math.sin on complex, imaginary part

Percentage Accurate: 54.2% → 99.4%

Time: 12.3s

Alternatives: 18

Speedup: 14.7×

• 49.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.6× speedup

Math

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

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 (- INFINITY))
      (* 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 <= -((double) INFINITY)) {
		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;
}

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 <= -Double.POSITIVE_INFINITY) {
		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 <= -math.inf:
		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 <= Float64(-Inf))
		tmp = Float64(t_0 * Float64(0.5 * cos(re)));
	else
		tmp = Float64(im_m * Float64(cos(re) * Float64(-1.0 + Float64(im_m * Float64(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 <= -Inf)
		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, (-Infinity)], 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[(im$95$m * N[(im$95$m * 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]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

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

   1. Initial program 41.9%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 96.3%

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

   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. unpow2 N/A

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

      6. associate-*l* N/A

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

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

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

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

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

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

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

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

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

      17. sub-neg N/A

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

      18. metadata-eval N/A

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

      19. +-commutative N/A

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

   8. Simplified 96.3%

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

4. Final simplification 97.3%

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

Alternative 2: 98.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := im\_m \cdot \left(\cos re \cdot \left(-1 + im\_m \cdot \left(im\_m \cdot \left(-0.16666666666666666 + im\_m \cdot \left(im\_m \cdot \left(-0.008333333333333333 + \left(im\_m \cdot im\_m\right) \cdot -0.0001984126984126984\right)\right)\right)\right)\right)\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 5.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im\_m \leq 4 \cdot 10^{+44}:\\ \;\;\;\;0.5 - e^{im\_m} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 5.5)
      t_0
      (if (<= im_m 4e+44) (- 0.5 (* (exp im_m) 0.5)) 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 <= 5.5) {
		tmp = t_0;
	} else if (im_m <= 4e+44) {
		tmp = 0.5 - (exp(im_m) * 0.5);
	} 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 <= 5.5d0) then
        tmp = t_0
    else if (im_m <= 4d+44) then
        tmp = 0.5d0 - (exp(im_m) * 0.5d0)
    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 <= 5.5) {
		tmp = t_0;
	} else if (im_m <= 4e+44) {
		tmp = 0.5 - (Math.exp(im_m) * 0.5);
	} 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 <= 5.5:
		tmp = t_0
	elif im_m <= 4e+44:
		tmp = 0.5 - (math.exp(im_m) * 0.5)
	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(im_m * Float64(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 <= 5.5)
		tmp = t_0;
	elseif (im_m <= 4e+44)
		tmp = Float64(0.5 - Float64(exp(im_m) * 0.5));
	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 <= 5.5)
		tmp = t_0;
	elseif (im_m <= 4e+44)
		tmp = 0.5 - (exp(im_m) * 0.5);
	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[(im$95$m * N[(im$95$m * 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]}, N[(im$95$s * If[LessEqual[im$95$m, 5.5], t$95$0, If[LessEqual[im$95$m, 4e+44], N[(0.5 - N[(N[Exp[im$95$m], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 5.5 or 4.0000000000000004e44 < im

   1. Initial program 55.9%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 97.2%

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

   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. unpow2 N/A

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

      6. associate-*l* N/A

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

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

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

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

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

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

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

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

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

      17. sub-neg N/A

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

      18. metadata-eval N/A

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

      19. +-commutative N/A

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

   8. Simplified 97.2%

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

   if 5.5 < im < 4.0000000000000004e44

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. sub-neg N/A

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

      2. distribute-rgt-in N/A

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

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

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

      4. unsub-neg N/A

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

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

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

      6. exp-neg N/A

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

      7. associate-*l/ N/A

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

      8. metadata-eval N/A

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

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

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

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

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

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

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

      12. exp-lowering-exp.f64 70.0%

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

   5. Simplified 70.0%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 70.0%

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

Alternative 3: 97.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := im\_m \cdot \left(\cos re \cdot \left(-1 + \left(im\_m \cdot im\_m\right) \cdot \left(-0.16666666666666666 + -0.008333333333333333 \cdot \left(im\_m \cdot im\_m\right)\right)\right)\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 4.8:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im\_m \leq 1.2 \cdot 10^{+62}:\\ \;\;\;\;0.5 - e^{im\_m} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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
              (* -0.008333333333333333 (* im_m im_m)))))))))
   (*
    im_s
    (if (<= im_m 4.8)
      t_0
      (if (<= im_m 1.2e+62) (- 0.5 (* (exp im_m) 0.5)) 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 + (-0.008333333333333333 * (im_m * im_m))))));
	double tmp;
	if (im_m <= 4.8) {
		tmp = t_0;
	} else if (im_m <= 1.2e+62) {
		tmp = 0.5 - (exp(im_m) * 0.5);
	} 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) + ((-0.008333333333333333d0) * (im_m * im_m))))))
    if (im_m <= 4.8d0) then
        tmp = t_0
    else if (im_m <= 1.2d+62) then
        tmp = 0.5d0 - (exp(im_m) * 0.5d0)
    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 + (-0.008333333333333333 * (im_m * im_m))))));
	double tmp;
	if (im_m <= 4.8) {
		tmp = t_0;
	} else if (im_m <= 1.2e+62) {
		tmp = 0.5 - (Math.exp(im_m) * 0.5);
	} 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 + (-0.008333333333333333 * (im_m * im_m))))))
	tmp = 0
	if im_m <= 4.8:
		tmp = t_0
	elif im_m <= 1.2e+62:
		tmp = 0.5 - (math.exp(im_m) * 0.5)
	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(-0.008333333333333333 * Float64(im_m * im_m)))))))
	tmp = 0.0
	if (im_m <= 4.8)
		tmp = t_0;
	elseif (im_m <= 1.2e+62)
		tmp = Float64(0.5 - Float64(exp(im_m) * 0.5));
	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 + (-0.008333333333333333 * (im_m * im_m))))));
	tmp = 0.0;
	if (im_m <= 4.8)
		tmp = t_0;
	elseif (im_m <= 1.2e+62)
		tmp = 0.5 - (exp(im_m) * 0.5);
	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[(-0.008333333333333333 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[im$95$m, 4.8], t$95$0, If[LessEqual[im$95$m, 1.2e+62], N[(0.5 - N[(N[Exp[im$95$m], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 4.79999999999999982 or 1.2e62 < im

   1. Initial program 55.1%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \cos re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \cos re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)} \]
   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. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-out N/A

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

      11. distribute-lft-out N/A

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

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

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

   5. Simplified 95.9%

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

   if 4.79999999999999982 < 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

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

      1. sub-neg N/A

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

      2. distribute-rgt-in N/A

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

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

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

      4. unsub-neg N/A

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

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

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

      6. exp-neg N/A

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

      7. associate-*l/ N/A

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

      8. metadata-eval N/A

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

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

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

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

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

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

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

      12. exp-lowering-exp.f64 71.4%

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

   5. Simplified 71.4%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 71.4%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.8:\\ \;\;\;\;im \cdot \left(\cos re \cdot \left(-1 + \left(im \cdot im\right) \cdot \left(-0.16666666666666666 + -0.008333333333333333 \cdot \left(im \cdot im\right)\right)\right)\right)\\ \mathbf{elif}\;im \leq 1.2 \cdot 10^{+62}:\\ \;\;\;\;0.5 - e^{im} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\cos re \cdot \left(-1 + \left(im \cdot im\right) \cdot \left(-0.16666666666666666 + -0.008333333333333333 \cdot \left(im \cdot im\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(im\_m \cdot \cos re\right) \cdot \left(-1 + im\_m \cdot \left(im\_m \cdot -0.16666666666666666\right)\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 4:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im\_m \leq 1.15 \cdot 10^{+103}:\\ \;\;\;\;0.5 - e^{im\_m} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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_s
    (if (<= im_m 4.0)
      t_0
      (if (<= im_m 1.15e+103) (- 0.5 (* (exp im_m) 0.5)) 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)));
	double tmp;
	if (im_m <= 4.0) {
		tmp = t_0;
	} else if (im_m <= 1.15e+103) {
		tmp = 0.5 - (exp(im_m) * 0.5);
	} 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))))
    if (im_m <= 4.0d0) then
        tmp = t_0
    else if (im_m <= 1.15d+103) then
        tmp = 0.5d0 - (exp(im_m) * 0.5d0)
    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)));
	double tmp;
	if (im_m <= 4.0) {
		tmp = t_0;
	} else if (im_m <= 1.15e+103) {
		tmp = 0.5 - (Math.exp(im_m) * 0.5);
	} 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)))
	tmp = 0
	if im_m <= 4.0:
		tmp = t_0
	elif im_m <= 1.15e+103:
		tmp = 0.5 - (math.exp(im_m) * 0.5)
	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(Float64(im_m * cos(re)) * Float64(-1.0 + Float64(im_m * Float64(im_m * -0.16666666666666666))))
	tmp = 0.0
	if (im_m <= 4.0)
		tmp = t_0;
	elseif (im_m <= 1.15e+103)
		tmp = Float64(0.5 - Float64(exp(im_m) * 0.5));
	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)));
	tmp = 0.0;
	if (im_m <= 4.0)
		tmp = t_0;
	elseif (im_m <= 1.15e+103)
		tmp = 0.5 - (exp(im_m) * 0.5);
	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[(im$95$m * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(im$95$m * N[(im$95$m * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[im$95$m, 4.0], t$95$0, If[LessEqual[im$95$m, 1.15e+103], N[(0.5 - N[(N[Exp[im$95$m], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 4 or 1.15000000000000004e103 < im

   1. Initial program 52.6%

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

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 93.1%

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

   5. Simplified 93.1%

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

   if 4 < im < 1.15000000000000004e103

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. sub-neg N/A

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

      2. distribute-rgt-in N/A

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

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

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

      4. unsub-neg N/A

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

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

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

      6. exp-neg N/A

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

      7. associate-*l/ N/A

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

      8. metadata-eval N/A

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

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

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

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

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

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

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

      12. exp-lowering-exp.f64 70.4%

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

   5. Simplified 70.4%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 70.4%

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

Alternative 5: 86.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 2.9:\\ \;\;\;\;im\_m \cdot \left(0 - \cos re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 - e^{im\_m} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 2.9) (* im_m (- 0.0 (cos re))) (- 0.5 (* (exp im_m) 0.5)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 2.9) {
		tmp = im_m * (0.0 - cos(re));
	} else {
		tmp = 0.5 - (exp(im_m) * 0.5);
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 2.9d0) then
        tmp = im_m * (0.0d0 - cos(re))
    else
        tmp = 0.5d0 - (exp(im_m) * 0.5d0)
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 2.9) {
		tmp = im_m * (0.0 - Math.cos(re));
	} else {
		tmp = 0.5 - (Math.exp(im_m) * 0.5);
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 2.9:
		tmp = im_m * (0.0 - math.cos(re))
	else:
		tmp = 0.5 - (math.exp(im_m) * 0.5)
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 2.9)
		tmp = Float64(im_m * Float64(0.0 - cos(re)));
	else
		tmp = Float64(0.5 - Float64(exp(im_m) * 0.5));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 2.9)
		tmp = im_m * (0.0 - cos(re));
	else
		tmp = 0.5 - (exp(im_m) * 0.5);
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 2.9], N[(im$95$m * N[(0.0 - N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 - N[(N[Exp[im$95$m], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 2.89999999999999991

