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

   if 2.2999999999999998 < im < 2.45e151

   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

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\frac{1}{2}}, \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, im\right)\right), \mathsf{exp.f64}\left(im\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 76.7%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 76.7%

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

   if 2.45e151 < 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

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \cos re\right)} \]
   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 7.7%

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

   5. Simplified 7.7%

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

      1. cancel-sign-sub-inv N/A

         \[\leadsto 0 + \color{blue}{\left(\mathsf{neg}\left(im\right)\right) \cdot \cos re} \]

      2. +-lft-identity N/A

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

      3. sub0-neg N/A

         \[\leadsto \left(0 - im\right) \cdot \cos \color{blue}{re} \]

      4. flip-- N/A

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

      5. +-lft-identity N/A

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

      6. associate-*l/ N/A

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

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

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

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

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

      9. metadata-eval N/A

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

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

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

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

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

      12. cos-lowering-cos.f64 100.0%

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

   7. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\left(0 - im \cdot im\right) \cdot \cos re}{im}} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

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

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

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

      3. *-lowering-*.f64 100.0%

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

   9. Applied egg-rr 100.0%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.3:\\ \;\;\;\;im \cdot \left(0 - \cos re\right)\\ \mathbf{elif}\;im \leq 2.45 \cdot 10^{+151}:\\ \;\;\;\;0.5 \cdot \left(1 - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos re \cdot \left(im \cdot im\right)}{0 - im}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 87.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 3.8:\\ \;\;\;\;im\_m \cdot \left(0 - \cos re\right)\\ \mathbf{elif}\;im\_m \leq 2 \cdot 10^{+130}:\\ \;\;\;\;0.5 \cdot \left(1 - e^{im\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + \left(re \cdot re\right) \cdot -0.25\right) \cdot \left(im\_m \cdot \left(-2 + im\_m \cdot \left(im\_m \cdot -0.3333333333333333\right)\right)\right)\\ \end{array} \end{array} \]

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 3.8)
    (* im_m (- 0.0 (cos re)))
    (if (<= im_m 2e+130)
      (* 0.5 (- 1.0 (exp im_m)))
      (*
       (+ 0.5 (* (* re re) -0.25))
       (* im_m (+ -2.0 (* im_m (* im_m -0.3333333333333333)))))))))

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 <= 3.8) {
		tmp = im_m * (0.0 - cos(re));
	} else if (im_m <= 2e+130) {
		tmp = 0.5 * (1.0 - exp(im_m));
	} else {
		tmp = (0.5 + ((re * re) * -0.25)) * (im_m * (-2.0 + (im_m * (im_m * -0.3333333333333333))));
	}
	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 <= 3.8d0) then
        tmp = im_m * (0.0d0 - cos(re))
    else if (im_m <= 2d+130) then
        tmp = 0.5d0 * (1.0d0 - exp(im_m))
    else
        tmp = (0.5d0 + ((re * re) * (-0.25d0))) * (im_m * ((-2.0d0) + (im_m * (im_m * (-0.3333333333333333d0)))))
    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 <= 3.8) {
		tmp = im_m * (0.0 - Math.cos(re));
	} else if (im_m <= 2e+130) {
		tmp = 0.5 * (1.0 - Math.exp(im_m));
	} else {
		tmp = (0.5 + ((re * re) * -0.25)) * (im_m * (-2.0 + (im_m * (im_m * -0.3333333333333333))));
	}
	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 <= 3.8:
		tmp = im_m * (0.0 - math.cos(re))
	elif im_m <= 2e+130:
		tmp = 0.5 * (1.0 - math.exp(im_m))
	else:
		tmp = (0.5 + ((re * re) * -0.25)) * (im_m * (-2.0 + (im_m * (im_m * -0.3333333333333333))))
	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 <= 3.8)
		tmp = Float64(im_m * Float64(0.0 - cos(re)));
	elseif (im_m <= 2e+130)
		tmp = Float64(0.5 * Float64(1.0 - exp(im_m)));
	else
		tmp = Float64(Float64(0.5 + Float64(Float64(re * re) * -0.25)) * Float64(im_m * Float64(-2.0 + Float64(im_m * Float64(im_m * -0.3333333333333333)))));
	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 <= 3.8)
		tmp = im_m * (0.0 - cos(re));
	elseif (im_m <= 2e+130)
		tmp = 0.5 * (1.0 - exp(im_m));
	else
		tmp = (0.5 + ((re * re) * -0.25)) * (im_m * (-2.0 + (im_m * (im_m * -0.3333333333333333))));
	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, 3.8], N[(im$95$m * N[(0.0 - N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 2e+130], N[(0.5 * N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 + N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision] * N[(im$95$m * N[(-2.0 + N[(im$95$m * N[(im$95$m * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 3.7999999999999998

   1. Initial program 40.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 66.7%

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

   5. Simplified 66.7%

      \[\leadsto \color{blue}{0 - im \cdot \cos re} \]
   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 66.7%

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

   7. Applied egg-rr 66.7%

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

   if 3.7999999999999998 < im < 2.0000000000000001e130

   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

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\frac{1}{2}}, \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, im\right)\right), \mathsf{exp.f64}\left(im\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 76.9%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 76.9%

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

   if 2.0000000000000001e130 < 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 re around 0

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

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

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

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left({re}^{2} \cdot \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{3}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({re}^{2}\right), \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{3}\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{3}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 80.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{3}\right)\right)\right)\right)\right) \]

   8. Simplified 80.0%

      \[\leadsto \color{blue}{\left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)} \cdot \left(im \cdot \left(-2 + im \cdot \left(im \cdot -0.3333333333333333\right)\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.8:\\ \;\;\;\;im \cdot \left(0 - \cos re\right)\\ \mathbf{elif}\;im \leq 2 \cdot 10^{+130}:\\ \;\;\;\;0.5 \cdot \left(1 - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + \left(re \cdot re\right) \cdot -0.25\right) \cdot \left(im \cdot \left(-2 + im \cdot \left(im \cdot -0.3333333333333333\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 0.0067:\\ \;\;\;\;im\_m \cdot \left(0 - \cos re\right)\\ \mathbf{elif}\;im\_m \leq 3.5 \cdot 10^{+130}:\\ \;\;\;\;im\_m \cdot \left(\left(-1 + \left(im\_m \cdot im\_m\right) \cdot \left(-0.16666666666666666 + im\_m \cdot \left(im\_m \cdot \left(-0.008333333333333333 + \left(im\_m \cdot im\_m\right) \cdot -0.0001984126984126984\right)\right)\right)\right) \cdot \left(1 + re \cdot \left(re \cdot \left(-0.5 + \left(re \cdot re\right) \cdot 0.041666666666666664\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + \left(re \cdot re\right) \cdot -0.25\right) \cdot \left(im\_m \cdot \left(-2 + im\_m \cdot \left(im\_m \cdot -0.3333333333333333\right)\right)\right)\\ \end{array} \end{array} \]

