math.cos on complex, imaginary part

Percentage Accurate: 65.3% → 99.6%

Time: 19.5s

Alternatives: 26

Speedup: 2.8×

• 49.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 65.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{0 - im\_m} - e^{im\_m}\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -200:\\ \;\;\;\;t\_0 \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im\_m \cdot \sin re\right) \cdot \left(-1 + -0.16666666666666666 \cdot \left(im\_m \cdot im\_m\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (- (exp (- 0.0 im_m)) (exp im_m))))
   (*
    im_s
    (if (<= t_0 -200.0)
      (* t_0 (* 0.5 (sin re)))
      (* (* im_m (sin re)) (+ -1.0 (* -0.16666666666666666 (* im_m im_m))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = exp((0.0 - im_m)) - exp(im_m);
	double tmp;
	if (t_0 <= -200.0) {
		tmp = t_0 * (0.5 * sin(re));
	} else {
		tmp = (im_m * sin(re)) * (-1.0 + (-0.16666666666666666 * (im_m * im_m)));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp((0.0d0 - im_m)) - exp(im_m)
    if (t_0 <= (-200.0d0)) then
        tmp = t_0 * (0.5d0 * sin(re))
    else
        tmp = (im_m * sin(re)) * ((-1.0d0) + ((-0.16666666666666666d0) * (im_m * im_m)))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double t_0 = Math.exp((0.0 - im_m)) - Math.exp(im_m);
	double tmp;
	if (t_0 <= -200.0) {
		tmp = t_0 * (0.5 * Math.sin(re));
	} else {
		tmp = (im_m * Math.sin(re)) * (-1.0 + (-0.16666666666666666 * (im_m * im_m)));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	t_0 = math.exp((0.0 - im_m)) - math.exp(im_m)
	tmp = 0
	if t_0 <= -200.0:
		tmp = t_0 * (0.5 * math.sin(re))
	else:
		tmp = (im_m * math.sin(re)) * (-1.0 + (-0.16666666666666666 * (im_m * im_m)))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(exp(Float64(0.0 - im_m)) - exp(im_m))
	tmp = 0.0
	if (t_0 <= -200.0)
		tmp = Float64(t_0 * Float64(0.5 * sin(re)));
	else
		tmp = Float64(Float64(im_m * sin(re)) * Float64(-1.0 + Float64(-0.16666666666666666 * Float64(im_m * im_m))));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	t_0 = exp((0.0 - im_m)) - exp(im_m);
	tmp = 0.0;
	if (t_0 <= -200.0)
		tmp = t_0 * (0.5 * sin(re));
	else
		tmp = (im_m * sin(re)) * (-1.0 + (-0.16666666666666666 * (im_m * im_m)));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[Exp[N[(0.0 - im$95$m), $MachinePrecision]], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -200.0], N[(t$95$0 * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im$95$m * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -200

   1. Initial program 100.0%

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

   if -200 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 54.3%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      6. exp-neg N/A

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

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

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

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

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

      9. exp-lowering-exp.f64 54.3%

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

   3. Simplified 54.3%

      \[\leadsto \color{blue}{0.5 \cdot \left(\sin re \cdot \left(\frac{1}{e^{im}} - e^{im}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

         \[\leadsto -1 \cdot \left(im \cdot \sin re\right) + \left(\left(\frac{-1}{6} \cdot \left(im \cdot im\right)\right) \cdot \sin re\right) \cdot im \]

      6. associate-*r* N/A

         \[\leadsto -1 \cdot \left(im \cdot \sin re\right) + \left(\left(\left(\frac{-1}{6} \cdot im\right) \cdot im\right) \cdot \sin re\right) \cdot im \]

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-out N/A

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

      10. +-commutative N/A

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

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

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

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

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

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

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

      14. +-commutative N/A

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

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

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

      16. associate-*r* N/A

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

      17. unpow2 N/A

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

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

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

      19. unpow2 N/A

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

      20. *-lowering-*.f64 88.8%

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

   7. Simplified 88.8%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{0 - im} - e^{im} \leq -200:\\ \;\;\;\;\left(e^{0 - im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \sin re\right) \cdot \left(-1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;e^{0 - im\_m} - e^{im\_m} \leq -200:\\ \;\;\;\;0.5 \cdot \left(\sin re \cdot \left(\frac{1}{1 + im\_m \cdot \left(1 + im\_m \cdot \left(0.5 + im\_m \cdot 0.16666666666666666\right)\right)} - e^{im\_m}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im\_m \cdot \sin re\right) \cdot \left(-1 + -0.16666666666666666 \cdot \left(im\_m \cdot im\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (- (exp (- 0.0 im_m)) (exp im_m)) -200.0)
    (*
     0.5
     (*
      (sin re)
      (-
       (/
        1.0
        (+ 1.0 (* im_m (+ 1.0 (* im_m (+ 0.5 (* im_m 0.16666666666666666)))))))
       (exp im_m))))
    (* (* im_m (sin re)) (+ -1.0 (* -0.16666666666666666 (* im_m im_m)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if ((exp((0.0 - im_m)) - exp(im_m)) <= -200.0) {
		tmp = 0.5 * (sin(re) * ((1.0 / (1.0 + (im_m * (1.0 + (im_m * (0.5 + (im_m * 0.16666666666666666))))))) - exp(im_m)));
	} else {
		tmp = (im_m * sin(re)) * (-1.0 + (-0.16666666666666666 * (im_m * im_m)));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if ((exp((0.0d0 - im_m)) - exp(im_m)) <= (-200.0d0)) then
        tmp = 0.5d0 * (sin(re) * ((1.0d0 / (1.0d0 + (im_m * (1.0d0 + (im_m * (0.5d0 + (im_m * 0.16666666666666666d0))))))) - exp(im_m)))
    else
        tmp = (im_m * sin(re)) * ((-1.0d0) + ((-0.16666666666666666d0) * (im_m * im_m)))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if ((Math.exp((0.0 - im_m)) - Math.exp(im_m)) <= -200.0) {
		tmp = 0.5 * (Math.sin(re) * ((1.0 / (1.0 + (im_m * (1.0 + (im_m * (0.5 + (im_m * 0.16666666666666666))))))) - Math.exp(im_m)));
	} else {
		tmp = (im_m * Math.sin(re)) * (-1.0 + (-0.16666666666666666 * (im_m * im_m)));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if (math.exp((0.0 - im_m)) - math.exp(im_m)) <= -200.0:
		tmp = 0.5 * (math.sin(re) * ((1.0 / (1.0 + (im_m * (1.0 + (im_m * (0.5 + (im_m * 0.16666666666666666))))))) - math.exp(im_m)))
	else:
		tmp = (im_m * math.sin(re)) * (-1.0 + (-0.16666666666666666 * (im_m * im_m)))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(exp(Float64(0.0 - im_m)) - exp(im_m)) <= -200.0)
		tmp = Float64(0.5 * Float64(sin(re) * Float64(Float64(1.0 / Float64(1.0 + Float64(im_m * Float64(1.0 + Float64(im_m * Float64(0.5 + Float64(im_m * 0.16666666666666666))))))) - exp(im_m))));
	else
		tmp = Float64(Float64(im_m * sin(re)) * Float64(-1.0 + Float64(-0.16666666666666666 * Float64(im_m * im_m))));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if ((exp((0.0 - im_m)) - exp(im_m)) <= -200.0)
		tmp = 0.5 * (sin(re) * ((1.0 / (1.0 + (im_m * (1.0 + (im_m * (0.5 + (im_m * 0.16666666666666666))))))) - exp(im_m)));
	else
		tmp = (im_m * sin(re)) * (-1.0 + (-0.16666666666666666 * (im_m * im_m)));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[Exp[N[(0.0 - im$95$m), $MachinePrecision]], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision], -200.0], N[(0.5 * N[(N[Sin[re], $MachinePrecision] * N[(N[(1.0 / N[(1.0 + N[(im$95$m * N[(1.0 + N[(im$95$m * N[(0.5 + N[(im$95$m * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im$95$m * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -200

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      6. exp-neg N/A

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

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

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

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

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

      9. exp-lowering-exp.f64 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(\sin re \cdot \left(\frac{1}{e^{im}} - e^{im}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\left(1 + im \cdot \left(1 + im \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot im\right)\right)\right)}\right), \mathsf{exp.f64}\left(im\right)\right)\right)\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(im \cdot \left(1 + im \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot im\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(im\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(1 + im \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot im\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(im\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \left(im \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot im\right)\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(im\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \left(\frac{1}{2} + \frac{1}{6} \cdot im\right)\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(im\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot im\right)\right)\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(im\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \left(im \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(im\right)\right)\right)\right) \]

      7. *-lowering-*.f64 98.9%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \frac{1}{6}\right)\right)\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(im\right)\right)\right)\right) \]

   7. Simplified 98.9%

      \[\leadsto 0.5 \cdot \left(\sin re \cdot \left(\frac{1}{\color{blue}{1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)}} - e^{im}\right)\right) \]

   if -200 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 54.3%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      6. exp-neg N/A

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

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

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

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

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

      9. exp-lowering-exp.f64 54.3%

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

   3. Simplified 54.3%

      \[\leadsto \color{blue}{0.5 \cdot \left(\sin re \cdot \left(\frac{1}{e^{im}} - e^{im}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

         \[\leadsto -1 \cdot \left(im \cdot \sin re\right) + \left(\left(\frac{-1}{6} \cdot \left(im \cdot im\right)\right) \cdot \sin re\right) \cdot im \]

      6. associate-*r* N/A

         \[\leadsto -1 \cdot \left(im \cdot \sin re\right) + \left(\left(\left(\frac{-1}{6} \cdot im\right) \cdot im\right) \cdot \sin re\right) \cdot im \]

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-out N/A

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

      10. +-commutative N/A

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

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

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

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

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

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

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

      14. +-commutative N/A

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

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

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

      16. associate-*r* N/A

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

      17. unpow2 N/A

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

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

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

      19. unpow2 N/A

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

      20. *-lowering-*.f64 88.8%

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

   7. Simplified 88.8%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{0 - im} - e^{im} \leq -200:\\ \;\;\;\;0.5 \cdot \left(\sin re \cdot \left(\frac{1}{1 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)} - e^{im}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \sin re\right) \cdot \left(-1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;e^{0 - im\_m} - e^{im\_m} \leq -200:\\ \;\;\;\;0.5 \cdot \left(\sin re \cdot \left(\frac{1}{1 + im\_m \cdot \left(1 + im\_m \cdot 0.5\right)} - e^{im\_m}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im\_m \cdot \sin re\right) \cdot \left(-1 + -0.16666666666666666 \cdot \left(im\_m \cdot im\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (- (exp (- 0.0 im_m)) (exp im_m)) -200.0)
    (*
     0.5
     (* (sin re) (- (/ 1.0 (+ 1.0 (* im_m (+ 1.0 (* im_m 0.5))))) (exp im_m))))
    (* (* im_m (sin re)) (+ -1.0 (* -0.16666666666666666 (* im_m im_m)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if ((exp((0.0 - im_m)) - exp(im_m)) <= -200.0) {
		tmp = 0.5 * (sin(re) * ((1.0 / (1.0 + (im_m * (1.0 + (im_m * 0.5))))) - exp(im_m)));
	} else {
		tmp = (im_m * sin(re)) * (-1.0 + (-0.16666666666666666 * (im_m * im_m)));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if ((exp((0.0d0 - im_m)) - exp(im_m)) <= (-200.0d0)) then
        tmp = 0.5d0 * (sin(re) * ((1.0d0 / (1.0d0 + (im_m * (1.0d0 + (im_m * 0.5d0))))) - exp(im_m)))
    else
        tmp = (im_m * sin(re)) * ((-1.0d0) + ((-0.16666666666666666d0) * (im_m * im_m)))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if ((Math.exp((0.0 - im_m)) - Math.exp(im_m)) <= -200.0) {
		tmp = 0.5 * (Math.sin(re) * ((1.0 / (1.0 + (im_m * (1.0 + (im_m * 0.5))))) - Math.exp(im_m)));
	} else {
		tmp = (im_m * Math.sin(re)) * (-1.0 + (-0.16666666666666666 * (im_m * im_m)));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if (math.exp((0.0 - im_m)) - math.exp(im_m)) <= -200.0:
		tmp = 0.5 * (math.sin(re) * ((1.0 / (1.0 + (im_m * (1.0 + (im_m * 0.5))))) - math.exp(im_m)))
	else:
		tmp = (im_m * math.sin(re)) * (-1.0 + (-0.16666666666666666 * (im_m * im_m)))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(exp(Float64(0.0 - im_m)) - exp(im_m)) <= -200.0)
		tmp = Float64(0.5 * Float64(sin(re) * Float64(Float64(1.0 / Float64(1.0 + Float64(im_m * Float64(1.0 + Float64(im_m * 0.5))))) - exp(im_m))));
	else
		tmp = Float64(Float64(im_m * sin(re)) * Float64(-1.0 + Float64(-0.16666666666666666 * Float64(im_m * im_m))));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if ((exp((0.0 - im_m)) - exp(im_m)) <= -200.0)
		tmp = 0.5 * (sin(re) * ((1.0 / (1.0 + (im_m * (1.0 + (im_m * 0.5))))) - exp(im_m)));
	else
		tmp = (im_m * sin(re)) * (-1.0 + (-0.16666666666666666 * (im_m * im_m)));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[Exp[N[(0.0 - im$95$m), $MachinePrecision]], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision], -200.0], N[(0.5 * N[(N[Sin[re], $MachinePrecision] * N[(N[(1.0 / N[(1.0 + N[(im$95$m * N[(1.0 + N[(im$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im$95$m * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -200

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      6. exp-neg N/A

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

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

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

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

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

      9. exp-lowering-exp.f64 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(\sin re \cdot \left(\frac{1}{e^{im}} - e^{im}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0

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

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

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

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

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

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

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

      4. *-lowering-*.f64 98.8%

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

   7. Simplified 98.8%

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

   if -200 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 54.3%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      6. exp-neg N/A

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

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

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

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

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

      9. exp-lowering-exp.f64 54.3%

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

   3. Simplified 54.3%

      \[\leadsto \color{blue}{0.5 \cdot \left(\sin re \cdot \left(\frac{1}{e^{im}} - e^{im}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

         \[\leadsto -1 \cdot \left(im \cdot \sin re\right) + \left(\left(\frac{-1}{6} \cdot \left(im \cdot im\right)\right) \cdot \sin re\right) \cdot im \]

      6. associate-*r* N/A

         \[\leadsto -1 \cdot \left(im \cdot \sin re\right) + \left(\left(\left(\frac{-1}{6} \cdot im\right) \cdot im\right) \cdot \sin re\right) \cdot im \]

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-out N/A

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

      10. +-commutative N/A

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

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

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

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

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

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

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

      14. +-commutative N/A

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

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

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

      16. associate-*r* N/A

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

      17. unpow2 N/A

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

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

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

      19. unpow2 N/A

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

      20. *-lowering-*.f64 88.8%

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

   7. Simplified 88.8%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{0 - im} - e^{im} \leq -200:\\ \;\;\;\;0.5 \cdot \left(\sin re \cdot \left(\frac{1}{1 + im \cdot \left(1 + im \cdot 0.5\right)} - e^{im}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \sin re\right) \cdot \left(-1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;e^{0 - im\_m} - e^{im\_m} \leq -200:\\ \;\;\;\;0.5 \cdot \left(\sin re \cdot \left(\frac{1}{im\_m + 1} - e^{im\_m}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im\_m \cdot \sin re\right) \cdot \left(-1 + -0.16666666666666666 \cdot \left(im\_m \cdot im\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (- (exp (- 0.0 im_m)) (exp im_m)) -200.0)
    (* 0.5 (* (sin re) (- (/ 1.0 (+ im_m 1.0)) (exp im_m))))
    (* (* im_m (sin re)) (+ -1.0 (* -0.16666666666666666 (* im_m im_m)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if ((exp((0.0 - im_m)) - exp(im_m)) <= -200.0) {
		tmp = 0.5 * (sin(re) * ((1.0 / (im_m + 1.0)) - exp(im_m)));
	} else {
		tmp = (im_m * sin(re)) * (-1.0 + (-0.16666666666666666 * (im_m * im_m)));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if ((exp((0.0d0 - im_m)) - exp(im_m)) <= (-200.0d0)) then
        tmp = 0.5d0 * (sin(re) * ((1.0d0 / (im_m + 1.0d0)) - exp(im_m)))
    else
        tmp = (im_m * sin(re)) * ((-1.0d0) + ((-0.16666666666666666d0) * (im_m * im_m)))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if ((Math.exp((0.0 - im_m)) - Math.exp(im_m)) <= -200.0) {
		tmp = 0.5 * (Math.sin(re) * ((1.0 / (im_m + 1.0)) - Math.exp(im_m)));
	} else {
		tmp = (im_m * Math.sin(re)) * (-1.0 + (-0.16666666666666666 * (im_m * im_m)));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if (math.exp((0.0 - im_m)) - math.exp(im_m)) <= -200.0:
		tmp = 0.5 * (math.sin(re) * ((1.0 / (im_m + 1.0)) - math.exp(im_m)))
	else:
		tmp = (im_m * math.sin(re)) * (-1.0 + (-0.16666666666666666 * (im_m * im_m)))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(exp(Float64(0.0 - im_m)) - exp(im_m)) <= -200.0)
		tmp = Float64(0.5 * Float64(sin(re) * Float64(Float64(1.0 / Float64(im_m + 1.0)) - exp(im_m))));
	else
		tmp = Float64(Float64(im_m * sin(re)) * Float64(-1.0 + Float64(-0.16666666666666666 * Float64(im_m * im_m))));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if ((exp((0.0 - im_m)) - exp(im_m)) <= -200.0)
		tmp = 0.5 * (sin(re) * ((1.0 / (im_m + 1.0)) - exp(im_m)));
	else
		tmp = (im_m * sin(re)) * (-1.0 + (-0.16666666666666666 * (im_m * im_m)));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[Exp[N[(0.0 - im$95$m), $MachinePrecision]], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision], -200.0], N[(0.5 * N[(N[Sin[re], $MachinePrecision] * N[(N[(1.0 / N[(im$95$m + 1.0), $MachinePrecision]), $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im$95$m * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -200

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      6. exp-neg N/A

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

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

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

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

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

      9. exp-lowering-exp.f64 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(\sin re \cdot \left(\frac{1}{e^{im}} - e^{im}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0

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

      1. +-lowering-+.f64 98.8%

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

   7. Simplified 98.8%

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

   if -200 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 54.3%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      6. exp-neg N/A

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

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

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

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

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

      9. exp-lowering-exp.f64 54.3%

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

   3. Simplified 54.3%

      \[\leadsto \color{blue}{0.5 \cdot \left(\sin re \cdot \left(\frac{1}{e^{im}} - e^{im}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

         \[\leadsto -1 \cdot \left(im \cdot \sin re\right) + \left(\left(\frac{-1}{6} \cdot \left(im \cdot im\right)\right) \cdot \sin re\right) \cdot im \]

      6. associate-*r* N/A

         \[\leadsto -1 \cdot \left(im \cdot \sin re\right) + \left(\left(\left(\frac{-1}{6} \cdot im\right) \cdot im\right) \cdot \sin re\right) \cdot im \]

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-out N/A

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

      10. +-commutative N/A

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

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

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

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

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

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

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

      14. +-commutative N/A

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

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

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

      16. associate-*r* N/A

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

      17. unpow2 N/A

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

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

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

      19. unpow2 N/A

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

      20. *-lowering-*.f64 88.8%

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

   7. Simplified 88.8%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{0 - im} - e^{im} \leq -200:\\ \;\;\;\;0.5 \cdot \left(\sin re \cdot \left(\frac{1}{im + 1} - e^{im}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \sin re\right) \cdot \left(-1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;e^{0 - im\_m} - e^{im\_m} \leq -200:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(1 - e^{im\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im\_m \cdot \sin re\right) \cdot \left(-1 + -0.16666666666666666 \cdot \left(im\_m \cdot im\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (- (exp (- 0.0 im_m)) (exp im_m)) -200.0)
    (* (* 0.5 (sin re)) (- 1.0 (exp im_m)))
    (* (* im_m (sin re)) (+ -1.0 (* -0.16666666666666666 (* im_m im_m)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if ((exp((0.0 - im_m)) - exp(im_m)) <= -200.0) {
		tmp = (0.5 * sin(re)) * (1.0 - exp(im_m));
	} else {
		tmp = (im_m * sin(re)) * (-1.0 + (-0.16666666666666666 * (im_m * im_m)));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if ((exp((0.0d0 - im_m)) - exp(im_m)) <= (-200.0d0)) then
        tmp = (0.5d0 * sin(re)) * (1.0d0 - exp(im_m))
    else
        tmp = (im_m * sin(re)) * ((-1.0d0) + ((-0.16666666666666666d0) * (im_m * im_m)))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if ((Math.exp((0.0 - im_m)) - Math.exp(im_m)) <= -200.0) {
		tmp = (0.5 * Math.sin(re)) * (1.0 - Math.exp(im_m));
	} else {
		tmp = (im_m * Math.sin(re)) * (-1.0 + (-0.16666666666666666 * (im_m * im_m)));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if (math.exp((0.0 - im_m)) - math.exp(im_m)) <= -200.0:
		tmp = (0.5 * math.sin(re)) * (1.0 - math.exp(im_m))
	else:
		tmp = (im_m * math.sin(re)) * (-1.0 + (-0.16666666666666666 * (im_m * im_m)))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(exp(Float64(0.0 - im_m)) - exp(im_m)) <= -200.0)
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(1.0 - exp(im_m)));
	else
		tmp = Float64(Float64(im_m * sin(re)) * Float64(-1.0 + Float64(-0.16666666666666666 * Float64(im_m * im_m))));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if ((exp((0.0 - im_m)) - exp(im_m)) <= -200.0)
		tmp = (0.5 * sin(re)) * (1.0 - exp(im_m));
	else
		tmp = (im_m * sin(re)) * (-1.0 + (-0.16666666666666666 * (im_m * im_m)));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[Exp[N[(0.0 - im$95$m), $MachinePrecision]], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision], -200.0], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im$95$m * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -200

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 98.7%

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

   if -200 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 54.3%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      6. exp-neg N/A