   1. Initial program 41.9%

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

   3. Taylor expanded in im around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

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

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

      5. cos-lowering-cos.f64 64.6%

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

   5. Simplified 64.6%

      \[\leadsto \color{blue}{0 - im \cdot \cos re} \]
   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 64.6%

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

   7. Applied egg-rr 64.6%

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

   if 2.89999999999999991 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. sub-neg N/A

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

      2. distribute-rgt-in N/A

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

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

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

      4. unsub-neg N/A

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

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

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

      6. exp-neg N/A

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

      7. associate-*l/ N/A

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

      8. metadata-eval N/A

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

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

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

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

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

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

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

      12. exp-lowering-exp.f64 73.9%

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

   5. Simplified 73.9%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 73.9%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.9:\\ \;\;\;\;im \cdot \left(0 - \cos re\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 - e^{im} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 132000:\\ \;\;\;\;im\_m \cdot \left(0 - \cos re\right)\\ \mathbf{elif}\;im\_m \leq 2.8 \cdot 10^{+41}:\\ \;\;\;\;re \cdot \left(re \cdot \left(im\_m \cdot \left(0.5 + \frac{-1}{re \cdot re}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(-1 + im\_m \cdot \left(im\_m \cdot \left(-0.16666666666666666 + im\_m \cdot \left(im\_m \cdot \left(-0.008333333333333333 + \left(im\_m \cdot im\_m\right) \cdot -0.0001984126984126984\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 132000.0)
    (* im_m (- 0.0 (cos re)))
    (if (<= im_m 2.8e+41)
      (* re (* re (* im_m (+ 0.5 (/ -1.0 (* re re))))))
      (*
       im_m
       (+
        -1.0
        (*
         im_m
         (*
          im_m
          (+
           -0.16666666666666666
           (*
            im_m
            (*
             im_m
             (+
              -0.008333333333333333
              (* (* im_m im_m) -0.0001984126984126984)))))))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 132000.0) {
		tmp = im_m * (0.0 - cos(re));
	} else if (im_m <= 2.8e+41) {
		tmp = re * (re * (im_m * (0.5 + (-1.0 / (re * re)))));
	} else {
		tmp = im_m * (-1.0 + (im_m * (im_m * (-0.16666666666666666 + (im_m * (im_m * (-0.008333333333333333 + ((im_m * im_m) * -0.0001984126984126984))))))));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 132000.0d0) then
        tmp = im_m * (0.0d0 - cos(re))
    else if (im_m <= 2.8d+41) then
        tmp = re * (re * (im_m * (0.5d0 + ((-1.0d0) / (re * re)))))
    else
        tmp = im_m * ((-1.0d0) + (im_m * (im_m * ((-0.16666666666666666d0) + (im_m * (im_m * ((-0.008333333333333333d0) + ((im_m * im_m) * (-0.0001984126984126984d0)))))))))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 132000.0) {
		tmp = im_m * (0.0 - Math.cos(re));
	} else if (im_m <= 2.8e+41) {
		tmp = re * (re * (im_m * (0.5 + (-1.0 / (re * re)))));
	} else {
		tmp = im_m * (-1.0 + (im_m * (im_m * (-0.16666666666666666 + (im_m * (im_m * (-0.008333333333333333 + ((im_m * im_m) * -0.0001984126984126984))))))));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 132000.0:
		tmp = im_m * (0.0 - math.cos(re))
	elif im_m <= 2.8e+41:
		tmp = re * (re * (im_m * (0.5 + (-1.0 / (re * re)))))
	else:
		tmp = im_m * (-1.0 + (im_m * (im_m * (-0.16666666666666666 + (im_m * (im_m * (-0.008333333333333333 + ((im_m * im_m) * -0.0001984126984126984))))))))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 132000.0)
		tmp = Float64(im_m * Float64(0.0 - cos(re)));
	elseif (im_m <= 2.8e+41)
		tmp = Float64(re * Float64(re * Float64(im_m * Float64(0.5 + Float64(-1.0 / Float64(re * re))))));
	else
		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(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)
	tmp = 0.0;
	if (im_m <= 132000.0)
		tmp = im_m * (0.0 - cos(re));
	elseif (im_m <= 2.8e+41)
		tmp = re * (re * (im_m * (0.5 + (-1.0 / (re * re)))));
	else
		tmp = im_m * (-1.0 + (im_m * (im_m * (-0.16666666666666666 + (im_m * (im_m * (-0.008333333333333333 + ((im_m * im_m) * -0.0001984126984126984))))))));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 132000.0], N[(im$95$m * N[(0.0 - N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 2.8e+41], N[(re * N[(re * N[(im$95$m * N[(0.5 + N[(-1.0 / N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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[(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 3 regimes

2. if im < 132000

   1. Initial program 41.9%

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

   3. Taylor expanded in im around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

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

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

      5. cos-lowering-cos.f64 64.6%

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

   5. Simplified 64.6%

      \[\leadsto \color{blue}{0 - im \cdot \cos re} \]
   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 64.6%

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

   7. Applied egg-rr 64.6%

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

   if 132000 < im < 2.7999999999999999e41

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

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

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

      5. cos-lowering-cos.f64 3.3%

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

   5. Simplified 3.3%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{{re}^{2} \cdot \left(\frac{-1}{24} \cdot \left(im \cdot {re}^{2}\right) - \frac{-1}{2} \cdot im\right) - im} \]
   7. Step-by-step derivation

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-out N/A

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

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

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

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

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 23.7%

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

   8. Simplified 23.7%

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

   9. Taylor expanded in re around 0

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 3.8%

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

   11. Simplified 3.8%

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

   12. Taylor expanded in re around inf

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

       3. *-commutative N/A

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

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

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

       5. *-commutative N/A

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

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

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

       7. associate-*r/ N/A

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

       8. *-commutative N/A

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

       9. associate-/l* N/A

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

       10. *-commutative N/A

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

       11. distribute-lft-out N/A

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

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

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

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

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

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

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

       15. unpow2 N/A

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

       16. *-lowering-*.f64 46.8%

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

   14. Simplified 46.8%

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

   if 2.7999999999999999e41 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. sub-neg N/A

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

      2. distribute-rgt-in N/A

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

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

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

      4. unsub-neg N/A

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

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

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

      6. exp-neg N/A

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

      7. associate-*l/ N/A

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

      8. metadata-eval N/A

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

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

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

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

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

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

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

      12. exp-lowering-exp.f64 75.0%

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

   5. Simplified 75.0%

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

   6. Taylor expanded in im around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

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

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

      6. 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

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

   8. Simplified 73.5%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 132000:\\ \;\;\;\;im \cdot \left(0 - \cos re\right)\\ \mathbf{elif}\;im \leq 2.8 \cdot 10^{+41}:\\ \;\;\;\;re \cdot \left(re \cdot \left(im \cdot \left(0.5 + \frac{-1}{re \cdot re}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-1 + im \cdot \left(im \cdot \left(-0.16666666666666666 + im \cdot \left(im \cdot \left(-0.008333333333333333 + \left(im \cdot im\right) \cdot -0.0001984126984126984\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.9% accurate, 4.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(im\_m \cdot im\_m\right) \cdot \left(-0.16666666666666666 + \left(im\_m \cdot im\_m\right) \cdot \left(-0.008333333333333333 + im\_m \cdot \left(im\_m \cdot -0.0001984126984126984\right)\right)\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 5 \cdot 10^{+44}:\\ \;\;\;\;im\_m \cdot \frac{-1 + t\_0 \cdot t\_0}{t\_0 - -1}\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(-1 + im\_m \cdot \left(im\_m \cdot \left(-0.16666666666666666 + im\_m \cdot \left(im\_m \cdot \left(-0.008333333333333333 + \left(im\_m \cdot im\_m\right) \cdot -0.0001984126984126984\right)\right)\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

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
         (*
          (* im_m im_m)
          (+
           -0.16666666666666666
           (*
            (* im_m im_m)
            (+
             -0.008333333333333333
             (* im_m (* im_m -0.0001984126984126984))))))))
   (*
    im_s
    (if (<= im_m 5e+44)
      (* im_m (/ (+ -1.0 (* t_0 t_0)) (- t_0 -1.0)))
      (*
       im_m
       (+
        -1.0
        (*
         im_m
         (*
          im_m
          (+
           -0.16666666666666666
           (*
            im_m
            (*
             im_m
             (+
              -0.008333333333333333
              (* (* im_m im_m) -0.0001984126984126984)))))))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = (im_m * im_m) * (-0.16666666666666666 + ((im_m * im_m) * (-0.008333333333333333 + (im_m * (im_m * -0.0001984126984126984)))));
	double tmp;
	if (im_m <= 5e+44) {
		tmp = im_m * ((-1.0 + (t_0 * t_0)) / (t_0 - -1.0));
	} else {
		tmp = im_m * (-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 = (im_m * im_m) * ((-0.16666666666666666d0) + ((im_m * im_m) * ((-0.008333333333333333d0) + (im_m * (im_m * (-0.0001984126984126984d0))))))
    if (im_m <= 5d+44) then
        tmp = im_m * (((-1.0d0) + (t_0 * t_0)) / (t_0 - (-1.0d0)))
    else
        tmp = im_m * ((-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 = (im_m * im_m) * (-0.16666666666666666 + ((im_m * im_m) * (-0.008333333333333333 + (im_m * (im_m * -0.0001984126984126984)))));
	double tmp;
	if (im_m <= 5e+44) {
		tmp = im_m * ((-1.0 + (t_0 * t_0)) / (t_0 - -1.0));
	} else {
		tmp = im_m * (-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 = (im_m * im_m) * (-0.16666666666666666 + ((im_m * im_m) * (-0.008333333333333333 + (im_m * (im_m * -0.0001984126984126984)))))
	tmp = 0
	if im_m <= 5e+44:
		tmp = im_m * ((-1.0 + (t_0 * t_0)) / (t_0 - -1.0))
	else:
		tmp = im_m * (-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(Float64(im_m * im_m) * Float64(-0.16666666666666666 + Float64(Float64(im_m * im_m) * Float64(-0.008333333333333333 + Float64(im_m * Float64(im_m * -0.0001984126984126984))))))
	tmp = 0.0
	if (im_m <= 5e+44)
		tmp = Float64(im_m * Float64(Float64(-1.0 + Float64(t_0 * t_0)) / Float64(t_0 - -1.0)));
	else
		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(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 = (im_m * im_m) * (-0.16666666666666666 + ((im_m * im_m) * (-0.008333333333333333 + (im_m * (im_m * -0.0001984126984126984)))));
	tmp = 0.0;
	if (im_m <= 5e+44)
		tmp = im_m * ((-1.0 + (t_0 * t_0)) / (t_0 - -1.0));
	else
		tmp = im_m * (-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[(im$95$m * im$95$m), $MachinePrecision] * N[(-0.16666666666666666 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(-0.008333333333333333 + N[(im$95$m * N[(im$95$m * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[im$95$m, 5e+44], N[(im$95$m * N[(N[(-1.0 + N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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[(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 im < 4.9999999999999996e44

   1. Initial program 44.9%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 91.6%

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

   6. Taylor expanded in re around 0

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

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

         \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(\left(\frac{-1}{6} \cdot {im}^{2} + {im}^{4} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)\right) - 1\right)}\right) \]