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 0.0067)
    (* im_m (- 0.0 (cos re)))
    (if (<= im_m 3.5e+130)
      (*
       im_m
       (*
        (+
         -1.0
         (*
          (* im_m im_m)
          (+
           -0.16666666666666666
           (*
            im_m
            (*
             im_m
             (+
              -0.008333333333333333
              (* (* im_m im_m) -0.0001984126984126984)))))))
        (+ 1.0 (* re (* re (+ -0.5 (* (* re re) 0.041666666666666664)))))))
      (*
       (+ 0.5 (* (* re re) -0.25))
       (* im_m (+ -2.0 (* im_m (* im_m -0.3333333333333333)))))))))

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 <= 0.0067) {
		tmp = im_m * (0.0 - cos(re));
	} else if (im_m <= 3.5e+130) {
		tmp = im_m * ((-1.0 + ((im_m * im_m) * (-0.16666666666666666 + (im_m * (im_m * (-0.008333333333333333 + ((im_m * im_m) * -0.0001984126984126984))))))) * (1.0 + (re * (re * (-0.5 + ((re * re) * 0.041666666666666664))))));
	} else {
		tmp = (0.5 + ((re * re) * -0.25)) * (im_m * (-2.0 + (im_m * (im_m * -0.3333333333333333))));
	}
	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 <= 0.0067d0) then
        tmp = im_m * (0.0d0 - cos(re))
    else if (im_m <= 3.5d+130) then
        tmp = im_m * (((-1.0d0) + ((im_m * im_m) * ((-0.16666666666666666d0) + (im_m * (im_m * ((-0.008333333333333333d0) + ((im_m * im_m) * (-0.0001984126984126984d0)))))))) * (1.0d0 + (re * (re * ((-0.5d0) + ((re * re) * 0.041666666666666664d0))))))
    else
        tmp = (0.5d0 + ((re * re) * (-0.25d0))) * (im_m * ((-2.0d0) + (im_m * (im_m * (-0.3333333333333333d0)))))
    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 <= 0.0067) {
		tmp = im_m * (0.0 - Math.cos(re));
	} else if (im_m <= 3.5e+130) {
		tmp = im_m * ((-1.0 + ((im_m * im_m) * (-0.16666666666666666 + (im_m * (im_m * (-0.008333333333333333 + ((im_m * im_m) * -0.0001984126984126984))))))) * (1.0 + (re * (re * (-0.5 + ((re * re) * 0.041666666666666664))))));
	} else {
		tmp = (0.5 + ((re * re) * -0.25)) * (im_m * (-2.0 + (im_m * (im_m * -0.3333333333333333))));
	}
	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 <= 0.0067:
		tmp = im_m * (0.0 - math.cos(re))
	elif im_m <= 3.5e+130:
		tmp = im_m * ((-1.0 + ((im_m * im_m) * (-0.16666666666666666 + (im_m * (im_m * (-0.008333333333333333 + ((im_m * im_m) * -0.0001984126984126984))))))) * (1.0 + (re * (re * (-0.5 + ((re * re) * 0.041666666666666664))))))
	else:
		tmp = (0.5 + ((re * re) * -0.25)) * (im_m * (-2.0 + (im_m * (im_m * -0.3333333333333333))))
	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 <= 0.0067)
		tmp = Float64(im_m * Float64(0.0 - cos(re)));
	elseif (im_m <= 3.5e+130)
		tmp = Float64(im_m * Float64(Float64(-1.0 + Float64(Float64(im_m * im_m) * Float64(-0.16666666666666666 + Float64(im_m * Float64(im_m * Float64(-0.008333333333333333 + Float64(Float64(im_m * im_m) * -0.0001984126984126984))))))) * Float64(1.0 + Float64(re * Float64(re * Float64(-0.5 + Float64(Float64(re * re) * 0.041666666666666664)))))));
	else
		tmp = Float64(Float64(0.5 + Float64(Float64(re * re) * -0.25)) * Float64(im_m * Float64(-2.0 + Float64(im_m * Float64(im_m * -0.3333333333333333)))));
	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 <= 0.0067)
		tmp = im_m * (0.0 - cos(re));
	elseif (im_m <= 3.5e+130)
		tmp = im_m * ((-1.0 + ((im_m * im_m) * (-0.16666666666666666 + (im_m * (im_m * (-0.008333333333333333 + ((im_m * im_m) * -0.0001984126984126984))))))) * (1.0 + (re * (re * (-0.5 + ((re * re) * 0.041666666666666664))))));
	else
		tmp = (0.5 + ((re * re) * -0.25)) * (im_m * (-2.0 + (im_m * (im_m * -0.3333333333333333))));
	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, 0.0067], N[(im$95$m * N[(0.0 - N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 3.5e+130], N[(im$95$m * N[(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[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(re * N[(re * N[(-0.5 + N[(N[(re * re), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 + N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision] * N[(im$95$m * N[(-2.0 + N[(im$95$m * N[(im$95$m * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 0.00670000000000000023

   1. Initial program 39.7%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \cos re\right)} \]
   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 66.8%

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

   5. Simplified 66.8%

      \[\leadsto \color{blue}{0 - im \cdot \cos re} \]
   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 66.8%

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

   7. Applied egg-rr 66.8%

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

   if 0.00670000000000000023 < im < 3.5000000000000001e130

   1. Initial program 99.8%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \cos re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \cos re + {im}^{2} \cdot \left(\frac{-1}{120} \cdot \cos re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)\right)} \]
   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 79.2%

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

   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. *-lowering-*.f64 N/A

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

      22. unpow2 N/A

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

      23. *-lowering-*.f64 79.2%

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

   8. Simplified 79.2%

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

   9. Taylor expanded in re around 0

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

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

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

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

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

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

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

       6. sub-neg N/A

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

       7. metadata-eval N/A

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

       8. +-commutative N/A

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

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

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

       10. *-commutative N/A

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

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

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

       12. unpow2 N/A

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

       13. *-lowering-*.f64 70.7%

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

   11. Simplified 70.7%

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

   if 3.5000000000000001e130 < 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 re around 0

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

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

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

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left({re}^{2} \cdot \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{3}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({re}^{2}\right), \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{3}\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{3}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 80.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{3}\right)\right)\right)\right)\right) \]