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

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

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

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

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

      9. exp-lowering-exp.f64 54.3%

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

   3. Simplified 54.3%

      \[\leadsto \color{blue}{0.5 \cdot \left(\sin re \cdot \left(\frac{1}{e^{im}} - e^{im}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

         \[\leadsto -1 \cdot \left(im \cdot \sin re\right) + \left(\left(\frac{-1}{6} \cdot \left(im \cdot im\right)\right) \cdot \sin re\right) \cdot im \]

      6. associate-*r* N/A

         \[\leadsto -1 \cdot \left(im \cdot \sin re\right) + \left(\left(\left(\frac{-1}{6} \cdot im\right) \cdot im\right) \cdot \sin re\right) \cdot im \]

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-out N/A

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

      10. +-commutative N/A

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

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

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

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

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

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

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

      14. +-commutative N/A

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

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

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

      16. associate-*r* N/A

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

      17. unpow2 N/A

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

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

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

      19. unpow2 N/A

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

      20. *-lowering-*.f64 88.8%

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

   7. Simplified 88.8%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{0 - im} - e^{im} \leq -200:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(1 - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \sin re\right) \cdot \left(-1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 10.5:\\ \;\;\;\;im\_m \cdot \left(\sin re \cdot \left(-1 + \left(im\_m \cdot im\_m\right) \cdot \left(-0.16666666666666666 + \left(im\_m \cdot im\_m\right) \cdot -0.008333333333333333\right)\right)\right)\\ \mathbf{elif}\;im\_m \leq 9.6 \cdot 10^{+54}:\\ \;\;\;\;\left(1 - e^{im\_m}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im\_m \cdot \left(-2 + im\_m \cdot \left(im\_m \cdot \left(-0.3333333333333333 + \left(im\_m \cdot im\_m\right) \cdot -0.016666666666666666\right)\right)\right)\right)\\ \end{array} \end{array} \]

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 10.5)
    (*
     im_m
     (*
      (sin re)
      (+
       -1.0
       (*
        (* im_m im_m)
        (+ -0.16666666666666666 (* (* im_m im_m) -0.008333333333333333))))))
    (if (<= im_m 9.6e+54)
      (* (- 1.0 (exp im_m)) (* 0.5 re))
      (*
       (* 0.5 (sin re))
       (*
        im_m
        (+
         -2.0
         (*
          im_m
          (*
           im_m
           (+
            -0.3333333333333333
            (* (* im_m im_m) -0.016666666666666666)))))))))))

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 <= 10.5) {
		tmp = im_m * (sin(re) * (-1.0 + ((im_m * im_m) * (-0.16666666666666666 + ((im_m * im_m) * -0.008333333333333333)))));
	} else if (im_m <= 9.6e+54) {
		tmp = (1.0 - exp(im_m)) * (0.5 * re);
	} else {
		tmp = (0.5 * sin(re)) * (im_m * (-2.0 + (im_m * (im_m * (-0.3333333333333333 + ((im_m * im_m) * -0.016666666666666666))))));
	}
	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 <= 10.5d0) then
        tmp = im_m * (sin(re) * ((-1.0d0) + ((im_m * im_m) * ((-0.16666666666666666d0) + ((im_m * im_m) * (-0.008333333333333333d0))))))
    else if (im_m <= 9.6d+54) then
        tmp = (1.0d0 - exp(im_m)) * (0.5d0 * re)
    else
        tmp = (0.5d0 * sin(re)) * (im_m * ((-2.0d0) + (im_m * (im_m * ((-0.3333333333333333d0) + ((im_m * im_m) * (-0.016666666666666666d0)))))))
    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 <= 10.5) {
		tmp = im_m * (Math.sin(re) * (-1.0 + ((im_m * im_m) * (-0.16666666666666666 + ((im_m * im_m) * -0.008333333333333333)))));
	} else if (im_m <= 9.6e+54) {
		tmp = (1.0 - Math.exp(im_m)) * (0.5 * re);
	} else {
		tmp = (0.5 * Math.sin(re)) * (im_m * (-2.0 + (im_m * (im_m * (-0.3333333333333333 + ((im_m * im_m) * -0.016666666666666666))))));
	}
	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 <= 10.5:
		tmp = im_m * (math.sin(re) * (-1.0 + ((im_m * im_m) * (-0.16666666666666666 + ((im_m * im_m) * -0.008333333333333333)))))
	elif im_m <= 9.6e+54:
		tmp = (1.0 - math.exp(im_m)) * (0.5 * re)
	else:
		tmp = (0.5 * math.sin(re)) * (im_m * (-2.0 + (im_m * (im_m * (-0.3333333333333333 + ((im_m * im_m) * -0.016666666666666666))))))
	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 <= 10.5)
		tmp = Float64(im_m * Float64(sin(re) * Float64(-1.0 + Float64(Float64(im_m * im_m) * Float64(-0.16666666666666666 + Float64(Float64(im_m * im_m) * -0.008333333333333333))))));
	elseif (im_m <= 9.6e+54)
		tmp = Float64(Float64(1.0 - exp(im_m)) * Float64(0.5 * re));
	else
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(im_m * Float64(-2.0 + Float64(im_m * Float64(im_m * Float64(-0.3333333333333333 + Float64(Float64(im_m * im_m) * -0.016666666666666666)))))));
	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 <= 10.5)
		tmp = im_m * (sin(re) * (-1.0 + ((im_m * im_m) * (-0.16666666666666666 + ((im_m * im_m) * -0.008333333333333333)))));
	elseif (im_m <= 9.6e+54)
		tmp = (1.0 - exp(im_m)) * (0.5 * re);
	else
		tmp = (0.5 * sin(re)) * (im_m * (-2.0 + (im_m * (im_m * (-0.3333333333333333 + ((im_m * im_m) * -0.016666666666666666))))));
	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, 10.5], N[(im$95$m * N[(N[Sin[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, 9.6e+54], N[(N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(im$95$m * N[(-2.0 + N[(im$95$m * N[(im$95$m * N[(-0.3333333333333333 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 10.5

   1. Initial program 54.5%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      6. exp-neg N/A

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

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

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

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

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

      9. exp-lowering-exp.f64 54.5%

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

   3. Simplified 54.5%

      \[\leadsto \color{blue}{0.5 \cdot \left(\sin re \cdot \left(\frac{1}{e^{im}} - e^{im}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0

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

   7. Simplified 93.5%

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

   if 10.5 < im < 9.59999999999999993e54

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

   5. Simplified 80.0%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 80.0%

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

   if 9.59999999999999993e54 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

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

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

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

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

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

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

      14. *-commutative N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 98.1%

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

   5. Simplified 98.1%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 10.5:\\ \;\;\;\;im \cdot \left(\sin re \cdot \left(-1 + \left(im \cdot im\right) \cdot \left(-0.16666666666666666 + \left(im \cdot im\right) \cdot -0.008333333333333333\right)\right)\right)\\ \mathbf{elif}\;im \leq 9.6 \cdot 10^{+54}:\\ \;\;\;\;\left(1 - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left(-2 + im \cdot \left(im \cdot \left(-0.3333333333333333 + \left(im \cdot im\right) \cdot -0.016666666666666666\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := im\_m \cdot \left(\sin re \cdot \left(-1 + \left(im\_m \cdot im\_m\right) \cdot \left(-0.16666666666666666 + \left(im\_m \cdot im\_m\right) \cdot -0.008333333333333333\right)\right)\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 10.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im\_m \leq 3.8 \cdot 10^{+77}:\\ \;\;\;\;\left(1 - e^{im\_m}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0
         (*
          im_m
          (*
           (sin re)
           (+
            -1.0
            (*
             (* im_m im_m)
             (+
              -0.16666666666666666
              (* (* im_m im_m) -0.008333333333333333))))))))
   (*
    im_s
    (if (<= im_m 10.5)
      t_0
      (if (<= im_m 3.8e+77) (* (- 1.0 (exp im_m)) (* 0.5 re)) t_0)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = im_m * (sin(re) * (-1.0 + ((im_m * im_m) * (-0.16666666666666666 + ((im_m * im_m) * -0.008333333333333333)))));
	double tmp;
	if (im_m <= 10.5) {
		tmp = t_0;
	} else if (im_m <= 3.8e+77) {
		tmp = (1.0 - exp(im_m)) * (0.5 * re);
	} else {
		tmp = t_0;
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = im_m * (sin(re) * ((-1.0d0) + ((im_m * im_m) * ((-0.16666666666666666d0) + ((im_m * im_m) * (-0.008333333333333333d0))))))
    if (im_m <= 10.5d0) then
        tmp = t_0
    else if (im_m <= 3.8d+77) then
        tmp = (1.0d0 - exp(im_m)) * (0.5d0 * re)
    else
        tmp = t_0
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double t_0 = im_m * (Math.sin(re) * (-1.0 + ((im_m * im_m) * (-0.16666666666666666 + ((im_m * im_m) * -0.008333333333333333)))));
	double tmp;
	if (im_m <= 10.5) {
		tmp = t_0;
	} else if (im_m <= 3.8e+77) {
		tmp = (1.0 - Math.exp(im_m)) * (0.5 * re);
	} else {
		tmp = t_0;
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	t_0 = im_m * (math.sin(re) * (-1.0 + ((im_m * im_m) * (-0.16666666666666666 + ((im_m * im_m) * -0.008333333333333333)))))
	tmp = 0
	if im_m <= 10.5:
		tmp = t_0
	elif im_m <= 3.8e+77:
		tmp = (1.0 - math.exp(im_m)) * (0.5 * re)
	else:
		tmp = t_0
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(im_m * Float64(sin(re) * Float64(-1.0 + Float64(Float64(im_m * im_m) * Float64(-0.16666666666666666 + Float64(Float64(im_m * im_m) * -0.008333333333333333))))))
	tmp = 0.0
	if (im_m <= 10.5)
		tmp = t_0;
	elseif (im_m <= 3.8e+77)
		tmp = Float64(Float64(1.0 - exp(im_m)) * Float64(0.5 * re));
	else
		tmp = t_0;
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	t_0 = im_m * (sin(re) * (-1.0 + ((im_m * im_m) * (-0.16666666666666666 + ((im_m * im_m) * -0.008333333333333333)))));
	tmp = 0.0;
	if (im_m <= 10.5)
		tmp = t_0;
	elseif (im_m <= 3.8e+77)
		tmp = (1.0 - exp(im_m)) * (0.5 * re);
	else
		tmp = t_0;
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(im$95$m * N[(N[Sin[re], $MachinePrecision] * N[(-1.0 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(-0.16666666666666666 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[im$95$m, 10.5], t$95$0, If[LessEqual[im$95$m, 3.8e+77], N[(N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 10.5 or 3.8000000000000001e77 < im

   1. Initial program 62.9%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      6. exp-neg N/A

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

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

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

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

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

      9. exp-lowering-exp.f64 62.9%

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

   3. Simplified 62.9%

      \[\leadsto \color{blue}{0.5 \cdot \left(\sin re \cdot \left(\frac{1}{e^{im}} - e^{im}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0

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

   7. Simplified 94.7%

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

   if 10.5 < im < 3.8000000000000001e77

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

   5. Simplified 71.4%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 71.4%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 10.5:\\ \;\;\;\;im \cdot \left(\sin re \cdot \left(-1 + \left(im \cdot im\right) \cdot \left(-0.16666666666666666 + \left(im \cdot im\right) \cdot -0.008333333333333333\right)\right)\right)\\ \mathbf{elif}\;im \leq 3.8 \cdot 10^{+77}:\\ \;\;\;\;\left(1 - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\sin re \cdot \left(-1 + \left(im \cdot im\right) \cdot \left(-0.16666666666666666 + \left(im \cdot im\right) \cdot -0.008333333333333333\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(\left(0.5 \cdot \sin re\right) \cdot \left(im\_m \cdot \left(-2 + \left(im\_m \cdot im\_m\right) \cdot \left(-0.3333333333333333 + \left(im\_m \cdot im\_m\right) \cdot \left(-0.016666666666666666 + \left(im\_m \cdot im\_m\right) \cdot -0.0003968253968253968\right)\right)\right)\right)\right) \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (*
   (* 0.5 (sin re))
   (*
    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) {
	return im_s * ((0.5 * sin(re)) * (im_m * (-2.0 + ((im_m * im_m) * (-0.3333333333333333 + ((im_m * im_m) * (-0.016666666666666666 + ((im_m * im_m) * -0.0003968253968253968))))))));
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = im_s * ((0.5d0 * sin(re)) * (im_m * ((-2.0d0) + ((im_m * im_m) * ((-0.3333333333333333d0) + ((im_m * im_m) * ((-0.016666666666666666d0) + ((im_m * im_m) * (-0.0003968253968253968d0)))))))))
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	return im_s * ((0.5 * Math.sin(re)) * (im_m * (-2.0 + ((im_m * im_m) * (-0.3333333333333333 + ((im_m * im_m) * (-0.016666666666666666 + ((im_m * im_m) * -0.0003968253968253968))))))));
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * ((0.5 * math.sin(re)) * (im_m * (-2.0 + ((im_m * im_m) * (-0.3333333333333333 + ((im_m * im_m) * (-0.016666666666666666 + ((im_m * im_m) * -0.0003968253968253968))))))))

Julia

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

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * ((0.5 * sin(re)) * (im_m * (-2.0 + ((im_m * im_m) * (-0.3333333333333333 + ((im_m * im_m) * (-0.016666666666666666 + ((im_m * im_m) * -0.0003968253968253968))))))));
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * N[(N[(0.5 * N[Sin[re], $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. Initial program 64.0%

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

3. Taylor expanded in im around 0

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

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

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

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

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

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

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

   7. unpow2 N/A

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

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

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

   9. sub-neg N/A

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

   10. metadata-eval N/A

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

   11. +-commutative N/A

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

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

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

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

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

   14. unpow2 N/A

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

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

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

   16. sub-neg N/A

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

   17. metadata-eval N/A

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

   18. +-commutative N/A

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

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

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

   20. *-commutative N/A

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

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

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

   22. unpow2 N/A

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

   23. *-lowering-*.f64 96.7%

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

5. Simplified 96.7%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left(-2 + \left(im \cdot im\right) \cdot \left(-0.3333333333333333 + \left(im \cdot im\right) \cdot \left(-0.016666666666666666 + \left(im \cdot im\right) \cdot -0.0003968253968253968\right)\right)\right)\right)} \]
6. Add Preprocessing

Alternative 9: 94.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 3.7:\\ \;\;\;\;\left(im\_m \cdot \sin re\right) \cdot \left(-1 + -0.16666666666666666 \cdot \left(im\_m \cdot im\_m\right)\right)\\ \mathbf{elif}\;im\_m \leq 9.6 \cdot 10^{+54}:\\ \;\;\;\;\left(1 - e^{im\_m}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;im\_m \leq 4.4 \cdot 10^{+102}:\\ \;\;\;\;\left(im\_m \cdot \left(-2 + im\_m \cdot \left(im\_m \cdot \left(-0.3333333333333333 + \left(im\_m \cdot im\_m\right) \cdot -0.016666666666666666\right)\right)\right)\right) \cdot \left(re \cdot \left(0.5 + re \cdot \left(re \cdot -0.08333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im\_m \cdot \left(im\_m \cdot im\_m\right)\right) \cdot \left(\sin re \cdot -0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 3.7)
    (* (* im_m (sin re)) (+ -1.0 (* -0.16666666666666666 (* im_m im_m))))
    (if (<= im_m 9.6e+54)
      (* (- 1.0 (exp im_m)) (* 0.5 re))
      (if (<= im_m 4.4e+102)
        (*
         (*
          im_m
          (+
           -2.0
           (*
            im_m
            (*
             im_m
             (+
              -0.3333333333333333
              (* (* im_m im_m) -0.016666666666666666))))))
         (* re (+ 0.5 (* re (* re -0.08333333333333333)))))
        (* (* im_m (* im_m im_m)) (* (sin re) -0.16666666666666666)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 3.7) {
		tmp = (im_m * sin(re)) * (-1.0 + (-0.16666666666666666 * (im_m * im_m)));
	} else if (im_m <= 9.6e+54) {
		tmp = (1.0 - exp(im_m)) * (0.5 * re);
	} else if (im_m <= 4.4e+102) {
		tmp = (im_m * (-2.0 + (im_m * (im_m * (-0.3333333333333333 + ((im_m * im_m) * -0.016666666666666666)))))) * (re * (0.5 + (re * (re * -0.08333333333333333))));
	} else {
		tmp = (im_m * (im_m * im_m)) * (sin(re) * -0.16666666666666666);
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 3.7d0) then
        tmp = (im_m * sin(re)) * ((-1.0d0) + ((-0.16666666666666666d0) * (im_m * im_m)))
    else if (im_m <= 9.6d+54) then
        tmp = (1.0d0 - exp(im_m)) * (0.5d0 * re)
    else if (im_m <= 4.4d+102) then
        tmp = (im_m * ((-2.0d0) + (im_m * (im_m * ((-0.3333333333333333d0) + ((im_m * im_m) * (-0.016666666666666666d0))))))) * (re * (0.5d0 + (re * (re * (-0.08333333333333333d0)))))
    else
        tmp = (im_m * (im_m * im_m)) * (sin(re) * (-0.16666666666666666d0))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 3.7) {
		tmp = (im_m * Math.sin(re)) * (-1.0 + (-0.16666666666666666 * (im_m * im_m)));
	} else if (im_m <= 9.6e+54) {
		tmp = (1.0 - Math.exp(im_m)) * (0.5 * re);
	} else if (im_m <= 4.4e+102) {
		tmp = (im_m * (-2.0 + (im_m * (im_m * (-0.3333333333333333 + ((im_m * im_m) * -0.016666666666666666)))))) * (re * (0.5 + (re * (re * -0.08333333333333333))));
	} else {
		tmp = (im_m * (im_m * im_m)) * (Math.sin(re) * -0.16666666666666666);
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 3.7:
		tmp = (im_m * math.sin(re)) * (-1.0 + (-0.16666666666666666 * (im_m * im_m)))
	elif im_m <= 9.6e+54:
		tmp = (1.0 - math.exp(im_m)) * (0.5 * re)
	elif im_m <= 4.4e+102:
		tmp = (im_m * (-2.0 + (im_m * (im_m * (-0.3333333333333333 + ((im_m * im_m) * -0.016666666666666666)))))) * (re * (0.5 + (re * (re * -0.08333333333333333))))
	else:
		tmp = (im_m * (im_m * im_m)) * (math.sin(re) * -0.16666666666666666)
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 3.7)
		tmp = Float64(Float64(im_m * sin(re)) * Float64(-1.0 + Float64(-0.16666666666666666 * Float64(im_m * im_m))));
	elseif (im_m <= 9.6e+54)
		tmp = Float64(Float64(1.0 - exp(im_m)) * Float64(0.5 * re));
	elseif (im_m <= 4.4e+102)
		tmp = Float64(Float64(im_m * Float64(-2.0 + Float64(im_m * Float64(im_m * Float64(-0.3333333333333333 + Float64(Float64(im_m * im_m) * -0.016666666666666666)))))) * Float64(re * Float64(0.5 + Float64(re * Float64(re * -0.08333333333333333)))));
	else
		tmp = Float64(Float64(im_m * Float64(im_m * im_m)) * Float64(sin(re) * -0.16666666666666666));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 3.7)
		tmp = (im_m * sin(re)) * (-1.0 + (-0.16666666666666666 * (im_m * im_m)));
	elseif (im_m <= 9.6e+54)
		tmp = (1.0 - exp(im_m)) * (0.5 * re);
	elseif (im_m <= 4.4e+102)
		tmp = (im_m * (-2.0 + (im_m * (im_m * (-0.3333333333333333 + ((im_m * im_m) * -0.016666666666666666)))))) * (re * (0.5 + (re * (re * -0.08333333333333333))));
	else
		tmp = (im_m * (im_m * im_m)) * (sin(re) * -0.16666666666666666);
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 3.7], N[(N[(im$95$m * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 9.6e+54], N[(N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 4.4e+102], N[(N[(im$95$m * N[(-2.0 + N[(im$95$m * N[(im$95$m * N[(-0.3333333333333333 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(re * N[(0.5 + N[(re * N[(re * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im$95$m * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[re], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if im < 3.7000000000000002

   1. Initial program 54.3%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      6. exp-neg N/A

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

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

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

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

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

      9. exp-lowering-exp.f64 54.3%

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

   3. Simplified 54.3%

      \[\leadsto \color{blue}{0.5 \cdot \left(\sin re \cdot \left(\frac{1}{e^{im}} - e^{im}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

         \[\leadsto -1 \cdot \left(im \cdot \sin re\right) + \left(\left(\frac{-1}{6} \cdot \left(im \cdot im\right)\right) \cdot \sin re\right) \cdot im \]

      6. associate-*r* N/A

         \[\leadsto -1 \cdot \left(im \cdot \sin re\right) + \left(\left(\left(\frac{-1}{6} \cdot im\right) \cdot im\right) \cdot \sin re\right) \cdot im \]

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-out N/A

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

      10. +-commutative N/A

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

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

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

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

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

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

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

      14. +-commutative N/A

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

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

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

      16. associate-*r* N/A

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

      17. unpow2 N/A

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

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

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

      19. unpow2 N/A

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

      20. *-lowering-*.f64 88.8%

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

   7. Simplified 88.8%

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

   if 3.7000000000000002 < im < 9.59999999999999993e54

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

   5. Simplified 83.3%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 71.6%

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

   if 9.59999999999999993e54 < im < 4.40000000000000015e102

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

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

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

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

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

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

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

      14. *-commutative N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 89.8%

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

   5. Simplified 89.8%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)}, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right), \mathsf{*.f64}\left(\color{blue}{im}, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{-1}{12} \cdot {re}^{2}\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left({re}^{2} \cdot \frac{-1}{12}\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(re \cdot re\right) \cdot \frac{-1}{12}\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \left(re \cdot \frac{-1}{12}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \left(re \cdot \frac{-1}{12}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 88.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \frac{-1}{12}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 88.9%

      \[\leadsto \color{blue}{\left(re \cdot \left(0.5 + re \cdot \left(re \cdot -0.08333333333333333\right)\right)\right)} \cdot \left(im \cdot \left(-2 + im \cdot \left(im \cdot \left(-0.3333333333333333 + \left(im \cdot im\right) \cdot -0.016666666666666666\right)\right)\right)\right) \]

   if 4.40000000000000015e102 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around inf

