math.cube on complex, imaginary part

Percentage Accurate: 82.7% → 99.2%

Time: 9.3s

Alternatives: 10

Speedup: 0.5×

• 43.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (+
  (* (- (* x.re x.re) (* x.im x.im)) x.im)
  (* (+ (* x.re x.im) (* x.im x.re)) x.re)))

C

double code(double x_46_re, double x_46_im) {
	return (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = (((x_46re * x_46re) - (x_46im * x_46im)) * x_46im) + (((x_46re * x_46im) + (x_46im * x_46re)) * x_46re)
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
}

Python

def code(x_46_re, x_46_im):
	return (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re)

Julia

function code(x_46_re, x_46_im)
	return Float64(Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im)) * x_46_im) + Float64(Float64(Float64(x_46_re * x_46_im) + Float64(x_46_im * x_46_re)) * x_46_re))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
end

Wolfram

code[x$46$re_, x$46$im_] := N[(N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision] + N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]

Initial Program: 82.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (+
  (* (- (* x.re x.re) (* x.im x.im)) x.im)
  (* (+ (* x.re x.im) (* x.im x.re)) x.re)))

C

double code(double x_46_re, double x_46_im) {
	return (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = (((x_46re * x_46re) - (x_46im * x_46im)) * x_46im) + (((x_46re * x_46im) + (x_46im * x_46re)) * x_46re)
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
}

Python

def code(x_46_re, x_46_im):
	return (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re)

Julia

function code(x_46_re, x_46_im)
	return Float64(Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im)) * x_46_im) + Float64(Float64(Float64(x_46_re * x_46_im) + Float64(x_46_im * x_46_re)) * x_46_re))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
end

Wolfram

code[x$46$re_, x$46$im_] := N[(N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision] + N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 2.55 \cdot 10^{-51}:\\ \;\;\;\;\left(\left(x.im\_m + x.im\_m\right) \cdot x.re\right) \cdot x.re + \frac{x.re - x.im\_m}{\frac{\frac{{\left(x.im\_m - x.re\right)}^{-1}}{x.im\_m}}{x.re + x.im\_m} \cdot \left(x.im\_m - x.re\right)}\\ \mathbf{else}:\\ \;\;\;\;{x.im\_m}^{3} \cdot \mathsf{fma}\left(\frac{3 \cdot x.re}{x.im\_m}, \frac{x.re}{x.im\_m}, -1\right)\\ \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.im_m 2.55e-51)
    (+
     (* (* (+ x.im_m x.im_m) x.re) x.re)
     (/
      (- x.re x.im_m)
      (*
       (/ (/ (pow (- x.im_m x.re) -1.0) x.im_m) (+ x.re x.im_m))
       (- x.im_m x.re))))
    (* (pow x.im_m 3.0) (fma (/ (* 3.0 x.re) x.im_m) (/ x.re x.im_m) -1.0)))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 2.55e-51) {
		tmp = (((x_46_im_m + x_46_im_m) * x_46_re) * x_46_re) + ((x_46_re - x_46_im_m) / (((pow((x_46_im_m - x_46_re), -1.0) / x_46_im_m) / (x_46_re + x_46_im_m)) * (x_46_im_m - x_46_re)));
	} else {
		tmp = pow(x_46_im_m, 3.0) * fma(((3.0 * x_46_re) / x_46_im_m), (x_46_re / x_46_im_m), -1.0);
	}
	return x_46_im_s * tmp;
}

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 2.55e-51)
		tmp = Float64(Float64(Float64(Float64(x_46_im_m + x_46_im_m) * x_46_re) * x_46_re) + Float64(Float64(x_46_re - x_46_im_m) / Float64(Float64(Float64((Float64(x_46_im_m - x_46_re) ^ -1.0) / x_46_im_m) / Float64(x_46_re + x_46_im_m)) * Float64(x_46_im_m - x_46_re))));
	else
		tmp = Float64((x_46_im_m ^ 3.0) * fma(Float64(Float64(3.0 * x_46_re) / x_46_im_m), Float64(x_46_re / x_46_im_m), -1.0));
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 2.55e-51], N[(N[(N[(N[(x$46$im$95$m + x$46$im$95$m), $MachinePrecision] * x$46$re), $MachinePrecision] * x$46$re), $MachinePrecision] + N[(N[(x$46$re - x$46$im$95$m), $MachinePrecision] / N[(N[(N[(N[Power[N[(x$46$im$95$m - x$46$re), $MachinePrecision], -1.0], $MachinePrecision] / x$46$im$95$m), $MachinePrecision] / N[(x$46$re + x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$46$im$95$m - x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x$46$im$95$m, 3.0], $MachinePrecision] * N[(N[(N[(3.0 * x$46$re), $MachinePrecision] / x$46$im$95$m), $MachinePrecision] * N[(x$46$re / x$46$im$95$m), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 2.5499999999999999e-51

   1. Initial program 81.5%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 93.9

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

   4. Applied rewrites 93.9%

      \[\leadsto \color{blue}{\left(x.im \cdot \left(x.im + x.re\right)\right) \cdot \left(x.re - x.im\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 93.9

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

   6. Applied rewrites 93.9%

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

   8. Applied rewrites 93.9%

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

   if 2.5499999999999999e-51 < x.im

   1. Initial program 90.8%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip3-- N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. pow-pow N/A

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

      11. lower-pow.f64 N/A

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

      12. metadata-eval N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

          \[\leadsto \frac{\left({x.re}^{6} - {\color{blue}{\left({x.im}^{2}\right)}}^{3}\right) \cdot x.im}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

      15. pow-pow N/A

          \[\leadsto \frac{\left({x.re}^{6} - \color{blue}{{x.im}^{\left(2 \cdot 3\right)}}\right) \cdot x.im}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

      16. lower-pow.f64 N/A

          \[\leadsto \frac{\left({x.re}^{6} - \color{blue}{{x.im}^{\left(2 \cdot 3\right)}}\right) \cdot x.im}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