   8. Simplified 80.0%

      \[\leadsto \color{blue}{\left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)} \cdot \left(im \cdot \left(-2 + im \cdot \left(im \cdot -0.3333333333333333\right)\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.0067:\\ \;\;\;\;im \cdot \left(0 - \cos re\right)\\ \mathbf{elif}\;im \leq 3.5 \cdot 10^{+130}:\\ \;\;\;\;im \cdot \left(\left(-1 + \left(im \cdot im\right) \cdot \left(-0.16666666666666666 + im \cdot \left(im \cdot \left(-0.008333333333333333 + \left(im \cdot im\right) \cdot -0.0001984126984126984\right)\right)\right)\right) \cdot \left(1 + re \cdot \left(re \cdot \left(-0.5 + \left(re \cdot re\right) \cdot 0.041666666666666664\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + \left(re \cdot re\right) \cdot -0.25\right) \cdot \left(im \cdot \left(-2 + im \cdot \left(im \cdot -0.3333333333333333\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.9% accurate, 7.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;re \leq 5.4 \cdot 10^{+60}:\\ \;\;\;\;im\_m \cdot \left(\left(-1 + \left(im\_m \cdot im\_m\right) \cdot \left(-0.16666666666666666 + im\_m \cdot \left(im\_m \cdot \left(-0.008333333333333333 + \left(im\_m \cdot im\_m\right) \cdot -0.0001984126984126984\right)\right)\right)\right) \cdot \left(1 + re \cdot \left(re \cdot \left(-0.5 + \left(re \cdot re\right) \cdot 0.041666666666666664\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + \left(re \cdot re\right) \cdot -0.25\right) \cdot \left(im\_m \cdot \left(-2 + im\_m \cdot \left(im\_m \cdot -0.3333333333333333\right)\right)\right)\\ \end{array} \end{array} \]

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 (<= re 5.4e+60)
    (*
     im_m
     (*
      (+
       -1.0
       (*
        (* im_m im_m)
        (+
         -0.16666666666666666
         (*
          im_m
          (*
           im_m
           (+
            -0.008333333333333333
            (* (* im_m im_m) -0.0001984126984126984)))))))
      (+ 1.0 (* re (* re (+ -0.5 (* (* re re) 0.041666666666666664)))))))
    (*
     (+ 0.5 (* (* re re) -0.25))
     (* im_m (+ -2.0 (* im_m (* im_m -0.3333333333333333))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (re <= 5.4e+60) {
		tmp = im_m * ((-1.0 + ((im_m * im_m) * (-0.16666666666666666 + (im_m * (im_m * (-0.008333333333333333 + ((im_m * im_m) * -0.0001984126984126984))))))) * (1.0 + (re * (re * (-0.5 + ((re * re) * 0.041666666666666664))))));
	} else {
		tmp = (0.5 + ((re * re) * -0.25)) * (im_m * (-2.0 + (im_m * (im_m * -0.3333333333333333))));
	}
	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 (re <= 5.4d+60) then
        tmp = im_m * (((-1.0d0) + ((im_m * im_m) * ((-0.16666666666666666d0) + (im_m * (im_m * ((-0.008333333333333333d0) + ((im_m * im_m) * (-0.0001984126984126984d0)))))))) * (1.0d0 + (re * (re * ((-0.5d0) + ((re * re) * 0.041666666666666664d0))))))
    else
        tmp = (0.5d0 + ((re * re) * (-0.25d0))) * (im_m * ((-2.0d0) + (im_m * (im_m * (-0.3333333333333333d0)))))
    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 (re <= 5.4e+60) {
		tmp = im_m * ((-1.0 + ((im_m * im_m) * (-0.16666666666666666 + (im_m * (im_m * (-0.008333333333333333 + ((im_m * im_m) * -0.0001984126984126984))))))) * (1.0 + (re * (re * (-0.5 + ((re * re) * 0.041666666666666664))))));
	} else {
		tmp = (0.5 + ((re * re) * -0.25)) * (im_m * (-2.0 + (im_m * (im_m * -0.3333333333333333))));
	}
	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 re <= 5.4e+60:
		tmp = im_m * ((-1.0 + ((im_m * im_m) * (-0.16666666666666666 + (im_m * (im_m * (-0.008333333333333333 + ((im_m * im_m) * -0.0001984126984126984))))))) * (1.0 + (re * (re * (-0.5 + ((re * re) * 0.041666666666666664))))))
	else:
		tmp = (0.5 + ((re * re) * -0.25)) * (im_m * (-2.0 + (im_m * (im_m * -0.3333333333333333))))
	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 (re <= 5.4e+60)
		tmp = Float64(im_m * Float64(Float64(-1.0 + Float64(Float64(im_m * im_m) * Float64(-0.16666666666666666 + Float64(im_m * Float64(im_m * Float64(-0.008333333333333333 + Float64(Float64(im_m * im_m) * -0.0001984126984126984))))))) * Float64(1.0 + Float64(re * Float64(re * Float64(-0.5 + Float64(Float64(re * re) * 0.041666666666666664)))))));
	else
		tmp = Float64(Float64(0.5 + Float64(Float64(re * re) * -0.25)) * Float64(im_m * Float64(-2.0 + Float64(im_m * Float64(im_m * -0.3333333333333333)))));
	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 (re <= 5.4e+60)
		tmp = im_m * ((-1.0 + ((im_m * im_m) * (-0.16666666666666666 + (im_m * (im_m * (-0.008333333333333333 + ((im_m * im_m) * -0.0001984126984126984))))))) * (1.0 + (re * (re * (-0.5 + ((re * re) * 0.041666666666666664))))));
	else
		tmp = (0.5 + ((re * re) * -0.25)) * (im_m * (-2.0 + (im_m * (im_m * -0.3333333333333333))));
	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[re, 5.4e+60], N[(im$95$m * N[(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[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(re * N[(re * N[(-0.5 + N[(N[(re * re), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 + N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision] * N[(im$95$m * N[(-2.0 + N[(im$95$m * N[(im$95$m * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if re < 5.3999999999999999e60

   1. Initial program 54.2%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \cos re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \cos re + {im}^{2} \cdot \left(\frac{-1}{120} \cdot \cos re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)\right)} \]
   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 93.7%

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

   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. *-lowering-*.f64 N/A

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

      22. unpow2 N/A

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

      23. *-lowering-*.f64 93.7%

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

   8. Simplified 93.7%

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

   9. Taylor expanded in re around 0

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

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

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

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

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

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

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

       6. sub-neg N/A

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

       7. metadata-eval N/A

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

       8. +-commutative N/A

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

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

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

       10. *-commutative N/A

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

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

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

       12. unpow2 N/A

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

       13. *-lowering-*.f64 69.8%

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

   11. Simplified 69.8%

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

   if 5.3999999999999999e60 < re

   1. Initial program 54.5%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 82.2%

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

   5. Simplified 82.2%

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

   6. Taylor expanded in re around 0

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

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

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

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left({re}^{2} \cdot \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{3}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({re}^{2}\right), \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{3}\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{3}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 30.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{3}\right)\right)\right)\right)\right) \]