      2. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left(\frac{-1}{6} \cdot {im}^{2} + \color{blue}{\left({im}^{4} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - 1\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\left(\frac{-1}{6} \cdot {im}^{2}\right), \color{blue}{\left({im}^{4} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - 1\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{6}, \left({im}^{2}\right)\right), \left(\color{blue}{{im}^{4} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)} - 1\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{6}, \left(im \cdot im\right)\right), \left({im}^{4} \cdot \color{blue}{\left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)} - 1\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right), \left({im}^{4} \cdot \color{blue}{\left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)} - 1\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right), \left({im}^{4} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right), \left({im}^{4} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) + -1\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(\left({im}^{4} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right)\right), \color{blue}{-1}\right)\right)\right) \]

   8. Simplified 61.3%

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

      1. associate-+r+ N/A

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

      2. flip-+ N/A

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

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

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

   10. Applied egg-rr 38.7%

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

   if 4.9999999999999996e44 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. sub-neg N/A

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

      2. distribute-rgt-in N/A

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

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

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

      4. unsub-neg N/A

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

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

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

      6. exp-neg N/A

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

      7. associate-*l/ N/A

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

      8. metadata-eval N/A

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

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

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

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

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

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

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

      12. exp-lowering-exp.f64 74.6%

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

   5. Simplified 74.6%

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

   6. Taylor expanded in im around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

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

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

      6. 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

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

   8. Simplified 74.6%

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

4. Final simplification 47.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 5 \cdot 10^{+44}:\\ \;\;\;\;im \cdot \frac{-1 + \left(\left(im \cdot im\right) \cdot \left(-0.16666666666666666 + \left(im \cdot im\right) \cdot \left(-0.008333333333333333 + im \cdot \left(im \cdot -0.0001984126984126984\right)\right)\right)\right) \cdot \left(\left(im \cdot im\right) \cdot \left(-0.16666666666666666 + \left(im \cdot im\right) \cdot \left(-0.008333333333333333 + im \cdot \left(im \cdot -0.0001984126984126984\right)\right)\right)\right)}{\left(im \cdot im\right) \cdot \left(-0.16666666666666666 + \left(im \cdot im\right) \cdot \left(-0.008333333333333333 + im \cdot \left(im \cdot -0.0001984126984126984\right)\right)\right) - -1}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-1 + im \cdot \left(im \cdot \left(-0.16666666666666666 + im \cdot \left(im \cdot \left(-0.008333333333333333 + \left(im \cdot im\right) \cdot -0.0001984126984126984\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 59.5% accurate, 11.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;re \leq 1.4 \cdot 10^{+219}:\\ \;\;\;\;im\_m \cdot \left(-1 + im\_m \cdot \left(im\_m \cdot \left(-0.16666666666666666 + im\_m \cdot \left(im\_m \cdot \left(-0.008333333333333333 + \left(im\_m \cdot im\_m\right) \cdot -0.0001984126984126984\right)\right)\right)\right)\right)\\ \mathbf{elif}\;re \leq 8.5 \cdot 10^{+269}:\\ \;\;\;\;re \cdot \left(re \cdot \left(im\_m \cdot 0.5\right)\right) - im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im\_m \cdot \left(\left(re \cdot re\right) \cdot -0.041666666666666664\right)\right) - im\_m\\ \end{array} \end{array} \]

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 1.4e+219)
    (*
     im_m
     (+
      -1.0
      (*
       im_m
       (*
        im_m
        (+
         -0.16666666666666666
         (*
          im_m
          (*
           im_m
           (+
            -0.008333333333333333
            (* (* im_m im_m) -0.0001984126984126984)))))))))
    (if (<= re 8.5e+269)
      (- (* re (* re (* im_m 0.5))) im_m)
      (- (* (* re re) (* im_m (* (* re re) -0.041666666666666664))) 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 <= 1.4e+219) {
		tmp = im_m * (-1.0 + (im_m * (im_m * (-0.16666666666666666 + (im_m * (im_m * (-0.008333333333333333 + ((im_m * im_m) * -0.0001984126984126984))))))));
	} else if (re <= 8.5e+269) {
		tmp = (re * (re * (im_m * 0.5))) - im_m;
	} else {
		tmp = ((re * re) * (im_m * ((re * re) * -0.041666666666666664))) - 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 <= 1.4d+219) 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 if (re <= 8.5d+269) then
        tmp = (re * (re * (im_m * 0.5d0))) - im_m
    else
        tmp = ((re * re) * (im_m * ((re * re) * (-0.041666666666666664d0)))) - 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 <= 1.4e+219) {
		tmp = im_m * (-1.0 + (im_m * (im_m * (-0.16666666666666666 + (im_m * (im_m * (-0.008333333333333333 + ((im_m * im_m) * -0.0001984126984126984))))))));
	} else if (re <= 8.5e+269) {
		tmp = (re * (re * (im_m * 0.5))) - im_m;
	} else {
		tmp = ((re * re) * (im_m * ((re * re) * -0.041666666666666664))) - 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 <= 1.4e+219:
		tmp = im_m * (-1.0 + (im_m * (im_m * (-0.16666666666666666 + (im_m * (im_m * (-0.008333333333333333 + ((im_m * im_m) * -0.0001984126984126984))))))))
	elif re <= 8.5e+269:
		tmp = (re * (re * (im_m * 0.5))) - im_m
	else:
		tmp = ((re * re) * (im_m * ((re * re) * -0.041666666666666664))) - 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 <= 1.4e+219)
		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(Float64(im_m * im_m) * -0.0001984126984126984)))))))));
	elseif (re <= 8.5e+269)
		tmp = Float64(Float64(re * Float64(re * Float64(im_m * 0.5))) - im_m);
	else
		tmp = Float64(Float64(Float64(re * re) * Float64(im_m * Float64(Float64(re * re) * -0.041666666666666664))) - 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 <= 1.4e+219)
		tmp = im_m * (-1.0 + (im_m * (im_m * (-0.16666666666666666 + (im_m * (im_m * (-0.008333333333333333 + ((im_m * im_m) * -0.0001984126984126984))))))));
	elseif (re <= 8.5e+269)
		tmp = (re * (re * (im_m * 0.5))) - im_m;
	else
		tmp = ((re * re) * (im_m * ((re * re) * -0.041666666666666664))) - 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, 1.4e+219], 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[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 8.5e+269], N[(N[(re * N[(re * N[(im$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im$95$m), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * N[(im$95$m * N[(N[(re * re), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if re < 1.40000000000000008e219

   1. Initial program 56.0%

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

   3. Taylor expanded in re around 0

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

      1. sub-neg N/A

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

      2. distribute-rgt-in N/A

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

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

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

      4. unsub-neg N/A

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

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

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

      6. exp-neg N/A

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

      7. associate-*l/ N/A

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

      8. metadata-eval N/A

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

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

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

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

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

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

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

      12. exp-lowering-exp.f64 46.1%

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

   5. Simplified 46.1%

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

   6. Taylor expanded in im around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

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

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

      6. 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

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

   8. Simplified 67.4%

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

   if 1.40000000000000008e219 < re < 8.5000000000000004e269

   1. Initial program 84.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 21.6%

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

   5. Simplified 21.6%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{{re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{720} \cdot \left(im \cdot {re}^{2}\right) - \frac{1}{24} \cdot im\right) - \frac{-1}{2} \cdot im\right) - im} \]
   7. Step-by-step derivation

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

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

   8. Simplified 58.6%

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

   9. Taylor expanded in re around 0

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 58.6%

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

   11. Simplified 58.6%

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

   if 8.5000000000000004e269 < re

   1. Initial program 64.5%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 39.5%

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

   5. Simplified 39.5%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{{re}^{2} \cdot \left(\frac{-1}{24} \cdot \left(im \cdot {re}^{2}\right) - \frac{-1}{2} \cdot im\right) - im} \]
   7. Step-by-step derivation

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-out N/A

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

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

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

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

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 50.6%

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

   8. Simplified 50.6%

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

   9. Taylor expanded in re around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

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

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

       5. *-commutative N/A

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

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

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

       7. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{24}\right)\right)\right), im\right) \]

       8. *-lowering-*.f64 50.6%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{24}\right)\right)\right), im\right) \]

   11. Simplified 50.6%

       \[\leadsto \left(re \cdot re\right) \cdot \color{blue}{\left(im \cdot \left(\left(re \cdot re\right) \cdot -0.041666666666666664\right)\right)} - im \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 57.6% accurate, 13.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;re \leq 1.4 \cdot 10^{+219}:\\ \;\;\;\;im\_m \cdot \left(-1 + im\_m \cdot \left(im\_m \cdot \left(-0.16666666666666666 + im\_m \cdot \left(im\_m \cdot -0.008333333333333333\right)\right)\right)\right)\\ \mathbf{elif}\;re \leq 8.5 \cdot 10^{+269}:\\ \;\;\;\;re \cdot \left(re \cdot \left(im\_m \cdot 0.5\right)\right) - im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im\_m \cdot \left(\left(re \cdot re\right) \cdot -0.041666666666666664\right)\right) - im\_m\\ \end{array} \end{array} \]

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 1.4e+219)
    (*
     im_m
     (+
      -1.0
      (*
       im_m
       (*
        im_m
        (+ -0.16666666666666666 (* im_m (* im_m -0.008333333333333333)))))))
    (if (<= re 8.5e+269)
      (- (* re (* re (* im_m 0.5))) im_m)
      (- (* (* re re) (* im_m (* (* re re) -0.041666666666666664))) 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 <= 1.4e+219) {
		tmp = im_m * (-1.0 + (im_m * (im_m * (-0.16666666666666666 + (im_m * (im_m * -0.008333333333333333))))));
	} else if (re <= 8.5e+269) {
		tmp = (re * (re * (im_m * 0.5))) - im_m;
	} else {
		tmp = ((re * re) * (im_m * ((re * re) * -0.041666666666666664))) - 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 <= 1.4d+219) then
        tmp = im_m * ((-1.0d0) + (im_m * (im_m * ((-0.16666666666666666d0) + (im_m * (im_m * (-0.008333333333333333d0)))))))
    else if (re <= 8.5d+269) then
        tmp = (re * (re * (im_m * 0.5d0))) - im_m
    else
        tmp = ((re * re) * (im_m * ((re * re) * (-0.041666666666666664d0)))) - 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 <= 1.4e+219) {
		tmp = im_m * (-1.0 + (im_m * (im_m * (-0.16666666666666666 + (im_m * (im_m * -0.008333333333333333))))));
	} else if (re <= 8.5e+269) {
		tmp = (re * (re * (im_m * 0.5))) - im_m;
	} else {
		tmp = ((re * re) * (im_m * ((re * re) * -0.041666666666666664))) - 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 <= 1.4e+219:
		tmp = im_m * (-1.0 + (im_m * (im_m * (-0.16666666666666666 + (im_m * (im_m * -0.008333333333333333))))))
	elif re <= 8.5e+269:
		tmp = (re * (re * (im_m * 0.5))) - im_m
	else:
		tmp = ((re * re) * (im_m * ((re * re) * -0.041666666666666664))) - 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 <= 1.4e+219)
		tmp = Float64(im_m * Float64(-1.0 + Float64(im_m * Float64(im_m * Float64(-0.16666666666666666 + Float64(im_m * Float64(im_m * -0.008333333333333333)))))));
	elseif (re <= 8.5e+269)
		tmp = Float64(Float64(re * Float64(re * Float64(im_m * 0.5))) - im_m);
	else
		tmp = Float64(Float64(Float64(re * re) * Float64(im_m * Float64(Float64(re * re) * -0.041666666666666664))) - 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 <= 1.4e+219)
		tmp = im_m * (-1.0 + (im_m * (im_m * (-0.16666666666666666 + (im_m * (im_m * -0.008333333333333333))))));
	elseif (re <= 8.5e+269)
		tmp = (re * (re * (im_m * 0.5))) - im_m;
	else
		tmp = ((re * re) * (im_m * ((re * re) * -0.041666666666666664))) - 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, 1.4e+219], 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 * -0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 8.5e+269], N[(N[(re * N[(re * N[(im$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im$95$m), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * N[(im$95$m * N[(N[(re * re), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if re < 1.40000000000000008e219