      17. metadata-eval N/A

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

   4. Applied rewrites 14.0%

      \[\leadsto \color{blue}{\frac{\left({x.re}^{6} - {x.im}^{6}\right) \cdot x.im}{\mathsf{fma}\left(x.im \cdot x.im, \mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right), {x.re}^{4}\right)}} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

   5. Taylor expanded in x.im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft1-in N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. times-frac N/A

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

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{3 \cdot x.re}{x.im}, \frac{x.re}{x.im}, -1\right)} \cdot {x.im}^{3} \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{3 \cdot x.re}{x.im}}, \frac{x.re}{x.im}, -1\right) \cdot {x.im}^{3} \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{3 \cdot x.re}}{x.im}, \frac{x.re}{x.im}, -1\right) \cdot {x.im}^{3} \]

      15. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{3 \cdot x.re}{x.im}, \color{blue}{\frac{x.re}{x.im}}, -1\right) \cdot {x.im}^{3} \]

      16. lower-pow.f64 99.8

          \[\leadsto \mathsf{fma}\left(\frac{3 \cdot x.re}{x.im}, \frac{x.re}{x.im}, -1\right) \cdot \color{blue}{{x.im}^{3}} \]

   7. Applied rewrites 99.8%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 2.55 \cdot 10^{-51}:\\ \;\;\;\;\left(\left(x.im + x.im\right) \cdot x.re\right) \cdot x.re + \frac{x.re - x.im}{\frac{\frac{{\left(x.im - x.re\right)}^{-1}}{x.im}}{x.re + x.im} \cdot \left(x.im - x.re\right)}\\ \mathbf{else}:\\ \;\;\;\;{x.im}^{3} \cdot \mathsf{fma}\left(\frac{3 \cdot x.re}{x.im}, \frac{x.re}{x.im}, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 2.55 \cdot 10^{-51}:\\ \;\;\;\;\left(\left(x.re + x.im\_m\right) \cdot x.im\_m\right) \cdot \left(x.re - x.im\_m\right) + \left(\left(x.im\_m + x.im\_m\right) \cdot x.re\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;{x.im\_m}^{3} \cdot \mathsf{fma}\left(\frac{3 \cdot x.re}{x.im\_m}, \frac{x.re}{x.im\_m}, -1\right)\\ \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.im_m 2.55e-51)
    (+
     (* (* (+ x.re x.im_m) x.im_m) (- x.re x.im_m))
     (* (* (+ x.im_m x.im_m) x.re) x.re))
    (* (pow x.im_m 3.0) (fma (/ (* 3.0 x.re) x.im_m) (/ x.re x.im_m) -1.0)))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 2.55e-51) {
		tmp = (((x_46_re + x_46_im_m) * x_46_im_m) * (x_46_re - x_46_im_m)) + (((x_46_im_m + x_46_im_m) * x_46_re) * x_46_re);
	} else {
		tmp = pow(x_46_im_m, 3.0) * fma(((3.0 * x_46_re) / x_46_im_m), (x_46_re / x_46_im_m), -1.0);
	}
	return x_46_im_s * tmp;
}

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 2.55e-51)
		tmp = Float64(Float64(Float64(Float64(x_46_re + x_46_im_m) * x_46_im_m) * Float64(x_46_re - x_46_im_m)) + Float64(Float64(Float64(x_46_im_m + x_46_im_m) * x_46_re) * x_46_re));
	else
		tmp = Float64((x_46_im_m ^ 3.0) * fma(Float64(Float64(3.0 * x_46_re) / x_46_im_m), Float64(x_46_re / x_46_im_m), -1.0));
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 2.55e-51], N[(N[(N[(N[(x$46$re + x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] * N[(x$46$re - x$46$im$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x$46$im$95$m + x$46$im$95$m), $MachinePrecision] * x$46$re), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision], N[(N[Power[x$46$im$95$m, 3.0], $MachinePrecision] * N[(N[(N[(3.0 * x$46$re), $MachinePrecision] / x$46$im$95$m), $MachinePrecision] * N[(x$46$re / x$46$im$95$m), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 2.5499999999999999e-51

   1. Initial program 81.5%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 93.9

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

   4. Applied rewrites 93.9%

      \[\leadsto \color{blue}{\left(x.im \cdot \left(x.im + x.re\right)\right) \cdot \left(x.re - x.im\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 93.9

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

   6. Applied rewrites 93.9%

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

   if 2.5499999999999999e-51 < x.im

   1. Initial program 90.8%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip3-- N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. pow-pow N/A

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

      11. lower-pow.f64 N/A

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

      12. metadata-eval N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

          \[\leadsto \frac{\left({x.re}^{6} - {\color{blue}{\left({x.im}^{2}\right)}}^{3}\right) \cdot x.im}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

      15. pow-pow N/A

          \[\leadsto \frac{\left({x.re}^{6} - \color{blue}{{x.im}^{\left(2 \cdot 3\right)}}\right) \cdot x.im}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

      16. lower-pow.f64 N/A

          \[\leadsto \frac{\left({x.re}^{6} - \color{blue}{{x.im}^{\left(2 \cdot 3\right)}}\right) \cdot x.im}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

      17. metadata-eval N/A

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

   4. Applied rewrites 14.0%

      \[\leadsto \color{blue}{\frac{\left({x.re}^{6} - {x.im}^{6}\right) \cdot x.im}{\mathsf{fma}\left(x.im \cdot x.im, \mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right), {x.re}^{4}\right)}} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

   5. Taylor expanded in x.im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft1-in N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. times-frac N/A

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

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{3 \cdot x.re}{x.im}, \frac{x.re}{x.im}, -1\right)} \cdot {x.im}^{3} \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{3 \cdot x.re}{x.im}}, \frac{x.re}{x.im}, -1\right) \cdot {x.im}^{3} \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{3 \cdot x.re}}{x.im}, \frac{x.re}{x.im}, -1\right) \cdot {x.im}^{3} \]