   8. Simplified 30.7%

      \[\leadsto \color{blue}{\left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)} \cdot \left(im \cdot \left(-2 + im \cdot \left(im \cdot -0.3333333333333333\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 5.4 \cdot 10^{+60}:\\ \;\;\;\;im \cdot \left(\left(-1 + \left(im \cdot im\right) \cdot \left(-0.16666666666666666 + im \cdot \left(im \cdot \left(-0.008333333333333333 + \left(im \cdot im\right) \cdot -0.0001984126984126984\right)\right)\right)\right) \cdot \left(1 + re \cdot \left(re \cdot \left(-0.5 + \left(re \cdot re\right) \cdot 0.041666666666666664\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + \left(re \cdot re\right) \cdot -0.25\right) \cdot \left(im \cdot \left(-2 + im \cdot \left(im \cdot -0.3333333333333333\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 60.4% accurate, 11.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;re \leq 9 \cdot 10^{+70}:\\ \;\;\;\;im\_m \cdot \left(-1 + im\_m \cdot \left(im\_m \cdot \left(-0.16666666666666666 + im\_m \cdot \left(im\_m \cdot \left(-0.008333333333333333 + im\_m \cdot \left(im\_m \cdot -0.0001984126984126984\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + \left(re \cdot re\right) \cdot -0.25\right) \cdot \left(im\_m \cdot \left(-2 + im\_m \cdot \left(im\_m \cdot -0.3333333333333333\right)\right)\right)\\ \end{array} \end{array} \]

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 (<= re 9e+70)
    (*
     im_m
     (+
      -1.0
      (*
       im_m
       (*
        im_m
        (+
         -0.16666666666666666
         (*
          im_m
          (*
           im_m
           (+
            -0.008333333333333333
            (* im_m (* im_m -0.0001984126984126984))))))))))
    (*
     (+ 0.5 (* (* re re) -0.25))
     (* im_m (+ -2.0 (* im_m (* im_m -0.3333333333333333))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (re <= 9e+70) {
		tmp = im_m * (-1.0 + (im_m * (im_m * (-0.16666666666666666 + (im_m * (im_m * (-0.008333333333333333 + (im_m * (im_m * -0.0001984126984126984)))))))));
	} else {
		tmp = (0.5 + ((re * re) * -0.25)) * (im_m * (-2.0 + (im_m * (im_m * -0.3333333333333333))));
	}
	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 (re <= 9d+70) then
        tmp = im_m * ((-1.0d0) + (im_m * (im_m * ((-0.16666666666666666d0) + (im_m * (im_m * ((-0.008333333333333333d0) + (im_m * (im_m * (-0.0001984126984126984d0))))))))))
    else
        tmp = (0.5d0 + ((re * re) * (-0.25d0))) * (im_m * ((-2.0d0) + (im_m * (im_m * (-0.3333333333333333d0)))))
    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 (re <= 9e+70) {
		tmp = im_m * (-1.0 + (im_m * (im_m * (-0.16666666666666666 + (im_m * (im_m * (-0.008333333333333333 + (im_m * (im_m * -0.0001984126984126984)))))))));
	} else {
		tmp = (0.5 + ((re * re) * -0.25)) * (im_m * (-2.0 + (im_m * (im_m * -0.3333333333333333))));
	}
	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 re <= 9e+70:
		tmp = im_m * (-1.0 + (im_m * (im_m * (-0.16666666666666666 + (im_m * (im_m * (-0.008333333333333333 + (im_m * (im_m * -0.0001984126984126984)))))))))
	else:
		tmp = (0.5 + ((re * re) * -0.25)) * (im_m * (-2.0 + (im_m * (im_m * -0.3333333333333333))))
	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 (re <= 9e+70)
		tmp = Float64(im_m * Float64(-1.0 + Float64(im_m * Float64(im_m * Float64(-0.16666666666666666 + Float64(im_m * Float64(im_m * Float64(-0.008333333333333333 + Float64(im_m * Float64(im_m * -0.0001984126984126984))))))))));
	else
		tmp = Float64(Float64(0.5 + Float64(Float64(re * re) * -0.25)) * Float64(im_m * Float64(-2.0 + Float64(im_m * Float64(im_m * -0.3333333333333333)))));
	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 (re <= 9e+70)
		tmp = im_m * (-1.0 + (im_m * (im_m * (-0.16666666666666666 + (im_m * (im_m * (-0.008333333333333333 + (im_m * (im_m * -0.0001984126984126984)))))))));
	else
		tmp = (0.5 + ((re * re) * -0.25)) * (im_m * (-2.0 + (im_m * (im_m * -0.3333333333333333))));
	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[re, 9e+70], N[(im$95$m * N[(-1.0 + N[(im$95$m * N[(im$95$m * 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], N[(N[(0.5 + N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision] * N[(im$95$m * N[(-2.0 + N[(im$95$m * N[(im$95$m * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if re < 8.9999999999999999e70

   1. Initial program 53.9%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \cos re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \cos re + {im}^{2} \cdot \left(\frac{-1}{120} \cdot \cos re + \frac{-1}{5040} \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)\right)} \]
   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 93.7%

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

   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. *-lowering-*.f64 N/A

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

      22. unpow2 N/A

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

      23. *-lowering-*.f64 93.7%

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

   8. Simplified 93.7%

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

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) - 1\right)} \]
   10. 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. unpow2 N/A

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

       7. associate-*l* N/A

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

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

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

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

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

       10. sub-neg N/A

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

       11. metadata-eval N/A

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

       12. +-commutative N/A

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

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

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

       14. unpow2 N/A

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

       15. associate-*l* N/A

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

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

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

   11. Simplified 70.1%

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

   if 8.9999999999999999e70 < re

   1. Initial program 55.5%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 81.9%

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

   5. Simplified 81.9%

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

   6. Taylor expanded in re around 0

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

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

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

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left({re}^{2} \cdot \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{3}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({re}^{2}\right), \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{3}\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{3}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 31.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{3}\right)\right)\right)\right)\right) \]

   8. Simplified 31.2%

      \[\leadsto \color{blue}{\left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)} \cdot \left(im \cdot \left(-2 + im \cdot \left(im \cdot -0.3333333333333333\right)\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 58.5% accurate, 14.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;re \leq 9 \cdot 10^{+70}:\\ \;\;\;\;im\_m \cdot \left(-1 + im\_m \cdot \left(im\_m \cdot \left(-0.16666666666666666 + \left(im\_m \cdot im\_m\right) \cdot -0.008333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + \left(re \cdot re\right) \cdot -0.25\right) \cdot \left(im\_m \cdot \left(-2 + im\_m \cdot \left(im\_m \cdot -0.3333333333333333\right)\right)\right)\\ \end{array} \end{array} \]