   1. Initial program 56.0%

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

   3. Taylor expanded in re around 0

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

      1. sub-neg N/A

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

      2. distribute-rgt-in N/A

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

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

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

      4. unsub-neg N/A

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

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

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

      6. exp-neg N/A

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

      7. associate-*l/ N/A

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

      8. metadata-eval N/A

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

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

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

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

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

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

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

      12. exp-lowering-exp.f64 46.1%

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

   5. Simplified 46.1%

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

   6. Taylor expanded in im around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

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

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

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

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

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

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

      14. *-commutative N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 65.8%

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

   8. Simplified 65.8%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-lowering-*.f64 65.8%

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

   10. Applied egg-rr 65.8%

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

   if 1.40000000000000008e219 < re < 8.5000000000000004e269

   1. Initial program 84.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 21.6%

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

   5. Simplified 21.6%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{{re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{720} \cdot \left(im \cdot {re}^{2}\right) - \frac{1}{24} \cdot im\right) - \frac{-1}{2} \cdot im\right) - im} \]
   7. Step-by-step derivation

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

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

   8. Simplified 58.6%

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

   9. Taylor expanded in re around 0

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 58.6%

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

   11. Simplified 58.6%

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

   if 8.5000000000000004e269 < re

   1. Initial program 64.5%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 39.5%

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

   5. Simplified 39.5%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{{re}^{2} \cdot \left(\frac{-1}{24} \cdot \left(im \cdot {re}^{2}\right) - \frac{-1}{2} \cdot im\right) - im} \]
   7. Step-by-step derivation

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-out N/A

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

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

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

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

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 50.6%

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

   8. Simplified 50.6%

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

   9. Taylor expanded in re around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

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

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

       5. *-commutative N/A

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

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

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

       7. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{24}\right)\right)\right), im\right) \]

       8. *-lowering-*.f64 50.6%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{24}\right)\right)\right), im\right) \]

   11. Simplified 50.6%

       \[\leadsto \left(re \cdot re\right) \cdot \color{blue}{\left(im \cdot \left(\left(re \cdot re\right) \cdot -0.041666666666666664\right)\right)} - im \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.4 \cdot 10^{+219}:\\ \;\;\;\;im \cdot \left(-1 + im \cdot \left(im \cdot \left(-0.16666666666666666 + im \cdot \left(im \cdot -0.008333333333333333\right)\right)\right)\right)\\ \mathbf{elif}\;re \leq 8.5 \cdot 10^{+269}:\\ \;\;\;\;re \cdot \left(re \cdot \left(im \cdot 0.5\right)\right) - im\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot \left(\left(re \cdot re\right) \cdot -0.041666666666666664\right)\right) - im\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.1% accurate, 13.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 0.0185:\\ \;\;\;\;im\_m \cdot \left(-1 + -0.16666666666666666 \cdot \left(im\_m \cdot im\_m\right)\right)\\ \mathbf{elif}\;im\_m \leq 1.16 \cdot 10^{+71}:\\ \;\;\;\;re \cdot \left(re \cdot \left(im\_m \cdot \left(0.5 + \frac{-1}{re \cdot re}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(-1 + im\_m \cdot \left(im\_m \cdot \left(-0.008333333333333333 \cdot \left(im\_m \cdot im\_m\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 0.0185)
    (* im_m (+ -1.0 (* -0.16666666666666666 (* im_m im_m))))
    (if (<= im_m 1.16e+71)
      (* re (* re (* im_m (+ 0.5 (/ -1.0 (* re re))))))
      (*
       im_m
       (+ -1.0 (* im_m (* im_m (* -0.008333333333333333 (* 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 <= 0.0185) {
		tmp = im_m * (-1.0 + (-0.16666666666666666 * (im_m * im_m)));
	} else if (im_m <= 1.16e+71) {
		tmp = re * (re * (im_m * (0.5 + (-1.0 / (re * re)))));
	} else {
		tmp = im_m * (-1.0 + (im_m * (im_m * (-0.008333333333333333 * (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 <= 0.0185d0) then
        tmp = im_m * ((-1.0d0) + ((-0.16666666666666666d0) * (im_m * im_m)))
    else if (im_m <= 1.16d+71) then
        tmp = re * (re * (im_m * (0.5d0 + ((-1.0d0) / (re * re)))))
    else
        tmp = im_m * ((-1.0d0) + (im_m * (im_m * ((-0.008333333333333333d0) * (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 <= 0.0185) {
		tmp = im_m * (-1.0 + (-0.16666666666666666 * (im_m * im_m)));
	} else if (im_m <= 1.16e+71) {
		tmp = re * (re * (im_m * (0.5 + (-1.0 / (re * re)))));
	} else {
		tmp = im_m * (-1.0 + (im_m * (im_m * (-0.008333333333333333 * (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 <= 0.0185:
		tmp = im_m * (-1.0 + (-0.16666666666666666 * (im_m * im_m)))
	elif im_m <= 1.16e+71:
		tmp = re * (re * (im_m * (0.5 + (-1.0 / (re * re)))))
	else:
		tmp = im_m * (-1.0 + (im_m * (im_m * (-0.008333333333333333 * (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 <= 0.0185)
		tmp = Float64(im_m * Float64(-1.0 + Float64(-0.16666666666666666 * Float64(im_m * im_m))));
	elseif (im_m <= 1.16e+71)
		tmp = Float64(re * Float64(re * Float64(im_m * Float64(0.5 + Float64(-1.0 / Float64(re * re))))));
	else
		tmp = Float64(im_m * Float64(-1.0 + Float64(im_m * Float64(im_m * Float64(-0.008333333333333333 * 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 <= 0.0185)
		tmp = im_m * (-1.0 + (-0.16666666666666666 * (im_m * im_m)));
	elseif (im_m <= 1.16e+71)
		tmp = re * (re * (im_m * (0.5 + (-1.0 / (re * re)))));
	else
		tmp = im_m * (-1.0 + (im_m * (im_m * (-0.008333333333333333 * (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, 0.0185], N[(im$95$m * N[(-1.0 + N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 1.16e+71], N[(re * N[(re * N[(im$95$m * N[(0.5 + N[(-1.0 / N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(-1.0 + N[(im$95$m * N[(im$95$m * N[(-0.008333333333333333 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 0.0184999999999999991

   1. Initial program 41.6%

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

   3. Taylor expanded in re around 0

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

      1. sub-neg N/A

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

      2. distribute-rgt-in N/A

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

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

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

      4. unsub-neg N/A

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

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

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

      6. exp-neg N/A

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

      7. associate-*l/ N/A

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

      8. metadata-eval N/A

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

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

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

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

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

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

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

      12. exp-lowering-exp.f64 35.0%

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

   5. Simplified 35.0%

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

   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 61.6%

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

   8. Simplified 61.6%

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

   if 0.0184999999999999991 < im < 1.1599999999999999e71

   1. Initial program 99.7%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 5.6%

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

   5. Simplified 5.6%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{{re}^{2} \cdot \left(\frac{-1}{24} \cdot \left(im \cdot {re}^{2}\right) - \frac{-1}{2} \cdot im\right) - im} \]
   7. Step-by-step derivation

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-out N/A

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

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

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

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

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 24.8%

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

   8. Simplified 24.8%

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

   9. Taylor expanded in re around 0

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 15.0%

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

   11. Simplified 15.0%

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

   12. Taylor expanded in re around inf

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

       3. *-commutative N/A

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

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

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

       5. *-commutative N/A

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

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

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

       7. associate-*r/ N/A

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

       8. *-commutative N/A

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

       9. associate-/l* N/A

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

       10. *-commutative N/A

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

       11. distribute-lft-out N/A

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

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

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

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

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

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

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

       15. unpow2 N/A

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

       16. *-lowering-*.f64 43.4%

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

   14. Simplified 43.4%

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

   if 1.1599999999999999e71 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. sub-neg N/A

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

      2. distribute-rgt-in N/A

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

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

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

      4. unsub-neg N/A

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

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

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

      6. exp-neg N/A

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

      7. associate-*l/ N/A

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

      8. metadata-eval N/A

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

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

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

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

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

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

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

      12. exp-lowering-exp.f64 77.4%

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

   5. Simplified 77.4%

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

   6. Taylor expanded in im around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

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

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

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

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

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

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

      14. *-commutative N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 77.4%

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

   8. Simplified 77.4%

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

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{120} \cdot {im}^{3}\right)}\right)\right)\right) \]
   10. Step-by-step derivation

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

       6. *-commutative N/A

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

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

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

       8. unpow2 N/A

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

       9. *-lowering-*.f64 77.4%

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

   11. Simplified 77.4%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.0185:\\ \;\;\;\;im \cdot \left(-1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 1.16 \cdot 10^{+71}:\\ \;\;\;\;re \cdot \left(re \cdot \left(im \cdot \left(0.5 + \frac{-1}{re \cdot re}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-1 + im \cdot \left(im \cdot \left(-0.008333333333333333 \cdot \left(im \cdot im\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.5% accurate, 14.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;re \leq 1.4 \cdot 10^{+219}:\\ \;\;\;\;im\_m \cdot \left(-1 + im\_m \cdot \left(im\_m \cdot \left(-0.008333333333333333 \cdot \left(im\_m \cdot im\_m\right)\right)\right)\right)\\ \mathbf{elif}\;re \leq 8.5 \cdot 10^{+269}:\\ \;\;\;\;re \cdot \left(re \cdot \left(im\_m \cdot 0.5\right)\right) - im\_m\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(re \cdot \left(re \cdot \left(\left(re \cdot re\right) \cdot -0.041666666666666664\right)\right)\right)\\ \end{array} \end{array} \]

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 1.4e+219)
    (* im_m (+ -1.0 (* im_m (* im_m (* -0.008333333333333333 (* im_m im_m))))))
    (if (<= re 8.5e+269)
      (- (* re (* re (* im_m 0.5))) im_m)
      (* im_m (* re (* re (* (* re re) -0.041666666666666664))))))))

