math.sin on complex, imaginary part

Percentage Accurate: 54.7% → 99.8%

Time: 12.2s

Alternatives: 21

Speedup: 15.4×

• 49.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{0 - im\_m} - e^{im\_m}\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -100:\\ \;\;\;\;t\_0 \cdot \left(0.5 \cdot \cos re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-1 + \left(im\_m \cdot im\_m\right) \cdot \left(-0.16666666666666666 + im\_m \cdot \left(im\_m \cdot \left(-0.008333333333333333 + im\_m \cdot \left(im\_m \cdot -0.0001984126984126984\right)\right)\right)\right)\right) \cdot \left(im\_m \cdot \cos re\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (- (exp (- 0.0 im_m)) (exp im_m))))
   (*
    im_s
    (if (<= t_0 -100.0)
      (* t_0 (* 0.5 (cos re)))
      (*
       (+
        -1.0
        (*
         (* im_m im_m)
         (+
          -0.16666666666666666
          (*
           im_m
           (*
            im_m
            (+
             -0.008333333333333333
             (* im_m (* im_m -0.0001984126984126984))))))))
       (* im_m (cos re)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = exp((0.0 - im_m)) - exp(im_m);
	double tmp;
	if (t_0 <= -100.0) {
		tmp = t_0 * (0.5 * cos(re));
	} else {
		tmp = (-1.0 + ((im_m * im_m) * (-0.16666666666666666 + (im_m * (im_m * (-0.008333333333333333 + (im_m * (im_m * -0.0001984126984126984)))))))) * (im_m * cos(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) :: t_0
    real(8) :: tmp
    t_0 = exp((0.0d0 - im_m)) - exp(im_m)
    if (t_0 <= (-100.0d0)) then
        tmp = t_0 * (0.5d0 * cos(re))
    else
        tmp = ((-1.0d0) + ((im_m * im_m) * ((-0.16666666666666666d0) + (im_m * (im_m * ((-0.008333333333333333d0) + (im_m * (im_m * (-0.0001984126984126984d0))))))))) * (im_m * cos(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 t_0 = Math.exp((0.0 - im_m)) - Math.exp(im_m);
	double tmp;
	if (t_0 <= -100.0) {
		tmp = t_0 * (0.5 * Math.cos(re));
	} else {
		tmp = (-1.0 + ((im_m * im_m) * (-0.16666666666666666 + (im_m * (im_m * (-0.008333333333333333 + (im_m * (im_m * -0.0001984126984126984)))))))) * (im_m * Math.cos(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):
	t_0 = math.exp((0.0 - im_m)) - math.exp(im_m)
	tmp = 0
	if t_0 <= -100.0:
		tmp = t_0 * (0.5 * math.cos(re))
	else:
		tmp = (-1.0 + ((im_m * im_m) * (-0.16666666666666666 + (im_m * (im_m * (-0.008333333333333333 + (im_m * (im_m * -0.0001984126984126984)))))))) * (im_m * math.cos(re))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(exp(Float64(0.0 - im_m)) - exp(im_m))
	tmp = 0.0
	if (t_0 <= -100.0)
		tmp = Float64(t_0 * Float64(0.5 * cos(re)));
	else
		tmp = Float64(Float64(-1.0 + Float64(Float64(im_m * im_m) * Float64(-0.16666666666666666 + Float64(im_m * Float64(im_m * Float64(-0.008333333333333333 + Float64(im_m * Float64(im_m * -0.0001984126984126984)))))))) * Float64(im_m * cos(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)
	t_0 = exp((0.0 - im_m)) - exp(im_m);
	tmp = 0.0;
	if (t_0 <= -100.0)
		tmp = t_0 * (0.5 * cos(re));
	else
		tmp = (-1.0 + ((im_m * im_m) * (-0.16666666666666666 + (im_m * (im_m * (-0.008333333333333333 + (im_m * (im_m * -0.0001984126984126984)))))))) * (im_m * cos(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_] := 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, -100.0], N[(t$95$0 * N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(-0.16666666666666666 + N[(im$95$m * N[(im$95$m * N[(-0.008333333333333333 + N[(im$95$m * N[(im$95$m * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(im$95$m * N[Cos[re], $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)) < -100

   1. Initial program 100.0%

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

   if -100 < (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))

   1. Initial program 39.6%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 97.9%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

   7. Applied egg-rr 97.9%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{0 - im} - e^{im} \leq -100:\\ \;\;\;\;\left(e^{0 - im} - e^{im}\right) \cdot \left(0.5 \cdot \cos re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-1 + \left(im \cdot im\right) \cdot \left(-0.16666666666666666 + im \cdot \left(im \cdot \left(-0.008333333333333333 + im \cdot \left(im \cdot -0.0001984126984126984\right)\right)\right)\right)\right) \cdot \left(im \cdot \cos re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := -0.016666666666666666 + \left(im\_m \cdot im\_m\right) \cdot -0.0003968253968253968\\ t_1 := \left(im\_m \cdot im\_m\right) \cdot \left(im\_m \cdot im\_m\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 2 \cdot 10^{+59}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(im\_m \cdot \left(-2 + \frac{\left(im\_m \cdot im\_m\right) \cdot \left(0.1111111111111111 - t\_0 \cdot \left(t\_0 \cdot t\_1\right)\right)}{-0.3333333333333333 - \left(im\_m \cdot im\_m\right) \cdot t\_0}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(\cos re \cdot \left(-1 + -0.008333333333333333 \cdot t\_1\right)\right)\\ \end{array} \end{array} \end{array} \]

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
         (+ -0.016666666666666666 (* (* im_m im_m) -0.0003968253968253968)))
        (t_1 (* (* im_m im_m) (* im_m im_m))))
   (*
    im_s
    (if (<= im_m 2e+59)
      (*
       (* 0.5 (cos re))
       (*
        im_m
        (+
         -2.0
         (/
          (* (* im_m im_m) (- 0.1111111111111111 (* t_0 (* t_0 t_1))))
          (- -0.3333333333333333 (* (* im_m im_m) t_0))))))
      (* im_m (* (cos re) (+ -1.0 (* -0.008333333333333333 t_1))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = -0.016666666666666666 + ((im_m * im_m) * -0.0003968253968253968);
	double t_1 = (im_m * im_m) * (im_m * im_m);
	double tmp;
	if (im_m <= 2e+59) {
		tmp = (0.5 * cos(re)) * (im_m * (-2.0 + (((im_m * im_m) * (0.1111111111111111 - (t_0 * (t_0 * t_1)))) / (-0.3333333333333333 - ((im_m * im_m) * t_0)))));
	} else {
		tmp = im_m * (cos(re) * (-1.0 + (-0.008333333333333333 * t_1)));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (-0.016666666666666666d0) + ((im_m * im_m) * (-0.0003968253968253968d0))
    t_1 = (im_m * im_m) * (im_m * im_m)
    if (im_m <= 2d+59) then
        tmp = (0.5d0 * cos(re)) * (im_m * ((-2.0d0) + (((im_m * im_m) * (0.1111111111111111d0 - (t_0 * (t_0 * t_1)))) / ((-0.3333333333333333d0) - ((im_m * im_m) * t_0)))))
    else
        tmp = im_m * (cos(re) * ((-1.0d0) + ((-0.008333333333333333d0) * t_1)))
    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 = -0.016666666666666666 + ((im_m * im_m) * -0.0003968253968253968);
	double t_1 = (im_m * im_m) * (im_m * im_m);
	double tmp;
	if (im_m <= 2e+59) {
		tmp = (0.5 * Math.cos(re)) * (im_m * (-2.0 + (((im_m * im_m) * (0.1111111111111111 - (t_0 * (t_0 * t_1)))) / (-0.3333333333333333 - ((im_m * im_m) * t_0)))));
	} else {
		tmp = im_m * (Math.cos(re) * (-1.0 + (-0.008333333333333333 * t_1)));
	}
	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 = -0.016666666666666666 + ((im_m * im_m) * -0.0003968253968253968)
	t_1 = (im_m * im_m) * (im_m * im_m)
	tmp = 0
	if im_m <= 2e+59:
		tmp = (0.5 * math.cos(re)) * (im_m * (-2.0 + (((im_m * im_m) * (0.1111111111111111 - (t_0 * (t_0 * t_1)))) / (-0.3333333333333333 - ((im_m * im_m) * t_0)))))
	else:
		tmp = im_m * (math.cos(re) * (-1.0 + (-0.008333333333333333 * t_1)))
	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(-0.016666666666666666 + Float64(Float64(im_m * im_m) * -0.0003968253968253968))
	t_1 = Float64(Float64(im_m * im_m) * Float64(im_m * im_m))
	tmp = 0.0
	if (im_m <= 2e+59)
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(im_m * Float64(-2.0 + Float64(Float64(Float64(im_m * im_m) * Float64(0.1111111111111111 - Float64(t_0 * Float64(t_0 * t_1)))) / Float64(-0.3333333333333333 - Float64(Float64(im_m * im_m) * t_0))))));
	else
		tmp = Float64(im_m * Float64(cos(re) * Float64(-1.0 + Float64(-0.008333333333333333 * t_1))));
	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 = -0.016666666666666666 + ((im_m * im_m) * -0.0003968253968253968);
	t_1 = (im_m * im_m) * (im_m * im_m);
	tmp = 0.0;
	if (im_m <= 2e+59)
		tmp = (0.5 * cos(re)) * (im_m * (-2.0 + (((im_m * im_m) * (0.1111111111111111 - (t_0 * (t_0 * t_1)))) / (-0.3333333333333333 - ((im_m * im_m) * t_0)))));
	else
		tmp = im_m * (cos(re) * (-1.0 + (-0.008333333333333333 * t_1)));
	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[(-0.016666666666666666 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0003968253968253968), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[im$95$m, 2e+59], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(im$95$m * N[(-2.0 + N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(0.1111111111111111 - N[(t$95$0 * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-0.3333333333333333 - N[(N[(im$95$m * im$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(N[Cos[re], $MachinePrecision] * N[(-1.0 + N[(-0.008333333333333333 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if im < 1.99999999999999994e59

   1. Initial program 41.7%

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

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)}\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)}\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) + -2\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left(-2 + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right)}\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{{im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right)} - \frac{1}{3}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{{im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right)} - \frac{1}{3}\right)\right)\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) + \frac{-1}{3}\right)\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{3} + \color{blue}{{im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right)\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right)}\right)\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{-1}{2520} \cdot {im}^{2}} - \frac{1}{60}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{-1}{2520} \cdot {im}^{2}} - \frac{1}{60}\right)\right)\right)\right)\right)\right)\right) \]

      16. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{2520} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{60}\right)\right)}\right)\right)\right)\right)\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{2520} \cdot {im}^{2} + \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]

      18. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{60} + \color{blue}{\frac{-1}{2520} \cdot {im}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{60}, \color{blue}{\left(\frac{-1}{2520} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{60}, \left({im}^{2} \cdot \color{blue}{\frac{-1}{2520}}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{60}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{-1}{2520}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      22. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{60}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{-1}{2520}\right)\right)\right)\right)\right)\right)\right)\right) \]

      23. *-lowering-*.f64 94.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{60}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{2520}\right)\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 94.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left(-2 + \left(im \cdot im\right) \cdot \left(-0.3333333333333333 + \left(im \cdot im\right) \cdot \left(-0.016666666666666666 + \left(im \cdot im\right) \cdot -0.0003968253968253968\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \left(\left(\frac{-1}{3} + \left(im \cdot im\right) \cdot \left(\frac{-1}{60} + \left(im \cdot im\right) \cdot \frac{-1}{2520}\right)\right) \cdot \color{blue}{\left(im \cdot im\right)}\right)\right)\right)\right) \]

      2. flip-+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \left(\frac{\frac{-1}{3} \cdot \frac{-1}{3} - \left(\left(im \cdot im\right) \cdot \left(\frac{-1}{60} + \left(im \cdot im\right) \cdot \frac{-1}{2520}\right)\right) \cdot \left(\left(im \cdot im\right) \cdot \left(\frac{-1}{60} + \left(im \cdot im\right) \cdot \frac{-1}{2520}\right)\right)}{\frac{-1}{3} - \left(im \cdot im\right) \cdot \left(\frac{-1}{60} + \left(im \cdot im\right) \cdot \frac{-1}{2520}\right)} \cdot \left(\color{blue}{im} \cdot im\right)\right)\right)\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \left(\frac{\left(\frac{-1}{3} \cdot \frac{-1}{3} - \left(\left(im \cdot im\right) \cdot \left(\frac{-1}{60} + \left(im \cdot im\right) \cdot \frac{-1}{2520}\right)\right) \cdot \left(\left(im \cdot im\right) \cdot \left(\frac{-1}{60} + \left(im \cdot im\right) \cdot \frac{-1}{2520}\right)\right)\right) \cdot \left(im \cdot im\right)}{\color{blue}{\frac{-1}{3} - \left(im \cdot im\right) \cdot \left(\frac{-1}{60} + \left(im \cdot im\right) \cdot \frac{-1}{2520}\right)}}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\left(\left(\frac{-1}{3} \cdot \frac{-1}{3} - \left(\left(im \cdot im\right) \cdot \left(\frac{-1}{60} + \left(im \cdot im\right) \cdot \frac{-1}{2520}\right)\right) \cdot \left(\left(im \cdot im\right) \cdot \left(\frac{-1}{60} + \left(im \cdot im\right) \cdot \frac{-1}{2520}\right)\right)\right) \cdot \left(im \cdot im\right)\right), \color{blue}{\left(\frac{-1}{3} - \left(im \cdot im\right) \cdot \left(\frac{-1}{60} + \left(im \cdot im\right) \cdot \frac{-1}{2520}\right)\right)}\right)\right)\right)\right) \]