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

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

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. metadata-eval N/A

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

      13. distribute-neg-frac N/A

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

      14. distribute-lft-out N/A

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

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

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

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

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

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

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

      18. distribute-neg-frac N/A

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

      19. metadata-eval N/A

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

      20. /-lowering-/.f64 N/A

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

      21. unpow2 N/A

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

      22. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in im around inf

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

       1. *-commutative N/A

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

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

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

       3. sin-lowering-sin.f64 100.0%

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

   11. Simplified 100.0%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.7:\\ \;\;\;\;\left(im \cdot \sin re\right) \cdot \left(-1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\\ \mathbf{elif}\;im \leq 9.6 \cdot 10^{+54}:\\ \;\;\;\;\left(1 - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;im \leq 4.4 \cdot 10^{+102}:\\ \;\;\;\;\left(im \cdot \left(-2 + im \cdot \left(im \cdot \left(-0.3333333333333333 + \left(im \cdot im\right) \cdot -0.016666666666666666\right)\right)\right)\right) \cdot \left(re \cdot \left(0.5 + re \cdot \left(re \cdot -0.08333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \left(im \cdot im\right)\right) \cdot \left(\sin re \cdot -0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 94.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 3.75:\\ \;\;\;\;0 - im\_m \cdot \sin re\\ \mathbf{elif}\;im\_m \leq 9.6 \cdot 10^{+54}:\\ \;\;\;\;\left(1 - e^{im\_m}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;im\_m \leq 4.4 \cdot 10^{+102}:\\ \;\;\;\;\left(im\_m \cdot \left(-2 + im\_m \cdot \left(im\_m \cdot \left(-0.3333333333333333 + \left(im\_m \cdot im\_m\right) \cdot -0.016666666666666666\right)\right)\right)\right) \cdot \left(re \cdot \left(0.5 + re \cdot \left(re \cdot -0.08333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im\_m \cdot \left(im\_m \cdot im\_m\right)\right) \cdot \left(\sin re \cdot -0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 3.75)
    (- 0.0 (* im_m (sin re)))
    (if (<= im_m 9.6e+54)
      (* (- 1.0 (exp im_m)) (* 0.5 re))
      (if (<= im_m 4.4e+102)
        (*
         (*
          im_m
          (+
           -2.0
           (*
            im_m
            (*
             im_m
             (+
              -0.3333333333333333
              (* (* im_m im_m) -0.016666666666666666))))))
         (* re (+ 0.5 (* re (* re -0.08333333333333333)))))
        (* (* im_m (* im_m im_m)) (* (sin re) -0.16666666666666666)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 3.75) {
		tmp = 0.0 - (im_m * sin(re));
	} else if (im_m <= 9.6e+54) {
		tmp = (1.0 - exp(im_m)) * (0.5 * re);
	} else if (im_m <= 4.4e+102) {
		tmp = (im_m * (-2.0 + (im_m * (im_m * (-0.3333333333333333 + ((im_m * im_m) * -0.016666666666666666)))))) * (re * (0.5 + (re * (re * -0.08333333333333333))));
	} else {
		tmp = (im_m * (im_m * im_m)) * (sin(re) * -0.16666666666666666);
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 3.75d0) then
        tmp = 0.0d0 - (im_m * sin(re))
    else if (im_m <= 9.6d+54) then
        tmp = (1.0d0 - exp(im_m)) * (0.5d0 * re)
    else if (im_m <= 4.4d+102) then
        tmp = (im_m * ((-2.0d0) + (im_m * (im_m * ((-0.3333333333333333d0) + ((im_m * im_m) * (-0.016666666666666666d0))))))) * (re * (0.5d0 + (re * (re * (-0.08333333333333333d0)))))
    else
        tmp = (im_m * (im_m * im_m)) * (sin(re) * (-0.16666666666666666d0))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 3.75) {
		tmp = 0.0 - (im_m * Math.sin(re));
	} else if (im_m <= 9.6e+54) {
		tmp = (1.0 - Math.exp(im_m)) * (0.5 * re);
	} else if (im_m <= 4.4e+102) {
		tmp = (im_m * (-2.0 + (im_m * (im_m * (-0.3333333333333333 + ((im_m * im_m) * -0.016666666666666666)))))) * (re * (0.5 + (re * (re * -0.08333333333333333))));
	} else {
		tmp = (im_m * (im_m * im_m)) * (Math.sin(re) * -0.16666666666666666);
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 3.75:
		tmp = 0.0 - (im_m * math.sin(re))
	elif im_m <= 9.6e+54:
		tmp = (1.0 - math.exp(im_m)) * (0.5 * re)
	elif im_m <= 4.4e+102:
		tmp = (im_m * (-2.0 + (im_m * (im_m * (-0.3333333333333333 + ((im_m * im_m) * -0.016666666666666666)))))) * (re * (0.5 + (re * (re * -0.08333333333333333))))
	else:
		tmp = (im_m * (im_m * im_m)) * (math.sin(re) * -0.16666666666666666)
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 3.75)
		tmp = Float64(0.0 - Float64(im_m * sin(re)));
	elseif (im_m <= 9.6e+54)
		tmp = Float64(Float64(1.0 - exp(im_m)) * Float64(0.5 * re));
	elseif (im_m <= 4.4e+102)
		tmp = Float64(Float64(im_m * Float64(-2.0 + Float64(im_m * Float64(im_m * Float64(-0.3333333333333333 + Float64(Float64(im_m * im_m) * -0.016666666666666666)))))) * Float64(re * Float64(0.5 + Float64(re * Float64(re * -0.08333333333333333)))));
	else
		tmp = Float64(Float64(im_m * Float64(im_m * im_m)) * Float64(sin(re) * -0.16666666666666666));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 3.75)
		tmp = 0.0 - (im_m * sin(re));
	elseif (im_m <= 9.6e+54)
		tmp = (1.0 - exp(im_m)) * (0.5 * re);
	elseif (im_m <= 4.4e+102)
		tmp = (im_m * (-2.0 + (im_m * (im_m * (-0.3333333333333333 + ((im_m * im_m) * -0.016666666666666666)))))) * (re * (0.5 + (re * (re * -0.08333333333333333))));
	else
		tmp = (im_m * (im_m * im_m)) * (sin(re) * -0.16666666666666666);
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 3.75], N[(0.0 - N[(im$95$m * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 9.6e+54], N[(N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 4.4e+102], N[(N[(im$95$m * N[(-2.0 + N[(im$95$m * N[(im$95$m * N[(-0.3333333333333333 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(re * N[(0.5 + N[(re * N[(re * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im$95$m * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[re], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if im < 3.75

   1. Initial program 54.3%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      6. exp-neg N/A

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

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

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

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

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

      9. exp-lowering-exp.f64 54.3%

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

   3. Simplified 54.3%

      \[\leadsto \color{blue}{0.5 \cdot \left(\sin re \cdot \left(\frac{1}{e^{im}} - e^{im}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

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

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

      5. sin-lowering-sin.f64 71.9%

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

   7. Simplified 71.9%

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

      1. sub0-neg N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

      6. sin-lowering-sin.f64 71.9%

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

   9. Applied egg-rr 71.9%

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

   if 3.75 < im < 9.59999999999999993e54

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

   5. Simplified 83.3%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 71.6%

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

   if 9.59999999999999993e54 < im < 4.40000000000000015e102

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

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

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

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

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

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

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

      14. *-commutative N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 89.8%

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

   5. Simplified 89.8%

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

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)}, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right), \mathsf{*.f64}\left(\color{blue}{im}, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{-1}{12} \cdot {re}^{2}\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left({re}^{2} \cdot \frac{-1}{12}\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(re \cdot re\right) \cdot \frac{-1}{12}\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \left(re \cdot \frac{-1}{12}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \left(re \cdot \frac{-1}{12}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 88.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \frac{-1}{12}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 88.9%

      \[\leadsto \color{blue}{\left(re \cdot \left(0.5 + re \cdot \left(re \cdot -0.08333333333333333\right)\right)\right)} \cdot \left(im \cdot \left(-2 + im \cdot \left(im \cdot \left(-0.3333333333333333 + \left(im \cdot im\right) \cdot -0.016666666666666666\right)\right)\right)\right) \]

   if 4.40000000000000015e102 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around inf

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

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

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. metadata-eval N/A

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

      13. distribute-neg-frac N/A

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

      14. distribute-lft-out N/A

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

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

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

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

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

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

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

      18. distribute-neg-frac N/A

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

      19. metadata-eval N/A

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

      20. /-lowering-/.f64 N/A

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

      21. unpow2 N/A

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

      22. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in im around inf

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

       1. *-commutative N/A

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

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

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

       3. sin-lowering-sin.f64 100.0%

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

   11. Simplified 100.0%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.75:\\ \;\;\;\;0 - im \cdot \sin re\\ \mathbf{elif}\;im \leq 9.6 \cdot 10^{+54}:\\ \;\;\;\;\left(1 - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;im \leq 4.4 \cdot 10^{+102}:\\ \;\;\;\;\left(im \cdot \left(-2 + im \cdot \left(im \cdot \left(-0.3333333333333333 + \left(im \cdot im\right) \cdot -0.016666666666666666\right)\right)\right)\right) \cdot \left(re \cdot \left(0.5 + re \cdot \left(re \cdot -0.08333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \left(im \cdot im\right)\right) \cdot \left(\sin re \cdot -0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 92.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(\left(0.5 \cdot \sin re\right) \cdot \left(im\_m \cdot \left(-2 + \left(im\_m \cdot im\_m\right) \cdot \left(-0.3333333333333333 + -0.0003968253968253968 \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(im\_m \cdot im\_m\right)\right)\right)\right)\right)\right) \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (*
   (* 0.5 (sin re))
   (*
    im_m
    (+
     -2.0
     (*
      (* im_m im_m)
      (+
       -0.3333333333333333
       (* -0.0003968253968253968 (* (* im_m im_m) (* im_m im_m))))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * ((0.5 * sin(re)) * (im_m * (-2.0 + ((im_m * im_m) * (-0.3333333333333333 + (-0.0003968253968253968 * ((im_m * im_m) * (im_m * im_m))))))));
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = im_s * ((0.5d0 * sin(re)) * (im_m * ((-2.0d0) + ((im_m * im_m) * ((-0.3333333333333333d0) + ((-0.0003968253968253968d0) * ((im_m * im_m) * (im_m * im_m))))))))
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	return im_s * ((0.5 * Math.sin(re)) * (im_m * (-2.0 + ((im_m * im_m) * (-0.3333333333333333 + (-0.0003968253968253968 * ((im_m * im_m) * (im_m * im_m))))))));
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * ((0.5 * math.sin(re)) * (im_m * (-2.0 + ((im_m * im_m) * (-0.3333333333333333 + (-0.0003968253968253968 * ((im_m * im_m) * (im_m * im_m))))))))

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * Float64(Float64(0.5 * sin(re)) * Float64(im_m * Float64(-2.0 + Float64(Float64(im_m * im_m) * Float64(-0.3333333333333333 + Float64(-0.0003968253968253968 * Float64(Float64(im_m * im_m) * Float64(im_m * im_m)))))))))
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * ((0.5 * sin(re)) * (im_m * (-2.0 + ((im_m * im_m) * (-0.3333333333333333 + (-0.0003968253968253968 * ((im_m * im_m) * (im_m * im_m))))))));
end

Wolfram

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

Derivation

1. Initial program 64.0%

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

3. Taylor expanded in im around 0

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

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

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

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

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

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

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

   7. unpow2 N/A

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

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

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

   9. sub-neg N/A

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

   10. metadata-eval N/A

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

   11. +-commutative N/A

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

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

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

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

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

   14. unpow2 N/A

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

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

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

   16. sub-neg N/A

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

   17. metadata-eval N/A

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

   18. +-commutative N/A

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

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

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

   20. *-commutative N/A

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

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

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

   22. unpow2 N/A

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

   23. *-lowering-*.f64 96.7%

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

5. Simplified 96.7%

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

6. Taylor expanded in im around inf

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

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

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

   2. metadata-eval N/A

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

   3. pow-sqr N/A

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

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

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

   5. unpow2 N/A

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

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

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

   7. unpow2 N/A

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

   8. *-lowering-*.f64 96.7%

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

8. Simplified 96.7%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left(-2 + \left(im \cdot im\right) \cdot \left(-0.3333333333333333 + \color{blue}{-0.0003968253968253968 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)}\right)\right)\right) \]
9. Add Preprocessing

Alternative 12: 93.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 3.8:\\ \;\;\;\;0 - im\_m \cdot \sin re\\ \mathbf{elif}\;im\_m \leq 3.8 \cdot 10^{+147}:\\ \;\;\;\;\left(1 - e^{im\_m}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im\_m \cdot im\_m\right) \cdot \left(im\_m \cdot \left(\sin re \cdot -0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 3.8)
    (- 0.0 (* im_m (sin re)))
    (if (<= im_m 3.8e+147)
      (* (- 1.0 (exp im_m)) (* 0.5 re))
      (* (* im_m im_m) (* im_m (* (sin re) -0.16666666666666666)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 3.8) {
		tmp = 0.0 - (im_m * sin(re));
	} else if (im_m <= 3.8e+147) {
		tmp = (1.0 - exp(im_m)) * (0.5 * re);
	} else {
		tmp = (im_m * im_m) * (im_m * (sin(re) * -0.16666666666666666));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 3.8d0) then
        tmp = 0.0d0 - (im_m * sin(re))
    else if (im_m <= 3.8d+147) then
        tmp = (1.0d0 - exp(im_m)) * (0.5d0 * re)
    else
        tmp = (im_m * im_m) * (im_m * (sin(re) * (-0.16666666666666666d0)))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 3.8) {
		tmp = 0.0 - (im_m * Math.sin(re));
	} else if (im_m <= 3.8e+147) {
		tmp = (1.0 - Math.exp(im_m)) * (0.5 * re);
	} else {
		tmp = (im_m * im_m) * (im_m * (Math.sin(re) * -0.16666666666666666));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 3.8:
		tmp = 0.0 - (im_m * math.sin(re))
	elif im_m <= 3.8e+147:
		tmp = (1.0 - math.exp(im_m)) * (0.5 * re)
	else:
		tmp = (im_m * im_m) * (im_m * (math.sin(re) * -0.16666666666666666))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 3.8)
		tmp = Float64(0.0 - Float64(im_m * sin(re)));
	elseif (im_m <= 3.8e+147)
		tmp = Float64(Float64(1.0 - exp(im_m)) * Float64(0.5 * re));
	else
		tmp = Float64(Float64(im_m * im_m) * Float64(im_m * Float64(sin(re) * -0.16666666666666666)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 3.8)
		tmp = 0.0 - (im_m * sin(re));
	elseif (im_m <= 3.8e+147)
		tmp = (1.0 - exp(im_m)) * (0.5 * re);
	else
		tmp = (im_m * im_m) * (im_m * (sin(re) * -0.16666666666666666));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 3.8], N[(0.0 - N[(im$95$m * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 3.8e+147], N[(N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * N[(N[Sin[re], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 3.7999999999999998

   1. Initial program 54.3%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      6. exp-neg N/A

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

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

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

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

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

      9. exp-lowering-exp.f64 54.3%

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

   3. Simplified 54.3%

      \[\leadsto \color{blue}{0.5 \cdot \left(\sin re \cdot \left(\frac{1}{e^{im}} - e^{im}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

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

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

      5. sin-lowering-sin.f64 71.9%

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

   7. Simplified 71.9%

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

      1. sub0-neg N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

      6. sin-lowering-sin.f64 71.9%

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

   9. Applied egg-rr 71.9%

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

   if 3.7999999999999998 < im < 3.7999999999999997e147

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

   5. Simplified 81.8%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 78.6%

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

   if 3.7999999999999997e147 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. unpow3 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

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

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

      8. unpow2 N/A

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

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

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

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

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

      11. *-commutative N/A

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

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

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

      13. sin-lowering-sin.f64 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.8:\\ \;\;\;\;0 - im \cdot \sin re\\ \mathbf{elif}\;im \leq 3.8 \cdot 10^{+147}:\\ \;\;\;\;\left(1 - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(im \cdot \left(\sin re \cdot -0.16666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 85.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 3.9:\\ \;\;\;\;0 - im\_m \cdot \sin re\\ \mathbf{elif}\;im\_m \leq 9.6 \cdot 10^{+54}:\\ \;\;\;\;\left(1 - e^{im\_m}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im\_m \cdot \left(-2 + im\_m \cdot \left(im\_m \cdot \left(-0.3333333333333333 + \left(im\_m \cdot im\_m\right) \cdot -0.016666666666666666\right)\right)\right)\right) \cdot \left(re \cdot \left(0.5 + re \cdot \left(re \cdot -0.08333333333333333\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 3.9)
    (- 0.0 (* im_m (sin re)))
    (if (<= im_m 9.6e+54)
      (* (- 1.0 (exp im_m)) (* 0.5 re))
      (*
       (*
        im_m
        (+
         -2.0
         (*
          im_m
          (*
           im_m
           (+ -0.3333333333333333 (* (* im_m im_m) -0.016666666666666666))))))
       (* re (+ 0.5 (* 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 <= 3.9) {
		tmp = 0.0 - (im_m * sin(re));
	} else if (im_m <= 9.6e+54) {
		tmp = (1.0 - exp(im_m)) * (0.5 * re);
	} else {
		tmp = (im_m * (-2.0 + (im_m * (im_m * (-0.3333333333333333 + ((im_m * im_m) * -0.016666666666666666)))))) * (re * (0.5 + (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 <= 3.9d0) then
        tmp = 0.0d0 - (im_m * sin(re))
    else if (im_m <= 9.6d+54) then
        tmp = (1.0d0 - exp(im_m)) * (0.5d0 * re)
    else
        tmp = (im_m * ((-2.0d0) + (im_m * (im_m * ((-0.3333333333333333d0) + ((im_m * im_m) * (-0.016666666666666666d0))))))) * (re * (0.5d0 + (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 <= 3.9) {
		tmp = 0.0 - (im_m * Math.sin(re));
	} else if (im_m <= 9.6e+54) {
		tmp = (1.0 - Math.exp(im_m)) * (0.5 * re);
	} else {
		tmp = (im_m * (-2.0 + (im_m * (im_m * (-0.3333333333333333 + ((im_m * im_m) * -0.016666666666666666)))))) * (re * (0.5 + (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 <= 3.9:
		tmp = 0.0 - (im_m * math.sin(re))
	elif im_m <= 9.6e+54:
		tmp = (1.0 - math.exp(im_m)) * (0.5 * re)
	else:
		tmp = (im_m * (-2.0 + (im_m * (im_m * (-0.3333333333333333 + ((im_m * im_m) * -0.016666666666666666)))))) * (re * (0.5 + (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 <= 3.9)
		tmp = Float64(0.0 - Float64(im_m * sin(re)));
	elseif (im_m <= 9.6e+54)
		tmp = Float64(Float64(1.0 - exp(im_m)) * Float64(0.5 * re));
	else
		tmp = Float64(Float64(im_m * Float64(-2.0 + Float64(im_m * Float64(im_m * Float64(-0.3333333333333333 + Float64(Float64(im_m * im_m) * -0.016666666666666666)))))) * Float64(re * Float64(0.5 + Float64(re * Float64(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 <= 3.9)
		tmp = 0.0 - (im_m * sin(re));
	elseif (im_m <= 9.6e+54)
		tmp = (1.0 - exp(im_m)) * (0.5 * re);
	else
		tmp = (im_m * (-2.0 + (im_m * (im_m * (-0.3333333333333333 + ((im_m * im_m) * -0.016666666666666666)))))) * (re * (0.5 + (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, 3.9], N[(0.0 - N[(im$95$m * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 9.6e+54], N[(N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], N[(N[(im$95$m * N[(-2.0 + N[(im$95$m * N[(im$95$m * N[(-0.3333333333333333 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(re * N[(0.5 + N[(re * N[(re * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 3.89999999999999991

   1. Initial program 54.3%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      6. exp-neg N/A

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

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

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

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

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

      9. exp-lowering-exp.f64 54.3%

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

   3. Simplified 54.3%

      \[\leadsto \color{blue}{0.5 \cdot \left(\sin re \cdot \left(\frac{1}{e^{im}} - e^{im}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