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 <= 1.4e+219) {
		tmp = im_m * (-1.0 + (im_m * (im_m * (-0.008333333333333333 * (im_m * im_m)))));
	} else if (re <= 8.5e+269) {
		tmp = (re * (re * (im_m * 0.5))) - im_m;
	} else {
		tmp = im_m * (re * (re * ((re * re) * -0.041666666666666664)));
	}
	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 <= 1.4d+219) then
        tmp = im_m * ((-1.0d0) + (im_m * (im_m * ((-0.008333333333333333d0) * (im_m * im_m)))))
    else if (re <= 8.5d+269) then
        tmp = (re * (re * (im_m * 0.5d0))) - im_m
    else
        tmp = im_m * (re * (re * ((re * re) * (-0.041666666666666664d0))))
    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 <= 1.4e+219) {
		tmp = im_m * (-1.0 + (im_m * (im_m * (-0.008333333333333333 * (im_m * im_m)))));
	} else if (re <= 8.5e+269) {
		tmp = (re * (re * (im_m * 0.5))) - im_m;
	} else {
		tmp = im_m * (re * (re * ((re * re) * -0.041666666666666664)));
	}
	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 <= 1.4e+219:
		tmp = im_m * (-1.0 + (im_m * (im_m * (-0.008333333333333333 * (im_m * im_m)))))
	elif re <= 8.5e+269:
		tmp = (re * (re * (im_m * 0.5))) - im_m
	else:
		tmp = im_m * (re * (re * ((re * re) * -0.041666666666666664)))
	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 <= 1.4e+219)
		tmp = Float64(im_m * Float64(-1.0 + Float64(im_m * Float64(im_m * Float64(-0.008333333333333333 * Float64(im_m * im_m))))));
	elseif (re <= 8.5e+269)
		tmp = Float64(Float64(re * Float64(re * Float64(im_m * 0.5))) - im_m);
	else
		tmp = Float64(im_m * Float64(re * Float64(re * Float64(Float64(re * re) * -0.041666666666666664))));
	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 <= 1.4e+219)
		tmp = im_m * (-1.0 + (im_m * (im_m * (-0.008333333333333333 * (im_m * im_m)))));
	elseif (re <= 8.5e+269)
		tmp = (re * (re * (im_m * 0.5))) - im_m;
	else
		tmp = im_m * (re * (re * ((re * re) * -0.041666666666666664)));
	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, 1.4e+219], N[(im$95$m * N[(-1.0 + N[(im$95$m * N[(im$95$m * N[(-0.008333333333333333 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 8.5e+269], N[(N[(re * N[(re * N[(im$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im$95$m), $MachinePrecision], N[(im$95$m * N[(re * N[(re * N[(N[(re * re), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if re < 1.40000000000000008e219

   1. Initial program 56.0%

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

   3. Taylor expanded in re around 0

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

      1. sub-neg N/A

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

      2. distribute-rgt-in N/A

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

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

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

      4. unsub-neg N/A

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

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

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

      6. exp-neg N/A

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

      7. associate-*l/ N/A

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

      8. metadata-eval N/A

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

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

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

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

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

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

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

      12. exp-lowering-exp.f64 46.1%

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

   5. Simplified 46.1%

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

   6. Taylor expanded in im around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

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

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

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

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

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

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

      14. *-commutative N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 65.8%

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

   8. Simplified 65.8%

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

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{120} \cdot {im}^{3}\right)}\right)\right)\right) \]
   10. Step-by-step derivation

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

       6. *-commutative N/A

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

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

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

       8. unpow2 N/A

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

       9. *-lowering-*.f64 65.6%

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

   11. Simplified 65.6%

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

   if 1.40000000000000008e219 < re < 8.5000000000000004e269

   1. Initial program 84.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 21.6%

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

   5. Simplified 21.6%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{{re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{720} \cdot \left(im \cdot {re}^{2}\right) - \frac{1}{24} \cdot im\right) - \frac{-1}{2} \cdot im\right) - im} \]
   7. Step-by-step derivation

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

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

   8. Simplified 58.6%

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

   9. Taylor expanded in re around 0

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 58.6%

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

   11. Simplified 58.6%

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

   if 8.5000000000000004e269 < re

   1. Initial program 64.5%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 39.5%

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

   5. Simplified 39.5%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{{re}^{2} \cdot \left(\frac{-1}{24} \cdot \left(im \cdot {re}^{2}\right) - \frac{-1}{2} \cdot im\right) - im} \]
   7. Step-by-step derivation

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-out N/A

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

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

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

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

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 50.6%

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

   8. Simplified 50.6%

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

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{-1}{24} \cdot \left(im \cdot {re}^{4}\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(im \cdot {re}^{4}\right) \cdot \color{blue}{\frac{-1}{24}} \]

       2. associate-*l* N/A

          \[\leadsto im \cdot \color{blue}{\left({re}^{4} \cdot \frac{-1}{24}\right)} \]

       3. metadata-eval N/A

          \[\leadsto im \cdot \left({re}^{\left(2 \cdot 2\right)} \cdot \frac{-1}{24}\right) \]

       4. pow-sqr N/A

          \[\leadsto im \cdot \left(\left({re}^{2} \cdot {re}^{2}\right) \cdot \frac{-1}{24}\right) \]

       5. associate-*r* N/A

          \[\leadsto im \cdot \left({re}^{2} \cdot \color{blue}{\left({re}^{2} \cdot \frac{-1}{24}\right)}\right) \]

       6. *-commutative N/A

          \[\leadsto im \cdot \left({re}^{2} \cdot \left(\frac{-1}{24} \cdot \color{blue}{{re}^{2}}\right)\right) \]

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

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

       8. unpow2 N/A

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

       9. associate-*l* N/A

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

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

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

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

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

       12. *-commutative N/A

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

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

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

       14. unpow2 N/A

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

       15. *-lowering-*.f64 50.6%

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

   11. Simplified 50.6%

       \[\leadsto \color{blue}{im \cdot \left(re \cdot \left(re \cdot \left(\left(re \cdot re\right) \cdot -0.041666666666666664\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.4 \cdot 10^{+219}:\\ \;\;\;\;im \cdot \left(-1 + im \cdot \left(im \cdot \left(-0.008333333333333333 \cdot \left(im \cdot im\right)\right)\right)\right)\\ \mathbf{elif}\;re \leq 8.5 \cdot 10^{+269}:\\ \;\;\;\;re \cdot \left(re \cdot \left(im \cdot 0.5\right)\right) - im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot \left(\left(re \cdot re\right) \cdot -0.041666666666666664\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 53.2% accurate, 14.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;re \leq 1.4 \cdot 10^{+219}:\\ \;\;\;\;im\_m \cdot \left(-1 + -0.16666666666666666 \cdot \left(im\_m \cdot im\_m\right)\right)\\ \mathbf{elif}\;re \leq 8.5 \cdot 10^{+269}:\\ \;\;\;\;re \cdot \left(re \cdot \left(im\_m \cdot 0.5\right)\right) - im\_m\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(re \cdot \left(re \cdot \left(\left(re \cdot re\right) \cdot -0.041666666666666664\right)\right)\right)\\ \end{array} \end{array} \]

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 1.4e+219)
    (* im_m (+ -1.0 (* -0.16666666666666666 (* im_m im_m))))
    (if (<= re 8.5e+269)
      (- (* re (* re (* im_m 0.5))) im_m)
      (* im_m (* re (* re (* (* re re) -0.041666666666666664))))))))

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 <= 1.4e+219) {
		tmp = im_m * (-1.0 + (-0.16666666666666666 * (im_m * im_m)));
	} else if (re <= 8.5e+269) {
		tmp = (re * (re * (im_m * 0.5))) - im_m;
	} else {
		tmp = im_m * (re * (re * ((re * re) * -0.041666666666666664)));
	}
	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 <= 1.4d+219) then
        tmp = im_m * ((-1.0d0) + ((-0.16666666666666666d0) * (im_m * im_m)))
    else if (re <= 8.5d+269) then
        tmp = (re * (re * (im_m * 0.5d0))) - im_m
    else
        tmp = im_m * (re * (re * ((re * re) * (-0.041666666666666664d0))))
    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 <= 1.4e+219) {
		tmp = im_m * (-1.0 + (-0.16666666666666666 * (im_m * im_m)));
	} else if (re <= 8.5e+269) {
		tmp = (re * (re * (im_m * 0.5))) - im_m;
	} else {
		tmp = im_m * (re * (re * ((re * re) * -0.041666666666666664)));
	}
	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 <= 1.4e+219:
		tmp = im_m * (-1.0 + (-0.16666666666666666 * (im_m * im_m)))
	elif re <= 8.5e+269:
		tmp = (re * (re * (im_m * 0.5))) - im_m
	else:
		tmp = im_m * (re * (re * ((re * re) * -0.041666666666666664)))
	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 <= 1.4e+219)
		tmp = Float64(im_m * Float64(-1.0 + Float64(-0.16666666666666666 * Float64(im_m * im_m))));
	elseif (re <= 8.5e+269)
		tmp = Float64(Float64(re * Float64(re * Float64(im_m * 0.5))) - im_m);
	else
		tmp = Float64(im_m * Float64(re * Float64(re * Float64(Float64(re * re) * -0.041666666666666664))));
	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 <= 1.4e+219)
		tmp = im_m * (-1.0 + (-0.16666666666666666 * (im_m * im_m)));
	elseif (re <= 8.5e+269)
		tmp = (re * (re * (im_m * 0.5))) - im_m;
	else
		tmp = im_m * (re * (re * ((re * re) * -0.041666666666666664)));
	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, 1.4e+219], N[(im$95$m * N[(-1.0 + N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 8.5e+269], N[(N[(re * N[(re * N[(im$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im$95$m), $MachinePrecision], N[(im$95$m * N[(re * N[(re * N[(N[(re * re), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if re < 1.40000000000000008e219

   1. Initial program 56.0%

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

   3. Taylor expanded in re around 0

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

      1. sub-neg N/A

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

      2. distribute-rgt-in N/A

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

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

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

      4. unsub-neg N/A

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

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

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

      6. exp-neg N/A

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

      7. associate-*l/ N/A

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

      8. metadata-eval N/A

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

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

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

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

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

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

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

      12. exp-lowering-exp.f64 46.1%

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

   5. Simplified 46.1%

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

   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 61.5%

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

   8. Simplified 61.5%

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

   if 1.40000000000000008e219 < re < 8.5000000000000004e269

   1. Initial program 84.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 21.6%

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

   5. Simplified 21.6%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{{re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{720} \cdot \left(im \cdot {re}^{2}\right) - \frac{1}{24} \cdot im\right) - \frac{-1}{2} \cdot im\right) - im} \]
   7. Step-by-step derivation

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

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

   8. Simplified 58.6%

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

   9. Taylor expanded in re around 0

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 58.6%

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

   11. Simplified 58.6%

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

   if 8.5000000000000004e269 < re

   1. Initial program 64.5%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 39.5%

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

   5. Simplified 39.5%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{{re}^{2} \cdot \left(\frac{-1}{24} \cdot \left(im \cdot {re}^{2}\right) - \frac{-1}{2} \cdot im\right) - im} \]
   7. Step-by-step derivation

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-out N/A

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

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

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

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\left({re}^{2} \cdot \frac{-1}{24}\right), \frac{1}{2}\right)\right)\right), im\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({re}^{2}\right), \frac{-1}{24}\right), \frac{1}{2}\right)\right)\right), im\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{24}\right), \frac{1}{2}\right)\right)\right), im\right) \]

      15. *-lowering-*.f64 50.6%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{24}\right), \frac{1}{2}\right)\right)\right), im\right) \]

   8. Simplified 50.6%

      \[\leadsto \color{blue}{\left(re \cdot re\right) \cdot \left(im \cdot \left(\left(re \cdot re\right) \cdot -0.041666666666666664 + 0.5\right)\right) - im} \]

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{-1}{24} \cdot \left(im \cdot {re}^{4}\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(im \cdot {re}^{4}\right) \cdot \color{blue}{\frac{-1}{24}} \]