   7. Applied egg-rr 70.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(im \cdot \left(-2 + \color{blue}{\frac{\left(0.1111111111111111 - \left(-0.016666666666666666 + \left(im \cdot im\right) \cdot -0.0003968253968253968\right) \cdot \left(\left(-0.016666666666666666 + \left(im \cdot im\right) \cdot -0.0003968253968253968\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right) \cdot \left(im \cdot im\right)}{-0.3333333333333333 - \left(im \cdot im\right) \cdot \left(-0.016666666666666666 + \left(im \cdot im\right) \cdot -0.0003968253968253968\right)}}\right)\right) \]

   if 1.99999999999999994e59 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

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

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

      2. *-commutative N/A

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

      3. 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 100.0%

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

   6. Taylor expanded in im around inf

      \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \color{blue}{\left(\frac{-1}{120} \cdot {im}^{4}\right)}\right)\right)\right) \]
   7. Step-by-step derivation

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

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2 \cdot 10^{+59}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(im \cdot \left(-2 + \frac{\left(im \cdot im\right) \cdot \left(0.1111111111111111 - \left(-0.016666666666666666 + \left(im \cdot im\right) \cdot -0.0003968253968253968\right) \cdot \left(\left(-0.016666666666666666 + \left(im \cdot im\right) \cdot -0.0003968253968253968\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right)}{-0.3333333333333333 - \left(im \cdot im\right) \cdot \left(-0.016666666666666666 + \left(im \cdot im\right) \cdot -0.0003968253968253968\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\cos re \cdot \left(-1 + -0.008333333333333333 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 7.6:\\ \;\;\;\;im\_m \cdot \left(\cos re \cdot \left(-1 + \left(im\_m \cdot im\_m\right) \cdot \left(-0.16666666666666666 + \left(im\_m \cdot im\_m\right) \cdot -0.008333333333333333\right)\right)\right)\\ \mathbf{elif}\;im\_m \leq 1.2 \cdot 10^{+62}:\\ \;\;\;\;\left(0.5 + im\_m \cdot \left(-0.5 + im\_m \cdot 0.25\right)\right) - e^{im\_m} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(\cos re \cdot \left(-1 + -0.008333333333333333 \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(im\_m \cdot im\_m\right)\right)\right)\right)\\ \end{array} \end{array} \]

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 7.6)
    (*
     im_m
     (*
      (cos re)
      (+
       -1.0
       (*
        (* im_m im_m)
        (+ -0.16666666666666666 (* (* im_m im_m) -0.008333333333333333))))))
    (if (<= im_m 1.2e+62)
      (- (+ 0.5 (* im_m (+ -0.5 (* im_m 0.25)))) (* (exp im_m) 0.5))
      (*
       im_m
       (*
        (cos re)
        (+
         -1.0
         (* -0.008333333333333333 (* (* im_m im_m) (* im_m im_m))))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 7.6) {
		tmp = im_m * (cos(re) * (-1.0 + ((im_m * im_m) * (-0.16666666666666666 + ((im_m * im_m) * -0.008333333333333333)))));
	} else if (im_m <= 1.2e+62) {
		tmp = (0.5 + (im_m * (-0.5 + (im_m * 0.25)))) - (exp(im_m) * 0.5);
	} else {
		tmp = im_m * (cos(re) * (-1.0 + (-0.008333333333333333 * ((im_m * 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 <= 7.6d0) then
        tmp = im_m * (cos(re) * ((-1.0d0) + ((im_m * im_m) * ((-0.16666666666666666d0) + ((im_m * im_m) * (-0.008333333333333333d0))))))
    else if (im_m <= 1.2d+62) then
        tmp = (0.5d0 + (im_m * ((-0.5d0) + (im_m * 0.25d0)))) - (exp(im_m) * 0.5d0)
    else
        tmp = im_m * (cos(re) * ((-1.0d0) + ((-0.008333333333333333d0) * ((im_m * 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 <= 7.6) {
		tmp = im_m * (Math.cos(re) * (-1.0 + ((im_m * im_m) * (-0.16666666666666666 + ((im_m * im_m) * -0.008333333333333333)))));
	} else if (im_m <= 1.2e+62) {
		tmp = (0.5 + (im_m * (-0.5 + (im_m * 0.25)))) - (Math.exp(im_m) * 0.5);
	} else {
		tmp = im_m * (Math.cos(re) * (-1.0 + (-0.008333333333333333 * ((im_m * 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 <= 7.6:
		tmp = im_m * (math.cos(re) * (-1.0 + ((im_m * im_m) * (-0.16666666666666666 + ((im_m * im_m) * -0.008333333333333333)))))
	elif im_m <= 1.2e+62:
		tmp = (0.5 + (im_m * (-0.5 + (im_m * 0.25)))) - (math.exp(im_m) * 0.5)
	else:
		tmp = im_m * (math.cos(re) * (-1.0 + (-0.008333333333333333 * ((im_m * 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 <= 7.6)
		tmp = Float64(im_m * Float64(cos(re) * Float64(-1.0 + Float64(Float64(im_m * im_m) * Float64(-0.16666666666666666 + Float64(Float64(im_m * im_m) * -0.008333333333333333))))));
	elseif (im_m <= 1.2e+62)
		tmp = Float64(Float64(0.5 + Float64(im_m * Float64(-0.5 + Float64(im_m * 0.25)))) - Float64(exp(im_m) * 0.5));
	else
		tmp = Float64(im_m * Float64(cos(re) * Float64(-1.0 + Float64(-0.008333333333333333 * Float64(Float64(im_m * im_m) * 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 <= 7.6)
		tmp = im_m * (cos(re) * (-1.0 + ((im_m * im_m) * (-0.16666666666666666 + ((im_m * im_m) * -0.008333333333333333)))));
	elseif (im_m <= 1.2e+62)
		tmp = (0.5 + (im_m * (-0.5 + (im_m * 0.25)))) - (exp(im_m) * 0.5);
	else
		tmp = im_m * (cos(re) * (-1.0 + (-0.008333333333333333 * ((im_m * 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, 7.6], N[(im$95$m * N[(N[Cos[re], $MachinePrecision] * N[(-1.0 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(-0.16666666666666666 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 1.2e+62], N[(N[(0.5 + N[(im$95$m * N[(-0.5 + N[(im$95$m * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Exp[im$95$m], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(N[Cos[re], $MachinePrecision] * N[(-1.0 + N[(-0.008333333333333333 * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 7.5999999999999996

   1. Initial program 39.9%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \cos re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \cos re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)} \]
   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 95.5%

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

   if 7.5999999999999996 < 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 83.3%

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

   5. Simplified 83.3%

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

   6. Taylor expanded in im around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 83.3%

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

   8. Simplified 83.3%

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

   if 1.2e62 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

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

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

      2. *-commutative N/A

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

      3. 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 100.0%

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

   6. Taylor expanded in im around inf

      \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \color{blue}{\left(\frac{-1}{120} \cdot {im}^{4}\right)}\right)\right)\right) \]
   7. Step-by-step derivation

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

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

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

Alternative 4: 96.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 6:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot -0.3333333333333333\right)\right) + im\_m \cdot -2\right)\\ \mathbf{elif}\;im\_m \leq 1.2 \cdot 10^{+62}:\\ \;\;\;\;\left(0.5 + im\_m \cdot \left(-0.5 + im\_m \cdot 0.25\right)\right) - e^{im\_m} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(\cos re \cdot \left(-1 + -0.008333333333333333 \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(im\_m \cdot im\_m\right)\right)\right)\right)\\ \end{array} \end{array} \]

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 6.0)
    (*
     (* 0.5 (cos re))
     (+ (* im_m (* im_m (* im_m -0.3333333333333333))) (* im_m -2.0)))
    (if (<= im_m 1.2e+62)
      (- (+ 0.5 (* im_m (+ -0.5 (* im_m 0.25)))) (* (exp im_m) 0.5))
      (*
       im_m
       (*
        (cos re)
        (+
         -1.0
         (* -0.008333333333333333 (* (* im_m im_m) (* im_m im_m))))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 6.0) {
		tmp = (0.5 * cos(re)) * ((im_m * (im_m * (im_m * -0.3333333333333333))) + (im_m * -2.0));
	} else if (im_m <= 1.2e+62) {
		tmp = (0.5 + (im_m * (-0.5 + (im_m * 0.25)))) - (exp(im_m) * 0.5);
	} else {
		tmp = im_m * (cos(re) * (-1.0 + (-0.008333333333333333 * ((im_m * 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 <= 6.0d0) then
        tmp = (0.5d0 * cos(re)) * ((im_m * (im_m * (im_m * (-0.3333333333333333d0)))) + (im_m * (-2.0d0)))
    else if (im_m <= 1.2d+62) then
        tmp = (0.5d0 + (im_m * ((-0.5d0) + (im_m * 0.25d0)))) - (exp(im_m) * 0.5d0)
    else
        tmp = im_m * (cos(re) * ((-1.0d0) + ((-0.008333333333333333d0) * ((im_m * 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 <= 6.0) {
		tmp = (0.5 * Math.cos(re)) * ((im_m * (im_m * (im_m * -0.3333333333333333))) + (im_m * -2.0));
	} else if (im_m <= 1.2e+62) {
		tmp = (0.5 + (im_m * (-0.5 + (im_m * 0.25)))) - (Math.exp(im_m) * 0.5);
	} else {
		tmp = im_m * (Math.cos(re) * (-1.0 + (-0.008333333333333333 * ((im_m * 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 <= 6.0:
		tmp = (0.5 * math.cos(re)) * ((im_m * (im_m * (im_m * -0.3333333333333333))) + (im_m * -2.0))
	elif im_m <= 1.2e+62:
		tmp = (0.5 + (im_m * (-0.5 + (im_m * 0.25)))) - (math.exp(im_m) * 0.5)
	else:
		tmp = im_m * (math.cos(re) * (-1.0 + (-0.008333333333333333 * ((im_m * 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 <= 6.0)
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(Float64(im_m * Float64(im_m * Float64(im_m * -0.3333333333333333))) + Float64(im_m * -2.0)));
	elseif (im_m <= 1.2e+62)
		tmp = Float64(Float64(0.5 + Float64(im_m * Float64(-0.5 + Float64(im_m * 0.25)))) - Float64(exp(im_m) * 0.5));
	else
		tmp = Float64(im_m * Float64(cos(re) * Float64(-1.0 + Float64(-0.008333333333333333 * Float64(Float64(im_m * im_m) * 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 <= 6.0)
		tmp = (0.5 * cos(re)) * ((im_m * (im_m * (im_m * -0.3333333333333333))) + (im_m * -2.0));
	elseif (im_m <= 1.2e+62)
		tmp = (0.5 + (im_m * (-0.5 + (im_m * 0.25)))) - (exp(im_m) * 0.5);
	else
		tmp = im_m * (cos(re) * (-1.0 + (-0.008333333333333333 * ((im_m * 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, 6.0], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[(im$95$m * N[(im$95$m * N[(im$95$m * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(im$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 1.2e+62], N[(N[(0.5 + N[(im$95$m * N[(-0.5 + N[(im$95$m * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Exp[im$95$m], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(N[Cos[re], $MachinePrecision] * N[(-1.0 + N[(-0.008333333333333333 * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 6

   1. Initial program 39.9%

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

   3. Taylor expanded in im around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 90.3%

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

   5. Simplified 90.3%

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. associate-*l* N/A

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 90.3%

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

   7. Applied egg-rr 90.3%

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

   if 6 < im < 1.2e62

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)} \]
   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 83.3%

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

   5. Simplified 83.3%

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

   6. Taylor expanded in im around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 83.3%

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

   8. Simplified 83.3%

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

   if 1.2e62 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

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

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

      2. *-commutative N/A

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

      3. 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 100.0%

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

   6. Taylor expanded in im around inf

      \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \color{blue}{\left(\frac{-1}{120} \cdot {im}^{4}\right)}\right)\right)\right) \]
   7. Step-by-step derivation

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

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

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

Alternative 5: 96.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 7.8:\\ \;\;\;\;\left(im\_m \cdot \cos re\right) \cdot \left(-1 + im\_m \cdot \left(im\_m \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;im\_m \leq 1.2 \cdot 10^{+62}:\\ \;\;\;\;\left(0.5 + im\_m \cdot \left(-0.5 + im\_m \cdot 0.25\right)\right) - e^{im\_m} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(\cos re \cdot \left(-1 + -0.008333333333333333 \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(im\_m \cdot im\_m\right)\right)\right)\right)\\ \end{array} \end{array} \]