      15. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{3 \cdot x.re}{x.im}, \color{blue}{\frac{x.re}{x.im}}, -1\right) \cdot {x.im}^{3} \]

      16. lower-pow.f64 99.8

          \[\leadsto \mathsf{fma}\left(\frac{3 \cdot x.re}{x.im}, \frac{x.re}{x.im}, -1\right) \cdot \color{blue}{{x.im}^{3}} \]

   7. Applied rewrites 99.8%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 2.55 \cdot 10^{-51}:\\ \;\;\;\;\left(\left(x.re + x.im\right) \cdot x.im\right) \cdot \left(x.re - x.im\right) + \left(\left(x.im + x.im\right) \cdot x.re\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;{x.im}^{3} \cdot \mathsf{fma}\left(\frac{3 \cdot x.re}{x.im}, \frac{x.re}{x.im}, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 10000000000000:\\ \;\;\;\;\left(\left(x.re + x.im\_m\right) \cdot x.im\_m\right) \cdot \left(x.re - x.im\_m\right) + \left(\left(x.im\_m + x.im\_m\right) \cdot x.re\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\left(x.im\_m \cdot x.im\_m\right) \cdot \left(\mathsf{fma}\left(3, {\left(\frac{x.im\_m}{x.re}\right)}^{-2}, -1\right) \cdot x.im\_m\right)\\ \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.im_m 10000000000000.0)
    (+
     (* (* (+ x.re x.im_m) x.im_m) (- x.re x.im_m))
     (* (* (+ x.im_m x.im_m) x.re) x.re))
    (*
     (* x.im_m x.im_m)
     (* (fma 3.0 (pow (/ x.im_m x.re) -2.0) -1.0) x.im_m)))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 10000000000000.0) {
		tmp = (((x_46_re + x_46_im_m) * x_46_im_m) * (x_46_re - x_46_im_m)) + (((x_46_im_m + x_46_im_m) * x_46_re) * x_46_re);
	} else {
		tmp = (x_46_im_m * x_46_im_m) * (fma(3.0, pow((x_46_im_m / x_46_re), -2.0), -1.0) * x_46_im_m);
	}
	return x_46_im_s * tmp;
}

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 10000000000000.0)
		tmp = Float64(Float64(Float64(Float64(x_46_re + x_46_im_m) * x_46_im_m) * Float64(x_46_re - x_46_im_m)) + Float64(Float64(Float64(x_46_im_m + x_46_im_m) * x_46_re) * x_46_re));
	else
		tmp = Float64(Float64(x_46_im_m * x_46_im_m) * Float64(fma(3.0, (Float64(x_46_im_m / x_46_re) ^ -2.0), -1.0) * x_46_im_m));
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 10000000000000.0], N[(N[(N[(N[(x$46$re + x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] * N[(x$46$re - x$46$im$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x$46$im$95$m + x$46$im$95$m), $MachinePrecision] * x$46$re), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision] * N[(N[(3.0 * N[Power[N[(x$46$im$95$m / x$46$re), $MachinePrecision], -2.0], $MachinePrecision] + -1.0), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 1e13

   1. Initial program 82.8%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 94.3

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

   4. Applied rewrites 94.3%

      \[\leadsto \color{blue}{\left(x.im \cdot \left(x.im + x.re\right)\right) \cdot \left(x.re - x.im\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 94.3

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

   6. Applied rewrites 94.3%

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

   if 1e13 < x.im

   1. Initial program 88.3%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip3-- N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. pow-pow N/A

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

      11. lower-pow.f64 N/A

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

      12. metadata-eval N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

          \[\leadsto \frac{\left({x.re}^{6} - {\color{blue}{\left({x.im}^{2}\right)}}^{3}\right) \cdot x.im}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

      15. pow-pow N/A

          \[\leadsto \frac{\left({x.re}^{6} - \color{blue}{{x.im}^{\left(2 \cdot 3\right)}}\right) \cdot x.im}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

      16. lower-pow.f64 N/A

          \[\leadsto \frac{\left({x.re}^{6} - \color{blue}{{x.im}^{\left(2 \cdot 3\right)}}\right) \cdot x.im}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

      17. metadata-eval N/A

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

   4. Applied rewrites 0.8%

      \[\leadsto \color{blue}{\frac{\left({x.re}^{6} - {x.im}^{6}\right) \cdot x.im}{\mathsf{fma}\left(x.im \cdot x.im, \mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right), {x.re}^{4}\right)}} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

   5. Taylor expanded in x.im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft1-in N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. times-frac N/A

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

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{3 \cdot x.re}{x.im}, \frac{x.re}{x.im}, -1\right)} \cdot {x.im}^{3} \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{3 \cdot x.re}{x.im}}, \frac{x.re}{x.im}, -1\right) \cdot {x.im}^{3} \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{3 \cdot x.re}}{x.im}, \frac{x.re}{x.im}, -1\right) \cdot {x.im}^{3} \]

      15. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{3 \cdot x.re}{x.im}, \color{blue}{\frac{x.re}{x.im}}, -1\right) \cdot {x.im}^{3} \]

      16. lower-pow.f64 99.9

          \[\leadsto \mathsf{fma}\left(\frac{3 \cdot x.re}{x.im}, \frac{x.re}{x.im}, -1\right) \cdot \color{blue}{{x.im}^{3}} \]

   7. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{3 \cdot x.re}{x.im}, \frac{x.re}{x.im}, -1\right) \cdot {x.im}^{3}} \]
   8. Step-by-step derivation