       2. associate-*l* N/A

          \[\leadsto im \cdot \color{blue}{\left({re}^{4} \cdot \frac{-1}{24}\right)} \]

       3. metadata-eval N/A

          \[\leadsto im \cdot \left({re}^{\left(2 \cdot 2\right)} \cdot \frac{-1}{24}\right) \]

       4. pow-sqr N/A

          \[\leadsto im \cdot \left(\left({re}^{2} \cdot {re}^{2}\right) \cdot \frac{-1}{24}\right) \]

       5. associate-*r* N/A

          \[\leadsto im \cdot \left({re}^{2} \cdot \color{blue}{\left({re}^{2} \cdot \frac{-1}{24}\right)}\right) \]

       6. *-commutative N/A

          \[\leadsto im \cdot \left({re}^{2} \cdot \left(\frac{-1}{24} \cdot \color{blue}{{re}^{2}}\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left({re}^{2} \cdot \left(\frac{-1}{24} \cdot {re}^{2}\right)\right)}\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left(\left(re \cdot re\right) \cdot \left(\color{blue}{\frac{-1}{24}} \cdot {re}^{2}\right)\right)\right) \]

       9. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left(re \cdot \color{blue}{\left(re \cdot \left(\frac{-1}{24} \cdot {re}^{2}\right)\right)}\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, \color{blue}{\left(re \cdot \left(\frac{-1}{24} \cdot {re}^{2}\right)\right)}\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{-1}{24} \cdot {re}^{2}\right)}\right)\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left({re}^{2} \cdot \color{blue}{\frac{-1}{24}}\right)\right)\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{-1}{24}}\right)\right)\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{24}\right)\right)\right)\right) \]

       15. *-lowering-*.f64 50.6%

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{24}\right)\right)\right)\right) \]

   11. Simplified 50.6%

       \[\leadsto \color{blue}{im \cdot \left(re \cdot \left(re \cdot \left(\left(re \cdot re\right) \cdot -0.041666666666666664\right)\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 48.8% accurate, 18.1× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 30500000:\\ \;\;\;\;0 - im\_m\\ \mathbf{elif}\;im\_m \leq 5.4 \cdot 10^{+106}:\\ \;\;\;\;im\_m \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0 - im\_m \cdot im\_m}{im\_m}\\ \end{array} \end{array} \]

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 30500000.0)
    (- 0.0 im_m)
    (if (<= im_m 5.4e+106)
      (* im_m (* 0.5 (* re re)))
      (/ (- 0.0 (* im_m im_m)) im_m)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 30500000.0) {
		tmp = 0.0 - im_m;
	} else if (im_m <= 5.4e+106) {
		tmp = im_m * (0.5 * (re * re));
	} else {
		tmp = (0.0 - (im_m * 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 <= 30500000.0d0) then
        tmp = 0.0d0 - im_m
    else if (im_m <= 5.4d+106) then
        tmp = im_m * (0.5d0 * (re * re))
    else
        tmp = (0.0d0 - (im_m * 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 <= 30500000.0) {
		tmp = 0.0 - im_m;
	} else if (im_m <= 5.4e+106) {
		tmp = im_m * (0.5 * (re * re));
	} else {
		tmp = (0.0 - (im_m * 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 <= 30500000.0:
		tmp = 0.0 - im_m
	elif im_m <= 5.4e+106:
		tmp = im_m * (0.5 * (re * re))
	else:
		tmp = (0.0 - (im_m * 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 <= 30500000.0)
		tmp = Float64(0.0 - im_m);
	elseif (im_m <= 5.4e+106)
		tmp = Float64(im_m * Float64(0.5 * Float64(re * re)));
	else
		tmp = Float64(Float64(0.0 - Float64(im_m * 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 <= 30500000.0)
		tmp = 0.0 - im_m;
	elseif (im_m <= 5.4e+106)
		tmp = im_m * (0.5 * (re * re));
	else
		tmp = (0.0 - (im_m * 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, 30500000.0], N[(0.0 - im$95$m), $MachinePrecision], If[LessEqual[im$95$m, 5.4e+106], N[(im$95$m * N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.0 - N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] / im$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 3.05e7

   1. Initial program 42.2%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 64.2%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \mathsf{cos.f64}\left(re\right)\right)\right) \]

   5. Simplified 64.2%

      \[\leadsto \color{blue}{0 - im \cdot \cos re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{-1 \cdot im} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(im\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{im} \]

      3. --lowering--.f64 38.3%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{im}\right) \]

   8. Simplified 38.3%

      \[\leadsto \color{blue}{0 - im} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(im\right) \]

      2. neg-lowering-neg.f64 38.3%

         \[\leadsto \mathsf{neg.f64}\left(im\right) \]

   10. Applied egg-rr 38.3%

       \[\leadsto \color{blue}{-im} \]

   if 3.05e7 < im < 5.40000000000000012e106

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \cos re\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(im \cdot \cos re\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{im \cdot \cos re} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(im \cdot \cos re\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \color{blue}{\cos re}\right)\right) \]

      5. cos-lowering-cos.f64 3.6%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \mathsf{cos.f64}\left(re\right)\right)\right) \]

   5. Simplified 3.6%

      \[\leadsto \color{blue}{0 - im \cdot \cos re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{{re}^{2} \cdot \left(\frac{-1}{24} \cdot \left(im \cdot {re}^{2}\right) - \frac{-1}{2} \cdot im\right) - im} \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({re}^{2} \cdot \left(\frac{-1}{24} \cdot \left(im \cdot {re}^{2}\right) - \frac{-1}{2} \cdot im\right)\right), \color{blue}{im}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left({re}^{2}\right), \left(\frac{-1}{24} \cdot \left(im \cdot {re}^{2}\right) - \frac{-1}{2} \cdot im\right)\right), im\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(re \cdot re\right), \left(\frac{-1}{24} \cdot \left(im \cdot {re}^{2}\right) - \frac{-1}{2} \cdot im\right)\right), im\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{-1}{24} \cdot \left(im \cdot {re}^{2}\right) - \frac{-1}{2} \cdot im\right)\right), im\right) \]

      5. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{-1}{24} \cdot \left(im \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot im\right)\right), im\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\left(im \cdot {re}^{2}\right) \cdot \frac{-1}{24} + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot im\right)\right), im\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(im \cdot \left({re}^{2} \cdot \frac{-1}{24}\right) + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot im\right)\right), im\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(im \cdot \left({re}^{2} \cdot \frac{-1}{24}\right) + \frac{1}{2} \cdot im\right)\right), im\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(im \cdot \left({re}^{2} \cdot \frac{-1}{24}\right) + im \cdot \frac{1}{2}\right)\right), im\right) \]

      10. distribute-lft-out N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(im \cdot \left({re}^{2} \cdot \frac{-1}{24} + \frac{1}{2}\right)\right)\right), im\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(im, \left({re}^{2} \cdot \frac{-1}{24} + \frac{1}{2}\right)\right)\right), im\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\left({re}^{2} \cdot \frac{-1}{24}\right), \frac{1}{2}\right)\right)\right), im\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({re}^{2}\right), \frac{-1}{24}\right), \frac{1}{2}\right)\right)\right), im\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{24}\right), \frac{1}{2}\right)\right)\right), im\right) \]

      15. *-lowering-*.f64 27.4%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{24}\right), \frac{1}{2}\right)\right)\right), im\right) \]

   8. Simplified 27.4%

      \[\leadsto \color{blue}{\left(re \cdot re\right) \cdot \left(im \cdot \left(\left(re \cdot re\right) \cdot -0.041666666666666664 + 0.5\right)\right) - im} \]

   9. Taylor expanded in re around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \color{blue}{\left(\frac{1}{2} \cdot im\right)}\right), im\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(im \cdot \frac{1}{2}\right)\right), im\right) \]

       2. *-lowering-*.f64 21.1%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(im, \frac{1}{2}\right)\right), im\right) \]

   11. Simplified 21.1%

       \[\leadsto \left(re \cdot re\right) \cdot \color{blue}{\left(im \cdot 0.5\right)} - im \]

   12. Taylor expanded in re around inf

       \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(im \cdot {re}^{2}\right)} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(im \cdot {re}^{2}\right) \cdot \color{blue}{\frac{1}{2}} \]

       2. associate-*l* N/A

          \[\leadsto im \cdot \color{blue}{\left({re}^{2} \cdot \frac{1}{2}\right)} \]

       3. *-commutative N/A

          \[\leadsto im \cdot \left(\frac{1}{2} \cdot \color{blue}{{re}^{2}}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{2} \cdot {re}^{2}\right)}\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left({re}^{2} \cdot \color{blue}{\frac{1}{2}}\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{1}{2}}\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{1}{2}\right)\right) \]

       8. *-lowering-*.f64 20.1%

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{1}{2}\right)\right) \]

   14. Simplified 20.1%

       \[\leadsto \color{blue}{im \cdot \left(\left(re \cdot re\right) \cdot 0.5\right)} \]

   if 5.40000000000000012e106 < 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 6.4%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \mathsf{cos.f64}\left(re\right)\right)\right) \]

   5. Simplified 6.4%

      \[\leadsto \color{blue}{0 - im \cdot \cos re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{-1 \cdot im} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(im\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{im} \]

      3. --lowering--.f64 4.9%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{im}\right) \]

   8. Simplified 4.9%

      \[\leadsto \color{blue}{0 - im} \]
   9. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \frac{0 \cdot 0 - im \cdot im}{\color{blue}{0 + im}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{0 - im \cdot im}{0 + im} \]

      3. sub0-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(im \cdot im\right)}{\color{blue}{0} + im} \]

      4. +-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(im \cdot im\right)}{im} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(im \cdot im\right)\right), \color{blue}{im}\right) \]

      6. sub0-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(0 - im \cdot im\right), im\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(im \cdot im\right)\right), im\right) \]

      8. *-lowering-*.f64 54.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, im\right)\right), im\right) \]

   10. Applied egg-rr 54.7%

       \[\leadsto \color{blue}{\frac{0 - im \cdot im}{im}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 39.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 30500000:\\ \;\;\;\;0 - im\\ \mathbf{elif}\;im \leq 5.4 \cdot 10^{+106}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0 - im \cdot im}{im}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 53.1% accurate, 22.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;re \leq 1.4 \cdot 10^{+219}:\\ \;\;\;\;im\_m \cdot \left(-1 + -0.16666666666666666 \cdot \left(im\_m \cdot im\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(re \cdot \left(im\_m \cdot 0.5\right)\right) - im\_m\\ \end{array} \end{array} \]

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 1.4e+219)
    (* im_m (+ -1.0 (* -0.16666666666666666 (* im_m im_m))))
    (- (* re (* re (* im_m 0.5))) 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 <= 1.4e+219) {
		tmp = im_m * (-1.0 + (-0.16666666666666666 * (im_m * im_m)));
	} else {
		tmp = (re * (re * (im_m * 0.5))) - 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 <= 1.4d+219) then
        tmp = im_m * ((-1.0d0) + ((-0.16666666666666666d0) * (im_m * im_m)))
    else
        tmp = (re * (re * (im_m * 0.5d0))) - 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 <= 1.4e+219) {
		tmp = im_m * (-1.0 + (-0.16666666666666666 * (im_m * im_m)));
	} else {
		tmp = (re * (re * (im_m * 0.5))) - 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 <= 1.4e+219:
		tmp = im_m * (-1.0 + (-0.16666666666666666 * (im_m * im_m)))
	else:
		tmp = (re * (re * (im_m * 0.5))) - 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 <= 1.4e+219)
		tmp = Float64(im_m * Float64(-1.0 + Float64(-0.16666666666666666 * Float64(im_m * im_m))));
	else
		tmp = Float64(Float64(re * Float64(re * Float64(im_m * 0.5))) - 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 <= 1.4e+219)
		tmp = im_m * (-1.0 + (-0.16666666666666666 * (im_m * im_m)));
	else
		tmp = (re * (re * (im_m * 0.5))) - 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, 1.4e+219], N[(im$95$m * N[(-1.0 + N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(re * N[(re * N[(im$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if re < 1.40000000000000008e219