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 7.8)
    (* (* im_m (cos re)) (+ -1.0 (* im_m (* im_m -0.16666666666666666))))
    (if (<= im_m 1.2e+62)
      (- (+ 0.5 (* im_m (+ -0.5 (* im_m 0.25)))) (* (exp im_m) 0.5))
      (*
       im_m
       (*
        (cos re)
        (+
         -1.0
         (* -0.008333333333333333 (* (* im_m im_m) (* im_m im_m))))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 7.8) {
		tmp = (im_m * cos(re)) * (-1.0 + (im_m * (im_m * -0.16666666666666666)));
	} else if (im_m <= 1.2e+62) {
		tmp = (0.5 + (im_m * (-0.5 + (im_m * 0.25)))) - (exp(im_m) * 0.5);
	} else {
		tmp = im_m * (cos(re) * (-1.0 + (-0.008333333333333333 * ((im_m * 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 <= 7.8d0) then
        tmp = (im_m * cos(re)) * ((-1.0d0) + (im_m * (im_m * (-0.16666666666666666d0))))
    else if (im_m <= 1.2d+62) then
        tmp = (0.5d0 + (im_m * ((-0.5d0) + (im_m * 0.25d0)))) - (exp(im_m) * 0.5d0)
    else
        tmp = im_m * (cos(re) * ((-1.0d0) + ((-0.008333333333333333d0) * ((im_m * 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 <= 7.8) {
		tmp = (im_m * Math.cos(re)) * (-1.0 + (im_m * (im_m * -0.16666666666666666)));
	} else if (im_m <= 1.2e+62) {
		tmp = (0.5 + (im_m * (-0.5 + (im_m * 0.25)))) - (Math.exp(im_m) * 0.5);
	} else {
		tmp = im_m * (Math.cos(re) * (-1.0 + (-0.008333333333333333 * ((im_m * 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 <= 7.8:
		tmp = (im_m * math.cos(re)) * (-1.0 + (im_m * (im_m * -0.16666666666666666)))
	elif im_m <= 1.2e+62:
		tmp = (0.5 + (im_m * (-0.5 + (im_m * 0.25)))) - (math.exp(im_m) * 0.5)
	else:
		tmp = im_m * (math.cos(re) * (-1.0 + (-0.008333333333333333 * ((im_m * 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 <= 7.8)
		tmp = Float64(Float64(im_m * cos(re)) * Float64(-1.0 + Float64(im_m * Float64(im_m * -0.16666666666666666))));
	elseif (im_m <= 1.2e+62)
		tmp = Float64(Float64(0.5 + Float64(im_m * Float64(-0.5 + Float64(im_m * 0.25)))) - Float64(exp(im_m) * 0.5));
	else
		tmp = Float64(im_m * Float64(cos(re) * Float64(-1.0 + Float64(-0.008333333333333333 * Float64(Float64(im_m * im_m) * 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 <= 7.8)
		tmp = (im_m * cos(re)) * (-1.0 + (im_m * (im_m * -0.16666666666666666)));
	elseif (im_m <= 1.2e+62)
		tmp = (0.5 + (im_m * (-0.5 + (im_m * 0.25)))) - (exp(im_m) * 0.5);
	else
		tmp = im_m * (cos(re) * (-1.0 + (-0.008333333333333333 * ((im_m * 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, 7.8], 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], If[LessEqual[im$95$m, 1.2e+62], N[(N[(0.5 + N[(im$95$m * N[(-0.5 + N[(im$95$m * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Exp[im$95$m], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(N[Cos[re], $MachinePrecision] * N[(-1.0 + N[(-0.008333333333333333 * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 7.79999999999999982

   1. Initial program 39.9%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \cos re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \cos re\right)\right)} \]
   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 89.9%

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

   5. Simplified 89.9%

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

   if 7.79999999999999982 < 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 83.3%

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

   5. Simplified 83.3%

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

   6. Taylor expanded in im around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 83.3%

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

   8. Simplified 83.3%

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

   if 1.2e62 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

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

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

      2. *-commutative N/A

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

      3. 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 100.0%

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

   6. Taylor expanded in im around inf

      \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{+.f64}\left(-1, \color{blue}{\left(\frac{-1}{120} \cdot {im}^{4}\right)}\right)\right)\right) \]
   7. Step-by-step derivation

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

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

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

Alternative 6: 94.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 6:\\ \;\;\;\;\left(im\_m \cdot \cos re\right) \cdot \left(-1 + im\_m \cdot \left(im\_m \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;im\_m \leq 5.8 \cdot 10^{+102}:\\ \;\;\;\;\left(0.5 + im\_m \cdot \left(-0.5 + im\_m \cdot 0.25\right)\right) - e^{im\_m} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(-0.16666666666666666 \cdot \left(im\_m \cdot \left(im\_m \cdot im\_m\right)\right)\right)\\ \end{array} \end{array} \]

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 6.0)
    (* (* im_m (cos re)) (+ -1.0 (* im_m (* im_m -0.16666666666666666))))
    (if (<= im_m 5.8e+102)
      (- (+ 0.5 (* im_m (+ -0.5 (* im_m 0.25)))) (* (exp im_m) 0.5))
      (* (cos re) (* -0.16666666666666666 (* 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 <= 6.0) {
		tmp = (im_m * cos(re)) * (-1.0 + (im_m * (im_m * -0.16666666666666666)));
	} else if (im_m <= 5.8e+102) {
		tmp = (0.5 + (im_m * (-0.5 + (im_m * 0.25)))) - (exp(im_m) * 0.5);
	} else {
		tmp = cos(re) * (-0.16666666666666666 * (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 <= 6.0d0) then
        tmp = (im_m * cos(re)) * ((-1.0d0) + (im_m * (im_m * (-0.16666666666666666d0))))
    else if (im_m <= 5.8d+102) then
        tmp = (0.5d0 + (im_m * ((-0.5d0) + (im_m * 0.25d0)))) - (exp(im_m) * 0.5d0)
    else
        tmp = cos(re) * ((-0.16666666666666666d0) * (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 <= 6.0) {
		tmp = (im_m * Math.cos(re)) * (-1.0 + (im_m * (im_m * -0.16666666666666666)));
	} else if (im_m <= 5.8e+102) {
		tmp = (0.5 + (im_m * (-0.5 + (im_m * 0.25)))) - (Math.exp(im_m) * 0.5);
	} else {
		tmp = Math.cos(re) * (-0.16666666666666666 * (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 <= 6.0:
		tmp = (im_m * math.cos(re)) * (-1.0 + (im_m * (im_m * -0.16666666666666666)))
	elif im_m <= 5.8e+102:
		tmp = (0.5 + (im_m * (-0.5 + (im_m * 0.25)))) - (math.exp(im_m) * 0.5)
	else:
		tmp = math.cos(re) * (-0.16666666666666666 * (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 <= 6.0)
		tmp = Float64(Float64(im_m * cos(re)) * Float64(-1.0 + Float64(im_m * Float64(im_m * -0.16666666666666666))));
	elseif (im_m <= 5.8e+102)
		tmp = Float64(Float64(0.5 + Float64(im_m * Float64(-0.5 + Float64(im_m * 0.25)))) - Float64(exp(im_m) * 0.5));
	else
		tmp = Float64(cos(re) * Float64(-0.16666666666666666 * Float64(im_m * 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 <= 6.0)
		tmp = (im_m * cos(re)) * (-1.0 + (im_m * (im_m * -0.16666666666666666)));
	elseif (im_m <= 5.8e+102)
		tmp = (0.5 + (im_m * (-0.5 + (im_m * 0.25)))) - (exp(im_m) * 0.5);
	else
		tmp = cos(re) * (-0.16666666666666666 * (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, 6.0], 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], If[LessEqual[im$95$m, 5.8e+102], N[(N[(0.5 + N[(im$95$m * N[(-0.5 + N[(im$95$m * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Exp[im$95$m], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(-0.16666666666666666 * N[(im$95$m * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 6

   1. Initial program 39.9%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \cos re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \cos re\right)\right)} \]
   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 89.9%

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

   5. Simplified 89.9%

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

   if 6 < im < 5.8000000000000005e102

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)} \]
   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 81.3%

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

   5. Simplified 81.3%

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

   6. Taylor expanded in im around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 81.3%

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

   8. Simplified 81.3%

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

   if 5.8000000000000005e102 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

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

Alternative 7: 92.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(\left(-1 + \left(im\_m \cdot im\_m\right) \cdot \left(-0.16666666666666666 + im\_m \cdot \left(im\_m \cdot \left(-0.008333333333333333 + im\_m \cdot \left(im\_m \cdot -0.0001984126984126984\right)\right)\right)\right)\right) \cdot \left(im\_m \cdot \cos re\right)\right) \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (*
   (+
    -1.0
    (*
     (* im_m im_m)
     (+
      -0.16666666666666666
      (*
       im_m
       (*
        im_m
        (+ -0.008333333333333333 (* im_m (* im_m -0.0001984126984126984))))))))
   (* im_m (cos re)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * ((-1.0 + ((im_m * im_m) * (-0.16666666666666666 + (im_m * (im_m * (-0.008333333333333333 + (im_m * (im_m * -0.0001984126984126984)))))))) * (im_m * cos(re)));
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = im_s * (((-1.0d0) + ((im_m * im_m) * ((-0.16666666666666666d0) + (im_m * (im_m * ((-0.008333333333333333d0) + (im_m * (im_m * (-0.0001984126984126984d0))))))))) * (im_m * cos(re)))
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	return im_s * ((-1.0 + ((im_m * im_m) * (-0.16666666666666666 + (im_m * (im_m * (-0.008333333333333333 + (im_m * (im_m * -0.0001984126984126984)))))))) * (im_m * Math.cos(re)));
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * ((-1.0 + ((im_m * im_m) * (-0.16666666666666666 + (im_m * (im_m * (-0.008333333333333333 + (im_m * (im_m * -0.0001984126984126984)))))))) * (im_m * math.cos(re)))

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * Float64(Float64(-1.0 + Float64(Float64(im_m * im_m) * Float64(-0.16666666666666666 + Float64(im_m * Float64(im_m * Float64(-0.008333333333333333 + Float64(im_m * Float64(im_m * -0.0001984126984126984)))))))) * Float64(im_m * cos(re))))
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * ((-1.0 + ((im_m * im_m) * (-0.16666666666666666 + (im_m * (im_m * (-0.008333333333333333 + (im_m * (im_m * -0.0001984126984126984)))))))) * (im_m * cos(re)));
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * N[(N[(-1.0 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(-0.16666666666666666 + N[(im$95$m * N[(im$95$m * N[(-0.008333333333333333 + N[(im$95$m * N[(im$95$m * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(im$95$m * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.2%

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

3. Taylor expanded in im around 0

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

5. Simplified 95.9%

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

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

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

7. Applied egg-rr 96.0%

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

Alternative 8: 94.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 5.6:\\ \;\;\;\;\left(im\_m \cdot \cos re\right) \cdot \left(-1 + im\_m \cdot \left(im\_m \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;im\_m \leq 5.8 \cdot 10^{+102}:\\ \;\;\;\;0.5 - e^{im\_m} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(-0.16666666666666666 \cdot \left(im\_m \cdot \left(im\_m \cdot im\_m\right)\right)\right)\\ \end{array} \end{array} \]

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 5.6)
    (* (* im_m (cos re)) (+ -1.0 (* im_m (* im_m -0.16666666666666666))))
    (if (<= im_m 5.8e+102)
      (- 0.5 (* (exp im_m) 0.5))
      (* (cos re) (* -0.16666666666666666 (* 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 <= 5.6) {
		tmp = (im_m * cos(re)) * (-1.0 + (im_m * (im_m * -0.16666666666666666)));
	} else if (im_m <= 5.8e+102) {
		tmp = 0.5 - (exp(im_m) * 0.5);
	} else {
		tmp = cos(re) * (-0.16666666666666666 * (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 <= 5.6d0) then
        tmp = (im_m * cos(re)) * ((-1.0d0) + (im_m * (im_m * (-0.16666666666666666d0))))
    else if (im_m <= 5.8d+102) then
        tmp = 0.5d0 - (exp(im_m) * 0.5d0)
    else
        tmp = cos(re) * ((-0.16666666666666666d0) * (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 <= 5.6) {
		tmp = (im_m * Math.cos(re)) * (-1.0 + (im_m * (im_m * -0.16666666666666666)));
	} else if (im_m <= 5.8e+102) {
		tmp = 0.5 - (Math.exp(im_m) * 0.5);
	} else {
		tmp = Math.cos(re) * (-0.16666666666666666 * (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 <= 5.6:
		tmp = (im_m * math.cos(re)) * (-1.0 + (im_m * (im_m * -0.16666666666666666)))
	elif im_m <= 5.8e+102:
		tmp = 0.5 - (math.exp(im_m) * 0.5)
	else:
		tmp = math.cos(re) * (-0.16666666666666666 * (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 <= 5.6)
		tmp = Float64(Float64(im_m * cos(re)) * Float64(-1.0 + Float64(im_m * Float64(im_m * -0.16666666666666666))));
	elseif (im_m <= 5.8e+102)
		tmp = Float64(0.5 - Float64(exp(im_m) * 0.5));
	else
		tmp = Float64(cos(re) * Float64(-0.16666666666666666 * Float64(im_m * 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 <= 5.6)
		tmp = (im_m * cos(re)) * (-1.0 + (im_m * (im_m * -0.16666666666666666)));
	elseif (im_m <= 5.8e+102)
		tmp = 0.5 - (exp(im_m) * 0.5);
	else
		tmp = cos(re) * (-0.16666666666666666 * (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, 5.6], 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], If[LessEqual[im$95$m, 5.8e+102], N[(0.5 - N[(N[Exp[im$95$m], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(-0.16666666666666666 * N[(im$95$m * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 5.5999999999999996