   9. Applied rewrites 99.9%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 10000000000000:\\ \;\;\;\;\left(\left(x.re + x.im\right) \cdot x.im\right) \cdot \left(x.re - x.im\right) + \left(\left(x.im + x.im\right) \cdot x.re\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\left(x.im \cdot x.im\right) \cdot \left(\mathsf{fma}\left(3, {\left(\frac{x.im}{x.re}\right)}^{-2}, -1\right) \cdot x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(x.re \cdot x.im\_m + x.re \cdot x.im\_m\right) \cdot x.re + \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 10^{+291}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im\_m, x.re + x.im\_m, 2 \cdot \left(x.re \cdot x.re\right)\right) \cdot x.im\_m\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left(\left(3 \cdot x.re\right) \cdot x.im\_m\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re + x.im\_m, \left(x.re - x.im\_m\right) \cdot x.im\_m, 2 \cdot x.im\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0
         (+
          (* (+ (* x.re x.im_m) (* x.re x.im_m)) x.re)
          (* (- (* x.re x.re) (* x.im_m x.im_m)) x.im_m))))
   (*
    x.im_s
    (if (<= t_0 1e+291)
      (* (fma (- x.re x.im_m) (+ x.re x.im_m) (* 2.0 (* x.re x.re))) x.im_m)
      (if (<= t_0 INFINITY)
        (* (* (* 3.0 x.re) x.im_m) x.re)
        (fma (+ x.re x.im_m) (* (- x.re x.im_m) x.im_m) (* 2.0 x.im_m)))))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = (((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)) * x_46_re) + (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m);
	double tmp;
	if (t_0 <= 1e+291) {
		tmp = fma((x_46_re - x_46_im_m), (x_46_re + x_46_im_m), (2.0 * (x_46_re * x_46_re))) * x_46_im_m;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = ((3.0 * x_46_re) * x_46_im_m) * x_46_re;
	} else {
		tmp = fma((x_46_re + x_46_im_m), ((x_46_re - x_46_im_m) * x_46_im_m), (2.0 * x_46_im_m));
	}
	return x_46_im_s * tmp;
}

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = Float64(Float64(Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_re * x_46_im_m)) * x_46_re) + Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m))
	tmp = 0.0
	if (t_0 <= 1e+291)
		tmp = Float64(fma(Float64(x_46_re - x_46_im_m), Float64(x_46_re + x_46_im_m), Float64(2.0 * Float64(x_46_re * x_46_re))) * x_46_im_m);
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(Float64(3.0 * x_46_re) * x_46_im_m) * x_46_re);
	else
		tmp = fma(Float64(x_46_re + x_46_im_m), Float64(Float64(x_46_re - x_46_im_m) * x_46_im_m), Float64(2.0 * x_46_im_m));
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$re * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision] + N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$0, 1e+291], N[(N[(N[(x$46$re - x$46$im$95$m), $MachinePrecision] * N[(x$46$re + x$46$im$95$m), $MachinePrecision] + N[(2.0 * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[(3.0 * x$46$re), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] * x$46$re), $MachinePrecision], N[(N[(x$46$re + x$46$im$95$m), $MachinePrecision] * N[(N[(x$46$re - x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] + N[(2.0 * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < 9.9999999999999996e290

   1. Initial program 95.6%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 99.7

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

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\left(x.im \cdot \left(x.im + x.re\right)\right) \cdot \left(x.re - x.im\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-+.f64 N/A

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

      9. flip-+ N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. div-inv N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 99.6%

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

   8. Applied rewrites 95.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.re \cdot x.re, 2 \cdot x.im, \left(\left(x.re - x.im\right) \cdot x.im\right) \cdot \left(x.im + x.re\right)\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x.re \cdot x.re}, 2 \cdot x.im, \left(\left(x.re - x.im\right) \cdot x.im\right) \cdot \left(x.im + x.re\right)\right) \]

      2. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\left(x.re \cdot x.re\right) \cdot \left(2 \cdot x.im\right) + \left(\left(x.re - x.im\right) \cdot x.im\right) \cdot \left(x.im + x.re\right)} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(x.re - x.im\right) \cdot x.im\right) \cdot \left(x.im + x.re\right) + \left(x.re \cdot x.re\right) \cdot \left(2 \cdot x.im\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x.re - x.im\right) \cdot x.im\right) \cdot \left(x.im + x.re\right)} + \left(x.re \cdot x.re\right) \cdot \left(2 \cdot x.im\right) \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(x.im + x.re\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right)} + \left(x.re \cdot x.re\right) \cdot \left(2 \cdot x.im\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(x.im + x.re\right) \cdot \color{blue}{\left(\left(x.re - x.im\right) \cdot x.im\right)} + \left(x.re \cdot x.re\right) \cdot \left(2 \cdot x.im\right) \]

      7. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re - x.im\right)\right) \cdot x.im} + \left(x.re \cdot x.re\right) \cdot \left(2 \cdot x.im\right) \]

      8. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(x.im + x.re\right)} \cdot \left(x.re - x.im\right)\right) \cdot x.im + \left(x.re \cdot x.re\right) \cdot \left(2 \cdot x.im\right) \]

      9. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(x.re + x.im\right)} \cdot \left(x.re - x.im\right)\right) \cdot x.im + \left(x.re \cdot x.re\right) \cdot \left(2 \cdot x.im\right) \]

      10. lift--.f64 N/A

          \[\leadsto \left(\left(x.re + x.im\right) \cdot \color{blue}{\left(x.re - x.im\right)}\right) \cdot x.im + \left(x.re \cdot x.re\right) \cdot \left(2 \cdot x.im\right) \]

      11. difference-of-squares N/A

          \[\leadsto \color{blue}{\left(x.re \cdot x.re - x.im \cdot x.im\right)} \cdot x.im + \left(x.re \cdot x.re\right) \cdot \left(2 \cdot x.im\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.re\right) \cdot \color{blue}{\left(2 \cdot x.im\right)} \]

      13. associate-*r* N/A

          \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{\left(\left(x.re \cdot x.re\right) \cdot 2\right) \cdot x.im} \]