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

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

      5. sin-lowering-sin.f64 71.9%

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

   7. Simplified 71.9%

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

      1. sub0-neg N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

      6. sin-lowering-sin.f64 71.9%

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

   9. Applied egg-rr 71.9%

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

   if 3.89999999999999991 < im < 9.59999999999999993e54

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

   5. Simplified 83.3%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 71.6%

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

   if 9.59999999999999993e54 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right)\right)}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{-1}{60} \cdot {im}^{2}} - \frac{1}{3}\right)\right)\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right)\right)}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right)\right)}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right)}\right)\right)\right)\right)\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{60} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{60} \cdot {im}^{2} + \frac{-1}{3}\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{3} + \color{blue}{\frac{-1}{60} \cdot {im}^{2}}\right)\right)\right)\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{\left(\frac{-1}{60} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \left({im}^{2} \cdot \color{blue}{\frac{-1}{60}}\right)\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{-1}{60}}\right)\right)\right)\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 98.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 98.1%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left(-2 + im \cdot \left(im \cdot \left(-0.3333333333333333 + \left(im \cdot im\right) \cdot -0.016666666666666666\right)\right)\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)}, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right), \mathsf{*.f64}\left(\color{blue}{im}, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{-1}{12} \cdot {re}^{2}\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left({re}^{2} \cdot \frac{-1}{12}\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(re \cdot re\right) \cdot \frac{-1}{12}\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \left(re \cdot \frac{-1}{12}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \left(re \cdot \frac{-1}{12}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 77.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \frac{-1}{12}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 77.1%

      \[\leadsto \color{blue}{\left(re \cdot \left(0.5 + re \cdot \left(re \cdot -0.08333333333333333\right)\right)\right)} \cdot \left(im \cdot \left(-2 + im \cdot \left(im \cdot \left(-0.3333333333333333 + \left(im \cdot im\right) \cdot -0.016666666666666666\right)\right)\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.9:\\ \;\;\;\;0 - im \cdot \sin re\\ \mathbf{elif}\;im \leq 9.6 \cdot 10^{+54}:\\ \;\;\;\;\left(1 - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \left(-2 + im \cdot \left(im \cdot \left(-0.3333333333333333 + \left(im \cdot im\right) \cdot -0.016666666666666666\right)\right)\right)\right) \cdot \left(re \cdot \left(0.5 + re \cdot \left(re \cdot -0.08333333333333333\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 82.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := im\_m \cdot \left(-0.3333333333333333 + \left(im\_m \cdot im\_m\right) \cdot -0.016666666666666666\right)\\ t_1 := \left(im\_m \cdot im\_m\right) \cdot t\_0\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 4.7 \cdot 10^{-5}:\\ \;\;\;\;0 - im\_m \cdot \sin re\\ \mathbf{elif}\;im\_m \leq 9.6 \cdot 10^{+54}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \frac{\left(im\_m \cdot -2\right) \cdot \left(im\_m \cdot -2\right) - t\_1 \cdot t\_1}{im\_m \cdot -2 - t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(im\_m \cdot \left(-2 + im\_m \cdot t\_0\right)\right) \cdot \left(re \cdot \left(0.5 + re \cdot \left(re \cdot -0.08333333333333333\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0
         (*
          im_m
          (+ -0.3333333333333333 (* (* im_m im_m) -0.016666666666666666))))
        (t_1 (* (* im_m im_m) t_0)))
   (*
    im_s
    (if (<= im_m 4.7e-5)
      (- 0.0 (* im_m (sin re)))
      (if (<= im_m 9.6e+54)
        (*
         (* 0.5 re)
         (/
          (- (* (* im_m -2.0) (* im_m -2.0)) (* t_1 t_1))
          (- (* im_m -2.0) t_1)))
        (*
         (* im_m (+ -2.0 (* im_m t_0)))
         (* re (+ 0.5 (* 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 t_0 = im_m * (-0.3333333333333333 + ((im_m * im_m) * -0.016666666666666666));
	double t_1 = (im_m * im_m) * t_0;
	double tmp;
	if (im_m <= 4.7e-5) {
		tmp = 0.0 - (im_m * sin(re));
	} else if (im_m <= 9.6e+54) {
		tmp = (0.5 * re) * ((((im_m * -2.0) * (im_m * -2.0)) - (t_1 * t_1)) / ((im_m * -2.0) - t_1));
	} else {
		tmp = (im_m * (-2.0 + (im_m * t_0))) * (re * (0.5 + (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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = im_m * ((-0.3333333333333333d0) + ((im_m * im_m) * (-0.016666666666666666d0)))
    t_1 = (im_m * im_m) * t_0
    if (im_m <= 4.7d-5) then
        tmp = 0.0d0 - (im_m * sin(re))
    else if (im_m <= 9.6d+54) then
        tmp = (0.5d0 * re) * ((((im_m * (-2.0d0)) * (im_m * (-2.0d0))) - (t_1 * t_1)) / ((im_m * (-2.0d0)) - t_1))
    else
        tmp = (im_m * ((-2.0d0) + (im_m * t_0))) * (re * (0.5d0 + (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 t_0 = im_m * (-0.3333333333333333 + ((im_m * im_m) * -0.016666666666666666));
	double t_1 = (im_m * im_m) * t_0;
	double tmp;
	if (im_m <= 4.7e-5) {
		tmp = 0.0 - (im_m * Math.sin(re));
	} else if (im_m <= 9.6e+54) {
		tmp = (0.5 * re) * ((((im_m * -2.0) * (im_m * -2.0)) - (t_1 * t_1)) / ((im_m * -2.0) - t_1));
	} else {
		tmp = (im_m * (-2.0 + (im_m * t_0))) * (re * (0.5 + (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):
	t_0 = im_m * (-0.3333333333333333 + ((im_m * im_m) * -0.016666666666666666))
	t_1 = (im_m * im_m) * t_0
	tmp = 0
	if im_m <= 4.7e-5:
		tmp = 0.0 - (im_m * math.sin(re))
	elif im_m <= 9.6e+54:
		tmp = (0.5 * re) * ((((im_m * -2.0) * (im_m * -2.0)) - (t_1 * t_1)) / ((im_m * -2.0) - t_1))
	else:
		tmp = (im_m * (-2.0 + (im_m * t_0))) * (re * (0.5 + (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)
	t_0 = Float64(im_m * Float64(-0.3333333333333333 + Float64(Float64(im_m * im_m) * -0.016666666666666666)))
	t_1 = Float64(Float64(im_m * im_m) * t_0)
	tmp = 0.0
	if (im_m <= 4.7e-5)
		tmp = Float64(0.0 - Float64(im_m * sin(re)));
	elseif (im_m <= 9.6e+54)
		tmp = Float64(Float64(0.5 * re) * Float64(Float64(Float64(Float64(im_m * -2.0) * Float64(im_m * -2.0)) - Float64(t_1 * t_1)) / Float64(Float64(im_m * -2.0) - t_1)));
	else
		tmp = Float64(Float64(im_m * Float64(-2.0 + Float64(im_m * t_0))) * Float64(re * Float64(0.5 + Float64(re * Float64(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)
	t_0 = im_m * (-0.3333333333333333 + ((im_m * im_m) * -0.016666666666666666));
	t_1 = (im_m * im_m) * t_0;
	tmp = 0.0;
	if (im_m <= 4.7e-5)
		tmp = 0.0 - (im_m * sin(re));
	elseif (im_m <= 9.6e+54)
		tmp = (0.5 * re) * ((((im_m * -2.0) * (im_m * -2.0)) - (t_1 * t_1)) / ((im_m * -2.0) - t_1));
	else
		tmp = (im_m * (-2.0 + (im_m * t_0))) * (re * (0.5 + (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_] := Block[{t$95$0 = N[(im$95$m * N[(-0.3333333333333333 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(im$95$m * im$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]}, N[(im$95$s * If[LessEqual[im$95$m, 4.7e-5], N[(0.0 - N[(im$95$m * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 9.6e+54], N[(N[(0.5 * re), $MachinePrecision] * N[(N[(N[(N[(im$95$m * -2.0), $MachinePrecision] * N[(im$95$m * -2.0), $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(N[(im$95$m * -2.0), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im$95$m * N[(-2.0 + N[(im$95$m * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(re * N[(0.5 + N[(re * N[(re * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 4.69999999999999972e-5

   1. Initial program 54.3%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sin re \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\sin re \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\sin re, \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)}\right)\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} - e^{im}\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\left(e^{\mathsf{neg}\left(im\right)}\right), \color{blue}{\left(e^{im}\right)}\right)\right)\right) \]

      6. exp-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\left(\frac{1}{e^{im}}\right), \left(e^{\color{blue}{im}}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(e^{im}\right)\right), \left(e^{\color{blue}{im}}\right)\right)\right)\right) \]

      8. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(im\right)\right), \left(e^{im}\right)\right)\right)\right) \]

      9. exp-lowering-exp.f64 54.3%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(im\right)\right), \mathsf{exp.f64}\left(im\right)\right)\right)\right) \]

   3. Simplified 54.3%

      \[\leadsto \color{blue}{0.5 \cdot \left(\sin re \cdot \left(\frac{1}{e^{im}} - e^{im}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(im \cdot \sin re\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{im \cdot \sin re} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(im \cdot \sin re\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \color{blue}{\sin re}\right)\right) \]

      5. sin-lowering-sin.f64 71.9%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \mathsf{sin.f64}\left(re\right)\right)\right) \]

   7. Simplified 71.9%

      \[\leadsto \color{blue}{0 - im \cdot \sin re} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(im \cdot \sin re\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\sin re \cdot im\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \left(\mathsf{neg}\left(\sin re\right)\right) \cdot \color{blue}{im} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{neg}\left(\sin re\right)\right), \color{blue}{im}\right) \]

      5. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{neg.f64}\left(\sin re\right), im\right) \]

      6. sin-lowering-sin.f64 71.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{sin.f64}\left(re\right)\right), im\right) \]

   9. Applied egg-rr 71.9%

      \[\leadsto \color{blue}{\left(-\sin re\right) \cdot im} \]

   if 4.69999999999999972e-5 < im < 9.59999999999999993e54

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)}\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) + -2\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left(-2 + \color{blue}{{im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right)}\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right)\right)}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{-1}{60} \cdot {im}^{2}} - \frac{1}{3}\right)\right)\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right)\right)}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right)\right)}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right)}\right)\right)\right)\right)\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{60} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{60} \cdot {im}^{2} + \frac{-1}{3}\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{3} + \color{blue}{\frac{-1}{60} \cdot {im}^{2}}\right)\right)\right)\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{\left(\frac{-1}{60} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \left({im}^{2} \cdot \color{blue}{\frac{-1}{60}}\right)\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{-1}{60}}\right)\right)\right)\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 6.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 6.8%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left(-2 + im \cdot \left(im \cdot \left(-0.3333333333333333 + \left(im \cdot im\right) \cdot -0.016666666666666666\right)\right)\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{1}{2} \cdot re\right)}, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 6.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, re\right), \mathsf{*.f64}\left(\color{blue}{im}, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 6.2%

      \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot \left(im \cdot \left(-2 + im \cdot \left(im \cdot \left(-0.3333333333333333 + \left(im \cdot im\right) \cdot -0.016666666666666666\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, re\right), \left(-2 \cdot im + \color{blue}{\left(im \cdot \left(im \cdot \left(\frac{-1}{3} + \left(im \cdot im\right) \cdot \frac{-1}{60}\right)\right)\right) \cdot im}\right)\right) \]

      2. flip-+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, re\right), \left(\frac{\left(-2 \cdot im\right) \cdot \left(-2 \cdot im\right) - \left(\left(im \cdot \left(im \cdot \left(\frac{-1}{3} + \left(im \cdot im\right) \cdot \frac{-1}{60}\right)\right)\right) \cdot im\right) \cdot \left(\left(im \cdot \left(im \cdot \left(\frac{-1}{3} + \left(im \cdot im\right) \cdot \frac{-1}{60}\right)\right)\right) \cdot im\right)}{\color{blue}{-2 \cdot im - \left(im \cdot \left(im \cdot \left(\frac{-1}{3} + \left(im \cdot im\right) \cdot \frac{-1}{60}\right)\right)\right) \cdot im}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, re\right), \mathsf{/.f64}\left(\left(\left(-2 \cdot im\right) \cdot \left(-2 \cdot im\right) - \left(\left(im \cdot \left(im \cdot \left(\frac{-1}{3} + \left(im \cdot im\right) \cdot \frac{-1}{60}\right)\right)\right) \cdot im\right) \cdot \left(\left(im \cdot \left(im \cdot \left(\frac{-1}{3} + \left(im \cdot im\right) \cdot \frac{-1}{60}\right)\right)\right) \cdot im\right)\right), \color{blue}{\left(-2 \cdot im - \left(im \cdot \left(im \cdot \left(\frac{-1}{3} + \left(im \cdot im\right) \cdot \frac{-1}{60}\right)\right)\right) \cdot im\right)}\right)\right) \]

   10. Applied egg-rr 53.9%

       \[\leadsto \left(0.5 \cdot re\right) \cdot \color{blue}{\frac{\left(im \cdot -2\right) \cdot \left(im \cdot -2\right) - \left(\left(im \cdot im\right) \cdot \left(im \cdot \left(-0.3333333333333333 + \left(im \cdot im\right) \cdot -0.016666666666666666\right)\right)\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot \left(-0.3333333333333333 + \left(im \cdot im\right) \cdot -0.016666666666666666\right)\right)\right)}{im \cdot -2 - \left(im \cdot im\right) \cdot \left(im \cdot \left(-0.3333333333333333 + \left(im \cdot im\right) \cdot -0.016666666666666666\right)\right)}} \]

   if 9.59999999999999993e54 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)}\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) + -2\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left(-2 + \color{blue}{{im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right)}\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right)\right)}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{-1}{60} \cdot {im}^{2}} - \frac{1}{3}\right)\right)\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right)\right)}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right)\right)}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right)}\right)\right)\right)\right)\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{60} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{60} \cdot {im}^{2} + \frac{-1}{3}\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{3} + \color{blue}{\frac{-1}{60} \cdot {im}^{2}}\right)\right)\right)\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{\left(\frac{-1}{60} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \left({im}^{2} \cdot \color{blue}{\frac{-1}{60}}\right)\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{-1}{60}}\right)\right)\right)\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 98.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 98.1%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left(-2 + im \cdot \left(im \cdot \left(-0.3333333333333333 + \left(im \cdot im\right) \cdot -0.016666666666666666\right)\right)\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)}, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right), \mathsf{*.f64}\left(\color{blue}{im}, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{-1}{12} \cdot {re}^{2}\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left({re}^{2} \cdot \frac{-1}{12}\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(re \cdot re\right) \cdot \frac{-1}{12}\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \left(re \cdot \left(re \cdot \frac{-1}{12}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \left(re \cdot \frac{-1}{12}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 77.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(re, \frac{-1}{12}\right)\right)\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 77.1%

      \[\leadsto \color{blue}{\left(re \cdot \left(0.5 + re \cdot \left(re \cdot -0.08333333333333333\right)\right)\right)} \cdot \left(im \cdot \left(-2 + im \cdot \left(im \cdot \left(-0.3333333333333333 + \left(im \cdot im\right) \cdot -0.016666666666666666\right)\right)\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.7 \cdot 10^{-5}:\\ \;\;\;\;0 - im \cdot \sin re\\ \mathbf{elif}\;im \leq 9.6 \cdot 10^{+54}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \frac{\left(im \cdot -2\right) \cdot \left(im \cdot -2\right) - \left(\left(im \cdot im\right) \cdot \left(im \cdot \left(-0.3333333333333333 + \left(im \cdot im\right) \cdot -0.016666666666666666\right)\right)\right) \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot \left(-0.3333333333333333 + \left(im \cdot im\right) \cdot -0.016666666666666666\right)\right)\right)}{im \cdot -2 - \left(im \cdot im\right) \cdot \left(im \cdot \left(-0.3333333333333333 + \left(im \cdot im\right) \cdot -0.016666666666666666\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot \left(-2 + im \cdot \left(im \cdot \left(-0.3333333333333333 + \left(im \cdot im\right) \cdot -0.016666666666666666\right)\right)\right)\right) \cdot \left(re \cdot \left(0.5 + re \cdot \left(re \cdot -0.08333333333333333\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 58.0% accurate, 10.3× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;re \leq 1.6 \cdot 10^{+251}:\\ \;\;\;\;\left(im\_m \cdot \left(-2 + \left(im\_m \cdot im\_m\right) \cdot \left(-0.3333333333333333 + \left(im\_m \cdot im\_m\right) \cdot \left(-0.016666666666666666 + \left(im\_m \cdot im\_m\right) \cdot -0.0003968253968253968\right)\right)\right)\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im\_m \cdot \left(-1 + 0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= re 1.6e+251)
    (*
     (*
      im_m
      (+
       -2.0
       (*
        (* im_m im_m)
        (+
         -0.3333333333333333
         (*
          (* im_m im_m)
          (+
           -0.016666666666666666
           (* (* im_m im_m) -0.0003968253968253968)))))))
     (* 0.5 re))
    (* re (* im_m (+ -1.0 (* 0.16666666666666666 (* re re))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (re <= 1.6e+251) {
		tmp = (im_m * (-2.0 + ((im_m * im_m) * (-0.3333333333333333 + ((im_m * im_m) * (-0.016666666666666666 + ((im_m * im_m) * -0.0003968253968253968))))))) * (0.5 * re);
	} else {
		tmp = re * (im_m * (-1.0 + (0.16666666666666666 * (re * re))));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= 1.6d+251) then
        tmp = (im_m * ((-2.0d0) + ((im_m * im_m) * ((-0.3333333333333333d0) + ((im_m * im_m) * ((-0.016666666666666666d0) + ((im_m * im_m) * (-0.0003968253968253968d0)))))))) * (0.5d0 * re)
    else
        tmp = re * (im_m * ((-1.0d0) + (0.16666666666666666d0 * (re * re))))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (re <= 1.6e+251) {
		tmp = (im_m * (-2.0 + ((im_m * im_m) * (-0.3333333333333333 + ((im_m * im_m) * (-0.016666666666666666 + ((im_m * im_m) * -0.0003968253968253968))))))) * (0.5 * re);
	} else {
		tmp = re * (im_m * (-1.0 + (0.16666666666666666 * (re * re))));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if re <= 1.6e+251:
		tmp = (im_m * (-2.0 + ((im_m * im_m) * (-0.3333333333333333 + ((im_m * im_m) * (-0.016666666666666666 + ((im_m * im_m) * -0.0003968253968253968))))))) * (0.5 * re)
	else:
		tmp = re * (im_m * (-1.0 + (0.16666666666666666 * (re * re))))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (re <= 1.6e+251)
		tmp = Float64(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))))))) * Float64(0.5 * re));
	else
		tmp = Float64(re * Float64(im_m * Float64(-1.0 + Float64(0.16666666666666666 * Float64(re * re)))));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (re <= 1.6e+251)
		tmp = (im_m * (-2.0 + ((im_m * im_m) * (-0.3333333333333333 + ((im_m * im_m) * (-0.016666666666666666 + ((im_m * im_m) * -0.0003968253968253968))))))) * (0.5 * re);
	else
		tmp = re * (im_m * (-1.0 + (0.16666666666666666 * (re * re))));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[re, 1.6e+251], N[(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] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], N[(re * N[(im$95$m * N[(-1.0 + N[(0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if re < 1.5999999999999999e251

   1. Initial program 64.5%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)}\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) + -2\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left(-2 + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right)}\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right)\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right)}\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{{im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right)} - \frac{1}{3}\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{{im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right)} - \frac{1}{3}\right)\right)\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) + \frac{-1}{3}\right)\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{3} + \color{blue}{{im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right)}\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right)\right)}\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right)}\right)\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{-1}{2520} \cdot {im}^{2}} - \frac{1}{60}\right)\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{-1}{2520} \cdot {im}^{2}} - \frac{1}{60}\right)\right)\right)\right)\right)\right)\right) \]

      16. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{2520} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{60}\right)\right)}\right)\right)\right)\right)\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{2520} \cdot {im}^{2} + \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]

      18. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{60} + \color{blue}{\frac{-1}{2520} \cdot {im}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

      19. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{60}, \color{blue}{\left(\frac{-1}{2520} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{60}, \left({im}^{2} \cdot \color{blue}{\frac{-1}{2520}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      21. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{60}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{-1}{2520}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      22. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{60}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{-1}{2520}\right)\right)\right)\right)\right)\right)\right)\right) \]

      23. *-lowering-*.f64 96.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{60}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{2520}\right)\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 96.6%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left(-2 + \left(im \cdot im\right) \cdot \left(-0.3333333333333333 + \left(im \cdot im\right) \cdot \left(-0.016666666666666666 + \left(im \cdot im\right) \cdot -0.0003968253968253968\right)\right)\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{1}{2} \cdot re\right)}, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{60}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{2520}\right)\right)\right)\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 65.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, re\right), \mathsf{*.f64}\left(\color{blue}{im}, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{60}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{2520}\right)\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 65.1%

      \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot \left(im \cdot \left(-2 + \left(im \cdot im\right) \cdot \left(-0.3333333333333333 + \left(im \cdot im\right) \cdot \left(-0.016666666666666666 + \left(im \cdot im\right) \cdot -0.0003968253968253968\right)\right)\right)\right) \]

   if 1.5999999999999999e251 < re

   1. Initial program 48.3%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sin re \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\sin re \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\sin re, \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)}\right)\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} - e^{im}\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\left(e^{\mathsf{neg}\left(im\right)}\right), \color{blue}{\left(e^{im}\right)}\right)\right)\right) \]

      6. exp-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\left(\frac{1}{e^{im}}\right), \left(e^{\color{blue}{im}}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(e^{im}\right)\right), \left(e^{\color{blue}{im}}\right)\right)\right)\right) \]

      8. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(im\right)\right), \left(e^{im}\right)\right)\right)\right) \]

      9. exp-lowering-exp.f64 48.3%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(im\right)\right), \mathsf{exp.f64}\left(im\right)\right)\right)\right) \]

   3. Simplified 48.3%

      \[\leadsto \color{blue}{0.5 \cdot \left(\sin re \cdot \left(\frac{1}{e^{im}} - e^{im}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(im \cdot \sin re\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{im \cdot \sin re} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(im \cdot \sin re\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \color{blue}{\sin re}\right)\right) \]

      5. sin-lowering-sin.f64 58.3%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \mathsf{sin.f64}\left(re\right)\right)\right) \]

   7. Simplified 58.3%

      \[\leadsto \color{blue}{0 - im \cdot \sin re} \]

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(\frac{1}{6} \cdot \left(im \cdot {re}^{2}\right) - im\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{1}{6} \cdot \left(im \cdot {re}^{2}\right) - im\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(re, \left(\frac{1}{6} \cdot \left(im \cdot {re}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(im\right)\right)}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(re, \left(\left(\mathsf{neg}\left(im\right)\right) + \color{blue}{\frac{1}{6} \cdot \left(im \cdot {re}^{2}\right)}\right)\right) \]

      4. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \left(-1 \cdot im + \color{blue}{\frac{1}{6}} \cdot \left(im \cdot {re}^{2}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(re, \left(im \cdot -1 + \color{blue}{\frac{1}{6}} \cdot \left(im \cdot {re}^{2}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(re, \left(im \cdot -1 + \left(im \cdot {re}^{2}\right) \cdot \color{blue}{\frac{1}{6}}\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(re, \left(im \cdot -1 + im \cdot \color{blue}{\left({re}^{2} \cdot \frac{1}{6}\right)}\right)\right) \]

      8. distribute-lft-out N/A

         \[\leadsto \mathsf{*.f64}\left(re, \left(im \cdot \color{blue}{\left(-1 + {re}^{2} \cdot \frac{1}{6}\right)}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \color{blue}{\left(-1 + {re}^{2} \cdot \frac{1}{6}\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \color{blue}{\left({re}^{2} \cdot \frac{1}{6}\right)}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{1}{6}}\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{1}{6}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 45.3%

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{1}{6}\right)\right)\right)\right) \]