   1. Initial program 39.9%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \cos re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \cos re\right)\right)} \]
   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 89.9%

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

   5. Simplified 89.9%

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

   if 5.5999999999999996 < im < 5.8000000000000005e102

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)} \]
   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 81.3%

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

   5. Simplified 81.3%

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

   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 81.3%

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

   if 5.8000000000000005e102 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

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

Alternative 9: 94.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 6.1:\\ \;\;\;\;im\_m \cdot \left(\cos re \cdot \left(-1 + \left(im\_m \cdot im\_m\right) \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;im\_m \leq 5.8 \cdot 10^{+102}:\\ \;\;\;\;0.5 - e^{im\_m} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(-0.16666666666666666 \cdot \left(im\_m \cdot \left(im\_m \cdot im\_m\right)\right)\right)\\ \end{array} \end{array} \]

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 6.1)
    (* im_m (* (cos re) (+ -1.0 (* (* im_m im_m) -0.16666666666666666))))
    (if (<= im_m 5.8e+102)
      (- 0.5 (* (exp im_m) 0.5))
      (* (cos re) (* -0.16666666666666666 (* 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 <= 6.1) {
		tmp = im_m * (cos(re) * (-1.0 + ((im_m * im_m) * -0.16666666666666666)));
	} else if (im_m <= 5.8e+102) {
		tmp = 0.5 - (exp(im_m) * 0.5);
	} else {
		tmp = cos(re) * (-0.16666666666666666 * (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 <= 6.1d0) then
        tmp = im_m * (cos(re) * ((-1.0d0) + ((im_m * im_m) * (-0.16666666666666666d0))))
    else if (im_m <= 5.8d+102) then
        tmp = 0.5d0 - (exp(im_m) * 0.5d0)
    else
        tmp = cos(re) * ((-0.16666666666666666d0) * (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 <= 6.1) {
		tmp = im_m * (Math.cos(re) * (-1.0 + ((im_m * im_m) * -0.16666666666666666)));
	} else if (im_m <= 5.8e+102) {
		tmp = 0.5 - (Math.exp(im_m) * 0.5);
	} else {
		tmp = Math.cos(re) * (-0.16666666666666666 * (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 <= 6.1:
		tmp = im_m * (math.cos(re) * (-1.0 + ((im_m * im_m) * -0.16666666666666666)))
	elif im_m <= 5.8e+102:
		tmp = 0.5 - (math.exp(im_m) * 0.5)
	else:
		tmp = math.cos(re) * (-0.16666666666666666 * (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 <= 6.1)
		tmp = Float64(im_m * Float64(cos(re) * Float64(-1.0 + Float64(Float64(im_m * im_m) * -0.16666666666666666))));
	elseif (im_m <= 5.8e+102)
		tmp = Float64(0.5 - Float64(exp(im_m) * 0.5));
	else
		tmp = Float64(cos(re) * Float64(-0.16666666666666666 * Float64(im_m * 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 <= 6.1)
		tmp = im_m * (cos(re) * (-1.0 + ((im_m * im_m) * -0.16666666666666666)));
	elseif (im_m <= 5.8e+102)
		tmp = 0.5 - (exp(im_m) * 0.5);
	else
		tmp = cos(re) * (-0.16666666666666666 * (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, 6.1], N[(im$95$m * N[(N[Cos[re], $MachinePrecision] * N[(-1.0 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 5.8e+102], N[(0.5 - N[(N[Exp[im$95$m], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(-0.16666666666666666 * N[(im$95$m * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 6.0999999999999996

   1. Initial program 39.9%

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

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)}\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)}\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) + -2\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left(-2 + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right)}\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{{im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right)} - \frac{1}{3}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{{im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right)} - \frac{1}{3}\right)\right)\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) + \frac{-1}{3}\right)\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{3} + \color{blue}{{im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right)\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right)}\right)\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{-1}{2520} \cdot {im}^{2}} - \frac{1}{60}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{-1}{2520} \cdot {im}^{2}} - \frac{1}{60}\right)\right)\right)\right)\right)\right)\right) \]

      16. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{2520} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{60}\right)\right)}\right)\right)\right)\right)\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{2520} \cdot {im}^{2} + \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]

      18. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{60} + \color{blue}{\frac{-1}{2520} \cdot {im}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{60}, \color{blue}{\left(\frac{-1}{2520} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{60}, \left({im}^{2} \cdot \color{blue}{\frac{-1}{2520}}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{60}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{-1}{2520}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      22. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{60}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{-1}{2520}\right)\right)\right)\right)\right)\right)\right)\right) \]

      23. *-lowering-*.f64 97.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{60}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{2520}\right)\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 97.5%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

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

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

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

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

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

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

      17. *-commutative N/A

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

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

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

      19. unpow2 N/A

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

      20. *-lowering-*.f64 89.8%

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

   8. Simplified 89.8%

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

   if 6.0999999999999996 < im < 5.8000000000000005e102

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)} \]
   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 81.3%

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

   5. Simplified 81.3%

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

   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 81.3%

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

   if 5.8000000000000005e102 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

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

Alternative 10: 94.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 5:\\ \;\;\;\;0 - im\_m \cdot \cos re\\ \mathbf{elif}\;im\_m \leq 5.8 \cdot 10^{+102}:\\ \;\;\;\;0.5 - e^{im\_m} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(-0.16666666666666666 \cdot \left(im\_m \cdot \left(im\_m \cdot im\_m\right)\right)\right)\\ \end{array} \end{array} \]

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 5.0)
    (- 0.0 (* im_m (cos re)))
    (if (<= im_m 5.8e+102)
      (- 0.5 (* (exp im_m) 0.5))
      (* (cos re) (* -0.16666666666666666 (* 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 <= 5.0) {
		tmp = 0.0 - (im_m * cos(re));
	} else if (im_m <= 5.8e+102) {
		tmp = 0.5 - (exp(im_m) * 0.5);
	} else {
		tmp = cos(re) * (-0.16666666666666666 * (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 <= 5.0d0) then
        tmp = 0.0d0 - (im_m * cos(re))
    else if (im_m <= 5.8d+102) then
        tmp = 0.5d0 - (exp(im_m) * 0.5d0)
    else
        tmp = cos(re) * ((-0.16666666666666666d0) * (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 <= 5.0) {
		tmp = 0.0 - (im_m * Math.cos(re));
	} else if (im_m <= 5.8e+102) {
		tmp = 0.5 - (Math.exp(im_m) * 0.5);
	} else {
		tmp = Math.cos(re) * (-0.16666666666666666 * (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 <= 5.0:
		tmp = 0.0 - (im_m * math.cos(re))
	elif im_m <= 5.8e+102:
		tmp = 0.5 - (math.exp(im_m) * 0.5)
	else:
		tmp = math.cos(re) * (-0.16666666666666666 * (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 <= 5.0)
		tmp = Float64(0.0 - Float64(im_m * cos(re)));
	elseif (im_m <= 5.8e+102)
		tmp = Float64(0.5 - Float64(exp(im_m) * 0.5));
	else
		tmp = Float64(cos(re) * Float64(-0.16666666666666666 * Float64(im_m * 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 <= 5.0)
		tmp = 0.0 - (im_m * cos(re));
	elseif (im_m <= 5.8e+102)
		tmp = 0.5 - (exp(im_m) * 0.5);
	else
		tmp = cos(re) * (-0.16666666666666666 * (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, 5.0], N[(0.0 - N[(im$95$m * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 5.8e+102], N[(0.5 - N[(N[Exp[im$95$m], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(-0.16666666666666666 * N[(im$95$m * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 5

   1. Initial program 39.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 67.5%

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

   5. Simplified 67.5%

      \[\leadsto \color{blue}{0 - im \cdot \cos re} \]
   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 67.5%

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

   7. Applied egg-rr 67.5%

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

   if 5 < im < 5.8000000000000005e102

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)} \]
   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 81.3%

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

   5. Simplified 81.3%

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

   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 81.3%

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

   if 5.8000000000000005e102 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 5:\\ \;\;\;\;0 - im \cdot \cos re\\ \mathbf{elif}\;im \leq 5.8 \cdot 10^{+102}:\\ \;\;\;\;0.5 - e^{im} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(-0.16666666666666666 \cdot \left(im \cdot \left(im \cdot im\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 85.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 5.2:\\ \;\;\;\;0 - im\_m \cdot \cos re\\ \mathbf{elif}\;im\_m \leq 2.05 \cdot 10^{+142}:\\ \;\;\;\;0.5 - e^{im\_m} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + \left(re \cdot re\right) \cdot \left(-0.25 + \left(re \cdot re\right) \cdot \left(0.020833333333333332 + \left(re \cdot re\right) \cdot -0.0006944444444444445\right)\right)\right) \cdot \left(im\_m \cdot \left(-2 + \left(im\_m \cdot im\_m\right) \cdot \left(-0.3333333333333333 + \left(im\_m \cdot im\_m\right) \cdot \left(-0.016666666666666666 + \left(im\_m \cdot im\_m\right) \cdot -0.0003968253968253968\right)\right)\right)\right)\\ \end{array} \end{array} \]

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 5.2)
    (- 0.0 (* im_m (cos re)))
    (if (<= im_m 2.05e+142)
      (- 0.5 (* (exp im_m) 0.5))
      (*
       (+
        0.5
        (*
         (* re re)
         (+
          -0.25
          (*
           (* re re)
           (+ 0.020833333333333332 (* (* re re) -0.0006944444444444445))))))
       (*
        im_m
        (+
         -2.0
         (*
          (* im_m im_m)
          (+
           -0.3333333333333333
           (*
            (* im_m im_m)
            (+
             -0.016666666666666666
             (* (* im_m im_m) -0.0003968253968253968))))))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 5.2) {
		tmp = 0.0 - (im_m * cos(re));
	} else if (im_m <= 2.05e+142) {
		tmp = 0.5 - (exp(im_m) * 0.5);
	} else {
		tmp = (0.5 + ((re * re) * (-0.25 + ((re * re) * (0.020833333333333332 + ((re * re) * -0.0006944444444444445)))))) * (im_m * (-2.0 + ((im_m * im_m) * (-0.3333333333333333 + ((im_m * im_m) * (-0.016666666666666666 + ((im_m * im_m) * -0.0003968253968253968)))))));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 5.2d0) then
        tmp = 0.0d0 - (im_m * cos(re))
    else if (im_m <= 2.05d+142) then
        tmp = 0.5d0 - (exp(im_m) * 0.5d0)
    else
        tmp = (0.5d0 + ((re * re) * ((-0.25d0) + ((re * re) * (0.020833333333333332d0 + ((re * re) * (-0.0006944444444444445d0))))))) * (im_m * ((-2.0d0) + ((im_m * im_m) * ((-0.3333333333333333d0) + ((im_m * im_m) * ((-0.016666666666666666d0) + ((im_m * im_m) * (-0.0003968253968253968d0))))))))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 5.2) {
		tmp = 0.0 - (im_m * Math.cos(re));
	} else if (im_m <= 2.05e+142) {
		tmp = 0.5 - (Math.exp(im_m) * 0.5);
	} else {
		tmp = (0.5 + ((re * re) * (-0.25 + ((re * re) * (0.020833333333333332 + ((re * re) * -0.0006944444444444445)))))) * (im_m * (-2.0 + ((im_m * im_m) * (-0.3333333333333333 + ((im_m * im_m) * (-0.016666666666666666 + ((im_m * im_m) * -0.0003968253968253968)))))));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 5.2:
		tmp = 0.0 - (im_m * math.cos(re))
	elif im_m <= 2.05e+142:
		tmp = 0.5 - (math.exp(im_m) * 0.5)
	else:
		tmp = (0.5 + ((re * re) * (-0.25 + ((re * re) * (0.020833333333333332 + ((re * re) * -0.0006944444444444445)))))) * (im_m * (-2.0 + ((im_m * im_m) * (-0.3333333333333333 + ((im_m * im_m) * (-0.016666666666666666 + ((im_m * im_m) * -0.0003968253968253968)))))))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 5.2)
		tmp = Float64(0.0 - Float64(im_m * cos(re)));
	elseif (im_m <= 2.05e+142)
		tmp = Float64(0.5 - Float64(exp(im_m) * 0.5));
	else
		tmp = Float64(Float64(0.5 + Float64(Float64(re * re) * Float64(-0.25 + Float64(Float64(re * re) * Float64(0.020833333333333332 + Float64(Float64(re * re) * -0.0006944444444444445)))))) * Float64(im_m * Float64(-2.0 + Float64(Float64(im_m * im_m) * Float64(-0.3333333333333333 + Float64(Float64(im_m * im_m) * Float64(-0.016666666666666666 + Float64(Float64(im_m * im_m) * -0.0003968253968253968))))))));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 5.2)
		tmp = 0.0 - (im_m * cos(re));
	elseif (im_m <= 2.05e+142)
		tmp = 0.5 - (exp(im_m) * 0.5);
	else
		tmp = (0.5 + ((re * re) * (-0.25 + ((re * re) * (0.020833333333333332 + ((re * re) * -0.0006944444444444445)))))) * (im_m * (-2.0 + ((im_m * im_m) * (-0.3333333333333333 + ((im_m * im_m) * (-0.016666666666666666 + ((im_m * im_m) * -0.0003968253968253968)))))));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 5.2], N[(0.0 - N[(im$95$m * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 2.05e+142], N[(0.5 - N[(N[Exp[im$95$m], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 + N[(N[(re * re), $MachinePrecision] * N[(-0.25 + N[(N[(re * re), $MachinePrecision] * N[(0.020833333333333332 + N[(N[(re * re), $MachinePrecision] * -0.0006944444444444445), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(im$95$m * N[(-2.0 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(-0.3333333333333333 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(-0.016666666666666666 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0003968253968253968), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 5.20000000000000018