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 (<= re 9e+70)
    (*
     im_m
     (+
      -1.0
      (*
       im_m
       (*
        im_m
        (+ -0.16666666666666666 (* (* im_m im_m) -0.008333333333333333))))))
    (*
     (+ 0.5 (* (* re re) -0.25))
     (* im_m (+ -2.0 (* im_m (* im_m -0.3333333333333333))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (re <= 9e+70) {
		tmp = im_m * (-1.0 + (im_m * (im_m * (-0.16666666666666666 + ((im_m * im_m) * -0.008333333333333333)))));
	} else {
		tmp = (0.5 + ((re * re) * -0.25)) * (im_m * (-2.0 + (im_m * (im_m * -0.3333333333333333))));
	}
	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 (re <= 9d+70) then
        tmp = im_m * ((-1.0d0) + (im_m * (im_m * ((-0.16666666666666666d0) + ((im_m * im_m) * (-0.008333333333333333d0))))))
    else
        tmp = (0.5d0 + ((re * re) * (-0.25d0))) * (im_m * ((-2.0d0) + (im_m * (im_m * (-0.3333333333333333d0)))))
    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 (re <= 9e+70) {
		tmp = im_m * (-1.0 + (im_m * (im_m * (-0.16666666666666666 + ((im_m * im_m) * -0.008333333333333333)))));
	} else {
		tmp = (0.5 + ((re * re) * -0.25)) * (im_m * (-2.0 + (im_m * (im_m * -0.3333333333333333))));
	}
	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 re <= 9e+70:
		tmp = im_m * (-1.0 + (im_m * (im_m * (-0.16666666666666666 + ((im_m * im_m) * -0.008333333333333333)))))
	else:
		tmp = (0.5 + ((re * re) * -0.25)) * (im_m * (-2.0 + (im_m * (im_m * -0.3333333333333333))))
	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 (re <= 9e+70)
		tmp = Float64(im_m * Float64(-1.0 + Float64(im_m * Float64(im_m * Float64(-0.16666666666666666 + Float64(Float64(im_m * im_m) * -0.008333333333333333))))));
	else
		tmp = Float64(Float64(0.5 + Float64(Float64(re * re) * -0.25)) * Float64(im_m * Float64(-2.0 + Float64(im_m * Float64(im_m * -0.3333333333333333)))));
	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 (re <= 9e+70)
		tmp = im_m * (-1.0 + (im_m * (im_m * (-0.16666666666666666 + ((im_m * im_m) * -0.008333333333333333)))));
	else
		tmp = (0.5 + ((re * re) * -0.25)) * (im_m * (-2.0 + (im_m * (im_m * -0.3333333333333333))));
	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[re, 9e+70], 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], N[(N[(0.5 + N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision] * N[(im$95$m * N[(-2.0 + N[(im$95$m * N[(im$95$m * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if re < 8.9999999999999999e70

   1. Initial program 53.9%

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

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)}\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)}\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) + -2\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left(-2 + \color{blue}{{im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right)}\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{-1}{60} \cdot {im}^{2}} - \frac{1}{3}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{-1}{60} \cdot {im}^{2}} - \frac{1}{3}\right)\right)\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{60} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{60} \cdot {im}^{2} + \frac{-1}{3}\right)\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{3} + \color{blue}{\frac{-1}{60} \cdot {im}^{2}}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{\left(\frac{-1}{60} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \left({im}^{2} \cdot \color{blue}{\frac{-1}{60}}\right)\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \left(\left(im \cdot im\right) \cdot \frac{-1}{60}\right)\right)\right)\right)\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \left(im \cdot \color{blue}{\left(im \cdot \frac{-1}{60}\right)}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \frac{-1}{60}\right)}\right)\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 89.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{60}}\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 89.8%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{2} \cdot im\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)} \]

      2. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot im\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right), \color{blue}{\left(\frac{1}{2} \cdot im\right)}\right) \]

   8. Simplified 66.2%

      \[\leadsto \color{blue}{\left(-2 + im \cdot \left(im \cdot \left(-0.3333333333333333 + \left(im \cdot im\right) \cdot -0.016666666666666666\right)\right)\right) \cdot \left(im \cdot 0.5\right)} \]

   9. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)} \]
   10. 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 66.2%

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

   11. Simplified 66.2%

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

   if 8.9999999999999999e70 < re

   1. Initial program 55.5%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 81.9%

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

   5. Simplified 81.9%

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

   6. Taylor expanded in re around 0

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

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

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

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left({re}^{2} \cdot \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{3}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({re}^{2}\right), \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{3}\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{3}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 31.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{3}\right)\right)\right)\right)\right) \]

   8. Simplified 31.2%

      \[\leadsto \color{blue}{\left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)} \cdot \left(im \cdot \left(-2 + im \cdot \left(im \cdot -0.3333333333333333\right)\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 58.4% accurate, 15.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;re \leq 9 \cdot 10^{+70}:\\ \;\;\;\;im\_m \cdot \left(-1 + im\_m \cdot \left(im\_m \cdot \left(-0.16666666666666666 + \left(im\_m \cdot im\_m\right) \cdot -0.008333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(im\_m \cdot im\_m\right) \cdot \left(-1 + 0.5 \cdot \left(re \cdot re\right)\right)}{im\_m}\\ \end{array} \end{array} \]

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 (<= re 9e+70)
    (*
     im_m
     (+
      -1.0
      (*
       im_m
       (*
        im_m
        (+ -0.16666666666666666 (* (* im_m im_m) -0.008333333333333333))))))
    (/ (* (* im_m im_m) (+ -1.0 (* 0.5 (* re re)))) 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 (re <= 9e+70) {
		tmp = im_m * (-1.0 + (im_m * (im_m * (-0.16666666666666666 + ((im_m * im_m) * -0.008333333333333333)))));
	} else {
		tmp = ((im_m * im_m) * (-1.0 + (0.5 * (re * re)))) / 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 (re <= 9d+70) then
        tmp = im_m * ((-1.0d0) + (im_m * (im_m * ((-0.16666666666666666d0) + ((im_m * im_m) * (-0.008333333333333333d0))))))
    else
        tmp = ((im_m * im_m) * ((-1.0d0) + (0.5d0 * (re * re)))) / 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 (re <= 9e+70) {
		tmp = im_m * (-1.0 + (im_m * (im_m * (-0.16666666666666666 + ((im_m * im_m) * -0.008333333333333333)))));
	} else {
		tmp = ((im_m * im_m) * (-1.0 + (0.5 * (re * re)))) / 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 re <= 9e+70:
		tmp = im_m * (-1.0 + (im_m * (im_m * (-0.16666666666666666 + ((im_m * im_m) * -0.008333333333333333)))))
	else:
		tmp = ((im_m * im_m) * (-1.0 + (0.5 * (re * re)))) / 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 (re <= 9e+70)
		tmp = Float64(im_m * Float64(-1.0 + Float64(im_m * Float64(im_m * Float64(-0.16666666666666666 + Float64(Float64(im_m * im_m) * -0.008333333333333333))))));
	else
		tmp = Float64(Float64(Float64(im_m * im_m) * Float64(-1.0 + Float64(0.5 * Float64(re * re)))) / 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 (re <= 9e+70)
		tmp = im_m * (-1.0 + (im_m * (im_m * (-0.16666666666666666 + ((im_m * im_m) * -0.008333333333333333)))));
	else
		tmp = ((im_m * im_m) * (-1.0 + (0.5 * (re * re)))) / 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[re, 9e+70], 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], N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(-1.0 + N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / im$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if re < 8.9999999999999999e70