   10. Simplified 45.3%

       \[\leadsto \color{blue}{re \cdot \left(im \cdot \left(-1 + \left(re \cdot re\right) \cdot 0.16666666666666666\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.6 \cdot 10^{+251}:\\ \;\;\;\;\left(im \cdot \left(-2 + \left(im \cdot im\right) \cdot \left(-0.3333333333333333 + \left(im \cdot im\right) \cdot \left(-0.016666666666666666 + \left(im \cdot im\right) \cdot -0.0003968253968253968\right)\right)\right)\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot \left(-1 + 0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 57.9% accurate, 11.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;re \leq 1.6 \cdot 10^{+251}:\\ \;\;\;\;\left(im\_m \cdot \left(-2 + \left(im\_m \cdot im\_m\right) \cdot \left(-0.3333333333333333 + -0.0003968253968253968 \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(im\_m \cdot im\_m\right)\right)\right)\right)\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im\_m \cdot \left(-1 + 0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= re 1.6e+251)
    (*
     (*
      im_m
      (+
       -2.0
       (*
        (* im_m im_m)
        (+
         -0.3333333333333333
         (* -0.0003968253968253968 (* (* im_m im_m) (* im_m im_m)))))))
     (* 0.5 re))
    (* re (* im_m (+ -1.0 (* 0.16666666666666666 (* re re))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (re <= 1.6e+251) {
		tmp = (im_m * (-2.0 + ((im_m * im_m) * (-0.3333333333333333 + (-0.0003968253968253968 * ((im_m * im_m) * (im_m * im_m))))))) * (0.5 * re);
	} else {
		tmp = re * (im_m * (-1.0 + (0.16666666666666666 * (re * re))));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= 1.6d+251) then
        tmp = (im_m * ((-2.0d0) + ((im_m * im_m) * ((-0.3333333333333333d0) + ((-0.0003968253968253968d0) * ((im_m * im_m) * (im_m * im_m))))))) * (0.5d0 * re)
    else
        tmp = re * (im_m * ((-1.0d0) + (0.16666666666666666d0 * (re * re))))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (re <= 1.6e+251) {
		tmp = (im_m * (-2.0 + ((im_m * im_m) * (-0.3333333333333333 + (-0.0003968253968253968 * ((im_m * im_m) * (im_m * im_m))))))) * (0.5 * re);
	} else {
		tmp = re * (im_m * (-1.0 + (0.16666666666666666 * (re * re))));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if re <= 1.6e+251:
		tmp = (im_m * (-2.0 + ((im_m * im_m) * (-0.3333333333333333 + (-0.0003968253968253968 * ((im_m * im_m) * (im_m * im_m))))))) * (0.5 * re)
	else:
		tmp = re * (im_m * (-1.0 + (0.16666666666666666 * (re * re))))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (re <= 1.6e+251)
		tmp = Float64(Float64(im_m * Float64(-2.0 + Float64(Float64(im_m * im_m) * Float64(-0.3333333333333333 + Float64(-0.0003968253968253968 * Float64(Float64(im_m * im_m) * Float64(im_m * im_m))))))) * Float64(0.5 * re));
	else
		tmp = Float64(re * Float64(im_m * Float64(-1.0 + Float64(0.16666666666666666 * Float64(re * re)))));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (re <= 1.6e+251)
		tmp = (im_m * (-2.0 + ((im_m * im_m) * (-0.3333333333333333 + (-0.0003968253968253968 * ((im_m * im_m) * (im_m * im_m))))))) * (0.5 * re);
	else
		tmp = re * (im_m * (-1.0 + (0.16666666666666666 * (re * re))));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[re, 1.6e+251], N[(N[(im$95$m * N[(-2.0 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(-0.3333333333333333 + N[(-0.0003968253968253968 * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], N[(re * N[(im$95$m * N[(-1.0 + N[(0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if re < 1.5999999999999999e251

   1. Initial program 64.5%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)}\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) + -2\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left(-2 + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right)}\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right)\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right)}\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{{im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right)} - \frac{1}{3}\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{{im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right)} - \frac{1}{3}\right)\right)\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) + \frac{-1}{3}\right)\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{3} + \color{blue}{{im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right)}\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right)\right)}\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right)}\right)\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{-1}{2520} \cdot {im}^{2}} - \frac{1}{60}\right)\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{-1}{2520} \cdot {im}^{2}} - \frac{1}{60}\right)\right)\right)\right)\right)\right)\right) \]

      16. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{2520} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{60}\right)\right)}\right)\right)\right)\right)\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{2520} \cdot {im}^{2} + \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]

      18. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{60} + \color{blue}{\frac{-1}{2520} \cdot {im}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

      19. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{60}, \color{blue}{\left(\frac{-1}{2520} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{60}, \left({im}^{2} \cdot \color{blue}{\frac{-1}{2520}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      21. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{60}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{-1}{2520}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      22. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{60}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{-1}{2520}\right)\right)\right)\right)\right)\right)\right)\right) \]

      23. *-lowering-*.f64 96.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{60}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{2520}\right)\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 96.6%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left(-2 + \left(im \cdot im\right) \cdot \left(-0.3333333333333333 + \left(im \cdot im\right) \cdot \left(-0.016666666666666666 + \left(im \cdot im\right) \cdot -0.0003968253968253968\right)\right)\right)\right)} \]

   6. Taylor expanded in im around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{\left(\frac{-1}{2520} \cdot {im}^{4}\right)}\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\frac{-1}{2520}, \color{blue}{\left({im}^{4}\right)}\right)\right)\right)\right)\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\frac{-1}{2520}, \left({im}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      3. pow-sqr N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\frac{-1}{2520}, \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\frac{-1}{2520}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left({im}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\frac{-1}{2520}, \mathsf{*.f64}\left(\left(im \cdot im\right), \left({\color{blue}{im}}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\frac{-1}{2520}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({\color{blue}{im}}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\frac{-1}{2520}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(im \cdot \color{blue}{im}\right)\right)\right)\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 96.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\frac{-1}{2520}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 96.6%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left(-2 + \left(im \cdot im\right) \cdot \left(-0.3333333333333333 + \color{blue}{-0.0003968253968253968 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)}\right)\right)\right) \]

   9. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{1}{2} \cdot re\right)}, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\frac{-1}{2520}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 65.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, re\right), \mathsf{*.f64}\left(\color{blue}{im}, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\frac{-1}{2520}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right)\right)\right)\right) \]

   11. Simplified 65.1%

       \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot \left(im \cdot \left(-2 + \left(im \cdot im\right) \cdot \left(-0.3333333333333333 + -0.0003968253968253968 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right)\right) \]

   if 1.5999999999999999e251 < re

   1. Initial program 48.3%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sin re \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\sin re \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\sin re, \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)}\right)\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} - e^{im}\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\left(e^{\mathsf{neg}\left(im\right)}\right), \color{blue}{\left(e^{im}\right)}\right)\right)\right) \]

      6. exp-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\left(\frac{1}{e^{im}}\right), \left(e^{\color{blue}{im}}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(e^{im}\right)\right), \left(e^{\color{blue}{im}}\right)\right)\right)\right) \]

      8. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(im\right)\right), \left(e^{im}\right)\right)\right)\right) \]

      9. exp-lowering-exp.f64 48.3%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(im\right)\right), \mathsf{exp.f64}\left(im\right)\right)\right)\right) \]

   3. Simplified 48.3%

      \[\leadsto \color{blue}{0.5 \cdot \left(\sin re \cdot \left(\frac{1}{e^{im}} - e^{im}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(im \cdot \sin re\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{im \cdot \sin re} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(im \cdot \sin re\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \color{blue}{\sin re}\right)\right) \]

      5. sin-lowering-sin.f64 58.3%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \mathsf{sin.f64}\left(re\right)\right)\right) \]

   7. Simplified 58.3%

      \[\leadsto \color{blue}{0 - im \cdot \sin re} \]

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(\frac{1}{6} \cdot \left(im \cdot {re}^{2}\right) - im\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{1}{6} \cdot \left(im \cdot {re}^{2}\right) - im\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(re, \left(\frac{1}{6} \cdot \left(im \cdot {re}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(im\right)\right)}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(re, \left(\left(\mathsf{neg}\left(im\right)\right) + \color{blue}{\frac{1}{6} \cdot \left(im \cdot {re}^{2}\right)}\right)\right) \]

      4. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \left(-1 \cdot im + \color{blue}{\frac{1}{6}} \cdot \left(im \cdot {re}^{2}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(re, \left(im \cdot -1 + \color{blue}{\frac{1}{6}} \cdot \left(im \cdot {re}^{2}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(re, \left(im \cdot -1 + \left(im \cdot {re}^{2}\right) \cdot \color{blue}{\frac{1}{6}}\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(re, \left(im \cdot -1 + im \cdot \color{blue}{\left({re}^{2} \cdot \frac{1}{6}\right)}\right)\right) \]

      8. distribute-lft-out N/A

         \[\leadsto \mathsf{*.f64}\left(re, \left(im \cdot \color{blue}{\left(-1 + {re}^{2} \cdot \frac{1}{6}\right)}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \color{blue}{\left(-1 + {re}^{2} \cdot \frac{1}{6}\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \color{blue}{\left({re}^{2} \cdot \frac{1}{6}\right)}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{1}{6}}\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{1}{6}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 45.3%

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{1}{6}\right)\right)\right)\right) \]

   10. Simplified 45.3%

       \[\leadsto \color{blue}{re \cdot \left(im \cdot \left(-1 + \left(re \cdot re\right) \cdot 0.16666666666666666\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.6 \cdot 10^{+251}:\\ \;\;\;\;\left(im \cdot \left(-2 + \left(im \cdot im\right) \cdot \left(-0.3333333333333333 + -0.0003968253968253968 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right)\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot \left(-1 + 0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 56.3% accurate, 12.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;re \leq 1.6 \cdot 10^{+251}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im\_m \cdot \left(-2 + im\_m \cdot \left(im\_m \cdot \left(-0.3333333333333333 + \left(im\_m \cdot im\_m\right) \cdot -0.016666666666666666\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im\_m \cdot \left(-1 + 0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= re 1.6e+251)
    (*
     (* 0.5 re)
     (*
      im_m
      (+
       -2.0
       (*
        im_m
        (*
         im_m
         (+ -0.3333333333333333 (* (* im_m im_m) -0.016666666666666666)))))))
    (* re (* im_m (+ -1.0 (* 0.16666666666666666 (* re re))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (re <= 1.6e+251) {
		tmp = (0.5 * re) * (im_m * (-2.0 + (im_m * (im_m * (-0.3333333333333333 + ((im_m * im_m) * -0.016666666666666666))))));
	} else {
		tmp = re * (im_m * (-1.0 + (0.16666666666666666 * (re * re))));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= 1.6d+251) then
        tmp = (0.5d0 * re) * (im_m * ((-2.0d0) + (im_m * (im_m * ((-0.3333333333333333d0) + ((im_m * im_m) * (-0.016666666666666666d0)))))))
    else
        tmp = re * (im_m * ((-1.0d0) + (0.16666666666666666d0 * (re * re))))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (re <= 1.6e+251) {
		tmp = (0.5 * re) * (im_m * (-2.0 + (im_m * (im_m * (-0.3333333333333333 + ((im_m * im_m) * -0.016666666666666666))))));
	} else {
		tmp = re * (im_m * (-1.0 + (0.16666666666666666 * (re * re))));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if re <= 1.6e+251:
		tmp = (0.5 * re) * (im_m * (-2.0 + (im_m * (im_m * (-0.3333333333333333 + ((im_m * im_m) * -0.016666666666666666))))))
	else:
		tmp = re * (im_m * (-1.0 + (0.16666666666666666 * (re * re))))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (re <= 1.6e+251)
		tmp = Float64(Float64(0.5 * re) * Float64(im_m * Float64(-2.0 + Float64(im_m * Float64(im_m * Float64(-0.3333333333333333 + Float64(Float64(im_m * im_m) * -0.016666666666666666)))))));
	else
		tmp = Float64(re * Float64(im_m * Float64(-1.0 + Float64(0.16666666666666666 * Float64(re * re)))));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (re <= 1.6e+251)
		tmp = (0.5 * re) * (im_m * (-2.0 + (im_m * (im_m * (-0.3333333333333333 + ((im_m * im_m) * -0.016666666666666666))))));
	else
		tmp = re * (im_m * (-1.0 + (0.16666666666666666 * (re * re))));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[re, 1.6e+251], N[(N[(0.5 * re), $MachinePrecision] * N[(im$95$m * N[(-2.0 + N[(im$95$m * N[(im$95$m * N[(-0.3333333333333333 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(im$95$m * N[(-1.0 + N[(0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if re < 1.5999999999999999e251

   1. Initial program 64.5%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)}\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) + -2\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left(-2 + \color{blue}{{im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right)}\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right)\right)}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{-1}{60} \cdot {im}^{2}} - \frac{1}{3}\right)\right)\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right)\right)}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right)\right)}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right)}\right)\right)\right)\right)\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{60} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{60} \cdot {im}^{2} + \frac{-1}{3}\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{3} + \color{blue}{\frac{-1}{60} \cdot {im}^{2}}\right)\right)\right)\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{\left(\frac{-1}{60} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \left({im}^{2} \cdot \color{blue}{\frac{-1}{60}}\right)\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{-1}{60}}\right)\right)\right)\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 93.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 93.9%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left(-2 + im \cdot \left(im \cdot \left(-0.3333333333333333 + \left(im \cdot im\right) \cdot -0.016666666666666666\right)\right)\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{1}{2} \cdot re\right)}, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 63.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, re\right), \mathsf{*.f64}\left(\color{blue}{im}, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 63.5%

      \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot \left(im \cdot \left(-2 + im \cdot \left(im \cdot \left(-0.3333333333333333 + \left(im \cdot im\right) \cdot -0.016666666666666666\right)\right)\right)\right) \]

   if 1.5999999999999999e251 < re

   1. Initial program 48.3%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sin re \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\sin re \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\sin re, \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)}\right)\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} - e^{im}\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\left(e^{\mathsf{neg}\left(im\right)}\right), \color{blue}{\left(e^{im}\right)}\right)\right)\right) \]

      6. exp-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\left(\frac{1}{e^{im}}\right), \left(e^{\color{blue}{im}}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(e^{im}\right)\right), \left(e^{\color{blue}{im}}\right)\right)\right)\right) \]

      8. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(im\right)\right), \left(e^{im}\right)\right)\right)\right) \]

      9. exp-lowering-exp.f64 48.3%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(im\right)\right), \mathsf{exp.f64}\left(im\right)\right)\right)\right) \]

   3. Simplified 48.3%

      \[\leadsto \color{blue}{0.5 \cdot \left(\sin re \cdot \left(\frac{1}{e^{im}} - e^{im}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(im \cdot \sin re\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{im \cdot \sin re} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(im \cdot \sin re\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \color{blue}{\sin re}\right)\right) \]

      5. sin-lowering-sin.f64 58.3%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \mathsf{sin.f64}\left(re\right)\right)\right) \]

   7. Simplified 58.3%

      \[\leadsto \color{blue}{0 - im \cdot \sin re} \]

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(\frac{1}{6} \cdot \left(im \cdot {re}^{2}\right) - im\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{1}{6} \cdot \left(im \cdot {re}^{2}\right) - im\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(re, \left(\frac{1}{6} \cdot \left(im \cdot {re}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(im\right)\right)}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(re, \left(\left(\mathsf{neg}\left(im\right)\right) + \color{blue}{\frac{1}{6} \cdot \left(im \cdot {re}^{2}\right)}\right)\right) \]

      4. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \left(-1 \cdot im + \color{blue}{\frac{1}{6}} \cdot \left(im \cdot {re}^{2}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(re, \left(im \cdot -1 + \color{blue}{\frac{1}{6}} \cdot \left(im \cdot {re}^{2}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(re, \left(im \cdot -1 + \left(im \cdot {re}^{2}\right) \cdot \color{blue}{\frac{1}{6}}\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(re, \left(im \cdot -1 + im \cdot \color{blue}{\left({re}^{2} \cdot \frac{1}{6}\right)}\right)\right) \]

      8. distribute-lft-out N/A

         \[\leadsto \mathsf{*.f64}\left(re, \left(im \cdot \color{blue}{\left(-1 + {re}^{2} \cdot \frac{1}{6}\right)}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \color{blue}{\left(-1 + {re}^{2} \cdot \frac{1}{6}\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \color{blue}{\left({re}^{2} \cdot \frac{1}{6}\right)}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{1}{6}}\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{1}{6}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 45.3%

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{1}{6}\right)\right)\right)\right) \]

   10. Simplified 45.3%

       \[\leadsto \color{blue}{re \cdot \left(im \cdot \left(-1 + \left(re \cdot re\right) \cdot 0.16666666666666666\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.6 \cdot 10^{+251}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im \cdot \left(-2 + im \cdot \left(im \cdot \left(-0.3333333333333333 + \left(im \cdot im\right) \cdot -0.016666666666666666\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot \left(-1 + 0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 56.0% accurate, 14.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;re \leq 1.6 \cdot 10^{+251}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im\_m \cdot \left(-2 + im\_m \cdot \left(im\_m \cdot \left(\left(im\_m \cdot im\_m\right) \cdot -0.016666666666666666\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im\_m \cdot \left(-1 + 0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= re 1.6e+251)
    (*
     (* 0.5 re)
     (*
      im_m
      (+ -2.0 (* im_m (* im_m (* (* im_m im_m) -0.016666666666666666))))))
    (* re (* im_m (+ -1.0 (* 0.16666666666666666 (* re re))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (re <= 1.6e+251) {
		tmp = (0.5 * re) * (im_m * (-2.0 + (im_m * (im_m * ((im_m * im_m) * -0.016666666666666666)))));
	} else {
		tmp = re * (im_m * (-1.0 + (0.16666666666666666 * (re * re))));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= 1.6d+251) then
        tmp = (0.5d0 * re) * (im_m * ((-2.0d0) + (im_m * (im_m * ((im_m * im_m) * (-0.016666666666666666d0))))))
    else
        tmp = re * (im_m * ((-1.0d0) + (0.16666666666666666d0 * (re * re))))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (re <= 1.6e+251) {
		tmp = (0.5 * re) * (im_m * (-2.0 + (im_m * (im_m * ((im_m * im_m) * -0.016666666666666666)))));
	} else {
		tmp = re * (im_m * (-1.0 + (0.16666666666666666 * (re * re))));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if re <= 1.6e+251:
		tmp = (0.5 * re) * (im_m * (-2.0 + (im_m * (im_m * ((im_m * im_m) * -0.016666666666666666)))))
	else:
		tmp = re * (im_m * (-1.0 + (0.16666666666666666 * (re * re))))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (re <= 1.6e+251)
		tmp = Float64(Float64(0.5 * re) * Float64(im_m * Float64(-2.0 + Float64(im_m * Float64(im_m * Float64(Float64(im_m * im_m) * -0.016666666666666666))))));
	else
		tmp = Float64(re * Float64(im_m * Float64(-1.0 + Float64(0.16666666666666666 * Float64(re * re)))));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (re <= 1.6e+251)
		tmp = (0.5 * re) * (im_m * (-2.0 + (im_m * (im_m * ((im_m * im_m) * -0.016666666666666666)))));
	else
		tmp = re * (im_m * (-1.0 + (0.16666666666666666 * (re * re))));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[re, 1.6e+251], N[(N[(0.5 * re), $MachinePrecision] * N[(im$95$m * N[(-2.0 + N[(im$95$m * N[(im$95$m * N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(im$95$m * N[(-1.0 + N[(0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if re < 1.5999999999999999e251

   1. Initial program 64.5%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)}\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) + -2\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left(-2 + \color{blue}{{im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right)}\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right)\right)}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{-1}{60} \cdot {im}^{2}} - \frac{1}{3}\right)\right)\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right)\right)}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right)\right)}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right)}\right)\right)\right)\right)\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{60} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{60} \cdot {im}^{2} + \frac{-1}{3}\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{3} + \color{blue}{\frac{-1}{60} \cdot {im}^{2}}\right)\right)\right)\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{\left(\frac{-1}{60} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \left({im}^{2} \cdot \color{blue}{\frac{-1}{60}}\right)\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{-1}{60}}\right)\right)\right)\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 93.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 93.9%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left(-2 + im \cdot \left(im \cdot \left(-0.3333333333333333 + \left(im \cdot im\right) \cdot -0.016666666666666666\right)\right)\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{1}{2} \cdot re\right)}, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 63.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, re\right), \mathsf{*.f64}\left(\color{blue}{im}, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 63.5%

      \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot \left(im \cdot \left(-2 + im \cdot \left(im \cdot \left(-0.3333333333333333 + \left(im \cdot im\right) \cdot -0.016666666666666666\right)\right)\right)\right) \]

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{60} \cdot {im}^{3}\right)}\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \left({im}^{3} \cdot \color{blue}{\frac{-1}{60}}\right)\right)\right)\right)\right) \]

       2. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{60}\right)\right)\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \left(\left(im \cdot {im}^{2}\right) \cdot \frac{-1}{60}\right)\right)\right)\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \left(im \cdot \color{blue}{\left({im}^{2} \cdot \frac{-1}{60}\right)}\right)\right)\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \left(im \cdot \left(\frac{-1}{60} \cdot \color{blue}{{im}^{2}}\right)\right)\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{60} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \color{blue}{\frac{-1}{60}}\right)\right)\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{-1}{60}}\right)\right)\right)\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right) \]

       10. *-lowering-*.f64 63.4%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right) \]

   11. Simplified 63.4%

       \[\leadsto \left(0.5 \cdot re\right) \cdot \left(im \cdot \left(-2 + im \cdot \color{blue}{\left(im \cdot \left(\left(im \cdot im\right) \cdot -0.016666666666666666\right)\right)}\right)\right) \]

   if 1.5999999999999999e251 < re

   1. Initial program 48.3%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sin re \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\sin re \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\sin re, \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)}\right)\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} - e^{im}\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\left(e^{\mathsf{neg}\left(im\right)}\right), \color{blue}{\left(e^{im}\right)}\right)\right)\right) \]

      6. exp-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\left(\frac{1}{e^{im}}\right), \left(e^{\color{blue}{im}}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(e^{im}\right)\right), \left(e^{\color{blue}{im}}\right)\right)\right)\right) \]

      8. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(im\right)\right), \left(e^{im}\right)\right)\right)\right) \]