      14. distribute-rgt-out N/A

          \[\leadsto \color{blue}{x.im \cdot \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot 2\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \color{blue}{x.im \cdot \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot 2\right)} \]

      16. difference-of-squares N/A

          \[\leadsto x.im \cdot \left(\color{blue}{\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)} + \left(x.re \cdot x.re\right) \cdot 2\right) \]

      17. +-commutative N/A

          \[\leadsto x.im \cdot \left(\color{blue}{\left(x.im + x.re\right)} \cdot \left(x.re - x.im\right) + \left(x.re \cdot x.re\right) \cdot 2\right) \]

      18. lift-+.f64 N/A

          \[\leadsto x.im \cdot \left(\color{blue}{\left(x.im + x.re\right)} \cdot \left(x.re - x.im\right) + \left(x.re \cdot x.re\right) \cdot 2\right) \]

      19. lift--.f64 N/A

          \[\leadsto x.im \cdot \left(\left(x.im + x.re\right) \cdot \color{blue}{\left(x.re - x.im\right)} + \left(x.re \cdot x.re\right) \cdot 2\right) \]

      20. *-commutative N/A

          \[\leadsto x.im \cdot \left(\color{blue}{\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)} + \left(x.re \cdot x.re\right) \cdot 2\right) \]

      21. lower-fma.f64 N/A

          \[\leadsto x.im \cdot \color{blue}{\mathsf{fma}\left(x.re - x.im, x.im + x.re, \left(x.re \cdot x.re\right) \cdot 2\right)} \]

   10. Applied rewrites 95.6%

       \[\leadsto \color{blue}{x.im \cdot \mathsf{fma}\left(x.re - x.im, x.re + x.im, 2 \cdot \left(x.re \cdot x.re\right)\right)} \]

   if 9.9999999999999996e290 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   1. Initial program 76.8%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Add Preprocessing

   3. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(2 \cdot {x.re}^{2} + {x.re}^{2}\right) \cdot x.im} \]

      2. *-rgt-identity N/A

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

      3. *-inverses N/A

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

      4. associate-/l* N/A

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

      5. unpow2 N/A

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

      6. cube-mult N/A

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

      7. associate-/l* N/A

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

      8. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {x.re}^{2} + {x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3}} \]

      9. distribute-lft1-in N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{3} \cdot {x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3} \]

      11. associate-*r/ N/A

          \[\leadsto \color{blue}{\left(3 \cdot \frac{{x.re}^{2}}{{x.im}^{2}}\right)} \cdot {x.im}^{3} \]

      12. associate-*l* N/A

          \[\leadsto \color{blue}{3 \cdot \left(\frac{{x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3}\right)} \]

      13. metadata-eval N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(-3\right)\right)} \cdot \left(\frac{{x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3}\right) \]

      14. metadata-eval N/A

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

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

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

      16. *-commutative N/A

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

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

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

      18. metadata-eval N/A

          \[\leadsto \left(\frac{{x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3}\right) \cdot \left(\mathsf{neg}\left(\color{blue}{-3}\right)\right) \]

      19. metadata-eval N/A

          \[\leadsto \left(\frac{{x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3}\right) \cdot \color{blue}{3} \]

   5. Applied rewrites 33.4%

      \[\leadsto \color{blue}{\left(\left(x.re \cdot x.re\right) \cdot x.im\right) \cdot 3} \]
   6. Step-by-step derivation

   7. Applied rewrites 56.3%

      \[\leadsto \left(\left(x.re \cdot x.im\right) \cdot x.re\right) \cdot 3 \]
   8. Step-by-step derivation

   9. Applied rewrites 56.4%

      \[\leadsto x.re \cdot \color{blue}{\left(x.im \cdot \left(3 \cdot x.re\right)\right)} \]

   if +inf.0 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re))

   1. Initial program 0.0%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 25.0

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

   4. Applied rewrites 25.0%

      \[\leadsto \color{blue}{\left(x.im \cdot \left(x.im + x.re\right)\right) \cdot \left(x.re - x.im\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-+.f64 N/A

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

      9. flip-+ N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. div-inv N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 25.0%

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

   8. Applied rewrites 100.0%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.im + x.re \cdot x.im\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im \leq 10^{+291}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im, x.re + x.im, 2 \cdot \left(x.re \cdot x.re\right)\right) \cdot x.im\\ \mathbf{elif}\;\left(x.re \cdot x.im + x.re \cdot x.im\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im \leq \infty:\\ \;\;\;\;\left(\left(3 \cdot x.re\right) \cdot x.im\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re + x.im, \left(x.re - x.im\right) \cdot x.im, 2 \cdot x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(x.re \cdot x.im\_m + x.re \cdot x.im\_m\right) \cdot x.re + \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 10^{+256}:\\ \;\;\;\;\left(-x.im\_m\right) \cdot \mathsf{fma}\left(-3, x.re \cdot x.re, x.im\_m \cdot x.im\_m\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left(\left(3 \cdot x.re\right) \cdot x.im\_m\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re + x.im\_m, \left(x.re - x.im\_m\right) \cdot x.im\_m, 2 \cdot x.im\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0
         (+
          (* (+ (* x.re x.im_m) (* x.re x.im_m)) x.re)
          (* (- (* x.re x.re) (* x.im_m x.im_m)) x.im_m))))
   (*
    x.im_s
    (if (<= t_0 1e+256)
      (* (- x.im_m) (fma -3.0 (* x.re x.re) (* x.im_m x.im_m)))
      (if (<= t_0 INFINITY)
        (* (* (* 3.0 x.re) x.im_m) x.re)
        (fma (+ x.re x.im_m) (* (- x.re x.im_m) x.im_m) (* 2.0 x.im_m)))))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = (((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)) * x_46_re) + (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m);
	double tmp;
	if (t_0 <= 1e+256) {
		tmp = -x_46_im_m * fma(-3.0, (x_46_re * x_46_re), (x_46_im_m * x_46_im_m));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = ((3.0 * x_46_re) * x_46_im_m) * x_46_re;
	} else {
		tmp = fma((x_46_re + x_46_im_m), ((x_46_re - x_46_im_m) * x_46_im_m), (2.0 * x_46_im_m));
	}
	return x_46_im_s * tmp;
}