   1. Initial program 56.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + \color{blue}{\left(\mathsf{neg}\left(e^{im}\right)\right)}\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(e^{im}\right)\right) \cdot \frac{1}{2}} \]

      3. distribute-lft-neg-out N/A

         \[\leadsto e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2} + \left(\mathsf{neg}\left(e^{im} \cdot \frac{1}{2}\right)\right) \]

      4. unsub-neg N/A

         \[\leadsto e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2} - \color{blue}{e^{im} \cdot \frac{1}{2}} \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right), \color{blue}{\left(e^{im} \cdot \frac{1}{2}\right)}\right) \]

      6. exp-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{1}{e^{im}} \cdot \frac{1}{2}\right), \left(e^{\color{blue}{im}} \cdot \frac{1}{2}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{1 \cdot \frac{1}{2}}{e^{im}}\right), \left(\color{blue}{e^{im}} \cdot \frac{1}{2}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\frac{1}{2}}{e^{im}}\right), \left(e^{\color{blue}{im}} \cdot \frac{1}{2}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(e^{im}\right)\right), \left(\color{blue}{e^{im}} \cdot \frac{1}{2}\right)\right) \]

      10. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(e^{im} \cdot \frac{1}{2}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{*.f64}\left(\left(e^{im}\right), \color{blue}{\frac{1}{2}}\right)\right) \]

      12. exp-lowering-exp.f64 46.1%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{1}{2}\right)\right) \]

   5. Simplified 46.1%

      \[\leadsto \color{blue}{\frac{0.5}{e^{im}} - e^{im} \cdot 0.5} \]

   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 61.5%

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   8. Simplified 61.5%

      \[\leadsto \color{blue}{im \cdot \left(-1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)} \]

   if 1.40000000000000008e219 < re

   1. Initial program 76.2%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 28.8%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \mathsf{cos.f64}\left(re\right)\right)\right) \]

   5. Simplified 28.8%

      \[\leadsto \color{blue}{0 - im \cdot \cos re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{{re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{720} \cdot \left(im \cdot {re}^{2}\right) - \frac{1}{24} \cdot im\right) - \frac{-1}{2} \cdot im\right) - im} \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{720} \cdot \left(im \cdot {re}^{2}\right) - \frac{1}{24} \cdot im\right) - \frac{-1}{2} \cdot im\right)\right), \color{blue}{im}\right) \]

   8. Simplified 40.4%

      \[\leadsto \color{blue}{re \cdot \left(re \cdot \left(\left(re \cdot re\right) \cdot \left(im \cdot \left(\left(re \cdot re\right) \cdot 0.001388888888888889 + -0.041666666666666664\right)\right) + im \cdot 0.5\right)\right) - im} \]

   9. Taylor expanded in re around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{1}{2} \cdot im\right)}\right)\right), im\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \left(im \cdot \frac{1}{2}\right)\right)\right), im\right) \]

       2. *-lowering-*.f64 40.4%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \frac{1}{2}\right)\right)\right), im\right) \]

   11. Simplified 40.4%

       \[\leadsto re \cdot \left(re \cdot \color{blue}{\left(im \cdot 0.5\right)}\right) - im \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 53.1% accurate, 22.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;re \leq 1.4 \cdot 10^{+219}:\\ \;\;\;\;im\_m \cdot \left(-1 + -0.16666666666666666 \cdot \left(im\_m \cdot im\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(-1 + 0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

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 1.4e+219)
    (* im_m (+ -1.0 (* -0.16666666666666666 (* im_m im_m))))
    (* 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 <= 1.4e+219) {
		tmp = im_m * (-1.0 + (-0.16666666666666666 * (im_m * im_m)));
	} 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 <= 1.4d+219) then
        tmp = im_m * ((-1.0d0) + ((-0.16666666666666666d0) * (im_m * im_m)))
    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 <= 1.4e+219) {
		tmp = im_m * (-1.0 + (-0.16666666666666666 * (im_m * im_m)));
	} 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 <= 1.4e+219:
		tmp = im_m * (-1.0 + (-0.16666666666666666 * (im_m * im_m)))
	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 <= 1.4e+219)
		tmp = Float64(im_m * Float64(-1.0 + Float64(-0.16666666666666666 * Float64(im_m * im_m))));
	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 <= 1.4e+219)
		tmp = im_m * (-1.0 + (-0.16666666666666666 * (im_m * im_m)));
	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, 1.4e+219], N[(im$95$m * N[(-1.0 + N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision]), $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 < 1.40000000000000008e219

   1. Initial program 56.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + \color{blue}{\left(\mathsf{neg}\left(e^{im}\right)\right)}\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(e^{im}\right)\right) \cdot \frac{1}{2}} \]

      3. distribute-lft-neg-out N/A

         \[\leadsto e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2} + \left(\mathsf{neg}\left(e^{im} \cdot \frac{1}{2}\right)\right) \]

      4. unsub-neg N/A

         \[\leadsto e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2} - \color{blue}{e^{im} \cdot \frac{1}{2}} \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right), \color{blue}{\left(e^{im} \cdot \frac{1}{2}\right)}\right) \]

      6. exp-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{1}{e^{im}} \cdot \frac{1}{2}\right), \left(e^{\color{blue}{im}} \cdot \frac{1}{2}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{1 \cdot \frac{1}{2}}{e^{im}}\right), \left(\color{blue}{e^{im}} \cdot \frac{1}{2}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\frac{1}{2}}{e^{im}}\right), \left(e^{\color{blue}{im}} \cdot \frac{1}{2}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(e^{im}\right)\right), \left(\color{blue}{e^{im}} \cdot \frac{1}{2}\right)\right) \]

      10. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(e^{im} \cdot \frac{1}{2}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{*.f64}\left(\left(e^{im}\right), \color{blue}{\frac{1}{2}}\right)\right) \]

      12. exp-lowering-exp.f64 46.1%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{1}{2}\right)\right) \]

   5. Simplified 46.1%

      \[\leadsto \color{blue}{\frac{0.5}{e^{im}} - e^{im} \cdot 0.5} \]

   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 61.5%

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   8. Simplified 61.5%

      \[\leadsto \color{blue}{im \cdot \left(-1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)} \]

   if 1.40000000000000008e219 < re

   1. Initial program 76.2%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 28.8%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \mathsf{cos.f64}\left(re\right)\right)\right) \]

   5. Simplified 28.8%

      \[\leadsto \color{blue}{0 - im \cdot \cos re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(im \cdot {re}^{2}\right) - im} \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \frac{1}{2} \cdot \left(im \cdot {re}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(im\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left({re}^{2} \cdot im\right) + \left(\mathsf{neg}\left(im\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\frac{1}{2} \cdot {re}^{2}\right) \cdot im + \left(\mathsf{neg}\left(\color{blue}{im}\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \left(\frac{1}{2} \cdot {re}^{2}\right) \cdot im + -1 \cdot \color{blue}{im} \]

      5. distribute-rgt-out N/A

         \[\leadsto im \cdot \color{blue}{\left(\frac{1}{2} \cdot {re}^{2} + -1\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{2} \cdot {re}^{2} + -1\right)}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot {re}^{2}\right), \color{blue}{-1}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({re}^{2}\right)\right), -1\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(re \cdot re\right)\right), -1\right)\right) \]

      10. *-lowering-*.f64 40.4%

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, re\right)\right), -1\right)\right) \]

   8. Simplified 40.4%

      \[\leadsto \color{blue}{im \cdot \left(0.5 \cdot \left(re \cdot re\right) + -1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.4 \cdot 10^{+219}:\\ \;\;\;\;im \cdot \left(-1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-1 + 0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 53.1% accurate, 22.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;re \leq 1.4 \cdot 10^{+219}:\\ \;\;\;\;im\_m \cdot \left(-1 + -0.16666666666666666 \cdot \left(im\_m \cdot im\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

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 1.4e+219)
    (* im_m (+ -1.0 (* -0.16666666666666666 (* im_m im_m))))
    (* im_m (* 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 <= 1.4e+219) {
		tmp = im_m * (-1.0 + (-0.16666666666666666 * (im_m * im_m)));
	} else {
		tmp = im_m * (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 <= 1.4d+219) then
        tmp = im_m * ((-1.0d0) + ((-0.16666666666666666d0) * (im_m * im_m)))
    else
        tmp = im_m * (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 <= 1.4e+219) {
		tmp = im_m * (-1.0 + (-0.16666666666666666 * (im_m * im_m)));
	} else {
		tmp = im_m * (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 <= 1.4e+219:
		tmp = im_m * (-1.0 + (-0.16666666666666666 * (im_m * im_m)))
	else:
		tmp = im_m * (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 <= 1.4e+219)
		tmp = Float64(im_m * Float64(-1.0 + Float64(-0.16666666666666666 * Float64(im_m * im_m))));
	else
		tmp = Float64(im_m * 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 <= 1.4e+219)
		tmp = im_m * (-1.0 + (-0.16666666666666666 * (im_m * im_m)));
	else
		tmp = im_m * (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, 1.4e+219], N[(im$95$m * N[(-1.0 + N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if re < 1.40000000000000008e219

   1. Initial program 56.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} + \color{blue}{\left(\mathsf{neg}\left(e^{im}\right)\right)}\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(e^{im}\right)\right) \cdot \frac{1}{2}} \]

      3. distribute-lft-neg-out N/A

         \[\leadsto e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2} + \left(\mathsf{neg}\left(e^{im} \cdot \frac{1}{2}\right)\right) \]

      4. unsub-neg N/A

         \[\leadsto e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2} - \color{blue}{e^{im} \cdot \frac{1}{2}} \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(e^{\mathsf{neg}\left(im\right)} \cdot \frac{1}{2}\right), \color{blue}{\left(e^{im} \cdot \frac{1}{2}\right)}\right) \]

      6. exp-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{1}{e^{im}} \cdot \frac{1}{2}\right), \left(e^{\color{blue}{im}} \cdot \frac{1}{2}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{1 \cdot \frac{1}{2}}{e^{im}}\right), \left(\color{blue}{e^{im}} \cdot \frac{1}{2}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\frac{1}{2}}{e^{im}}\right), \left(e^{\color{blue}{im}} \cdot \frac{1}{2}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(e^{im}\right)\right), \left(\color{blue}{e^{im}} \cdot \frac{1}{2}\right)\right) \]

      10. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \left(e^{im} \cdot \frac{1}{2}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{*.f64}\left(\left(e^{im}\right), \color{blue}{\frac{1}{2}}\right)\right) \]

      12. exp-lowering-exp.f64 46.1%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{exp.f64}\left(im\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(im\right), \frac{1}{2}\right)\right) \]