      9. exp-lowering-exp.f64 48.3%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(im\right)\right), \mathsf{exp.f64}\left(im\right)\right)\right)\right) \]

   3. Simplified 48.3%

      \[\leadsto \color{blue}{0.5 \cdot \left(\sin re \cdot \left(\frac{1}{e^{im}} - e^{im}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(im \cdot \sin re\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{im \cdot \sin re} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(im \cdot \sin re\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \color{blue}{\sin re}\right)\right) \]

      5. sin-lowering-sin.f64 58.3%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \mathsf{sin.f64}\left(re\right)\right)\right) \]

   7. Simplified 58.3%

      \[\leadsto \color{blue}{0 - im \cdot \sin re} \]

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(\frac{1}{6} \cdot \left(im \cdot {re}^{2}\right) - im\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{1}{6} \cdot \left(im \cdot {re}^{2}\right) - im\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(re, \left(\frac{1}{6} \cdot \left(im \cdot {re}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(im\right)\right)}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(re, \left(\left(\mathsf{neg}\left(im\right)\right) + \color{blue}{\frac{1}{6} \cdot \left(im \cdot {re}^{2}\right)}\right)\right) \]

      4. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \left(-1 \cdot im + \color{blue}{\frac{1}{6}} \cdot \left(im \cdot {re}^{2}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(re, \left(im \cdot -1 + \color{blue}{\frac{1}{6}} \cdot \left(im \cdot {re}^{2}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(re, \left(im \cdot -1 + \left(im \cdot {re}^{2}\right) \cdot \color{blue}{\frac{1}{6}}\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(re, \left(im \cdot -1 + im \cdot \color{blue}{\left({re}^{2} \cdot \frac{1}{6}\right)}\right)\right) \]

      8. distribute-lft-out N/A

         \[\leadsto \mathsf{*.f64}\left(re, \left(im \cdot \color{blue}{\left(-1 + {re}^{2} \cdot \frac{1}{6}\right)}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \color{blue}{\left(-1 + {re}^{2} \cdot \frac{1}{6}\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \color{blue}{\left({re}^{2} \cdot \frac{1}{6}\right)}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{1}{6}}\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{1}{6}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 45.3%

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{1}{6}\right)\right)\right)\right) \]

   10. Simplified 45.3%

       \[\leadsto \color{blue}{re \cdot \left(im \cdot \left(-1 + \left(re \cdot re\right) \cdot 0.16666666666666666\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.6 \cdot 10^{+251}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im \cdot \left(-2 + im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot -0.016666666666666666\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot \left(-1 + 0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 54.0% accurate, 14.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;re \leq 1.6 \cdot 10^{+251}:\\ \;\;\;\;im\_m \cdot \left(\left(-0.16666666666666666 + \left(im\_m \cdot im\_m\right) \cdot -0.008333333333333333\right) \cdot \left(im\_m \cdot \left(im\_m \cdot re\right)\right) - re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im\_m \cdot \left(-1 + 0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= re 1.6e+251)
    (*
     im_m
     (-
      (*
       (+ -0.16666666666666666 (* (* im_m im_m) -0.008333333333333333))
       (* im_m (* im_m re)))
      re))
    (* re (* im_m (+ -1.0 (* 0.16666666666666666 (* re re))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (re <= 1.6e+251) {
		tmp = im_m * (((-0.16666666666666666 + ((im_m * im_m) * -0.008333333333333333)) * (im_m * (im_m * re))) - re);
	} else {
		tmp = re * (im_m * (-1.0 + (0.16666666666666666 * (re * re))));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= 1.6d+251) then
        tmp = im_m * ((((-0.16666666666666666d0) + ((im_m * im_m) * (-0.008333333333333333d0))) * (im_m * (im_m * re))) - re)
    else
        tmp = re * (im_m * ((-1.0d0) + (0.16666666666666666d0 * (re * re))))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (re <= 1.6e+251) {
		tmp = im_m * (((-0.16666666666666666 + ((im_m * im_m) * -0.008333333333333333)) * (im_m * (im_m * re))) - re);
	} else {
		tmp = re * (im_m * (-1.0 + (0.16666666666666666 * (re * re))));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if re <= 1.6e+251:
		tmp = im_m * (((-0.16666666666666666 + ((im_m * im_m) * -0.008333333333333333)) * (im_m * (im_m * re))) - re)
	else:
		tmp = re * (im_m * (-1.0 + (0.16666666666666666 * (re * re))))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (re <= 1.6e+251)
		tmp = Float64(im_m * Float64(Float64(Float64(-0.16666666666666666 + Float64(Float64(im_m * im_m) * -0.008333333333333333)) * Float64(im_m * Float64(im_m * re))) - re));
	else
		tmp = Float64(re * Float64(im_m * Float64(-1.0 + Float64(0.16666666666666666 * Float64(re * re)))));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (re <= 1.6e+251)
		tmp = im_m * (((-0.16666666666666666 + ((im_m * im_m) * -0.008333333333333333)) * (im_m * (im_m * re))) - re);
	else
		tmp = re * (im_m * (-1.0 + (0.16666666666666666 * (re * re))));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[re, 1.6e+251], N[(im$95$m * N[(N[(N[(-0.16666666666666666 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.008333333333333333), $MachinePrecision]), $MachinePrecision] * N[(im$95$m * N[(im$95$m * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision], N[(re * N[(im$95$m * N[(-1.0 + N[(0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if re < 1.5999999999999999e251

   1. Initial program 64.5%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{re}\right), \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(im\right)\right), \mathsf{exp.f64}\left(im\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 52.5%

      \[\leadsto \left(0.5 \cdot \color{blue}{re}\right) \cdot \left(e^{-im} - e^{im}\right) \]

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot re + \frac{-1}{120} \cdot \left({im}^{2} \cdot re\right)\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(-1 \cdot re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot re + \frac{-1}{120} \cdot \left({im}^{2} \cdot re\right)\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left(\frac{-1}{6} \cdot re + \frac{-1}{120} \cdot \left({im}^{2} \cdot re\right)\right) + \color{blue}{-1 \cdot re}\right)\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left(\frac{-1}{6} \cdot re + \frac{-1}{120} \cdot \left({im}^{2} \cdot re\right)\right) + \left(\mathsf{neg}\left(re\right)\right)\right)\right) \]

      4. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left(\frac{-1}{6} \cdot re + \frac{-1}{120} \cdot \left({im}^{2} \cdot re\right)\right) - \color{blue}{re}\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \mathsf{\_.f64}\left(\left({im}^{2} \cdot \left(\frac{-1}{6} \cdot re + \frac{-1}{120} \cdot \left({im}^{2} \cdot re\right)\right)\right), \color{blue}{re}\right)\right) \]

   8. Simplified 61.6%

      \[\leadsto \color{blue}{im \cdot \left(\left(im \cdot \left(im \cdot re\right)\right) \cdot \left(-0.16666666666666666 + \left(im \cdot im\right) \cdot -0.008333333333333333\right) - re\right)} \]

   if 1.5999999999999999e251 < re

   1. Initial program 48.3%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sin re \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\sin re \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\sin re, \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)}\right)\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} - e^{im}\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\left(e^{\mathsf{neg}\left(im\right)}\right), \color{blue}{\left(e^{im}\right)}\right)\right)\right) \]

      6. exp-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\left(\frac{1}{e^{im}}\right), \left(e^{\color{blue}{im}}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(e^{im}\right)\right), \left(e^{\color{blue}{im}}\right)\right)\right)\right) \]

      8. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(im\right)\right), \left(e^{im}\right)\right)\right)\right) \]

      9. exp-lowering-exp.f64 48.3%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(im\right)\right), \mathsf{exp.f64}\left(im\right)\right)\right)\right) \]

   3. Simplified 48.3%

      \[\leadsto \color{blue}{0.5 \cdot \left(\sin re \cdot \left(\frac{1}{e^{im}} - e^{im}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(im \cdot \sin re\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{im \cdot \sin re} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(im \cdot \sin re\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \color{blue}{\sin re}\right)\right) \]

      5. sin-lowering-sin.f64 58.3%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \mathsf{sin.f64}\left(re\right)\right)\right) \]

   7. Simplified 58.3%

      \[\leadsto \color{blue}{0 - im \cdot \sin re} \]

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(\frac{1}{6} \cdot \left(im \cdot {re}^{2}\right) - im\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{1}{6} \cdot \left(im \cdot {re}^{2}\right) - im\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(re, \left(\frac{1}{6} \cdot \left(im \cdot {re}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(im\right)\right)}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(re, \left(\left(\mathsf{neg}\left(im\right)\right) + \color{blue}{\frac{1}{6} \cdot \left(im \cdot {re}^{2}\right)}\right)\right) \]

      4. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \left(-1 \cdot im + \color{blue}{\frac{1}{6}} \cdot \left(im \cdot {re}^{2}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(re, \left(im \cdot -1 + \color{blue}{\frac{1}{6}} \cdot \left(im \cdot {re}^{2}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(re, \left(im \cdot -1 + \left(im \cdot {re}^{2}\right) \cdot \color{blue}{\frac{1}{6}}\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(re, \left(im \cdot -1 + im \cdot \color{blue}{\left({re}^{2} \cdot \frac{1}{6}\right)}\right)\right) \]

      8. distribute-lft-out N/A

         \[\leadsto \mathsf{*.f64}\left(re, \left(im \cdot \color{blue}{\left(-1 + {re}^{2} \cdot \frac{1}{6}\right)}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \color{blue}{\left(-1 + {re}^{2} \cdot \frac{1}{6}\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \color{blue}{\left({re}^{2} \cdot \frac{1}{6}\right)}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{1}{6}}\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{1}{6}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 45.3%

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{1}{6}\right)\right)\right)\right) \]

   10. Simplified 45.3%

       \[\leadsto \color{blue}{re \cdot \left(im \cdot \left(-1 + \left(re \cdot re\right) \cdot 0.16666666666666666\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.6 \cdot 10^{+251}:\\ \;\;\;\;im \cdot \left(\left(-0.16666666666666666 + \left(im \cdot im\right) \cdot -0.008333333333333333\right) \cdot \left(im \cdot \left(im \cdot re\right)\right) - re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot \left(-1 + 0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 56.8% accurate, 15.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 1.35 \cdot 10^{+60}:\\ \;\;\;\;re \cdot \left(im\_m \cdot \left(-1 + 0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im\_m \cdot \left(-0.016666666666666666 \cdot \left(\left(im\_m \cdot im\_m\right) \cdot \left(im\_m \cdot im\_m\right)\right)\right)\right)\\ \end{array} \end{array} \]

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 1.35e+60)
    (* re (* im_m (+ -1.0 (* 0.16666666666666666 (* re re)))))
    (*
     (* 0.5 re)
     (* im_m (* -0.016666666666666666 (* (* 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 <= 1.35e+60) {
		tmp = re * (im_m * (-1.0 + (0.16666666666666666 * (re * re))));
	} else {
		tmp = (0.5 * re) * (im_m * (-0.016666666666666666 * ((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 <= 1.35d+60) then
        tmp = re * (im_m * ((-1.0d0) + (0.16666666666666666d0 * (re * re))))
    else
        tmp = (0.5d0 * re) * (im_m * ((-0.016666666666666666d0) * ((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 <= 1.35e+60) {
		tmp = re * (im_m * (-1.0 + (0.16666666666666666 * (re * re))));
	} else {
		tmp = (0.5 * re) * (im_m * (-0.016666666666666666 * ((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 <= 1.35e+60:
		tmp = re * (im_m * (-1.0 + (0.16666666666666666 * (re * re))))
	else:
		tmp = (0.5 * re) * (im_m * (-0.016666666666666666 * ((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 <= 1.35e+60)
		tmp = Float64(re * Float64(im_m * Float64(-1.0 + Float64(0.16666666666666666 * Float64(re * re)))));
	else
		tmp = Float64(Float64(0.5 * re) * Float64(im_m * Float64(-0.016666666666666666 * 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 <= 1.35e+60)
		tmp = re * (im_m * (-1.0 + (0.16666666666666666 * (re * re))));
	else
		tmp = (0.5 * re) * (im_m * (-0.016666666666666666 * ((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, 1.35e+60], N[(re * N[(im$95$m * N[(-1.0 + N[(0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(im$95$m * N[(-0.016666666666666666 * N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 1.35e60

   1. Initial program 55.9%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sin re \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\sin re \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\sin re, \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)}\right)\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} - e^{im}\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\left(e^{\mathsf{neg}\left(im\right)}\right), \color{blue}{\left(e^{im}\right)}\right)\right)\right) \]

      6. exp-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\left(\frac{1}{e^{im}}\right), \left(e^{\color{blue}{im}}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(e^{im}\right)\right), \left(e^{\color{blue}{im}}\right)\right)\right)\right) \]

      8. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(im\right)\right), \left(e^{im}\right)\right)\right)\right) \]

      9. exp-lowering-exp.f64 55.8%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(im\right)\right), \mathsf{exp.f64}\left(im\right)\right)\right)\right) \]

   3. Simplified 55.8%

      \[\leadsto \color{blue}{0.5 \cdot \left(\sin re \cdot \left(\frac{1}{e^{im}} - e^{im}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(im \cdot \sin re\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{im \cdot \sin re} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(im \cdot \sin re\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \color{blue}{\sin re}\right)\right) \]

      5. sin-lowering-sin.f64 69.7%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \mathsf{sin.f64}\left(re\right)\right)\right) \]

   7. Simplified 69.7%

      \[\leadsto \color{blue}{0 - im \cdot \sin re} \]

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(\frac{1}{6} \cdot \left(im \cdot {re}^{2}\right) - im\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{1}{6} \cdot \left(im \cdot {re}^{2}\right) - im\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(re, \left(\frac{1}{6} \cdot \left(im \cdot {re}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(im\right)\right)}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(re, \left(\left(\mathsf{neg}\left(im\right)\right) + \color{blue}{\frac{1}{6} \cdot \left(im \cdot {re}^{2}\right)}\right)\right) \]

      4. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \left(-1 \cdot im + \color{blue}{\frac{1}{6}} \cdot \left(im \cdot {re}^{2}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(re, \left(im \cdot -1 + \color{blue}{\frac{1}{6}} \cdot \left(im \cdot {re}^{2}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(re, \left(im \cdot -1 + \left(im \cdot {re}^{2}\right) \cdot \color{blue}{\frac{1}{6}}\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(re, \left(im \cdot -1 + im \cdot \color{blue}{\left({re}^{2} \cdot \frac{1}{6}\right)}\right)\right) \]

      8. distribute-lft-out N/A

         \[\leadsto \mathsf{*.f64}\left(re, \left(im \cdot \color{blue}{\left(-1 + {re}^{2} \cdot \frac{1}{6}\right)}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \color{blue}{\left(-1 + {re}^{2} \cdot \frac{1}{6}\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \color{blue}{\left({re}^{2} \cdot \frac{1}{6}\right)}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{1}{6}}\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{1}{6}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 48.4%

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{1}{6}\right)\right)\right)\right) \]

   10. Simplified 48.4%

       \[\leadsto \color{blue}{re \cdot \left(im \cdot \left(-1 + \left(re \cdot re\right) \cdot 0.16666666666666666\right)\right)} \]

   if 1.35e60 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)}\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) + -2\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left(-2 + \color{blue}{{im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right)}\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right)\right)}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{-1}{60} \cdot {im}^{2}} - \frac{1}{3}\right)\right)\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right)\right)}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right)\right)}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right)}\right)\right)\right)\right)\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{60} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{60} \cdot {im}^{2} + \frac{-1}{3}\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{3} + \color{blue}{\frac{-1}{60} \cdot {im}^{2}}\right)\right)\right)\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{\left(\frac{-1}{60} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \left({im}^{2} \cdot \color{blue}{\frac{-1}{60}}\right)\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{-1}{60}}\right)\right)\right)\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left(-2 + im \cdot \left(im \cdot \left(-0.3333333333333333 + \left(im \cdot im\right) \cdot -0.016666666666666666\right)\right)\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{1}{2} \cdot re\right)}, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 76.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, re\right), \mathsf{*.f64}\left(\color{blue}{im}, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 76.6%

      \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot \left(im \cdot \left(-2 + im \cdot \left(im \cdot \left(-0.3333333333333333 + \left(im \cdot im\right) \cdot -0.016666666666666666\right)\right)\right)\right) \]

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, re\right), \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{60} \cdot {im}^{4}\right)}\right)\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, re\right), \mathsf{*.f64}\left(im, \left({im}^{4} \cdot \color{blue}{\frac{-1}{60}}\right)\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, re\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left({im}^{4}\right), \color{blue}{\frac{-1}{60}}\right)\right)\right) \]

       3. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, re\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left({im}^{\left(2 \cdot 2\right)}\right), \frac{-1}{60}\right)\right)\right) \]

       4. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, re\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left({im}^{2} \cdot {im}^{2}\right), \frac{-1}{60}\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, re\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({im}^{2}\right), \left({im}^{2}\right)\right), \frac{-1}{60}\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, re\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(im \cdot im\right), \left({im}^{2}\right)\right), \frac{-1}{60}\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, re\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2}\right)\right), \frac{-1}{60}\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, re\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(im \cdot im\right)\right), \frac{-1}{60}\right)\right)\right) \]

       9. *-lowering-*.f64 76.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, re\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \frac{-1}{60}\right)\right)\right) \]

   11. Simplified 76.6%

       \[\leadsto \left(0.5 \cdot re\right) \cdot \left(im \cdot \color{blue}{\left(\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot -0.016666666666666666\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.35 \cdot 10^{+60}:\\ \;\;\;\;re \cdot \left(im \cdot \left(-1 + 0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im \cdot \left(-0.016666666666666666 \cdot \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 52.3% accurate, 17.1× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;re \leq 1.6 \cdot 10^{+251}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im\_m \cdot \left(-2 + \left(im\_m \cdot im\_m\right) \cdot -0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im\_m \cdot \left(-1 + 0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= re 1.6e+251)
    (* (* 0.5 re) (* im_m (+ -2.0 (* (* im_m im_m) -0.3333333333333333))))
    (* re (* im_m (+ -1.0 (* 0.16666666666666666 (* re re))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (re <= 1.6e+251) {
		tmp = (0.5 * re) * (im_m * (-2.0 + ((im_m * im_m) * -0.3333333333333333)));
	} else {
		tmp = re * (im_m * (-1.0 + (0.16666666666666666 * (re * re))));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= 1.6d+251) then
        tmp = (0.5d0 * re) * (im_m * ((-2.0d0) + ((im_m * im_m) * (-0.3333333333333333d0))))
    else
        tmp = re * (im_m * ((-1.0d0) + (0.16666666666666666d0 * (re * re))))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (re <= 1.6e+251) {
		tmp = (0.5 * re) * (im_m * (-2.0 + ((im_m * im_m) * -0.3333333333333333)));
	} else {
		tmp = re * (im_m * (-1.0 + (0.16666666666666666 * (re * re))));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if re <= 1.6e+251:
		tmp = (0.5 * re) * (im_m * (-2.0 + ((im_m * im_m) * -0.3333333333333333)))
	else:
		tmp = re * (im_m * (-1.0 + (0.16666666666666666 * (re * re))))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (re <= 1.6e+251)
		tmp = Float64(Float64(0.5 * re) * Float64(im_m * Float64(-2.0 + Float64(Float64(im_m * im_m) * -0.3333333333333333))));
	else
		tmp = Float64(re * Float64(im_m * Float64(-1.0 + Float64(0.16666666666666666 * Float64(re * re)))));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (re <= 1.6e+251)
		tmp = (0.5 * re) * (im_m * (-2.0 + ((im_m * im_m) * -0.3333333333333333)));
	else
		tmp = re * (im_m * (-1.0 + (0.16666666666666666 * (re * re))));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[re, 1.6e+251], N[(N[(0.5 * re), $MachinePrecision] * N[(im$95$m * N[(-2.0 + N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(im$95$m * N[(-1.0 + N[(0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if re < 1.5999999999999999e251

   1. Initial program 64.5%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \color{blue}{\left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{3} \cdot {im}^{2} - 2\right)}\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left(\frac{-1}{3} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left(\frac{-1}{3} \cdot {im}^{2} + -2\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left(-2 + \color{blue}{\frac{-1}{3} \cdot {im}^{2}}\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \color{blue}{\left(\frac{-1}{3} \cdot {im}^{2}\right)}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \left({im}^{2} \cdot \color{blue}{\frac{-1}{3}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{-1}{3}}\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{-1}{3}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 87.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{3}\right)\right)\right)\right) \]

   5. Simplified 87.4%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left(-2 + \left(im \cdot im\right) \cdot -0.3333333333333333\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{1}{2} \cdot re\right)}, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{3}\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 59.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, re\right), \mathsf{*.f64}\left(\color{blue}{im}, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{3}\right)\right)\right)\right) \]

   8. Simplified 59.3%

      \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot \left(im \cdot \left(-2 + \left(im \cdot im\right) \cdot -0.3333333333333333\right)\right) \]

   if 1.5999999999999999e251 < re

   1. Initial program 48.3%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sin re \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\sin re \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\sin re, \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)}\right)\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} - e^{im}\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\left(e^{\mathsf{neg}\left(im\right)}\right), \color{blue}{\left(e^{im}\right)}\right)\right)\right) \]

      6. exp-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\left(\frac{1}{e^{im}}\right), \left(e^{\color{blue}{im}}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(e^{im}\right)\right), \left(e^{\color{blue}{im}}\right)\right)\right)\right) \]

      8. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(im\right)\right), \left(e^{im}\right)\right)\right)\right) \]

      9. exp-lowering-exp.f64 48.3%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(im\right)\right), \mathsf{exp.f64}\left(im\right)\right)\right)\right) \]

   3. Simplified 48.3%

      \[\leadsto \color{blue}{0.5 \cdot \left(\sin re \cdot \left(\frac{1}{e^{im}} - e^{im}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(im \cdot \sin re\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{im \cdot \sin re} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(im \cdot \sin re\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \color{blue}{\sin re}\right)\right) \]

      5. sin-lowering-sin.f64 58.3%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \mathsf{sin.f64}\left(re\right)\right)\right) \]

   7. Simplified 58.3%

      \[\leadsto \color{blue}{0 - im \cdot \sin re} \]