   1. Initial program 53.9%

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

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)}\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)}\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) + -2\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left(-2 + \color{blue}{{im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right)}\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{-1}{60} \cdot {im}^{2}} - \frac{1}{3}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{-1}{60} \cdot {im}^{2}} - \frac{1}{3}\right)\right)\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{60} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{60} \cdot {im}^{2} + \frac{-1}{3}\right)\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{3} + \color{blue}{\frac{-1}{60} \cdot {im}^{2}}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{\left(\frac{-1}{60} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \left({im}^{2} \cdot \color{blue}{\frac{-1}{60}}\right)\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \left(\left(im \cdot im\right) \cdot \frac{-1}{60}\right)\right)\right)\right)\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \left(im \cdot \color{blue}{\left(im \cdot \frac{-1}{60}\right)}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \frac{-1}{60}\right)}\right)\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 89.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{60}}\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 89.8%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{2} \cdot im\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)} \]

      2. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot im\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right), \color{blue}{\left(\frac{1}{2} \cdot im\right)}\right) \]

   8. Simplified 66.2%

      \[\leadsto \color{blue}{\left(-2 + im \cdot \left(im \cdot \left(-0.3333333333333333 + \left(im \cdot im\right) \cdot -0.016666666666666666\right)\right)\right) \cdot \left(im \cdot 0.5\right)} \]

   9. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)} \]
   10. 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 66.2%

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

   11. Simplified 66.2%

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

   if 8.9999999999999999e70 < re

   1. Initial program 55.5%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 51.5%

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

   5. Simplified 51.5%

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

      1. cancel-sign-sub-inv N/A

         \[\leadsto 0 + \color{blue}{\left(\mathsf{neg}\left(im\right)\right) \cdot \cos re} \]

      2. +-lft-identity N/A

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

      3. sub0-neg N/A

         \[\leadsto \left(0 - im\right) \cdot \cos \color{blue}{re} \]

      4. flip-- N/A

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

      5. +-lft-identity N/A

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

      6. associate-*l/ N/A

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

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

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

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

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

      9. metadata-eval N/A

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

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

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

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

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

      12. cos-lowering-cos.f64 54.4%

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

   7. Applied egg-rr 54.4%

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

   8. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-out N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. sub-neg N/A

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

      13. metadata-eval N/A

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

      14. +-commutative N/A

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

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

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

      16. *-commutative N/A

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

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

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

      18. unpow2 N/A

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

      19. *-lowering-*.f64 31.1%

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

   10. Simplified 31.1%

       \[\leadsto \frac{\color{blue}{\left(im \cdot im\right) \cdot \left(-1 + \left(re \cdot re\right) \cdot 0.5\right)}}{im} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 9 \cdot 10^{+70}:\\ \;\;\;\;im \cdot \left(-1 + im \cdot \left(im \cdot \left(-0.16666666666666666 + \left(im \cdot im\right) \cdot -0.008333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(im \cdot im\right) \cdot \left(-1 + 0.5 \cdot \left(re \cdot re\right)\right)}{im}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 54.3% accurate, 17.2× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;re \leq 9 \cdot 10^{+70}:\\ \;\;\;\;im\_m \cdot \left(-1 + \left(im\_m \cdot im\_m\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(im\_m \cdot im\_m\right) \cdot \left(-1 + 0.5 \cdot \left(re \cdot re\right)\right)}{im\_m}\\ \end{array} \end{array} \]

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 (<= re 9e+70)
    (* im_m (+ -1.0 (* (* im_m im_m) -0.16666666666666666)))
    (/ (* (* im_m im_m) (+ -1.0 (* 0.5 (* re re)))) 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 (re <= 9e+70) {
		tmp = im_m * (-1.0 + ((im_m * im_m) * -0.16666666666666666));
	} else {
		tmp = ((im_m * im_m) * (-1.0 + (0.5 * (re * re)))) / 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 (re <= 9d+70) then
        tmp = im_m * ((-1.0d0) + ((im_m * im_m) * (-0.16666666666666666d0)))
    else
        tmp = ((im_m * im_m) * ((-1.0d0) + (0.5d0 * (re * re)))) / 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 (re <= 9e+70) {
		tmp = im_m * (-1.0 + ((im_m * im_m) * -0.16666666666666666));
	} else {
		tmp = ((im_m * im_m) * (-1.0 + (0.5 * (re * re)))) / 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 re <= 9e+70:
		tmp = im_m * (-1.0 + ((im_m * im_m) * -0.16666666666666666))
	else:
		tmp = ((im_m * im_m) * (-1.0 + (0.5 * (re * re)))) / 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 (re <= 9e+70)
		tmp = Float64(im_m * Float64(-1.0 + Float64(Float64(im_m * im_m) * -0.16666666666666666)));
	else
		tmp = Float64(Float64(Float64(im_m * im_m) * Float64(-1.0 + Float64(0.5 * Float64(re * re)))) / 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 (re <= 9e+70)
		tmp = im_m * (-1.0 + ((im_m * im_m) * -0.16666666666666666));
	else
		tmp = ((im_m * im_m) * (-1.0 + (0.5 * (re * re)))) / 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[re, 9e+70], N[(im$95$m * N[(-1.0 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(-1.0 + N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / im$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if re < 8.9999999999999999e70

   1. Initial program 53.9%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\frac{1}{2}}, \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, im\right)\right), \mathsf{exp.f64}\left(im\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 45.7%

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

   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 60.8%

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

   8. Simplified 60.8%

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

   if 8.9999999999999999e70 < re

   1. Initial program 55.5%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 51.5%

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

   5. Simplified 51.5%

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

      1. cancel-sign-sub-inv N/A

         \[\leadsto 0 + \color{blue}{\left(\mathsf{neg}\left(im\right)\right) \cdot \cos re} \]