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = Float64(Float64(Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_re * x_46_im_m)) * x_46_re) + Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m))
	tmp = 0.0
	if (t_0 <= 1e+256)
		tmp = Float64(Float64(-x_46_im_m) * fma(-3.0, Float64(x_46_re * x_46_re), Float64(x_46_im_m * x_46_im_m)));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(Float64(3.0 * x_46_re) * x_46_im_m) * x_46_re);
	else
		tmp = fma(Float64(x_46_re + x_46_im_m), Float64(Float64(x_46_re - x_46_im_m) * x_46_im_m), Float64(2.0 * x_46_im_m));
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$re * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision] + N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$0, 1e+256], N[((-x$46$im$95$m) * N[(-3.0 * N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[(3.0 * x$46$re), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] * x$46$re), $MachinePrecision], N[(N[(x$46$re + x$46$im$95$m), $MachinePrecision] * N[(N[(x$46$re - x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] + N[(2.0 * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < 1e256

   1. Initial program 95.6%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Add Preprocessing

   3. Taylor expanded in x.im around 0

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

   5. Applied rewrites 95.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-3, x.re \cdot x.re, x.im \cdot x.im\right) \cdot \left(-x.im\right)} \]

   if 1e256 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   1. Initial program 77.6%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Add Preprocessing

   3. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(2 \cdot {x.re}^{2} + {x.re}^{2}\right) \cdot x.im} \]

      2. *-rgt-identity N/A

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

      3. *-inverses N/A

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

      4. associate-/l* N/A

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

      5. unpow2 N/A

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

      6. cube-mult N/A

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

      7. associate-/l* N/A

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

      8. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {x.re}^{2} + {x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3}} \]

      9. distribute-lft1-in N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{3} \cdot {x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3} \]

      11. associate-*r/ N/A

          \[\leadsto \color{blue}{\left(3 \cdot \frac{{x.re}^{2}}{{x.im}^{2}}\right)} \cdot {x.im}^{3} \]

      12. associate-*l* N/A

          \[\leadsto \color{blue}{3 \cdot \left(\frac{{x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3}\right)} \]

      13. metadata-eval N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(-3\right)\right)} \cdot \left(\frac{{x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3}\right) \]

      14. metadata-eval N/A

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

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

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

      16. *-commutative N/A

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

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

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

      18. metadata-eval N/A

          \[\leadsto \left(\frac{{x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3}\right) \cdot \left(\mathsf{neg}\left(\color{blue}{-3}\right)\right) \]

      19. metadata-eval N/A

          \[\leadsto \left(\frac{{x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3}\right) \cdot \color{blue}{3} \]

   5. Applied rewrites 35.6%

      \[\leadsto \color{blue}{\left(\left(x.re \cdot x.re\right) \cdot x.im\right) \cdot 3} \]
   6. Step-by-step derivation

   7. Applied rewrites 57.8%

      \[\leadsto \left(\left(x.re \cdot x.im\right) \cdot x.re\right) \cdot 3 \]
   8. Step-by-step derivation

   9. Applied rewrites 57.8%

      \[\leadsto x.re \cdot \color{blue}{\left(x.im \cdot \left(3 \cdot x.re\right)\right)} \]

   if +inf.0 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re))

   1. Initial program 0.0%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 25.0

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

   4. Applied rewrites 25.0%

      \[\leadsto \color{blue}{\left(x.im \cdot \left(x.im + x.re\right)\right) \cdot \left(x.re - x.im\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-+.f64 N/A

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

      9. flip-+ N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. div-inv N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 25.0%

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

   8. Applied rewrites 100.0%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.im + x.re \cdot x.im\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im \leq 10^{+256}:\\ \;\;\;\;\left(-x.im\right) \cdot \mathsf{fma}\left(-3, x.re \cdot x.re, x.im \cdot x.im\right)\\ \mathbf{elif}\;\left(x.re \cdot x.im + x.re \cdot x.im\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im \leq \infty:\\ \;\;\;\;\left(\left(3 \cdot x.re\right) \cdot x.im\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re + x.im, \left(x.re - x.im\right) \cdot x.im, 2 \cdot x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(x.re \cdot x.im\_m + x.re \cdot x.im\_m\right) \cdot x.re + \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-305}:\\ \;\;\;\;\left(\left(-x.im\_m\right) \cdot x.im\_m\right) \cdot x.im\_m\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left(\left(3 \cdot x.re\right) \cdot x.im\_m\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re + x.im\_m, \left(x.re - x.im\_m\right) \cdot x.im\_m, 2 \cdot x.im\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0
         (+
          (* (+ (* x.re x.im_m) (* x.re x.im_m)) x.re)
          (* (- (* x.re x.re) (* x.im_m x.im_m)) x.im_m))))
   (*
    x.im_s
    (if (<= t_0 -5e-305)
      (* (* (- x.im_m) x.im_m) x.im_m)
      (if (<= t_0 INFINITY)
        (* (* (* 3.0 x.re) x.im_m) x.re)
        (fma (+ x.re x.im_m) (* (- x.re x.im_m) x.im_m) (* 2.0 x.im_m)))))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = (((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)) * x_46_re) + (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m);
	double tmp;
	if (t_0 <= -5e-305) {
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = ((3.0 * x_46_re) * x_46_im_m) * x_46_re;
	} else {
		tmp = fma((x_46_re + x_46_im_m), ((x_46_re - x_46_im_m) * x_46_im_m), (2.0 * x_46_im_m));
	}
	return x_46_im_s * tmp;
}