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(\frac{1}{6} \cdot \left(im \cdot {re}^{2}\right) - im\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{1}{6} \cdot \left(im \cdot {re}^{2}\right) - im\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(re, \left(\frac{1}{6} \cdot \left(im \cdot {re}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(im\right)\right)}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(re, \left(\left(\mathsf{neg}\left(im\right)\right) + \color{blue}{\frac{1}{6} \cdot \left(im \cdot {re}^{2}\right)}\right)\right) \]

      4. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \left(-1 \cdot im + \color{blue}{\frac{1}{6}} \cdot \left(im \cdot {re}^{2}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(re, \left(im \cdot -1 + \color{blue}{\frac{1}{6}} \cdot \left(im \cdot {re}^{2}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(re, \left(im \cdot -1 + \left(im \cdot {re}^{2}\right) \cdot \color{blue}{\frac{1}{6}}\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(re, \left(im \cdot -1 + im \cdot \color{blue}{\left({re}^{2} \cdot \frac{1}{6}\right)}\right)\right) \]

      8. distribute-lft-out N/A

         \[\leadsto \mathsf{*.f64}\left(re, \left(im \cdot \color{blue}{\left(-1 + {re}^{2} \cdot \frac{1}{6}\right)}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \color{blue}{\left(-1 + {re}^{2} \cdot \frac{1}{6}\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \color{blue}{\left({re}^{2} \cdot \frac{1}{6}\right)}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{1}{6}}\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{1}{6}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 45.3%

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{1}{6}\right)\right)\right)\right) \]

   10. Simplified 45.3%

       \[\leadsto \color{blue}{re \cdot \left(im \cdot \left(-1 + \left(re \cdot re\right) \cdot 0.16666666666666666\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.6 \cdot 10^{+251}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im \cdot \left(-2 + \left(im \cdot im\right) \cdot -0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot \left(-1 + 0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 49.5% accurate, 19.2× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;re \leq 1.6 \cdot 10^{+251}:\\ \;\;\;\;im\_m \cdot \left(re \cdot \left(-1 + -0.16666666666666666 \cdot \left(im\_m \cdot im\_m\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im\_m \cdot \left(-1 + 0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= re 1.6e+251)
    (* im_m (* re (+ -1.0 (* -0.16666666666666666 (* im_m im_m)))))
    (* re (* im_m (+ -1.0 (* 0.16666666666666666 (* re re))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (re <= 1.6e+251) {
		tmp = im_m * (re * (-1.0 + (-0.16666666666666666 * (im_m * im_m))));
	} else {
		tmp = re * (im_m * (-1.0 + (0.16666666666666666 * (re * re))));
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= 1.6d+251) then
        tmp = im_m * (re * ((-1.0d0) + ((-0.16666666666666666d0) * (im_m * im_m))))
    else
        tmp = re * (im_m * ((-1.0d0) + (0.16666666666666666d0 * (re * re))))
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (re <= 1.6e+251) {
		tmp = im_m * (re * (-1.0 + (-0.16666666666666666 * (im_m * im_m))));
	} else {
		tmp = re * (im_m * (-1.0 + (0.16666666666666666 * (re * re))));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if re <= 1.6e+251:
		tmp = im_m * (re * (-1.0 + (-0.16666666666666666 * (im_m * im_m))))
	else:
		tmp = re * (im_m * (-1.0 + (0.16666666666666666 * (re * re))))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (re <= 1.6e+251)
		tmp = Float64(im_m * Float64(re * Float64(-1.0 + Float64(-0.16666666666666666 * Float64(im_m * im_m)))));
	else
		tmp = Float64(re * Float64(im_m * Float64(-1.0 + Float64(0.16666666666666666 * Float64(re * re)))));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (re <= 1.6e+251)
		tmp = im_m * (re * (-1.0 + (-0.16666666666666666 * (im_m * im_m))));
	else
		tmp = re * (im_m * (-1.0 + (0.16666666666666666 * (re * re))));
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[re, 1.6e+251], N[(im$95$m * N[(re * N[(-1.0 + N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(im$95$m * N[(-1.0 + N[(0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if re < 1.5999999999999999e251

   1. Initial program 64.5%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)}\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) + -2\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left(-2 + \color{blue}{{im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right)}\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right)\right)}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{-1}{60} \cdot {im}^{2}} - \frac{1}{3}\right)\right)\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right)\right)}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right)\right)}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right)}\right)\right)\right)\right)\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{60} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{60} \cdot {im}^{2} + \frac{-1}{3}\right)\right)\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{3} + \color{blue}{\frac{-1}{60} \cdot {im}^{2}}\right)\right)\right)\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{\left(\frac{-1}{60} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \left({im}^{2} \cdot \color{blue}{\frac{-1}{60}}\right)\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{-1}{60}}\right)\right)\right)\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 93.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 93.9%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left(-2 + im \cdot \left(im \cdot \left(-0.3333333333333333 + \left(im \cdot im\right) \cdot -0.016666666666666666\right)\right)\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{1}{2} \cdot re\right)}, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 63.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, re\right), \mathsf{*.f64}\left(\color{blue}{im}, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 63.5%

      \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot \left(im \cdot \left(-2 + im \cdot \left(im \cdot \left(-0.3333333333333333 + \left(im \cdot im\right) \cdot -0.016666666666666666\right)\right)\right)\right) \]

   9. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot re + \frac{-1}{6} \cdot \left({im}^{2} \cdot re\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto im \cdot \left(-1 \cdot re + \left({im}^{2} \cdot re\right) \cdot \color{blue}{\frac{-1}{6}}\right) \]

       2. associate-*r* N/A

          \[\leadsto im \cdot \left(-1 \cdot re + {im}^{2} \cdot \color{blue}{\left(re \cdot \frac{-1}{6}\right)}\right) \]

       3. *-commutative N/A

          \[\leadsto im \cdot \left(-1 \cdot re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \color{blue}{re}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(-1 \cdot re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot re\right)\right)}\right) \]

       5. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left(\frac{-1}{6} \cdot re\right) + \color{blue}{-1 \cdot re}\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left(re \cdot \frac{-1}{6}\right) + -1 \cdot re\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left(\left({im}^{2} \cdot re\right) \cdot \frac{-1}{6} + \color{blue}{-1} \cdot re\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot re\right) + \color{blue}{-1} \cdot re\right)\right) \]

       9. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left(\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot re + \color{blue}{-1} \cdot re\right)\right) \]

       10. distribute-rgt-out N/A

           \[\leadsto \mathsf{*.f64}\left(im, \left(re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2} + -1\right)}\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2} + -1\right)}\right)\right) \]

       12. +-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, \left(-1 + \color{blue}{\frac{-1}{6} \cdot {im}^{2}}\right)\right)\right) \]

       13. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(-1, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)}\right)\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(-1, \left({im}^{2} \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

       16. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{-1}{6}\right)\right)\right)\right) \]

       17. *-lowering-*.f64 56.6%

           \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{6}\right)\right)\right)\right) \]

   11. Simplified 56.6%

       \[\leadsto \color{blue}{im \cdot \left(re \cdot \left(-1 + \left(im \cdot im\right) \cdot -0.16666666666666666\right)\right)} \]

   if 1.5999999999999999e251 < re

   1. Initial program 48.3%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sin re \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\sin re \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\sin re, \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)}\right)\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} - e^{im}\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\left(e^{\mathsf{neg}\left(im\right)}\right), \color{blue}{\left(e^{im}\right)}\right)\right)\right) \]

      6. exp-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\left(\frac{1}{e^{im}}\right), \left(e^{\color{blue}{im}}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(e^{im}\right)\right), \left(e^{\color{blue}{im}}\right)\right)\right)\right) \]

      8. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(im\right)\right), \left(e^{im}\right)\right)\right)\right) \]

      9. exp-lowering-exp.f64 48.3%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(im\right)\right), \mathsf{exp.f64}\left(im\right)\right)\right)\right) \]

   3. Simplified 48.3%

      \[\leadsto \color{blue}{0.5 \cdot \left(\sin re \cdot \left(\frac{1}{e^{im}} - e^{im}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(im \cdot \sin re\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{im \cdot \sin re} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(im \cdot \sin re\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \color{blue}{\sin re}\right)\right) \]

      5. sin-lowering-sin.f64 58.3%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \mathsf{sin.f64}\left(re\right)\right)\right) \]

   7. Simplified 58.3%

      \[\leadsto \color{blue}{0 - im \cdot \sin re} \]

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(\frac{1}{6} \cdot \left(im \cdot {re}^{2}\right) - im\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{1}{6} \cdot \left(im \cdot {re}^{2}\right) - im\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(re, \left(\frac{1}{6} \cdot \left(im \cdot {re}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(im\right)\right)}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(re, \left(\left(\mathsf{neg}\left(im\right)\right) + \color{blue}{\frac{1}{6} \cdot \left(im \cdot {re}^{2}\right)}\right)\right) \]

      4. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \left(-1 \cdot im + \color{blue}{\frac{1}{6}} \cdot \left(im \cdot {re}^{2}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(re, \left(im \cdot -1 + \color{blue}{\frac{1}{6}} \cdot \left(im \cdot {re}^{2}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(re, \left(im \cdot -1 + \left(im \cdot {re}^{2}\right) \cdot \color{blue}{\frac{1}{6}}\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(re, \left(im \cdot -1 + im \cdot \color{blue}{\left({re}^{2} \cdot \frac{1}{6}\right)}\right)\right) \]

      8. distribute-lft-out N/A

         \[\leadsto \mathsf{*.f64}\left(re, \left(im \cdot \color{blue}{\left(-1 + {re}^{2} \cdot \frac{1}{6}\right)}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \color{blue}{\left(-1 + {re}^{2} \cdot \frac{1}{6}\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \color{blue}{\left({re}^{2} \cdot \frac{1}{6}\right)}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\frac{1}{6}}\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{1}{6}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 45.3%

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{1}{6}\right)\right)\right)\right) \]

   10. Simplified 45.3%

       \[\leadsto \color{blue}{re \cdot \left(im \cdot \left(-1 + \left(re \cdot re\right) \cdot 0.16666666666666666\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.6 \cdot 10^{+251}:\\ \;\;\;\;im \cdot \left(re \cdot \left(-1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(im \cdot \left(-1 + 0.16666666666666666 \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 41.3% accurate, 22.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 6.5 \cdot 10^{-13}:\\ \;\;\;\;\left(0 - im\_m\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\frac{im\_m \cdot \left(im\_m \cdot re\right)}{0 - im\_m}\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 6.5e-13)
    (* (- 0.0 im_m) re)
    (/ (* im_m (* im_m re)) (- 0.0 im_m)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 6.5e-13) {
		tmp = (0.0 - im_m) * re;
	} else {
		tmp = (im_m * (im_m * re)) / (0.0 - im_m);
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 6.5d-13) then
        tmp = (0.0d0 - im_m) * re
    else
        tmp = (im_m * (im_m * re)) / (0.0d0 - im_m)
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 6.5e-13) {
		tmp = (0.0 - im_m) * re;
	} else {
		tmp = (im_m * (im_m * re)) / (0.0 - im_m);
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 6.5e-13:
		tmp = (0.0 - im_m) * re
	else:
		tmp = (im_m * (im_m * re)) / (0.0 - im_m)
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 6.5e-13)
		tmp = Float64(Float64(0.0 - im_m) * re);
	else
		tmp = Float64(Float64(im_m * Float64(im_m * re)) / Float64(0.0 - im_m));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 6.5e-13)
		tmp = (0.0 - im_m) * re;
	else
		tmp = (im_m * (im_m * re)) / (0.0 - im_m);
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 6.5e-13], N[(N[(0.0 - im$95$m), $MachinePrecision] * re), $MachinePrecision], N[(N[(im$95$m * N[(im$95$m * re), $MachinePrecision]), $MachinePrecision] / N[(0.0 - im$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 6.49999999999999957e-13

   1. Initial program 54.4%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sin re \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\sin re \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\sin re, \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)}\right)\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} - e^{im}\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\left(e^{\mathsf{neg}\left(im\right)}\right), \color{blue}{\left(e^{im}\right)}\right)\right)\right) \]

      6. exp-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\left(\frac{1}{e^{im}}\right), \left(e^{\color{blue}{im}}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(e^{im}\right)\right), \left(e^{\color{blue}{im}}\right)\right)\right)\right) \]

      8. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(im\right)\right), \left(e^{im}\right)\right)\right)\right) \]

      9. exp-lowering-exp.f64 54.4%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(im\right)\right), \mathsf{exp.f64}\left(im\right)\right)\right)\right) \]

   3. Simplified 54.4%

      \[\leadsto \color{blue}{0.5 \cdot \left(\sin re \cdot \left(\frac{1}{e^{im}} - e^{im}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(im \cdot \sin re\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{im \cdot \sin re} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(im \cdot \sin re\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \color{blue}{\sin re}\right)\right) \]

      5. sin-lowering-sin.f64 71.7%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \mathsf{sin.f64}\left(re\right)\right)\right) \]

   7. Simplified 71.7%

      \[\leadsto \color{blue}{0 - im \cdot \sin re} \]

   8. Taylor expanded in re around 0

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \color{blue}{re}\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 46.2%

       \[\leadsto 0 - im \cdot \color{blue}{re} \]
   11. Step-by-step derivation

       1. sub0-neg N/A

          \[\leadsto \mathsf{neg}\left(im \cdot re\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(re \cdot im\right) \]

       3. distribute-lft-neg-in N/A

          \[\leadsto \left(\mathsf{neg}\left(re\right)\right) \cdot \color{blue}{im} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{neg}\left(re\right)\right), \color{blue}{im}\right) \]

       5. neg-lowering-neg.f64 46.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{neg.f64}\left(re\right), im\right) \]

   12. Applied egg-rr 46.2%

       \[\leadsto \color{blue}{\left(-re\right) \cdot im} \]

   if 6.49999999999999957e-13 < im

   1. Initial program 98.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sin re \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\sin re \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\sin re, \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)}\right)\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} - e^{im}\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\left(e^{\mathsf{neg}\left(im\right)}\right), \color{blue}{\left(e^{im}\right)}\right)\right)\right) \]

      6. exp-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\left(\frac{1}{e^{im}}\right), \left(e^{\color{blue}{im}}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(e^{im}\right)\right), \left(e^{\color{blue}{im}}\right)\right)\right)\right) \]

      8. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(im\right)\right), \left(e^{im}\right)\right)\right)\right) \]

      9. exp-lowering-exp.f64 98.0%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(im\right)\right), \mathsf{exp.f64}\left(im\right)\right)\right)\right) \]

   3. Simplified 98.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(\sin re \cdot \left(\frac{1}{e^{im}} - e^{im}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(im \cdot \sin re\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{im \cdot \sin re} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(im \cdot \sin re\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \color{blue}{\sin re}\right)\right) \]

      5. sin-lowering-sin.f64 8.7%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \mathsf{sin.f64}\left(re\right)\right)\right) \]

   7. Simplified 8.7%

      \[\leadsto \color{blue}{0 - im \cdot \sin re} \]

   8. Taylor expanded in re around 0

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \color{blue}{re}\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 15.3%

       \[\leadsto 0 - im \cdot \color{blue}{re} \]
   11. Step-by-step derivation

       1. +-lft-identity N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \left(0 + \color{blue}{im \cdot re}\right)\right) \]

       2. flip-+ N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \left(\frac{0 \cdot 0 - \left(im \cdot re\right) \cdot \left(im \cdot re\right)}{\color{blue}{0 - im \cdot re}}\right)\right) \]

       3. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \left(\frac{0 - \left(im \cdot re\right) \cdot \left(im \cdot re\right)}{0 - im \cdot re}\right)\right) \]

       4. sub0-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \left(\frac{\mathsf{neg}\left(\left(im \cdot re\right) \cdot \left(im \cdot re\right)\right)}{\color{blue}{0} - im \cdot re}\right)\right) \]

       5. +-lft-identity N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \left(\frac{\mathsf{neg}\left(\left(im \cdot re\right) \cdot \left(0 + im \cdot re\right)\right)}{0 - im \cdot re}\right)\right) \]

       6. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \left(\frac{\mathsf{neg}\left(\left(im \cdot re\right) \cdot \left(im \cdot re + 0\right)\right)}{0 - im \cdot re}\right)\right) \]

       7. distribute-rgt-out N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \left(\frac{\mathsf{neg}\left(\left(\left(im \cdot re\right) \cdot \left(im \cdot re\right) + 0 \cdot \left(im \cdot re\right)\right)\right)}{0 - im \cdot re}\right)\right) \]

       8. +-lft-identity N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \left(\frac{\mathsf{neg}\left(\left(0 + \left(\left(im \cdot re\right) \cdot \left(im \cdot re\right) + 0 \cdot \left(im \cdot re\right)\right)\right)\right)}{0 - im \cdot re}\right)\right) \]

       9. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \left(\frac{\mathsf{neg}\left(\left(0 \cdot 0 + \left(\left(im \cdot re\right) \cdot \left(im \cdot re\right) + 0 \cdot \left(im \cdot re\right)\right)\right)\right)}{0 - im \cdot re}\right)\right) \]

       10. sub0-neg N/A

           \[\leadsto \mathsf{\_.f64}\left(0, \left(\frac{\mathsf{neg}\left(\left(0 \cdot 0 + \left(\left(im \cdot re\right) \cdot \left(im \cdot re\right) + 0 \cdot \left(im \cdot re\right)\right)\right)\right)}{\mathsf{neg}\left(im \cdot re\right)}\right)\right) \]

       11. frac-2neg N/A

           \[\leadsto \mathsf{\_.f64}\left(0, \left(\frac{0 \cdot 0 + \left(\left(im \cdot re\right) \cdot \left(im \cdot re\right) + 0 \cdot \left(im \cdot re\right)\right)}{\color{blue}{im \cdot re}}\right)\right) \]

       12. metadata-eval N/A

           \[\leadsto \mathsf{\_.f64}\left(0, \left(\frac{0 + \left(\left(im \cdot re\right) \cdot \left(im \cdot re\right) + 0 \cdot \left(im \cdot re\right)\right)}{im \cdot re}\right)\right) \]

       13. +-lft-identity N/A

           \[\leadsto \mathsf{\_.f64}\left(0, \left(\frac{\left(im \cdot re\right) \cdot \left(im \cdot re\right) + 0 \cdot \left(im \cdot re\right)}{\color{blue}{im} \cdot re}\right)\right) \]

       14. distribute-rgt-out N/A

           \[\leadsto \mathsf{\_.f64}\left(0, \left(\frac{\left(im \cdot re\right) \cdot \left(im \cdot re + 0\right)}{\color{blue}{im} \cdot re}\right)\right) \]

       15. +-commutative N/A

           \[\leadsto \mathsf{\_.f64}\left(0, \left(\frac{\left(im \cdot re\right) \cdot \left(0 + im \cdot re\right)}{im \cdot re}\right)\right) \]

       16. +-lft-identity N/A

           \[\leadsto \mathsf{\_.f64}\left(0, \left(\frac{\left(im \cdot re\right) \cdot \left(im \cdot re\right)}{im \cdot re}\right)\right) \]

       17. *-commutative N/A

           \[\leadsto \mathsf{\_.f64}\left(0, \left(\frac{\left(im \cdot re\right) \cdot \left(re \cdot im\right)}{im \cdot re}\right)\right) \]

       18. associate-*r* N/A

           \[\leadsto \mathsf{\_.f64}\left(0, \left(\frac{\left(\left(im \cdot re\right) \cdot re\right) \cdot im}{\color{blue}{im} \cdot re}\right)\right) \]

       19. *-commutative N/A

           \[\leadsto \mathsf{\_.f64}\left(0, \left(\frac{\left(\left(im \cdot re\right) \cdot re\right) \cdot im}{re \cdot \color{blue}{im}}\right)\right) \]

   12. Applied egg-rr 18.4%

       \[\leadsto 0 - \color{blue}{\frac{\left(re \cdot im\right) \cdot re}{re} \cdot \frac{im}{im}} \]
   13. Step-by-step derivation

       1. sub0-neg N/A

          \[\leadsto \mathsf{neg}\left(\frac{\left(re \cdot im\right) \cdot re}{re} \cdot \frac{im}{im}\right) \]

       2. associate-*r/ N/A

          \[\leadsto \mathsf{neg}\left(\frac{\frac{\left(re \cdot im\right) \cdot re}{re} \cdot im}{im}\right) \]

       3. distribute-neg-frac N/A

          \[\leadsto \frac{\mathsf{neg}\left(\frac{\left(re \cdot im\right) \cdot re}{re} \cdot im\right)}{\color{blue}{im}} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{\left(re \cdot im\right) \cdot re}{re} \cdot im\right)\right), \color{blue}{im}\right) \]

       5. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\left(\frac{\left(re \cdot im\right) \cdot re}{re} \cdot im\right)\right), im\right) \]

       6. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\left(\left(\left(re \cdot im\right) \cdot \frac{re}{re}\right) \cdot im\right)\right), im\right) \]

       7. *-inverses N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\left(\left(\left(re \cdot im\right) \cdot 1\right) \cdot im\right)\right), im\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\left(\left(\left(re \cdot im\right) \cdot \left(\mathsf{neg}\left(-1\right)\right)\right) \cdot im\right)\right), im\right) \]

       9. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\left(\left(re \cdot im\right) \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot im\right)\right)\right), im\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(re \cdot im\right), \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot im\right)\right)\right), im\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(im \cdot re\right), \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot im\right)\right)\right), im\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, re\right), \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot im\right)\right)\right), im\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, re\right), \mathsf{*.f64}\left(\left(\mathsf{neg}\left(-1\right)\right), im\right)\right)\right), im\right) \]

       14. metadata-eval 32.1%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, re\right), \mathsf{*.f64}\left(1, im\right)\right)\right), im\right) \]

   14. Applied egg-rr 32.1%

       \[\leadsto \color{blue}{\frac{-\left(im \cdot re\right) \cdot \left(1 \cdot im\right)}{im}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 6.5 \cdot 10^{-13}:\\ \;\;\;\;\left(0 - im\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot \left(im \cdot re\right)}{0 - im}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 36.4% accurate, 22.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 8 \cdot 10^{+50}:\\ \;\;\;\;\left(0 - im\_m\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\frac{re \cdot re}{-1} \cdot \frac{im\_m}{re}\\ \end{array} \end{array} \]