      2. +-lft-identity N/A

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

      3. sub0-neg N/A

         \[\leadsto \left(0 - im\right) \cdot \cos \color{blue}{re} \]

      4. flip-- N/A

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

      5. +-lft-identity N/A

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

      6. associate-*l/ N/A

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

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

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

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

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

      9. metadata-eval N/A

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

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

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

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

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

      12. cos-lowering-cos.f64 54.4%

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

   7. Applied egg-rr 54.4%

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

   8. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-out N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. sub-neg N/A

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

      13. metadata-eval N/A

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

      14. +-commutative N/A

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

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

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

      16. *-commutative N/A

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

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

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

      18. unpow2 N/A

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

      19. *-lowering-*.f64 31.1%

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

   10. Simplified 31.1%

       \[\leadsto \frac{\color{blue}{\left(im \cdot im\right) \cdot \left(-1 + \left(re \cdot re\right) \cdot 0.5\right)}}{im} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 9 \cdot 10^{+70}:\\ \;\;\;\;im \cdot \left(-1 + \left(im \cdot im\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(im \cdot im\right) \cdot \left(-1 + 0.5 \cdot \left(re \cdot re\right)\right)}{im}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 58.6% accurate, 17.2× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 5:\\ \;\;\;\;im\_m \cdot \left(-1 + \left(im\_m \cdot im\_m\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(im\_m \cdot im\_m\right) \cdot \left(im\_m \cdot im\_m\right)\right) \cdot -0.016666666666666666\right) \cdot \left(im\_m \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 5.0)
    (* im_m (+ -1.0 (* (* im_m im_m) -0.16666666666666666)))
    (*
     (* (* (* im_m im_m) (* im_m im_m)) -0.016666666666666666)
     (* 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 <= 5.0) {
		tmp = im_m * (-1.0 + ((im_m * im_m) * -0.16666666666666666));
	} else {
		tmp = (((im_m * im_m) * (im_m * im_m)) * -0.016666666666666666) * (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 <= 5.0d0) then
        tmp = im_m * ((-1.0d0) + ((im_m * im_m) * (-0.16666666666666666d0)))
    else
        tmp = (((im_m * im_m) * (im_m * im_m)) * (-0.016666666666666666d0)) * (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 <= 5.0) {
		tmp = im_m * (-1.0 + ((im_m * im_m) * -0.16666666666666666));
	} else {
		tmp = (((im_m * im_m) * (im_m * im_m)) * -0.016666666666666666) * (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 <= 5.0:
		tmp = im_m * (-1.0 + ((im_m * im_m) * -0.16666666666666666))
	else:
		tmp = (((im_m * im_m) * (im_m * im_m)) * -0.016666666666666666) * (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 <= 5.0)
		tmp = Float64(im_m * Float64(-1.0 + Float64(Float64(im_m * im_m) * -0.16666666666666666)));
	else
		tmp = Float64(Float64(Float64(Float64(im_m * im_m) * Float64(im_m * im_m)) * -0.016666666666666666) * Float64(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 <= 5.0)
		tmp = im_m * (-1.0 + ((im_m * im_m) * -0.16666666666666666));
	else
		tmp = (((im_m * im_m) * (im_m * im_m)) * -0.016666666666666666) * (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, 5.0], N[(im$95$m * N[(-1.0 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] * -0.016666666666666666), $MachinePrecision] * N[(im$95$m * 0.5), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 5

   1. Initial program 40.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\frac{1}{2}}, \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, im\right)\right), \mathsf{exp.f64}\left(im\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 29.3%

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

   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 49.8%

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

   8. Simplified 49.8%

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

   if 5 < 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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)}\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)}\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) + -2\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left(-2 + \color{blue}{{im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right)}\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{-1}{60} \cdot {im}^{2}} - \frac{1}{3}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{-1}{60} \cdot {im}^{2}} - \frac{1}{3}\right)\right)\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{60} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{60} \cdot {im}^{2} + \frac{-1}{3}\right)\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{3} + \color{blue}{\frac{-1}{60} \cdot {im}^{2}}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{\left(\frac{-1}{60} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \left({im}^{2} \cdot \color{blue}{\frac{-1}{60}}\right)\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \left(\left(im \cdot im\right) \cdot \frac{-1}{60}\right)\right)\right)\right)\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \left(im \cdot \color{blue}{\left(im \cdot \frac{-1}{60}\right)}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \frac{-1}{60}\right)}\right)\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 81.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{60}}\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 81.5%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{2} \cdot im\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)} \]

      2. *-commutative N/A

         \[\leadsto \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot im\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right), \color{blue}{\left(\frac{1}{2} \cdot im\right)}\right) \]

   8. Simplified 61.8%

      \[\leadsto \color{blue}{\left(-2 + im \cdot \left(im \cdot \left(-0.3333333333333333 + \left(im \cdot im\right) \cdot -0.016666666666666666\right)\right)\right) \cdot \left(im \cdot 0.5\right)} \]

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{-1}{60} \cdot {im}^{4}\right)}, \mathsf{*.f64}\left(im, \frac{1}{2}\right)\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left({im}^{4} \cdot \frac{-1}{60}\right), \mathsf{*.f64}\left(\color{blue}{im}, \frac{1}{2}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({im}^{4}\right), \frac{-1}{60}\right), \mathsf{*.f64}\left(\color{blue}{im}, \frac{1}{2}\right)\right) \]

       3. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({im}^{\left(2 \cdot 2\right)}\right), \frac{-1}{60}\right), \mathsf{*.f64}\left(im, \frac{1}{2}\right)\right) \]

       4. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({im}^{2} \cdot {im}^{2}\right), \frac{-1}{60}\right), \mathsf{*.f64}\left(im, \frac{1}{2}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({im}^{2}\right), \left({im}^{2}\right)\right), \frac{-1}{60}\right), \mathsf{*.f64}\left(im, \frac{1}{2}\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(im \cdot im\right), \left({im}^{2}\right)\right), \frac{-1}{60}\right), \mathsf{*.f64}\left(im, \frac{1}{2}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2}\right)\right), \frac{-1}{60}\right), \mathsf{*.f64}\left(im, \frac{1}{2}\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(im \cdot im\right)\right), \frac{-1}{60}\right), \mathsf{*.f64}\left(im, \frac{1}{2}\right)\right) \]

       9. *-lowering-*.f64 61.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \frac{-1}{60}\right), \mathsf{*.f64}\left(im, \frac{1}{2}\right)\right) \]

   11. Simplified 61.8%

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

4. Final simplification 52.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 5:\\ \;\;\;\;im \cdot \left(-1 + \left(im \cdot im\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot -0.016666666666666666\right) \cdot \left(im \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 54.0% accurate, 22.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;re \leq 9 \cdot 10^{+70}:\\ \;\;\;\;im\_m \cdot \left(-1 + \left(im\_m \cdot im\_m\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(-1 + 0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