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = Float64(Float64(Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_re * x_46_im_m)) * x_46_re) + Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m))
	tmp = 0.0
	if (t_0 <= -5e-305)
		tmp = Float64(Float64(Float64(-x_46_im_m) * x_46_im_m) * x_46_im_m);
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(Float64(3.0 * x_46_re) * x_46_im_m) * x_46_re);
	else
		tmp = fma(Float64(x_46_re + x_46_im_m), Float64(Float64(x_46_re - x_46_im_m) * x_46_im_m), Float64(2.0 * x_46_im_m));
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$re * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision] + N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$0, -5e-305], N[(N[((-x$46$im$95$m) * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[(3.0 * x$46$re), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] * x$46$re), $MachinePrecision], N[(N[(x$46$re + x$46$im$95$m), $MachinePrecision] * N[(N[(x$46$re - x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] + N[(2.0 * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < -4.99999999999999985e-305

   1. Initial program 92.5%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Add Preprocessing

   3. Taylor expanded in x.im around inf

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left({x.im}^{3}\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{-{x.im}^{3}} \]

      3. lower-pow.f64 52.2

         \[\leadsto -\color{blue}{{x.im}^{3}} \]

   5. Applied rewrites 52.2%

      \[\leadsto \color{blue}{-{x.im}^{3}} \]
   6. Step-by-step derivation

   7. Applied rewrites 52.1%

      \[\leadsto \left(\left(-x.im\right) \cdot x.im\right) \cdot \color{blue}{x.im} \]

   if -4.99999999999999985e-305 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   1. Initial program 90.0%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Add Preprocessing

   3. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(2 \cdot {x.re}^{2} + {x.re}^{2}\right) \cdot x.im} \]

      2. *-rgt-identity N/A

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

      3. *-inverses N/A

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

      4. associate-/l* N/A

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

      5. unpow2 N/A

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

      6. cube-mult N/A

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

      7. associate-/l* N/A

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

      8. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {x.re}^{2} + {x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3}} \]

      9. distribute-lft1-in N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{3} \cdot {x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3} \]

      11. associate-*r/ N/A

          \[\leadsto \color{blue}{\left(3 \cdot \frac{{x.re}^{2}}{{x.im}^{2}}\right)} \cdot {x.im}^{3} \]

      12. associate-*l* N/A

          \[\leadsto \color{blue}{3 \cdot \left(\frac{{x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3}\right)} \]

      13. metadata-eval N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(-3\right)\right)} \cdot \left(\frac{{x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3}\right) \]

      14. metadata-eval N/A

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

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

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

      16. *-commutative N/A

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

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

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

      18. metadata-eval N/A

          \[\leadsto \left(\frac{{x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3}\right) \cdot \left(\mathsf{neg}\left(\color{blue}{-3}\right)\right) \]

      19. metadata-eval N/A

          \[\leadsto \left(\frac{{x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3}\right) \cdot \color{blue}{3} \]

   5. Applied rewrites 59.2%

      \[\leadsto \color{blue}{\left(\left(x.re \cdot x.re\right) \cdot x.im\right) \cdot 3} \]
   6. Step-by-step derivation

   7. Applied rewrites 69.0%

      \[\leadsto \left(\left(x.re \cdot x.im\right) \cdot x.re\right) \cdot 3 \]
   8. Step-by-step derivation

   9. Applied rewrites 69.1%

      \[\leadsto x.re \cdot \color{blue}{\left(x.im \cdot \left(3 \cdot x.re\right)\right)} \]

   if +inf.0 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re))

   1. Initial program 0.0%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 25.0

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

   4. Applied rewrites 25.0%

      \[\leadsto \color{blue}{\left(x.im \cdot \left(x.im + x.re\right)\right) \cdot \left(x.re - x.im\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-+.f64 N/A

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

      9. flip-+ N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. div-inv N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 25.0%

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

   8. Applied rewrites 100.0%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.im + x.re \cdot x.im\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im \leq -5 \cdot 10^{-305}:\\ \;\;\;\;\left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \mathbf{elif}\;\left(x.re \cdot x.im + x.re \cdot x.im\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im \leq \infty:\\ \;\;\;\;\left(\left(3 \cdot x.re\right) \cdot x.im\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re + x.im, \left(x.re - x.im\right) \cdot x.im, 2 \cdot x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(\left(-x.im\_m\right) \cdot x.im\_m\right) \cdot x.im\_m\\ t_1 := \left(x.re \cdot x.im\_m + x.re \cdot x.im\_m\right) \cdot x.re + \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-305}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left(\left(3 \cdot x.re\right) \cdot x.im\_m\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0 (* (* (- x.im_m) x.im_m) x.im_m))
        (t_1
         (+
          (* (+ (* x.re x.im_m) (* x.re x.im_m)) x.re)
          (* (- (* x.re x.re) (* x.im_m x.im_m)) x.im_m))))
   (*
    x.im_s
    (if (<= t_1 -5e-305)
      t_0
      (if (<= t_1 INFINITY) (* (* (* 3.0 x.re) x.im_m) x.re) t_0)))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	double t_1 = (((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)) * x_46_re) + (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m);
	double tmp;
	if (t_1 <= -5e-305) {
		tmp = t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = ((3.0 * x_46_re) * x_46_im_m) * x_46_re;
	} else {
		tmp = t_0;
	}
	return x_46_im_s * tmp;
}

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	double t_1 = (((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)) * x_46_re) + (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m);
	double tmp;
	if (t_1 <= -5e-305) {
		tmp = t_0;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = ((3.0 * x_46_re) * x_46_im_m) * x_46_re;
	} else {
		tmp = t_0;
	}
	return x_46_im_s * tmp;
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	t_0 = (-x_46_im_m * x_46_im_m) * x_46_im_m
	t_1 = (((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)) * x_46_re) + (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m)
	tmp = 0
	if t_1 <= -5e-305:
		tmp = t_0
	elif t_1 <= math.inf:
		tmp = ((3.0 * x_46_re) * x_46_im_m) * x_46_re
	else:
		tmp = t_0
	return x_46_im_s * tmp