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 8e+50) (* (- 0.0 im_m) re) (* (/ (* re re) -1.0) (/ im_m re)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 8e+50) {
		tmp = (0.0 - im_m) * re;
	} else {
		tmp = ((re * re) / -1.0) * (im_m / re);
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 8d+50) then
        tmp = (0.0d0 - im_m) * re
    else
        tmp = ((re * re) / (-1.0d0)) * (im_m / re)
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 8e+50) {
		tmp = (0.0 - im_m) * re;
	} else {
		tmp = ((re * re) / -1.0) * (im_m / re);
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 8e+50:
		tmp = (0.0 - im_m) * re
	else:
		tmp = ((re * re) / -1.0) * (im_m / re)
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 8e+50)
		tmp = Float64(Float64(0.0 - im_m) * re);
	else
		tmp = Float64(Float64(Float64(re * re) / -1.0) * Float64(im_m / re));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 8e+50)
		tmp = (0.0 - im_m) * re;
	else
		tmp = ((re * re) / -1.0) * (im_m / re);
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 8e+50], N[(N[(0.0 - im$95$m), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] / -1.0), $MachinePrecision] * N[(im$95$m / re), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 8.0000000000000006e50

   1. Initial program 55.6%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sin re \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\sin re \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\sin re, \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)}\right)\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} - e^{im}\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\left(e^{\mathsf{neg}\left(im\right)}\right), \color{blue}{\left(e^{im}\right)}\right)\right)\right) \]

      6. exp-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\left(\frac{1}{e^{im}}\right), \left(e^{\color{blue}{im}}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(e^{im}\right)\right), \left(e^{\color{blue}{im}}\right)\right)\right)\right) \]

      8. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(im\right)\right), \left(e^{im}\right)\right)\right)\right) \]

      9. exp-lowering-exp.f64 55.6%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(im\right)\right), \mathsf{exp.f64}\left(im\right)\right)\right)\right) \]

   3. Simplified 55.6%

      \[\leadsto \color{blue}{0.5 \cdot \left(\sin re \cdot \left(\frac{1}{e^{im}} - e^{im}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(im \cdot \sin re\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{im \cdot \sin re} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(im \cdot \sin re\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \color{blue}{\sin re}\right)\right) \]

      5. sin-lowering-sin.f64 70.0%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \mathsf{sin.f64}\left(re\right)\right)\right) \]

   7. Simplified 70.0%

      \[\leadsto \color{blue}{0 - im \cdot \sin re} \]

   8. Taylor expanded in re around 0

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \color{blue}{re}\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 44.6%

       \[\leadsto 0 - im \cdot \color{blue}{re} \]
   11. Step-by-step derivation

       1. sub0-neg N/A

          \[\leadsto \mathsf{neg}\left(im \cdot re\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(re \cdot im\right) \]

       3. distribute-lft-neg-in N/A

          \[\leadsto \left(\mathsf{neg}\left(re\right)\right) \cdot \color{blue}{im} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{neg}\left(re\right)\right), \color{blue}{im}\right) \]

       5. neg-lowering-neg.f64 44.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{neg.f64}\left(re\right), im\right) \]

   12. Applied egg-rr 44.6%

       \[\leadsto \color{blue}{\left(-re\right) \cdot im} \]

   if 8.0000000000000006e50 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sin re \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\sin re \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\sin re, \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)}\right)\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} - e^{im}\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\left(e^{\mathsf{neg}\left(im\right)}\right), \color{blue}{\left(e^{im}\right)}\right)\right)\right) \]

      6. exp-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\left(\frac{1}{e^{im}}\right), \left(e^{\color{blue}{im}}\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(e^{im}\right)\right), \left(e^{\color{blue}{im}}\right)\right)\right)\right) \]

      8. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(im\right)\right), \left(e^{im}\right)\right)\right)\right) \]

      9. exp-lowering-exp.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(im\right)\right), \mathsf{exp.f64}\left(im\right)\right)\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(\sin re \cdot \left(\frac{1}{e^{im}} - e^{im}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(im \cdot \sin re\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{im \cdot \sin re} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(im \cdot \sin re\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \color{blue}{\sin re}\right)\right) \]

      5. sin-lowering-sin.f64 5.4%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \mathsf{sin.f64}\left(re\right)\right)\right) \]

   7. Simplified 5.4%

      \[\leadsto \color{blue}{0 - im \cdot \sin re} \]

   8. Taylor expanded in re around 0

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \color{blue}{re}\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 17.2%

       \[\leadsto 0 - im \cdot \color{blue}{re} \]
   11. Step-by-step derivation

       1. +-lft-identity N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \left(0 + \color{blue}{im \cdot re}\right)\right) \]

       2. flip-+ N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \left(\frac{0 \cdot 0 - \left(im \cdot re\right) \cdot \left(im \cdot re\right)}{\color{blue}{0 - im \cdot re}}\right)\right) \]

       3. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \left(\frac{0 - \left(im \cdot re\right) \cdot \left(im \cdot re\right)}{0 - im \cdot re}\right)\right) \]

       4. sub0-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \left(\frac{\mathsf{neg}\left(\left(im \cdot re\right) \cdot \left(im \cdot re\right)\right)}{\color{blue}{0} - im \cdot re}\right)\right) \]

       5. +-lft-identity N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \left(\frac{\mathsf{neg}\left(\left(im \cdot re\right) \cdot \left(0 + im \cdot re\right)\right)}{0 - im \cdot re}\right)\right) \]

       6. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \left(\frac{\mathsf{neg}\left(\left(im \cdot re\right) \cdot \left(im \cdot re + 0\right)\right)}{0 - im \cdot re}\right)\right) \]

       7. distribute-rgt-out N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \left(\frac{\mathsf{neg}\left(\left(\left(im \cdot re\right) \cdot \left(im \cdot re\right) + 0 \cdot \left(im \cdot re\right)\right)\right)}{0 - im \cdot re}\right)\right) \]

       8. +-lft-identity N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \left(\frac{\mathsf{neg}\left(\left(0 + \left(\left(im \cdot re\right) \cdot \left(im \cdot re\right) + 0 \cdot \left(im \cdot re\right)\right)\right)\right)}{0 - im \cdot re}\right)\right) \]

       9. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \left(\frac{\mathsf{neg}\left(\left(0 \cdot 0 + \left(\left(im \cdot re\right) \cdot \left(im \cdot re\right) + 0 \cdot \left(im \cdot re\right)\right)\right)\right)}{0 - im \cdot re}\right)\right) \]

       10. sub0-neg N/A

           \[\leadsto \mathsf{\_.f64}\left(0, \left(\frac{\mathsf{neg}\left(\left(0 \cdot 0 + \left(\left(im \cdot re\right) \cdot \left(im \cdot re\right) + 0 \cdot \left(im \cdot re\right)\right)\right)\right)}{\mathsf{neg}\left(im \cdot re\right)}\right)\right) \]

       11. frac-2neg N/A

           \[\leadsto \mathsf{\_.f64}\left(0, \left(\frac{0 \cdot 0 + \left(\left(im \cdot re\right) \cdot \left(im \cdot re\right) + 0 \cdot \left(im \cdot re\right)\right)}{\color{blue}{im \cdot re}}\right)\right) \]

       12. metadata-eval N/A

           \[\leadsto \mathsf{\_.f64}\left(0, \left(\frac{0 + \left(\left(im \cdot re\right) \cdot \left(im \cdot re\right) + 0 \cdot \left(im \cdot re\right)\right)}{im \cdot re}\right)\right) \]

       13. +-lft-identity N/A

           \[\leadsto \mathsf{\_.f64}\left(0, \left(\frac{\left(im \cdot re\right) \cdot \left(im \cdot re\right) + 0 \cdot \left(im \cdot re\right)}{\color{blue}{im} \cdot re}\right)\right) \]

       14. distribute-rgt-out N/A

           \[\leadsto \mathsf{\_.f64}\left(0, \left(\frac{\left(im \cdot re\right) \cdot \left(im \cdot re + 0\right)}{\color{blue}{im} \cdot re}\right)\right) \]

       15. +-commutative N/A

           \[\leadsto \mathsf{\_.f64}\left(0, \left(\frac{\left(im \cdot re\right) \cdot \left(0 + im \cdot re\right)}{im \cdot re}\right)\right) \]

       16. +-lft-identity N/A

           \[\leadsto \mathsf{\_.f64}\left(0, \left(\frac{\left(im \cdot re\right) \cdot \left(im \cdot re\right)}{im \cdot re}\right)\right) \]

       17. *-commutative N/A

           \[\leadsto \mathsf{\_.f64}\left(0, \left(\frac{\left(im \cdot re\right) \cdot \left(re \cdot im\right)}{im \cdot re}\right)\right) \]

       18. associate-*r* N/A

           \[\leadsto \mathsf{\_.f64}\left(0, \left(\frac{\left(\left(im \cdot re\right) \cdot re\right) \cdot im}{\color{blue}{im} \cdot re}\right)\right) \]

       19. *-commutative N/A

           \[\leadsto \mathsf{\_.f64}\left(0, \left(\frac{\left(\left(im \cdot re\right) \cdot re\right) \cdot im}{re \cdot \color{blue}{im}}\right)\right) \]

   12. Applied egg-rr 20.8%

       \[\leadsto 0 - \color{blue}{\frac{\left(re \cdot im\right) \cdot re}{re} \cdot \frac{im}{im}} \]
   13. Step-by-step derivation

       1. *-inverses N/A

          \[\leadsto 0 - \frac{\left(re \cdot im\right) \cdot re}{re} \cdot 1 \]

       2. metadata-eval N/A

          \[\leadsto 0 - \frac{\left(re \cdot im\right) \cdot re}{re} \cdot \left(\mathsf{neg}\left(-1\right)\right) \]

       3. cancel-sign-sub-inv N/A

          \[\leadsto 0 + \color{blue}{\left(\mathsf{neg}\left(\frac{\left(re \cdot im\right) \cdot re}{re}\right)\right) \cdot \left(\mathsf{neg}\left(-1\right)\right)} \]

       4. +-lft-identity N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{\left(re \cdot im\right) \cdot re}{re}\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \]

       5. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{\left(re \cdot im\right) \cdot re}{re}\right)\right) \cdot 1 \]

       6. *-rgt-identity N/A

          \[\leadsto \mathsf{neg}\left(\frac{\left(re \cdot im\right) \cdot re}{re}\right) \]

       7. distribute-neg-frac2 N/A

          \[\leadsto \frac{\left(re \cdot im\right) \cdot re}{\color{blue}{\mathsf{neg}\left(re\right)}} \]

       8. *-commutative N/A

          \[\leadsto \frac{re \cdot \left(re \cdot im\right)}{\mathsf{neg}\left(\color{blue}{re}\right)} \]

       9. associate-*r* N/A

          \[\leadsto \frac{\left(re \cdot re\right) \cdot im}{\mathsf{neg}\left(\color{blue}{re}\right)} \]

       10. neg-mul-1 N/A

           \[\leadsto \frac{\left(re \cdot re\right) \cdot im}{-1 \cdot \color{blue}{re}} \]

       11. times-frac N/A

           \[\leadsto \frac{re \cdot re}{-1} \cdot \color{blue}{\frac{im}{re}} \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\left(\frac{re \cdot re}{-1}\right), \color{blue}{\left(\frac{im}{re}\right)}\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(re \cdot re\right), -1\right), \left(\frac{\color{blue}{im}}{re}\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(re, re\right), -1\right), \left(\frac{im}{re}\right)\right) \]

       15. /-lowering-/.f64 26.5%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(re, re\right), -1\right), \mathsf{/.f64}\left(im, \color{blue}{re}\right)\right) \]

   14. Applied egg-rr 26.5%

       \[\leadsto \color{blue}{\frac{re \cdot re}{-1} \cdot \frac{im}{re}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 8 \cdot 10^{+50}:\\ \;\;\;\;\left(0 - im\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\frac{re \cdot re}{-1} \cdot \frac{im}{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 49.9% accurate, 28.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(im\_m \cdot \left(re \cdot \left(-1 + -0.16666666666666666 \cdot \left(im\_m \cdot im\_m\right)\right)\right)\right) \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (* im_s (* im_m (* re (+ -1.0 (* -0.16666666666666666 (* im_m im_m)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * (im_m * (re * (-1.0 + (-0.16666666666666666 * (im_m * im_m)))));
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = im_s * (im_m * (re * ((-1.0d0) + ((-0.16666666666666666d0) * (im_m * im_m)))))
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	return im_s * (im_m * (re * (-1.0 + (-0.16666666666666666 * (im_m * im_m)))));
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * (im_m * (re * (-1.0 + (-0.16666666666666666 * (im_m * im_m)))))

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * Float64(im_m * Float64(re * Float64(-1.0 + Float64(-0.16666666666666666 * Float64(im_m * im_m))))))
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * (im_m * (re * (-1.0 + (-0.16666666666666666 * (im_m * im_m)))));
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * N[(im$95$m * N[(re * N[(-1.0 + N[(-0.16666666666666666 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in im around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)}\right) \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)}\right)\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) + -2\right)\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left(-2 + \color{blue}{{im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right)}\right)\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right)\right)}\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{-1}{60} \cdot {im}^{2}} - \frac{1}{3}\right)\right)\right)\right)\right) \]

   7. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right)\right)}\right)\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right)\right)}\right)\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right)}\right)\right)\right)\right)\right) \]

   10. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{60} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}\right)\right)\right)\right)\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{60} \cdot {im}^{2} + \frac{-1}{3}\right)\right)\right)\right)\right)\right) \]

   12. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left(\frac{-1}{3} + \color{blue}{\frac{-1}{60} \cdot {im}^{2}}\right)\right)\right)\right)\right)\right) \]

   13. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{\left(\frac{-1}{60} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \left({im}^{2} \cdot \color{blue}{\frac{-1}{60}}\right)\right)\right)\right)\right)\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{-1}{60}}\right)\right)\right)\right)\right)\right)\right) \]

   16. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]

   17. *-lowering-*.f64 93.4%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]

5. Simplified 93.4%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left(-2 + im \cdot \left(im \cdot \left(-0.3333333333333333 + \left(im \cdot im\right) \cdot -0.016666666666666666\right)\right)\right)\right)} \]

6. Taylor expanded in re around 0

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{1}{2} \cdot re\right)}, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]
7. Step-by-step derivation

   1. *-lowering-*.f64 61.3%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, re\right), \mathsf{*.f64}\left(\color{blue}{im}, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]

8. Simplified 61.3%

   \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot \left(im \cdot \left(-2 + im \cdot \left(im \cdot \left(-0.3333333333333333 + \left(im \cdot im\right) \cdot -0.016666666666666666\right)\right)\right)\right) \]

9. Taylor expanded in im around 0

   \[\leadsto \color{blue}{im \cdot \left(-1 \cdot re + \frac{-1}{6} \cdot \left({im}^{2} \cdot re\right)\right)} \]
10. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto im \cdot \left(-1 \cdot re + \left({im}^{2} \cdot re\right) \cdot \color{blue}{\frac{-1}{6}}\right) \]

    2. associate-*r* N/A

       \[\leadsto im \cdot \left(-1 \cdot re + {im}^{2} \cdot \color{blue}{\left(re \cdot \frac{-1}{6}\right)}\right) \]

    3. *-commutative N/A

       \[\leadsto im \cdot \left(-1 \cdot re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \color{blue}{re}\right)\right) \]

    4. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(-1 \cdot re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot re\right)\right)}\right) \]

    5. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left(\frac{-1}{6} \cdot re\right) + \color{blue}{-1 \cdot re}\right)\right) \]

    6. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left(re \cdot \frac{-1}{6}\right) + -1 \cdot re\right)\right) \]

    7. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(im, \left(\left({im}^{2} \cdot re\right) \cdot \frac{-1}{6} + \color{blue}{-1} \cdot re\right)\right) \]

    8. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(im, \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot re\right) + \color{blue}{-1} \cdot re\right)\right) \]

    9. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(im, \left(\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot re + \color{blue}{-1} \cdot re\right)\right) \]

    10. distribute-rgt-out N/A

        \[\leadsto \mathsf{*.f64}\left(im, \left(re \cdot \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2} + -1\right)}\right)\right) \]

    11. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2} + -1\right)}\right)\right) \]

    12. +-commutative N/A

        \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, \left(-1 + \color{blue}{\frac{-1}{6} \cdot {im}^{2}}\right)\right)\right) \]

    13. +-lowering-+.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(-1, \color{blue}{\left(\frac{-1}{6} \cdot {im}^{2}\right)}\right)\right)\right) \]

    14. *-commutative N/A

        \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(-1, \left({im}^{2} \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

    15. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

    16. unpow2 N/A

        \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{-1}{6}\right)\right)\right)\right) \]

    17. *-lowering-*.f64 54.7%

        \[\leadsto \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{6}\right)\right)\right)\right) \]

11. Simplified 54.7%

    \[\leadsto \color{blue}{im \cdot \left(re \cdot \left(-1 + \left(im \cdot im\right) \cdot -0.16666666666666666\right)\right)} \]

12. Final simplification 54.7%

    \[\leadsto im \cdot \left(re \cdot \left(-1 + -0.16666666666666666 \cdot \left(im \cdot im\right)\right)\right) \]
13. Add Preprocessing

Alternative 26: 33.0% accurate, 61.6× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(\left(0 - im\_m\right) \cdot re\right) \end{array} \]

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) 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 * ((0.0 - im_m) * re);
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = im_s * ((0.0d0 - im_m) * re)
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	return im_s * ((0.0 - im_m) * re);
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * ((0.0 - im_m) * re)

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * Float64(Float64(0.0 - im_m) * 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 * ((0.0 - im_m) * re);
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * N[(N[(0.0 - im$95$m), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
2. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sin re \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right)} \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\sin re \cdot \left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)\right)}\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\sin re, \color{blue}{\left(e^{\mathsf{neg}\left(im\right)} - e^{im}\right)}\right)\right) \]

   4. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(\color{blue}{e^{\mathsf{neg}\left(im\right)}} - e^{im}\right)\right)\right) \]

   5. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\left(e^{\mathsf{neg}\left(im\right)}\right), \color{blue}{\left(e^{im}\right)}\right)\right)\right) \]

   6. exp-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\left(\frac{1}{e^{im}}\right), \left(e^{\color{blue}{im}}\right)\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(e^{im}\right)\right), \left(e^{\color{blue}{im}}\right)\right)\right)\right) \]

   8. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(im\right)\right), \left(e^{im}\right)\right)\right)\right) \]

   9. exp-lowering-exp.f64 63.9%

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(im\right)\right), \mathsf{exp.f64}\left(im\right)\right)\right)\right) \]

3. Simplified 63.9%

   \[\leadsto \color{blue}{0.5 \cdot \left(\sin re \cdot \left(\frac{1}{e^{im}} - e^{im}\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in im around 0

   \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right)} \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(im \cdot \sin re\right) \]

   2. neg-sub0 N/A

      \[\leadsto 0 - \color{blue}{im \cdot \sin re} \]

   3. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(im \cdot \sin re\right)}\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \color{blue}{\sin re}\right)\right) \]

   5. sin-lowering-sin.f64 57.9%

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \mathsf{sin.f64}\left(re\right)\right)\right) \]

7. Simplified 57.9%

   \[\leadsto \color{blue}{0 - im \cdot \sin re} \]

8. Taylor expanded in re around 0

   \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \color{blue}{re}\right)\right) \]
9. Step-by-step derivation

10. Simplified 39.5%

    \[\leadsto 0 - im \cdot \color{blue}{re} \]
11. Step-by-step derivation

    1. sub0-neg N/A

       \[\leadsto \mathsf{neg}\left(im \cdot re\right) \]

    2. *-commutative N/A

       \[\leadsto \mathsf{neg}\left(re \cdot im\right) \]

    3. distribute-lft-neg-in N/A

       \[\leadsto \left(\mathsf{neg}\left(re\right)\right) \cdot \color{blue}{im} \]

    4. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{neg}\left(re\right)\right), \color{blue}{im}\right) \]

    5. neg-lowering-neg.f64 39.5%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{neg.f64}\left(re\right), im\right) \]

12. Applied egg-rr 39.5%

    \[\leadsto \color{blue}{\left(-re\right) \cdot im} \]

13. Final simplification 39.5%

    \[\leadsto \left(0 - im\right) \cdot re \]
14. Add Preprocessing

Developer Target 1: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|im\right| < 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(0.16666666666666666 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (< (fabs im) 1.0)
   (-
    (*
     (sin re)
     (+
      (+ im (* (* (* 0.16666666666666666 im) im) im))
      (* (* (* (* (* 0.008333333333333333 im) im) im) im) im))))
   (* (* 0.5 (sin re)) (- (exp (- im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if (fabs(im) < 1.0) {
		tmp = -(sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * sin(re)) * (exp(-im) - exp(im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (abs(im) < 1.0d0) then
        tmp = -(sin(re) * ((im + (((0.16666666666666666d0 * im) * im) * im)) + (((((0.008333333333333333d0 * im) * im) * im) * im) * im)))
    else
        tmp = (0.5d0 * sin(re)) * (exp(-im) - exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.abs(im) < 1.0) {
		tmp = -(Math.sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * Math.sin(re)) * (Math.exp(-im) - Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.fabs(im) < 1.0:
		tmp = -(math.sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)))
	else:
		tmp = (0.5 * math.sin(re)) * (math.exp(-im) - math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (abs(im) < 1.0)
		tmp = Float64(-Float64(sin(re) * Float64(Float64(im + Float64(Float64(Float64(0.16666666666666666 * im) * im) * im)) + Float64(Float64(Float64(Float64(Float64(0.008333333333333333 * im) * im) * im) * im) * im))));
	else
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (abs(im) < 1.0)
		tmp = -(sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	else
		tmp = (0.5 * sin(re)) * (exp(-im) - exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Less[N[Abs[im], $MachinePrecision], 1.0], (-N[(N[Sin[re], $MachinePrecision] * N[(N[(im + N[(N[(N[(0.16666666666666666 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[(0.008333333333333333 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024191 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (fabs im) 1) (- (* (sin re) (+ im (* 1/6 im im im) (* 1/120 im im im im im)))) (* (* 1/2 (sin re)) (- (exp (- im)) (exp im)))))

  (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))