   5. Simplified 46.1%

      \[\leadsto \color{blue}{\frac{0.5}{e^{im}} - e^{im} \cdot 0.5} \]

   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 61.5%

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right) \]

   8. Simplified 61.5%

      \[\leadsto \color{blue}{im \cdot \left(-1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)} \]

   if 1.40000000000000008e219 < re

   1. Initial program 76.2%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 28.8%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \mathsf{cos.f64}\left(re\right)\right)\right) \]

   5. Simplified 28.8%

      \[\leadsto \color{blue}{0 - im \cdot \cos re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{{re}^{2} \cdot \left(\frac{-1}{24} \cdot \left(im \cdot {re}^{2}\right) - \frac{-1}{2} \cdot im\right) - im} \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({re}^{2} \cdot \left(\frac{-1}{24} \cdot \left(im \cdot {re}^{2}\right) - \frac{-1}{2} \cdot im\right)\right), \color{blue}{im}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left({re}^{2}\right), \left(\frac{-1}{24} \cdot \left(im \cdot {re}^{2}\right) - \frac{-1}{2} \cdot im\right)\right), im\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(re \cdot re\right), \left(\frac{-1}{24} \cdot \left(im \cdot {re}^{2}\right) - \frac{-1}{2} \cdot im\right)\right), im\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{-1}{24} \cdot \left(im \cdot {re}^{2}\right) - \frac{-1}{2} \cdot im\right)\right), im\right) \]

      5. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{-1}{24} \cdot \left(im \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot im\right)\right), im\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\left(im \cdot {re}^{2}\right) \cdot \frac{-1}{24} + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot im\right)\right), im\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(im \cdot \left({re}^{2} \cdot \frac{-1}{24}\right) + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot im\right)\right), im\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(im \cdot \left({re}^{2} \cdot \frac{-1}{24}\right) + \frac{1}{2} \cdot im\right)\right), im\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(im \cdot \left({re}^{2} \cdot \frac{-1}{24}\right) + im \cdot \frac{1}{2}\right)\right), im\right) \]

      10. distribute-lft-out N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(im \cdot \left({re}^{2} \cdot \frac{-1}{24} + \frac{1}{2}\right)\right)\right), im\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(im, \left({re}^{2} \cdot \frac{-1}{24} + \frac{1}{2}\right)\right)\right), im\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\left({re}^{2} \cdot \frac{-1}{24}\right), \frac{1}{2}\right)\right)\right), im\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({re}^{2}\right), \frac{-1}{24}\right), \frac{1}{2}\right)\right)\right), im\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{24}\right), \frac{1}{2}\right)\right)\right), im\right) \]

      15. *-lowering-*.f64 35.4%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{24}\right), \frac{1}{2}\right)\right)\right), im\right) \]

   8. Simplified 35.4%

      \[\leadsto \color{blue}{\left(re \cdot re\right) \cdot \left(im \cdot \left(\left(re \cdot re\right) \cdot -0.041666666666666664 + 0.5\right)\right) - im} \]

   9. Taylor expanded in re around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \color{blue}{\left(\frac{1}{2} \cdot im\right)}\right), im\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(im \cdot \frac{1}{2}\right)\right), im\right) \]

       2. *-lowering-*.f64 40.4%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(im, \frac{1}{2}\right)\right), im\right) \]

   11. Simplified 40.4%

       \[\leadsto \left(re \cdot re\right) \cdot \color{blue}{\left(im \cdot 0.5\right)} - im \]

   12. Taylor expanded in re around inf

       \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(im \cdot {re}^{2}\right)} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(im \cdot {re}^{2}\right) \cdot \color{blue}{\frac{1}{2}} \]

       2. associate-*l* N/A

          \[\leadsto im \cdot \color{blue}{\left({re}^{2} \cdot \frac{1}{2}\right)} \]

       3. *-commutative N/A

          \[\leadsto im \cdot \left(\frac{1}{2} \cdot \color{blue}{{re}^{2}}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{2} \cdot {re}^{2}\right)}\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left({re}^{2} \cdot \color{blue}{\frac{1}{2}}\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{1}{2}}\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{1}{2}\right)\right) \]

       8. *-lowering-*.f64 40.4%

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{1}{2}\right)\right) \]

   14. Simplified 40.4%

       \[\leadsto \color{blue}{im \cdot \left(\left(re \cdot re\right) \cdot 0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.4 \cdot 10^{+219}:\\ \;\;\;\;im \cdot \left(-1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 37.6% accurate, 25.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 3800000:\\ \;\;\;\;0 - im\_m\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

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 (<= im_m 3800000.0) (- 0.0 im_m) (* im_m (* 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 (im_m <= 3800000.0) {
		tmp = 0.0 - im_m;
	} else {
		tmp = im_m * (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 (im_m <= 3800000.0d0) then
        tmp = 0.0d0 - im_m
    else
        tmp = im_m * (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 (im_m <= 3800000.0) {
		tmp = 0.0 - im_m;
	} else {
		tmp = im_m * (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 im_m <= 3800000.0:
		tmp = 0.0 - im_m
	else:
		tmp = im_m * (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 (im_m <= 3800000.0)
		tmp = Float64(0.0 - im_m);
	else
		tmp = Float64(im_m * 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 (im_m <= 3800000.0)
		tmp = 0.0 - im_m;
	else
		tmp = im_m * (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[im$95$m, 3800000.0], N[(0.0 - im$95$m), $MachinePrecision], N[(im$95$m * N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 3.8e6

   1. Initial program 42.2%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 64.2%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \mathsf{cos.f64}\left(re\right)\right)\right) \]

   5. Simplified 64.2%

      \[\leadsto \color{blue}{0 - im \cdot \cos re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{-1 \cdot im} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(im\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{im} \]

      3. --lowering--.f64 38.3%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{im}\right) \]

   8. Simplified 38.3%

      \[\leadsto \color{blue}{0 - im} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(im\right) \]

      2. neg-lowering-neg.f64 38.3%

         \[\leadsto \mathsf{neg.f64}\left(im\right) \]

   10. Applied egg-rr 38.3%

       \[\leadsto \color{blue}{-im} \]

   if 3.8e6 < 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 5.3%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \mathsf{cos.f64}\left(re\right)\right)\right) \]

   5. Simplified 5.3%

      \[\leadsto \color{blue}{0 - im \cdot \cos re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{{re}^{2} \cdot \left(\frac{-1}{24} \cdot \left(im \cdot {re}^{2}\right) - \frac{-1}{2} \cdot im\right) - im} \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({re}^{2} \cdot \left(\frac{-1}{24} \cdot \left(im \cdot {re}^{2}\right) - \frac{-1}{2} \cdot im\right)\right), \color{blue}{im}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left({re}^{2}\right), \left(\frac{-1}{24} \cdot \left(im \cdot {re}^{2}\right) - \frac{-1}{2} \cdot im\right)\right), im\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(re \cdot re\right), \left(\frac{-1}{24} \cdot \left(im \cdot {re}^{2}\right) - \frac{-1}{2} \cdot im\right)\right), im\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{-1}{24} \cdot \left(im \cdot {re}^{2}\right) - \frac{-1}{2} \cdot im\right)\right), im\right) \]

      5. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{-1}{24} \cdot \left(im \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot im\right)\right), im\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\left(im \cdot {re}^{2}\right) \cdot \frac{-1}{24} + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot im\right)\right), im\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(im \cdot \left({re}^{2} \cdot \frac{-1}{24}\right) + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot im\right)\right), im\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(im \cdot \left({re}^{2} \cdot \frac{-1}{24}\right) + \frac{1}{2} \cdot im\right)\right), im\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(im \cdot \left({re}^{2} \cdot \frac{-1}{24}\right) + im \cdot \frac{1}{2}\right)\right), im\right) \]

      10. distribute-lft-out N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(im \cdot \left({re}^{2} \cdot \frac{-1}{24} + \frac{1}{2}\right)\right)\right), im\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(im, \left({re}^{2} \cdot \frac{-1}{24} + \frac{1}{2}\right)\right)\right), im\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\left({re}^{2} \cdot \frac{-1}{24}\right), \frac{1}{2}\right)\right)\right), im\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({re}^{2}\right), \frac{-1}{24}\right), \frac{1}{2}\right)\right)\right), im\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{24}\right), \frac{1}{2}\right)\right)\right), im\right) \]

      15. *-lowering-*.f64 26.3%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{24}\right), \frac{1}{2}\right)\right)\right), im\right) \]

   8. Simplified 26.3%

      \[\leadsto \color{blue}{\left(re \cdot re\right) \cdot \left(im \cdot \left(\left(re \cdot re\right) \cdot -0.041666666666666664 + 0.5\right)\right) - im} \]

   9. Taylor expanded in re around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \color{blue}{\left(\frac{1}{2} \cdot im\right)}\right), im\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(im \cdot \frac{1}{2}\right)\right), im\right) \]

       2. *-lowering-*.f64 22.2%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(im, \frac{1}{2}\right)\right), im\right) \]

   11. Simplified 22.2%

       \[\leadsto \left(re \cdot re\right) \cdot \color{blue}{\left(im \cdot 0.5\right)} - im \]

   12. Taylor expanded in re around inf

       \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(im \cdot {re}^{2}\right)} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(im \cdot {re}^{2}\right) \cdot \color{blue}{\frac{1}{2}} \]

       2. associate-*l* N/A

          \[\leadsto im \cdot \color{blue}{\left({re}^{2} \cdot \frac{1}{2}\right)} \]

       3. *-commutative N/A

          \[\leadsto im \cdot \left(\frac{1}{2} \cdot \color{blue}{{re}^{2}}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{2} \cdot {re}^{2}\right)}\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left({re}^{2} \cdot \color{blue}{\frac{1}{2}}\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{1}{2}}\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{1}{2}\right)\right) \]

       8. *-lowering-*.f64 20.2%

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{1}{2}\right)\right) \]

   14. Simplified 20.2%

       \[\leadsto \color{blue}{im \cdot \left(\left(re \cdot re\right) \cdot 0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3800000:\\ \;\;\;\;0 - im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 29.5% 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 57.6%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in im around 0

   \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \cos re\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(im \cdot \cos re\right) \]

   2. neg-sub0 N/A

      \[\leadsto 0 - \color{blue}{im \cdot \cos re} \]

   3. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(im \cdot \cos re\right)}\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \color{blue}{\cos re}\right)\right) \]

   5. cos-lowering-cos.f64 48.6%

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \mathsf{cos.f64}\left(re\right)\right)\right) \]

5. Simplified 48.6%

   \[\leadsto \color{blue}{0 - im \cdot \cos re} \]

6. Taylor expanded in re around 0

   \[\leadsto \color{blue}{-1 \cdot im} \]
7. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(im\right) \]

   2. neg-sub0 N/A

      \[\leadsto 0 - \color{blue}{im} \]

   3. --lowering--.f64 29.2%

      \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{im}\right) \]

8. Simplified 29.2%

   \[\leadsto \color{blue}{0 - im} \]
9. Step-by-step derivation

   1. sub0-neg N/A

      \[\leadsto \mathsf{neg}\left(im\right) \]

   2. neg-lowering-neg.f64 29.2%

      \[\leadsto \mathsf{neg.f64}\left(im\right) \]

10. Applied egg-rr 29.2%

    \[\leadsto \color{blue}{-im} \]

11. Final simplification 29.2%

    \[\leadsto 0 - im \]
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 2024144 
(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))))