   1. Initial program 39.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 67.5%

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

   5. Simplified 67.5%

      \[\leadsto \color{blue}{0 - im \cdot \cos re} \]
   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 67.5%

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

   7. Applied egg-rr 67.5%

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

   if 5.20000000000000018 < im < 2.04999999999999991e142

   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 88.5%

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

   5. Simplified 88.5%

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

   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 88.5%

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

   if 2.04999999999999991e142 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)}\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)}\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) + -2\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left(-2 + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right)}\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{{im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right)} - \frac{1}{3}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{{im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right)} - \frac{1}{3}\right)\right)\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) + \frac{-1}{3}\right)\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{3} + \color{blue}{{im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right)\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right)}\right)\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{-1}{2520} \cdot {im}^{2}} - \frac{1}{60}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{-1}{2520} \cdot {im}^{2}} - \frac{1}{60}\right)\right)\right)\right)\right)\right)\right) \]

      16. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{2520} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{60}\right)\right)}\right)\right)\right)\right)\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{2520} \cdot {im}^{2} + \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]

      18. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{60} + \color{blue}{\frac{-1}{2520} \cdot {im}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{60}, \color{blue}{\left(\frac{-1}{2520} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{60}, \left({im}^{2} \cdot \color{blue}{\frac{-1}{2520}}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{60}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{-1}{2520}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      22. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{60}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{-1}{2520}\right)\right)\right)\right)\right)\right)\right)\right) \]

      23. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{60}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{2520}\right)\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)}, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{60}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{2520}\right)\right)\right)\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{im}, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{60}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{2520}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({re}^{2}\right), \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{60}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{2520}\right)\right)\right)\right)\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(re \cdot re\right), \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{60}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{2520}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{60}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{2520}\right)\right)\right)\right)\right)\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{60}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{2520}\right)\right)\right)\right)\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) + \frac{-1}{4}\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{60}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{2520}\right)\right)\right)\right)\right)\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{-1}{4} + {re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{60}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{2520}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{4}, \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{60}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{2520}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(\left({re}^{2}\right), \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{60}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{2520}\right)\right)\right)\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(\left(re \cdot re\right), \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{60}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{2520}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{60}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{2520}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{1}{48}, \left(\frac{-1}{1440} \cdot {re}^{2}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{60}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{2520}\right)\right)\right)\right)\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{1}{48}, \left({re}^{2} \cdot \frac{-1}{1440}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{60}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{2520}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{1}{48}, \mathsf{*.f64}\left(\left({re}^{2}\right), \frac{-1}{1440}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{60}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{2520}\right)\right)\right)\right)\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{1}{48}, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{1440}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{60}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{2520}\right)\right)\right)\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 76.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{+.f64}\left(\frac{1}{48}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{1440}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{60}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{2520}\right)\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 76.9%

      \[\leadsto \color{blue}{\left(0.5 + \left(re \cdot re\right) \cdot \left(-0.25 + \left(re \cdot re\right) \cdot \left(0.020833333333333332 + \left(re \cdot re\right) \cdot -0.0006944444444444445\right)\right)\right)} \cdot \left(im \cdot \left(-2 + \left(im \cdot im\right) \cdot \left(-0.3333333333333333 + \left(im \cdot im\right) \cdot \left(-0.016666666666666666 + \left(im \cdot im\right) \cdot -0.0003968253968253968\right)\right)\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 5.2:\\ \;\;\;\;0 - im \cdot \cos re\\ \mathbf{elif}\;im \leq 2.05 \cdot 10^{+142}:\\ \;\;\;\;0.5 - e^{im} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + \left(re \cdot re\right) \cdot \left(-0.25 + \left(re \cdot re\right) \cdot \left(0.020833333333333332 + \left(re \cdot re\right) \cdot -0.0006944444444444445\right)\right)\right) \cdot \left(im \cdot \left(-2 + \left(im \cdot im\right) \cdot \left(-0.3333333333333333 + \left(im \cdot im\right) \cdot \left(-0.016666666666666666 + \left(im \cdot im\right) \cdot -0.0003968253968253968\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 80.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 0.0028:\\ \;\;\;\;0 - im\_m \cdot \cos re\\ \mathbf{elif}\;im\_m \leq 3.5 \cdot 10^{+132}:\\ \;\;\;\;im\_m \cdot \left(-1 + \frac{im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot im\_m\right)\right)}{im\_m \cdot im\_m} \cdot \left(-0.16666666666666666 + \left(im\_m \cdot im\_m\right) \cdot \left(-0.008333333333333333 + \left(im\_m \cdot im\_m\right) \cdot -0.0001984126984126984\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im\_m \cdot im\_m\right) \cdot \left(im\_m \cdot \left(-0.16666666666666666 + \left(re \cdot re\right) \cdot 0.08333333333333333\right)\right)\\ \end{array} \end{array} \]

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 0.0028)
    (- 0.0 (* im_m (cos re)))
    (if (<= im_m 3.5e+132)
      (*
       im_m
       (+
        -1.0
        (*
         (/ (* im_m (* im_m (* im_m im_m))) (* im_m im_m))
         (+
          -0.16666666666666666
          (*
           (* im_m im_m)
           (+
            -0.008333333333333333
            (* (* im_m im_m) -0.0001984126984126984)))))))
      (*
       (* im_m im_m)
       (* im_m (+ -0.16666666666666666 (* (* re re) 0.08333333333333333))))))))

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 <= 0.0028) {
		tmp = 0.0 - (im_m * cos(re));
	} else if (im_m <= 3.5e+132) {
		tmp = im_m * (-1.0 + (((im_m * (im_m * (im_m * im_m))) / (im_m * im_m)) * (-0.16666666666666666 + ((im_m * im_m) * (-0.008333333333333333 + ((im_m * im_m) * -0.0001984126984126984))))));
	} else {
		tmp = (im_m * im_m) * (im_m * (-0.16666666666666666 + ((re * re) * 0.08333333333333333)));
	}
	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 <= 0.0028d0) then
        tmp = 0.0d0 - (im_m * cos(re))
    else if (im_m <= 3.5d+132) then
        tmp = im_m * ((-1.0d0) + (((im_m * (im_m * (im_m * im_m))) / (im_m * im_m)) * ((-0.16666666666666666d0) + ((im_m * im_m) * ((-0.008333333333333333d0) + ((im_m * im_m) * (-0.0001984126984126984d0)))))))
    else
        tmp = (im_m * im_m) * (im_m * ((-0.16666666666666666d0) + ((re * re) * 0.08333333333333333d0)))
    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 <= 0.0028) {
		tmp = 0.0 - (im_m * Math.cos(re));
	} else if (im_m <= 3.5e+132) {
		tmp = im_m * (-1.0 + (((im_m * (im_m * (im_m * im_m))) / (im_m * im_m)) * (-0.16666666666666666 + ((im_m * im_m) * (-0.008333333333333333 + ((im_m * im_m) * -0.0001984126984126984))))));
	} else {
		tmp = (im_m * im_m) * (im_m * (-0.16666666666666666 + ((re * re) * 0.08333333333333333)));
	}
	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 <= 0.0028:
		tmp = 0.0 - (im_m * math.cos(re))
	elif im_m <= 3.5e+132:
		tmp = im_m * (-1.0 + (((im_m * (im_m * (im_m * im_m))) / (im_m * im_m)) * (-0.16666666666666666 + ((im_m * im_m) * (-0.008333333333333333 + ((im_m * im_m) * -0.0001984126984126984))))))
	else:
		tmp = (im_m * im_m) * (im_m * (-0.16666666666666666 + ((re * re) * 0.08333333333333333)))
	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 <= 0.0028)
		tmp = Float64(0.0 - Float64(im_m * cos(re)));
	elseif (im_m <= 3.5e+132)
		tmp = Float64(im_m * Float64(-1.0 + Float64(Float64(Float64(im_m * Float64(im_m * Float64(im_m * im_m))) / Float64(im_m * im_m)) * Float64(-0.16666666666666666 + Float64(Float64(im_m * im_m) * Float64(-0.008333333333333333 + Float64(Float64(im_m * im_m) * -0.0001984126984126984)))))));
	else
		tmp = Float64(Float64(im_m * im_m) * Float64(im_m * Float64(-0.16666666666666666 + Float64(Float64(re * re) * 0.08333333333333333))));
	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 <= 0.0028)
		tmp = 0.0 - (im_m * cos(re));
	elseif (im_m <= 3.5e+132)
		tmp = im_m * (-1.0 + (((im_m * (im_m * (im_m * im_m))) / (im_m * im_m)) * (-0.16666666666666666 + ((im_m * im_m) * (-0.008333333333333333 + ((im_m * im_m) * -0.0001984126984126984))))));
	else
		tmp = (im_m * im_m) * (im_m * (-0.16666666666666666 + ((re * re) * 0.08333333333333333)));
	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, 0.0028], N[(0.0 - N[(im$95$m * N[Cos[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 3.5e+132], N[(im$95$m * N[(-1.0 + N[(N[(N[(im$95$m * N[(im$95$m * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] * N[(-0.16666666666666666 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(-0.008333333333333333 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * N[(-0.16666666666666666 + N[(N[(re * re), $MachinePrecision] * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 0.00279999999999999997

   1. Initial program 39.3%

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

   3. Taylor expanded in im around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

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

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

      5. cos-lowering-cos.f64 67.9%

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

   5. Simplified 67.9%

      \[\leadsto \color{blue}{0 - im \cdot \cos re} \]
   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 67.9%

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

   7. Applied egg-rr 67.9%

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

   if 0.00279999999999999997 < im < 3.5000000000000002e132

   1. Initial program 99.6%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 84.3%

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

   5. Simplified 84.3%

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

   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. *-lowering-*.f64 N/A

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

      7. unpow2 N/A

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

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

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

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

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

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

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

      14. unpow2 N/A

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

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

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

      16. sub-neg N/A

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

      17. metadata-eval N/A

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

      18. +-commutative N/A

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

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

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

      20. *-commutative N/A

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

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

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

      22. unpow2 N/A

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

      23. *-lowering-*.f64 66.5%

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

   8. Simplified 66.5%

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

      1. +-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left(im \cdot \left(0 + im\right)\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left(im \cdot \left(im + 0\right)\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right) \]

      3. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left(im \cdot im + 0 \cdot im\right), \mathsf{+.f64}\left(\color{blue}{\frac{-1}{6}}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right) \]

      4. mul0-lft N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left(im \cdot im + 0\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right) \]

      5. flip-+ N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left(\frac{\left(im \cdot im\right) \cdot \left(im \cdot im\right) - 0 \cdot 0}{im \cdot im - 0}\right), \mathsf{+.f64}\left(\color{blue}{\frac{-1}{6}}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right) \]

      6. fmm-def N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left(\frac{\left(im \cdot im\right) \cdot \left(im \cdot im\right) - 0 \cdot 0}{\mathsf{fma}\left(im, im, \mathsf{neg}\left(0\right)\right)}\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left(\frac{\left(im \cdot im\right) \cdot \left(im \cdot im\right) - 0 \cdot 0}{\mathsf{fma}\left(im, im, 0\right)}\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right) \]

      8. mul0-lft N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left(\frac{\left(im \cdot im\right) \cdot \left(im \cdot im\right) - 0 \cdot 0}{\mathsf{fma}\left(im, im, 0 \cdot im\right)}\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right) \]

      9. fma-define N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left(\frac{\left(im \cdot im\right) \cdot \left(im \cdot im\right) - 0 \cdot 0}{im \cdot im + 0 \cdot im}\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left(\frac{\left(im \cdot im\right) \cdot \left(im \cdot im\right) - 0 \cdot 0}{im \cdot \left(im + 0\right)}\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left(\frac{\left(im \cdot im\right) \cdot \left(im \cdot im\right) - 0 \cdot 0}{im \cdot \left(0 + im\right)}\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right) \]

      12. +-lft-identity N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left(\frac{\left(im \cdot im\right) \cdot \left(im \cdot im\right) - 0 \cdot 0}{im \cdot im}\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right) - 0 \cdot 0\right), \left(im \cdot im\right)\right), \mathsf{+.f64}\left(\color{blue}{\frac{-1}{6}}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right) \]

      14. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(im \cdot \left(im \cdot \left(im \cdot im\right)\right) - 0 \cdot 0\right), \left(im \cdot im\right)\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(im \cdot \left(im \cdot \left(im \cdot im\right)\right) - 0\right), \left(im \cdot im\right)\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(im \cdot \left(im \cdot \left(im \cdot im\right)\right)\right), 0\right), \left(im \cdot im\right)\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(im, \left(im \cdot \left(im \cdot im\right)\right)\right), 0\right), \left(im \cdot im\right)\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(im \cdot im\right)\right)\right), 0\right), \left(im \cdot im\right)\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, im\right)\right)\right), 0\right), \left(im \cdot im\right)\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right) \]