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 (<= re 9e+70)
    (* im_m (+ -1.0 (* (* im_m im_m) -0.16666666666666666)))
    (* im_m (+ -1.0 (* 0.5 (* re re)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (re <= 9e+70) {
		tmp = im_m * (-1.0 + ((im_m * im_m) * -0.16666666666666666));
	} else {
		tmp = im_m * (-1.0 + (0.5 * (re * re)));
	}
	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 (re <= 9d+70) then
        tmp = im_m * ((-1.0d0) + ((im_m * im_m) * (-0.16666666666666666d0)))
    else
        tmp = im_m * ((-1.0d0) + (0.5d0 * (re * re)))
    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 (re <= 9e+70) {
		tmp = im_m * (-1.0 + ((im_m * im_m) * -0.16666666666666666));
	} else {
		tmp = im_m * (-1.0 + (0.5 * (re * re)));
	}
	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 re <= 9e+70:
		tmp = im_m * (-1.0 + ((im_m * im_m) * -0.16666666666666666))
	else:
		tmp = im_m * (-1.0 + (0.5 * (re * re)))
	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 (re <= 9e+70)
		tmp = Float64(im_m * Float64(-1.0 + Float64(Float64(im_m * im_m) * -0.16666666666666666)));
	else
		tmp = Float64(im_m * Float64(-1.0 + Float64(0.5 * Float64(re * re))));
	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 (re <= 9e+70)
		tmp = im_m * (-1.0 + ((im_m * im_m) * -0.16666666666666666));
	else
		tmp = im_m * (-1.0 + (0.5 * (re * re)));
	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[re, 9e+70], N[(im$95$m * N[(-1.0 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(-1.0 + N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if re < 8.9999999999999999e70

   1. Initial program 53.9%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\frac{1}{2}}, \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, im\right)\right), \mathsf{exp.f64}\left(im\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 45.7%

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

   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 60.8%

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

   8. Simplified 60.8%

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

   if 8.9999999999999999e70 < re

   1. Initial program 55.5%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 51.5%

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

   5. Simplified 51.5%

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

   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. neg-mul-1 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 29.4%

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

   8. Simplified 29.4%

      \[\leadsto \color{blue}{im \cdot \left(0.5 \cdot \left(re \cdot re\right) + -1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 9 \cdot 10^{+70}:\\ \;\;\;\;im \cdot \left(-1 + \left(im \cdot im\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-1 + 0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 47.4% accurate, 25.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 2.25 \cdot 10^{+154}:\\ \;\;\;\;0 - im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(im\_m \cdot im\_m\right) \cdot \frac{-1}{im\_m}\\ \end{array} \end{array} \]

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 2.25e+154) (- 0.0 im_m) (* (* im_m im_m) (/ -1.0 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 <= 2.25e+154) {
		tmp = 0.0 - im_m;
	} else {
		tmp = (im_m * im_m) * (-1.0 / 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 <= 2.25d+154) then
        tmp = 0.0d0 - im_m
    else
        tmp = (im_m * im_m) * ((-1.0d0) / 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 <= 2.25e+154) {
		tmp = 0.0 - im_m;
	} else {
		tmp = (im_m * im_m) * (-1.0 / 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 <= 2.25e+154:
		tmp = 0.0 - im_m
	else:
		tmp = (im_m * im_m) * (-1.0 / 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 <= 2.25e+154)
		tmp = Float64(0.0 - im_m);
	else
		tmp = Float64(Float64(im_m * im_m) * Float64(-1.0 / 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 <= 2.25e+154)
		tmp = 0.0 - im_m;
	else
		tmp = (im_m * im_m) * (-1.0 / 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, 2.25e+154], N[(0.0 - im$95$m), $MachinePrecision], N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(-1.0 / im$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 2.25000000000000005e154

   1. Initial program 48.2%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 58.1%

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

   5. Simplified 58.1%

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

   6. Taylor expanded in re around 0

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

      1. neg-mul-1 N/A

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 32.9%

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

   8. Simplified 32.9%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 32.9%

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

   10. Applied egg-rr 32.9%

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

   if 2.25000000000000005e154 < 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

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \cos re\right)} \]
   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 7.8%

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

   5. Simplified 7.8%

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

   6. Taylor expanded in re around 0

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

      1. neg-mul-1 N/A

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 6.5%

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

   8. Simplified 6.5%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 6.5%

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

   10. Applied egg-rr 6.5%

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

       1. unpow1 N/A

          \[\leadsto \mathsf{neg}\left({im}^{1}\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left({im}^{\left(3 - 2\right)}\right) \]

       3. pow-div N/A

          \[\leadsto \mathsf{neg}\left(\frac{{im}^{3}}{{im}^{2}}\right) \]

       4. cube-unmult N/A

          \[\leadsto \mathsf{neg}\left(\frac{im \cdot \left(im \cdot im\right)}{{im}^{2}}\right) \]

       5. pow2 N/A

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

       6. distribute-frac-neg N/A

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

       7. sub0-neg N/A

          \[\leadsto \frac{0 - im \cdot \left(im \cdot im\right)}{\color{blue}{im} \cdot im} \]

       8. cancel-sign-sub-inv N/A

          \[\leadsto \frac{0 + \left(\mathsf{neg}\left(im\right)\right) \cdot \left(im \cdot im\right)}{\color{blue}{im} \cdot im} \]

       9. +-lft-identity N/A

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

       10. *-rgt-identity N/A

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

       11. times-frac N/A

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

   12. Applied egg-rr 83.3%

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

4. Final simplification 38.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.25 \cdot 10^{+154}:\\ \;\;\;\;0 - im\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \frac{-1}{im}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 53.8% accurate, 34.3× speedup

Math

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

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)))))

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)));
}

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))))
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)));
}

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)))

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) * -0.16666666666666666))))
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)));
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] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.3%

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

3. Taylor expanded in re around 0

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{\frac{1}{2}}, \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, im\right)\right), \mathsf{exp.f64}\left(im\right)\right)\right) \]
4. Step-by-step derivation

5. Simplified 41.1%

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

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 51.9%

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

8. Simplified 51.9%

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

9. Final simplification 51.9%

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

Alternative 19: 29.7% 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 54.3%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
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 52.2%

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

5. Simplified 52.2%

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

6. Taylor expanded in re around 0

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

   1. neg-mul-1 N/A

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

   2. neg-sub0 N/A

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

   3. --lowering--.f64 29.8%

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

8. Simplified 29.8%

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

   1. sub0-neg N/A

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

   2. neg-lowering-neg.f64 29.8%

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

10. Applied egg-rr 29.8%

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

11. Final simplification 29.8%

    \[\leadsto 0 - im \]
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 2024158 
(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))))