      20. *-lowering-*.f64 66.5%

          \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, im\right)\right)\right), 0\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 66.5%

       \[\leadsto im \cdot \left(-1 + \color{blue}{\frac{im \cdot \left(im \cdot \left(im \cdot im\right)\right) - 0}{im \cdot im}} \cdot \left(-0.16666666666666666 + \left(im \cdot im\right) \cdot \left(-0.008333333333333333 + \left(im \cdot im\right) \cdot -0.0001984126984126984\right)\right)\right) \]

   if 3.5000000000000002e132 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 75.6%

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

   8. Simplified 75.6%

      \[\leadsto \color{blue}{\left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)} \cdot \left(im \cdot \left(-2 + \left(im \cdot im\right) \cdot -0.3333333333333333\right)\right) \]

   9. Taylor expanded in im around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. unpow3 N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

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

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

       8. unpow2 N/A

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

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

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

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

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

       11. distribute-lft-in N/A

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

       12. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \left(\frac{-1}{6} + \color{blue}{\frac{-1}{3}} \cdot \left(\frac{-1}{4} \cdot {re}^{2}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left(\frac{-1}{3} \cdot \left(\frac{-1}{4} \cdot {re}^{2}\right)\right)}\right)\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \left(\left(\frac{-1}{4} \cdot {re}^{2}\right) \cdot \color{blue}{\frac{-1}{3}}\right)\right)\right)\right) \]

       15. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \left(\left({re}^{2} \cdot \frac{-1}{4}\right) \cdot \frac{-1}{3}\right)\right)\right)\right) \]

       16. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \left({re}^{2} \cdot \color{blue}{\left(\frac{-1}{4} \cdot \frac{-1}{3}\right)}\right)\right)\right)\right) \]

       17. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \left({re}^{2} \cdot \frac{1}{12}\right)\right)\right)\right) \]

       18. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \left({re}^{2} \cdot \left(\frac{-1}{3} \cdot \color{blue}{\frac{-1}{4}}\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\left(\frac{-1}{3} \cdot \frac{-1}{4}\right)}\right)\right)\right)\right) \]

       20. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left(re \cdot re\right), \left(\color{blue}{\frac{-1}{3}} \cdot \frac{-1}{4}\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\color{blue}{\frac{-1}{3}} \cdot \frac{-1}{4}\right)\right)\right)\right)\right) \]

       22. metadata-eval 75.6%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{1}{12}\right)\right)\right)\right) \]

   11. Simplified 75.6%

       \[\leadsto \color{blue}{\left(im \cdot im\right) \cdot \left(im \cdot \left(-0.16666666666666666 + \left(re \cdot re\right) \cdot 0.08333333333333333\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.0028:\\ \;\;\;\;0 - im \cdot \cos re\\ \mathbf{elif}\;im \leq 3.5 \cdot 10^{+132}:\\ \;\;\;\;im \cdot \left(-1 + \frac{im \cdot \left(im \cdot \left(im \cdot im\right)\right)}{im \cdot im} \cdot \left(-0.16666666666666666 + \left(im \cdot im\right) \cdot \left(-0.008333333333333333 + \left(im \cdot im\right) \cdot -0.0001984126984126984\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(im \cdot \left(-0.16666666666666666 + \left(re \cdot re\right) \cdot 0.08333333333333333\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 59.5% accurate, 11.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 2.65 \cdot 10^{+132}:\\ \;\;\;\;im\_m \cdot \left(-1 + \left(im\_m \cdot im\_m\right) \cdot \left(-0.16666666666666666 + \left(im\_m \cdot im\_m\right) \cdot \left(-0.008333333333333333 + \left(im\_m \cdot im\_m\right) \cdot -0.0001984126984126984\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im\_m \cdot im\_m\right) \cdot \left(im\_m \cdot \left(-0.16666666666666666 + \left(re \cdot re\right) \cdot 0.08333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 2.65e+132)
    (*
     im_m
     (+
      -1.0
      (*
       (* im_m im_m)
       (+
        -0.16666666666666666
        (*
         (* im_m im_m)
         (+
          -0.008333333333333333
          (* (* im_m im_m) -0.0001984126984126984)))))))
    (*
     (* im_m im_m)
     (* im_m (+ -0.16666666666666666 (* (* re re) 0.08333333333333333)))))))

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 <= 2.65e+132) {
		tmp = im_m * (-1.0 + ((im_m * im_m) * (-0.16666666666666666 + ((im_m * im_m) * (-0.008333333333333333 + ((im_m * im_m) * -0.0001984126984126984))))));
	} else {
		tmp = (im_m * im_m) * (im_m * (-0.16666666666666666 + ((re * re) * 0.08333333333333333)));
	}
	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 <= 2.65d+132) then
        tmp = im_m * ((-1.0d0) + ((im_m * im_m) * ((-0.16666666666666666d0) + ((im_m * im_m) * ((-0.008333333333333333d0) + ((im_m * im_m) * (-0.0001984126984126984d0)))))))
    else
        tmp = (im_m * im_m) * (im_m * ((-0.16666666666666666d0) + ((re * re) * 0.08333333333333333d0)))
    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 <= 2.65e+132) {
		tmp = im_m * (-1.0 + ((im_m * im_m) * (-0.16666666666666666 + ((im_m * im_m) * (-0.008333333333333333 + ((im_m * im_m) * -0.0001984126984126984))))));
	} else {
		tmp = (im_m * im_m) * (im_m * (-0.16666666666666666 + ((re * re) * 0.08333333333333333)));
	}
	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 <= 2.65e+132:
		tmp = im_m * (-1.0 + ((im_m * im_m) * (-0.16666666666666666 + ((im_m * im_m) * (-0.008333333333333333 + ((im_m * im_m) * -0.0001984126984126984))))))
	else:
		tmp = (im_m * im_m) * (im_m * (-0.16666666666666666 + ((re * re) * 0.08333333333333333)))
	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 <= 2.65e+132)
		tmp = Float64(im_m * Float64(-1.0 + Float64(Float64(im_m * im_m) * Float64(-0.16666666666666666 + Float64(Float64(im_m * im_m) * Float64(-0.008333333333333333 + Float64(Float64(im_m * im_m) * -0.0001984126984126984)))))));
	else
		tmp = Float64(Float64(im_m * im_m) * Float64(im_m * Float64(-0.16666666666666666 + Float64(Float64(re * re) * 0.08333333333333333))));
	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 <= 2.65e+132)
		tmp = im_m * (-1.0 + ((im_m * im_m) * (-0.16666666666666666 + ((im_m * im_m) * (-0.008333333333333333 + ((im_m * im_m) * -0.0001984126984126984))))));
	else
		tmp = (im_m * im_m) * (im_m * (-0.16666666666666666 + ((re * re) * 0.08333333333333333)));
	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, 2.65e+132], N[(im$95$m * N[(-1.0 + 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[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * N[(-0.16666666666666666 + N[(N[(re * re), $MachinePrecision] * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 2.65e132

   1. Initial program 46.6%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 34.1%

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

   5. Simplified 34.1%

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

   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. *-lowering-*.f64 N/A

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

      7. unpow2 N/A

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

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

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

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

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

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

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

      14. unpow2 N/A

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

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

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

      16. sub-neg N/A

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

      17. metadata-eval N/A

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

      18. +-commutative N/A

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

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

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

      20. *-commutative N/A

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

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

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

      22. unpow2 N/A

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

      23. *-lowering-*.f64 55.4%

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

   8. Simplified 55.4%

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

   if 2.65e132 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 75.6%

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

   8. Simplified 75.6%

      \[\leadsto \color{blue}{\left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)} \cdot \left(im \cdot \left(-2 + \left(im \cdot im\right) \cdot -0.3333333333333333\right)\right) \]

   9. Taylor expanded in im around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. unpow3 N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

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

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

       8. unpow2 N/A

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

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

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

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

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

       11. distribute-lft-in N/A

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

       12. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \left(\frac{-1}{6} + \color{blue}{\frac{-1}{3}} \cdot \left(\frac{-1}{4} \cdot {re}^{2}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left(\frac{-1}{3} \cdot \left(\frac{-1}{4} \cdot {re}^{2}\right)\right)}\right)\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \left(\left(\frac{-1}{4} \cdot {re}^{2}\right) \cdot \color{blue}{\frac{-1}{3}}\right)\right)\right)\right) \]

       15. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \left(\left({re}^{2} \cdot \frac{-1}{4}\right) \cdot \frac{-1}{3}\right)\right)\right)\right) \]

       16. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \left({re}^{2} \cdot \color{blue}{\left(\frac{-1}{4} \cdot \frac{-1}{3}\right)}\right)\right)\right)\right) \]

       17. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \left({re}^{2} \cdot \frac{1}{12}\right)\right)\right)\right) \]

       18. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \left({re}^{2} \cdot \left(\frac{-1}{3} \cdot \color{blue}{\frac{-1}{4}}\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\left(\frac{-1}{3} \cdot \frac{-1}{4}\right)}\right)\right)\right)\right) \]

       20. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left(re \cdot re\right), \left(\color{blue}{\frac{-1}{3}} \cdot \frac{-1}{4}\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\color{blue}{\frac{-1}{3}} \cdot \frac{-1}{4}\right)\right)\right)\right)\right) \]

       22. metadata-eval 75.6%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{1}{12}\right)\right)\right)\right) \]

   11. Simplified 75.6%

       \[\leadsto \color{blue}{\left(im \cdot im\right) \cdot \left(im \cdot \left(-0.16666666666666666 + \left(re \cdot re\right) \cdot 0.08333333333333333\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 59.5% accurate, 12.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 3 \cdot 10^{+132}:\\ \;\;\;\;im\_m \cdot \left(-1 + \left(im\_m \cdot im\_m\right) \cdot \left(-0.16666666666666666 + -0.0001984126984126984 \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(im\_m \cdot im\_m\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im\_m \cdot im\_m\right) \cdot \left(im\_m \cdot \left(-0.16666666666666666 + \left(re \cdot re\right) \cdot 0.08333333333333333\right)\right)\\ \end{array} \end{array} \]

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 3e+132)
    (*
     im_m
     (+
      -1.0
      (*
       (* im_m im_m)
       (+
        -0.16666666666666666
        (* -0.0001984126984126984 (* (* im_m im_m) (* im_m im_m)))))))
    (*
     (* im_m im_m)
     (* im_m (+ -0.16666666666666666 (* (* re re) 0.08333333333333333)))))))

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 <= 3e+132) {
		tmp = im_m * (-1.0 + ((im_m * im_m) * (-0.16666666666666666 + (-0.0001984126984126984 * ((im_m * im_m) * (im_m * im_m))))));
	} else {
		tmp = (im_m * im_m) * (im_m * (-0.16666666666666666 + ((re * re) * 0.08333333333333333)));
	}
	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 <= 3d+132) then
        tmp = im_m * ((-1.0d0) + ((im_m * im_m) * ((-0.16666666666666666d0) + ((-0.0001984126984126984d0) * ((im_m * im_m) * (im_m * im_m))))))
    else
        tmp = (im_m * im_m) * (im_m * ((-0.16666666666666666d0) + ((re * re) * 0.08333333333333333d0)))
    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 <= 3e+132) {
		tmp = im_m * (-1.0 + ((im_m * im_m) * (-0.16666666666666666 + (-0.0001984126984126984 * ((im_m * im_m) * (im_m * im_m))))));
	} else {
		tmp = (im_m * im_m) * (im_m * (-0.16666666666666666 + ((re * re) * 0.08333333333333333)));
	}
	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 <= 3e+132:
		tmp = im_m * (-1.0 + ((im_m * im_m) * (-0.16666666666666666 + (-0.0001984126984126984 * ((im_m * im_m) * (im_m * im_m))))))
	else:
		tmp = (im_m * im_m) * (im_m * (-0.16666666666666666 + ((re * re) * 0.08333333333333333)))
	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 <= 3e+132)
		tmp = Float64(im_m * Float64(-1.0 + Float64(Float64(im_m * im_m) * Float64(-0.16666666666666666 + Float64(-0.0001984126984126984 * Float64(Float64(im_m * im_m) * Float64(im_m * im_m)))))));
	else
		tmp = Float64(Float64(im_m * im_m) * Float64(im_m * Float64(-0.16666666666666666 + Float64(Float64(re * re) * 0.08333333333333333))));
	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 <= 3e+132)
		tmp = im_m * (-1.0 + ((im_m * im_m) * (-0.16666666666666666 + (-0.0001984126984126984 * ((im_m * im_m) * (im_m * im_m))))));
	else
		tmp = (im_m * im_m) * (im_m * (-0.16666666666666666 + ((re * re) * 0.08333333333333333)));
	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, 3e+132], N[(im$95$m * N[(-1.0 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(-0.16666666666666666 + N[(-0.0001984126984126984 * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * N[(-0.16666666666666666 + N[(N[(re * re), $MachinePrecision] * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 2.9999999999999998e132

   1. Initial program 46.6%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 34.1%

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

   5. Simplified 34.1%

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

   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. *-lowering-*.f64 N/A

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

      7. unpow2 N/A

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

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

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

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

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

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

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

      14. unpow2 N/A

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

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

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

      16. sub-neg N/A

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

      17. metadata-eval N/A

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

      18. +-commutative N/A

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

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

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

      20. *-commutative N/A

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

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

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

      22. unpow2 N/A

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

      23. *-lowering-*.f64 55.4%

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

   8. Simplified 55.4%

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

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left(\frac{-1}{5040} \cdot {im}^{4}\right)}\right)\right)\right)\right) \]
   10. Step-by-step derivation

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

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

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

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

       5. unpow2 N/A

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 55.2%

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

   11. Simplified 55.2%

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

   if 2.9999999999999998e132 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 75.6%

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

   8. Simplified 75.6%

      \[\leadsto \color{blue}{\left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)} \cdot \left(im \cdot \left(-2 + \left(im \cdot im\right) \cdot -0.3333333333333333\right)\right) \]

   9. Taylor expanded in im around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. unpow3 N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

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

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

       8. unpow2 N/A

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

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

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

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

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

       11. distribute-lft-in N/A

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

       12. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \left(\frac{-1}{6} + \color{blue}{\frac{-1}{3}} \cdot \left(\frac{-1}{4} \cdot {re}^{2}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left(\frac{-1}{3} \cdot \left(\frac{-1}{4} \cdot {re}^{2}\right)\right)}\right)\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \left(\left(\frac{-1}{4} \cdot {re}^{2}\right) \cdot \color{blue}{\frac{-1}{3}}\right)\right)\right)\right) \]

       15. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \left(\left({re}^{2} \cdot \frac{-1}{4}\right) \cdot \frac{-1}{3}\right)\right)\right)\right) \]

       16. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \left({re}^{2} \cdot \color{blue}{\left(\frac{-1}{4} \cdot \frac{-1}{3}\right)}\right)\right)\right)\right) \]

       17. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \left({re}^{2} \cdot \frac{1}{12}\right)\right)\right)\right) \]

       18. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \left({re}^{2} \cdot \left(\frac{-1}{3} \cdot \color{blue}{\frac{-1}{4}}\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\left(\frac{-1}{3} \cdot \frac{-1}{4}\right)}\right)\right)\right)\right) \]

       20. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left(re \cdot re\right), \left(\color{blue}{\frac{-1}{3}} \cdot \frac{-1}{4}\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\color{blue}{\frac{-1}{3}} \cdot \frac{-1}{4}\right)\right)\right)\right)\right) \]

       22. metadata-eval 75.6%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{1}{12}\right)\right)\right)\right) \]

   11. Simplified 75.6%

       \[\leadsto \color{blue}{\left(im \cdot im\right) \cdot \left(im \cdot \left(-0.16666666666666666 + \left(re \cdot re\right) \cdot 0.08333333333333333\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 57.7% accurate, 15.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 4.8 \cdot 10^{+132}:\\ \;\;\;\;im\_m \cdot \left(-1 + im\_m \cdot \left(im\_m \cdot \left(-0.16666666666666666 + \left(im\_m \cdot im\_m\right) \cdot -0.008333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im\_m \cdot im\_m\right) \cdot \left(im\_m \cdot \left(-0.16666666666666666 + \left(re \cdot re\right) \cdot 0.08333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 4.8e+132)
    (*
     im_m
     (+
      -1.0
      (*
       im_m
       (*
        im_m
        (+ -0.16666666666666666 (* (* im_m im_m) -0.008333333333333333))))))
    (*
     (* im_m im_m)
     (* im_m (+ -0.16666666666666666 (* (* re re) 0.08333333333333333)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 4.8e+132) {
		tmp = im_m * (-1.0 + (im_m * (im_m * (-0.16666666666666666 + ((im_m * im_m) * -0.008333333333333333)))));
	} else {
		tmp = (im_m * im_m) * (im_m * (-0.16666666666666666 + ((re * re) * 0.08333333333333333)));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 4.8d+132) then
        tmp = im_m * ((-1.0d0) + (im_m * (im_m * ((-0.16666666666666666d0) + ((im_m * im_m) * (-0.008333333333333333d0))))))
    else
        tmp = (im_m * im_m) * (im_m * ((-0.16666666666666666d0) + ((re * re) * 0.08333333333333333d0)))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 4.8e+132) {
		tmp = im_m * (-1.0 + (im_m * (im_m * (-0.16666666666666666 + ((im_m * im_m) * -0.008333333333333333)))));
	} else {
		tmp = (im_m * im_m) * (im_m * (-0.16666666666666666 + ((re * re) * 0.08333333333333333)));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 4.8e+132:
		tmp = im_m * (-1.0 + (im_m * (im_m * (-0.16666666666666666 + ((im_m * im_m) * -0.008333333333333333)))))
	else:
		tmp = (im_m * im_m) * (im_m * (-0.16666666666666666 + ((re * re) * 0.08333333333333333)))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 4.8e+132)
		tmp = Float64(im_m * Float64(-1.0 + Float64(im_m * Float64(im_m * Float64(-0.16666666666666666 + Float64(Float64(im_m * im_m) * -0.008333333333333333))))));
	else
		tmp = Float64(Float64(im_m * im_m) * Float64(im_m * Float64(-0.16666666666666666 + Float64(Float64(re * re) * 0.08333333333333333))));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 4.8e+132)
		tmp = im_m * (-1.0 + (im_m * (im_m * (-0.16666666666666666 + ((im_m * im_m) * -0.008333333333333333)))));
	else
		tmp = (im_m * im_m) * (im_m * (-0.16666666666666666 + ((re * re) * 0.08333333333333333)));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 4.8e+132], N[(im$95$m * N[(-1.0 + N[(im$95$m * N[(im$95$m * N[(-0.16666666666666666 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * N[(-0.16666666666666666 + N[(N[(re * re), $MachinePrecision] * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 4.8000000000000002e132

   1. Initial program 46.6%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 34.1%

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

   5. Simplified 34.1%

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

   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. *-lowering-*.f64 N/A

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

      7. unpow2 N/A

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

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

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

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

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

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

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

      14. unpow2 N/A

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

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

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

      16. sub-neg N/A

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

      17. metadata-eval N/A

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

      18. +-commutative N/A

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

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

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

      20. *-commutative N/A

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

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

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

      22. unpow2 N/A

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

      23. *-lowering-*.f64 55.4%

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

   8. Simplified 55.4%

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

   9. Taylor expanded in im around 0

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

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

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

       2. sub-neg N/A

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

       3. metadata-eval N/A

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

       4. +-commutative N/A

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

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

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

       6. unpow2 N/A

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

       7. associate-*l* N/A

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

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

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

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

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

       10. sub-neg N/A

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

       11. metadata-eval N/A

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

       12. +-commutative N/A

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

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

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

       14. *-commutative N/A

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

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

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

       16. unpow2 N/A

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

       17. *-lowering-*.f64 53.6%

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

   11. Simplified 53.6%

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

   if 4.8000000000000002e132 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 75.6%

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

   8. Simplified 75.6%

      \[\leadsto \color{blue}{\left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)} \cdot \left(im \cdot \left(-2 + \left(im \cdot im\right) \cdot -0.3333333333333333\right)\right) \]

   9. Taylor expanded in im around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. unpow3 N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

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

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

       8. unpow2 N/A

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

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

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

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

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

       11. distribute-lft-in N/A

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

       12. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \left(\frac{-1}{6} + \color{blue}{\frac{-1}{3}} \cdot \left(\frac{-1}{4} \cdot {re}^{2}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left(\frac{-1}{3} \cdot \left(\frac{-1}{4} \cdot {re}^{2}\right)\right)}\right)\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \left(\left(\frac{-1}{4} \cdot {re}^{2}\right) \cdot \color{blue}{\frac{-1}{3}}\right)\right)\right)\right) \]

       15. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \left(\left({re}^{2} \cdot \frac{-1}{4}\right) \cdot \frac{-1}{3}\right)\right)\right)\right) \]

       16. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \left({re}^{2} \cdot \color{blue}{\left(\frac{-1}{4} \cdot \frac{-1}{3}\right)}\right)\right)\right)\right) \]

       17. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \left({re}^{2} \cdot \frac{1}{12}\right)\right)\right)\right) \]

       18. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \left({re}^{2} \cdot \left(\frac{-1}{3} \cdot \color{blue}{\frac{-1}{4}}\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\left(\frac{-1}{3} \cdot \frac{-1}{4}\right)}\right)\right)\right)\right) \]

       20. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left(re \cdot re\right), \left(\color{blue}{\frac{-1}{3}} \cdot \frac{-1}{4}\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\color{blue}{\frac{-1}{3}} \cdot \frac{-1}{4}\right)\right)\right)\right)\right) \]

       22. metadata-eval 75.6%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{1}{12}\right)\right)\right)\right) \]

   11. Simplified 75.6%

       \[\leadsto \color{blue}{\left(im \cdot im\right) \cdot \left(im \cdot \left(-0.16666666666666666 + \left(re \cdot re\right) \cdot 0.08333333333333333\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 53.1% accurate, 17.2× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 5 \cdot 10^{+132}:\\ \;\;\;\;0.5 \cdot \left(im\_m \cdot \left(-2 + \left(im\_m \cdot im\_m\right) \cdot -0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im\_m \cdot im\_m\right) \cdot \left(im\_m \cdot \left(-0.16666666666666666 + \left(re \cdot re\right) \cdot 0.08333333333333333\right)\right)\\ \end{array} \end{array} \]

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 5e+132)
    (* 0.5 (* im_m (+ -2.0 (* (* im_m im_m) -0.3333333333333333))))
    (*
     (* im_m im_m)
     (* im_m (+ -0.16666666666666666 (* (* re re) 0.08333333333333333)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 5e+132) {
		tmp = 0.5 * (im_m * (-2.0 + ((im_m * im_m) * -0.3333333333333333)));
	} else {
		tmp = (im_m * im_m) * (im_m * (-0.16666666666666666 + ((re * re) * 0.08333333333333333)));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 5d+132) then
        tmp = 0.5d0 * (im_m * ((-2.0d0) + ((im_m * im_m) * (-0.3333333333333333d0))))
    else
        tmp = (im_m * im_m) * (im_m * ((-0.16666666666666666d0) + ((re * re) * 0.08333333333333333d0)))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 5e+132) {
		tmp = 0.5 * (im_m * (-2.0 + ((im_m * im_m) * -0.3333333333333333)));
	} else {
		tmp = (im_m * im_m) * (im_m * (-0.16666666666666666 + ((re * re) * 0.08333333333333333)));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 5e+132:
		tmp = 0.5 * (im_m * (-2.0 + ((im_m * im_m) * -0.3333333333333333)))
	else:
		tmp = (im_m * im_m) * (im_m * (-0.16666666666666666 + ((re * re) * 0.08333333333333333)))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 5e+132)
		tmp = Float64(0.5 * Float64(im_m * Float64(-2.0 + Float64(Float64(im_m * im_m) * -0.3333333333333333))));
	else
		tmp = Float64(Float64(im_m * im_m) * Float64(im_m * Float64(-0.16666666666666666 + Float64(Float64(re * re) * 0.08333333333333333))));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 5e+132)
		tmp = 0.5 * (im_m * (-2.0 + ((im_m * im_m) * -0.3333333333333333)));
	else
		tmp = (im_m * im_m) * (im_m * (-0.16666666666666666 + ((re * re) * 0.08333333333333333)));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 5e+132], N[(0.5 * N[(im$95$m * N[(-2.0 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * N[(-0.16666666666666666 + N[(N[(re * re), $MachinePrecision] * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 5.0000000000000001e132

   1. Initial program 46.6%

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

   3. Taylor expanded in im around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 84.3%

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

   5. Simplified 84.3%

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

   6. Taylor expanded in re around 0

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

   8. Simplified 46.4%

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

   if 5.0000000000000001e132 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 75.6%

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

   8. Simplified 75.6%

      \[\leadsto \color{blue}{\left(0.5 + -0.25 \cdot \left(re \cdot re\right)\right)} \cdot \left(im \cdot \left(-2 + \left(im \cdot im\right) \cdot -0.3333333333333333\right)\right) \]

   9. Taylor expanded in im around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. unpow3 N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

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

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

       8. unpow2 N/A

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

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

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

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

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

       11. distribute-lft-in N/A

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

       12. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \left(\frac{-1}{6} + \color{blue}{\frac{-1}{3}} \cdot \left(\frac{-1}{4} \cdot {re}^{2}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left(\frac{-1}{3} \cdot \left(\frac{-1}{4} \cdot {re}^{2}\right)\right)}\right)\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \left(\left(\frac{-1}{4} \cdot {re}^{2}\right) \cdot \color{blue}{\frac{-1}{3}}\right)\right)\right)\right) \]

       15. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \left(\left({re}^{2} \cdot \frac{-1}{4}\right) \cdot \frac{-1}{3}\right)\right)\right)\right) \]

       16. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \left({re}^{2} \cdot \color{blue}{\left(\frac{-1}{4} \cdot \frac{-1}{3}\right)}\right)\right)\right)\right) \]

       17. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \left({re}^{2} \cdot \frac{1}{12}\right)\right)\right)\right) \]

       18. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \left({re}^{2} \cdot \left(\frac{-1}{3} \cdot \color{blue}{\frac{-1}{4}}\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\left(\frac{-1}{3} \cdot \frac{-1}{4}\right)}\right)\right)\right)\right) \]

       20. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left(re \cdot re\right), \left(\color{blue}{\frac{-1}{3}} \cdot \frac{-1}{4}\right)\right)\right)\right)\right) \]

       21. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\color{blue}{\frac{-1}{3}} \cdot \frac{-1}{4}\right)\right)\right)\right)\right) \]

       22. metadata-eval 75.6%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{1}{12}\right)\right)\right)\right) \]

   11. Simplified 75.6%

       \[\leadsto \color{blue}{\left(im \cdot im\right) \cdot \left(im \cdot \left(-0.16666666666666666 + \left(re \cdot re\right) \cdot 0.08333333333333333\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 54.0% accurate, 19.3× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;re \leq 1.6 \cdot 10^{+173}:\\ \;\;\;\;0.5 \cdot \left(im\_m \cdot \left(-2 + \left(im\_m \cdot im\_m\right) \cdot -0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im\_m \cdot \left(re \cdot re\right)\right) - im\_m\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= re 1.6e+173)
    (* 0.5 (* im_m (+ -2.0 (* (* im_m im_m) -0.3333333333333333))))
    (- (* 0.5 (* im_m (* re re))) im_m))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (re <= 1.6e+173) {
		tmp = 0.5 * (im_m * (-2.0 + ((im_m * im_m) * -0.3333333333333333)));
	} else {
		tmp = (0.5 * (im_m * (re * re))) - im_m;
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= 1.6d+173) then
        tmp = 0.5d0 * (im_m * ((-2.0d0) + ((im_m * im_m) * (-0.3333333333333333d0))))
    else
        tmp = (0.5d0 * (im_m * (re * re))) - im_m
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (re <= 1.6e+173) {
		tmp = 0.5 * (im_m * (-2.0 + ((im_m * im_m) * -0.3333333333333333)));
	} else {
		tmp = (0.5 * (im_m * (re * re))) - im_m;
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if re <= 1.6e+173:
		tmp = 0.5 * (im_m * (-2.0 + ((im_m * im_m) * -0.3333333333333333)))
	else:
		tmp = (0.5 * (im_m * (re * re))) - im_m
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (re <= 1.6e+173)
		tmp = Float64(0.5 * Float64(im_m * Float64(-2.0 + Float64(Float64(im_m * im_m) * -0.3333333333333333))));
	else
		tmp = Float64(Float64(0.5 * Float64(im_m * Float64(re * re))) - im_m);
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (re <= 1.6e+173)
		tmp = 0.5 * (im_m * (-2.0 + ((im_m * im_m) * -0.3333333333333333)));
	else
		tmp = (0.5 * (im_m * (re * re))) - im_m;
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[re, 1.6e+173], N[(0.5 * N[(im$95$m * N[(-2.0 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(im$95$m * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if re < 1.6000000000000001e173

   1. Initial program 54.2%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \color{blue}{\left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{3} \cdot {im}^{2} - 2\right)}\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left(\frac{-1}{3} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left(\frac{-1}{3} \cdot {im}^{2} + -2\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left(-2 + \color{blue}{\frac{-1}{3} \cdot {im}^{2}}\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \color{blue}{\left(\frac{-1}{3} \cdot {im}^{2}\right)}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \left({im}^{2} \cdot \color{blue}{\frac{-1}{3}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{-1}{3}}\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{-1}{3}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 86.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{3}\right)\right)\right)\right) \]

   5. Simplified 86.2%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left(-2 + \left(im \cdot im\right) \cdot -0.3333333333333333\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\frac{1}{2}}, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{3}\right)\right)\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 54.1%

      \[\leadsto \color{blue}{0.5} \cdot \left(im \cdot \left(-2 + \left(im \cdot im\right) \cdot -0.3333333333333333\right)\right) \]

   if 1.6000000000000001e173 < re

   1. Initial program 63.7%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 44.0%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \mathsf{cos.f64}\left(re\right)\right)\right) \]

   5. Simplified 44.0%

      \[\leadsto \color{blue}{0 - im \cdot \cos re} \]

   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. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot \left(im \cdot {re}^{2}\right)\right), \color{blue}{im}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot {re}^{2}\right)\right), im\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \left({re}^{2}\right)\right)\right), im\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \left(re \cdot re\right)\right)\right), im\right) \]

      5. *-lowering-*.f64 43.0%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, re\right)\right)\right), im\right) \]

   8. Simplified 43.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(im \cdot \left(re \cdot re\right)\right) - im} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 54.0% accurate, 22.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;re \leq 1.6 \cdot 10^{+173}:\\ \;\;\;\;im\_m \cdot \left(-1 + im\_m \cdot \left(im\_m \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im\_m \cdot \left(re \cdot re\right)\right) - im\_m\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= re 1.6e+173)
    (* im_m (+ -1.0 (* im_m (* im_m -0.16666666666666666))))
    (- (* 0.5 (* im_m (* re re))) im_m))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (re <= 1.6e+173) {
		tmp = im_m * (-1.0 + (im_m * (im_m * -0.16666666666666666)));
	} else {
		tmp = (0.5 * (im_m * (re * re))) - im_m;
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= 1.6d+173) then
        tmp = im_m * ((-1.0d0) + (im_m * (im_m * (-0.16666666666666666d0))))
    else
        tmp = (0.5d0 * (im_m * (re * re))) - im_m
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (re <= 1.6e+173) {
		tmp = im_m * (-1.0 + (im_m * (im_m * -0.16666666666666666)));
	} else {
		tmp = (0.5 * (im_m * (re * re))) - im_m;
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if re <= 1.6e+173:
		tmp = im_m * (-1.0 + (im_m * (im_m * -0.16666666666666666)))
	else:
		tmp = (0.5 * (im_m * (re * re))) - im_m
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (re <= 1.6e+173)
		tmp = Float64(im_m * Float64(-1.0 + Float64(im_m * Float64(im_m * -0.16666666666666666))));
	else
		tmp = Float64(Float64(0.5 * Float64(im_m * Float64(re * re))) - im_m);
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (re <= 1.6e+173)
		tmp = im_m * (-1.0 + (im_m * (im_m * -0.16666666666666666)));
	else
		tmp = (0.5 * (im_m * (re * re))) - im_m;
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[re, 1.6e+173], N[(im$95$m * N[(-1.0 + N[(im$95$m * N[(im$95$m * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(im$95$m * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if re < 1.6000000000000001e173

   1. Initial program 54.2%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \cos re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \cos re\right)\right)} \]
   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 85.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

   5. Simplified 85.9%

      \[\leadsto \color{blue}{\left(im \cdot \cos re\right) \cdot \left(-1 + im \cdot \left(im \cdot -0.16666666666666666\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{im}, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{6}\right)\right)\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 53.8%

      \[\leadsto \color{blue}{im} \cdot \left(-1 + im \cdot \left(im \cdot -0.16666666666666666\right)\right) \]

   if 1.6000000000000001e173 < re

   1. Initial program 63.7%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 44.0%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \mathsf{cos.f64}\left(re\right)\right)\right) \]

   5. Simplified 44.0%

      \[\leadsto \color{blue}{0 - im \cdot \cos re} \]

   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. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot \left(im \cdot {re}^{2}\right)\right), \color{blue}{im}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot {re}^{2}\right)\right), im\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \left({re}^{2}\right)\right)\right), im\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \left(re \cdot re\right)\right)\right), im\right) \]

      5. *-lowering-*.f64 43.0%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, re\right)\right)\right), im\right) \]

   8. Simplified 43.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(im \cdot \left(re \cdot re\right)\right) - im} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 47.4% accurate, 25.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 3.1 \cdot 10^{+155}:\\ \;\;\;\;0 - im\_m\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{im\_m \cdot im\_m}{im\_m}\\ \end{array} \end{array} \]

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 3.1e+155) (- 0.0 im_m) (- 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 <= 3.1e+155) {
		tmp = 0.0 - im_m;
	} 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 <= 3.1d+155) then
        tmp = 0.0d0 - im_m
    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 <= 3.1e+155) {
		tmp = 0.0 - im_m;
	} 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 <= 3.1e+155:
		tmp = 0.0 - im_m
	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 <= 3.1e+155)
		tmp = Float64(0.0 - im_m);
	else
		tmp = Float64(0.0 - Float64(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 <= 3.1e+155)
		tmp = 0.0 - im_m;
	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, 3.1e+155], N[(0.0 - im$95$m), $MachinePrecision], N[(0.0 - N[(N[(im$95$m * im$95$m), $MachinePrecision] / im$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 3.09999999999999989e155

   1. Initial program 47.6%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   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 59.3%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \mathsf{cos.f64}\left(re\right)\right)\right) \]

   5. Simplified 59.3%

      \[\leadsto \color{blue}{0 - im \cdot \cos re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{-1 \cdot im} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(im\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{im} \]

      3. --lowering--.f64 29.1%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{im}\right) \]

   8. Simplified 29.1%

      \[\leadsto \color{blue}{0 - im} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(im\right) \]

      2. neg-lowering-neg.f64 29.1%

         \[\leadsto \mathsf{neg.f64}\left(im\right) \]

   10. Applied egg-rr 29.1%

       \[\leadsto \color{blue}{-im} \]

   if 3.09999999999999989e155 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \cos re\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(im \cdot \cos re\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{im \cdot \cos re} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(im \cdot \cos re\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \color{blue}{\cos re}\right)\right) \]

      5. cos-lowering-cos.f64 7.1%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \mathsf{cos.f64}\left(re\right)\right)\right) \]

   5. Simplified 7.1%

      \[\leadsto \color{blue}{0 - im \cdot \cos re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{-1 \cdot im} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(im\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{im} \]

      3. --lowering--.f64 5.1%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{im}\right) \]

   8. Simplified 5.1%

      \[\leadsto \color{blue}{0 - im} \]
   9. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \frac{0 \cdot 0 - im \cdot im}{\color{blue}{0 + im}} \]

      2. +-lft-identity N/A

         \[\leadsto \frac{0 \cdot 0 - im \cdot im}{im} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(0 \cdot 0 - im \cdot im\right), \color{blue}{im}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(0 - im \cdot im\right), im\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(im \cdot im\right)\right), im\right) \]

      6. *-lowering-*.f64 73.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, im\right)\right), im\right) \]

   10. Applied egg-rr 73.0%

       \[\leadsto \color{blue}{\frac{0 - im \cdot im}{im}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.1 \cdot 10^{+155}:\\ \;\;\;\;0 - im\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{im \cdot im}{im}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 53.7% accurate, 34.3× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(im\_m \cdot \left(-1 + im\_m \cdot \left(im\_m \cdot -0.16666666666666666\right)\right)\right) \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (* im_s (* im_m (+ -1.0 (* im_m (* im_m -0.16666666666666666))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * (im_m * (-1.0 + (im_m * (im_m * -0.16666666666666666))));
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = im_s * (im_m * ((-1.0d0) + (im_m * (im_m * (-0.16666666666666666d0)))))
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	return im_s * (im_m * (-1.0 + (im_m * (im_m * -0.16666666666666666))));
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * (im_m * (-1.0 + (im_m * (im_m * -0.16666666666666666))))

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * Float64(im_m * Float64(-1.0 + Float64(im_m * Float64(im_m * -0.16666666666666666)))))
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * (im_m * (-1.0 + (im_m * (im_m * -0.16666666666666666))));
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * N[(im$95$m * N[(-1.0 + N[(im$95$m * N[(im$95$m * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.2%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in im around 0

   \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \cos re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \cos re\right)\right)} \]
4. Step-by-step derivation

   1. 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 86.5%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, \mathsf{cos.f64}\left(re\right)\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

5. Simplified 86.5%

   \[\leadsto \color{blue}{\left(im \cdot \cos re\right) \cdot \left(-1 + im \cdot \left(im \cdot -0.16666666666666666\right)\right)} \]

6. Taylor expanded in re around 0

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{im}, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{6}\right)\right)\right)\right) \]
7. Step-by-step derivation

8. Simplified 50.0%

   \[\leadsto \color{blue}{im} \cdot \left(-1 + im \cdot \left(im \cdot -0.16666666666666666\right)\right) \]
9. Add Preprocessing

Alternative 21: 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 55.2%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
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 51.8%

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \mathsf{cos.f64}\left(re\right)\right)\right) \]

5. Simplified 51.8%

   \[\leadsto \color{blue}{0 - im \cdot \cos re} \]

6. Taylor expanded in re around 0

   \[\leadsto \color{blue}{-1 \cdot im} \]
7. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(im\right) \]

   2. neg-sub0 N/A

      \[\leadsto 0 - \color{blue}{im} \]

   3. --lowering--.f64 25.6%

      \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{im}\right) \]

8. Simplified 25.6%

   \[\leadsto \color{blue}{0 - im} \]
9. Step-by-step derivation

   1. sub0-neg N/A

      \[\leadsto \mathsf{neg}\left(im\right) \]

   2. neg-lowering-neg.f64 25.6%

      \[\leadsto \mathsf{neg.f64}\left(im\right) \]

10. Applied egg-rr 25.6%

    \[\leadsto \color{blue}{-im} \]

11. Final simplification 25.6%

    \[\leadsto 0 - im \]
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 2024138 
(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))))