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = Float64(Float64(Float64(-x_46_im_m) * x_46_im_m) * x_46_im_m)
	t_1 = Float64(Float64(Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_re * x_46_im_m)) * x_46_re) + Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m))
	tmp = 0.0
	if (t_1 <= -5e-305)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(3.0 * x_46_re) * x_46_im_m) * x_46_re);
	else
		tmp = t_0;
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	t_1 = (((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)) * x_46_re) + (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m);
	tmp = 0.0;
	if (t_1 <= -5e-305)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = ((3.0 * x_46_re) * x_46_im_m) * x_46_re;
	else
		tmp = t_0;
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[((-x$46$im$95$m) * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$re * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision] + N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$1, -5e-305], t$95$0, If[LessEqual[t$95$1, Infinity], N[(N[(N[(3.0 * x$46$re), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] * x$46$re), $MachinePrecision], t$95$0]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < -4.99999999999999985e-305 or +inf.0 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re))

   1. Initial program 77.3%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Add Preprocessing

   3. Taylor expanded in x.im around inf

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left({x.im}^{3}\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{-{x.im}^{3}} \]

      3. lower-pow.f64 55.9

         \[\leadsto -\color{blue}{{x.im}^{3}} \]

   5. Applied rewrites 55.9%

      \[\leadsto \color{blue}{-{x.im}^{3}} \]
   6. Step-by-step derivation

   7. Applied rewrites 55.8%

      \[\leadsto \left(\left(-x.im\right) \cdot x.im\right) \cdot \color{blue}{x.im} \]

   if -4.99999999999999985e-305 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   1. Initial program 90.0%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Add Preprocessing

   3. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(2 \cdot {x.re}^{2} + {x.re}^{2}\right) \cdot x.im} \]

      2. *-rgt-identity N/A

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

      3. *-inverses N/A

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

      4. associate-/l* N/A

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

      5. unpow2 N/A

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

      6. cube-mult N/A

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

      7. associate-/l* N/A

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

      8. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {x.re}^{2} + {x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3}} \]

      9. distribute-lft1-in N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{3} \cdot {x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3} \]

      11. associate-*r/ N/A

          \[\leadsto \color{blue}{\left(3 \cdot \frac{{x.re}^{2}}{{x.im}^{2}}\right)} \cdot {x.im}^{3} \]

      12. associate-*l* N/A

          \[\leadsto \color{blue}{3 \cdot \left(\frac{{x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3}\right)} \]

      13. metadata-eval N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(-3\right)\right)} \cdot \left(\frac{{x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3}\right) \]

      14. metadata-eval N/A

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

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

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

      16. *-commutative N/A

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

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

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

      18. metadata-eval N/A

          \[\leadsto \left(\frac{{x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3}\right) \cdot \left(\mathsf{neg}\left(\color{blue}{-3}\right)\right) \]

      19. metadata-eval N/A

          \[\leadsto \left(\frac{{x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3}\right) \cdot \color{blue}{3} \]

   5. Applied rewrites 59.2%

      \[\leadsto \color{blue}{\left(\left(x.re \cdot x.re\right) \cdot x.im\right) \cdot 3} \]
   6. Step-by-step derivation

   7. Applied rewrites 69.0%

      \[\leadsto \left(\left(x.re \cdot x.im\right) \cdot x.re\right) \cdot 3 \]
   8. Step-by-step derivation

   9. Applied rewrites 69.1%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.im + x.re \cdot x.im\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im \leq -5 \cdot 10^{-305}:\\ \;\;\;\;\left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \mathbf{elif}\;\left(x.re \cdot x.im + x.re \cdot x.im\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im \leq \infty:\\ \;\;\;\;\left(\left(3 \cdot x.re\right) \cdot x.im\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(x.re \cdot x.im\_m + x.re \cdot x.im\_m\right) \cdot x.re\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 + \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m \leq \infty:\\ \;\;\;\;t\_0 + \left(\left(x.re + x.im\_m\right) \cdot x.im\_m\right) \cdot \left(x.re - x.im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re + x.im\_m, \left(x.re - x.im\_m\right) \cdot x.im\_m, 2 \cdot x.im\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0 (* (+ (* x.re x.im_m) (* x.re x.im_m)) x.re)))
   (*
    x.im_s
    (if (<= (+ t_0 (* (- (* x.re x.re) (* x.im_m x.im_m)) x.im_m)) INFINITY)
      (+ t_0 (* (* (+ x.re x.im_m) x.im_m) (- x.re x.im_m)))
      (fma (+ x.re x.im_m) (* (- x.re x.im_m) x.im_m) (* 2.0 x.im_m))))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)) * x_46_re;
	double tmp;
	if ((t_0 + (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m)) <= ((double) INFINITY)) {
		tmp = t_0 + (((x_46_re + x_46_im_m) * x_46_im_m) * (x_46_re - x_46_im_m));
	} else {
		tmp = fma((x_46_re + x_46_im_m), ((x_46_re - x_46_im_m) * x_46_im_m), (2.0 * x_46_im_m));
	}
	return x_46_im_s * tmp;
}

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = Float64(Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_re * x_46_im_m)) * x_46_re)
	tmp = 0.0
	if (Float64(t_0 + Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m)) <= Inf)
		tmp = Float64(t_0 + Float64(Float64(Float64(x_46_re + x_46_im_m) * x_46_im_m) * Float64(x_46_re - x_46_im_m)));
	else
		tmp = fma(Float64(x_46_re + x_46_im_m), Float64(Float64(x_46_re - x_46_im_m) * x_46_im_m), Float64(2.0 * x_46_im_m));
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$re * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[N[(t$95$0 + N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$0 + N[(N[(N[(x$46$re + x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] * N[(x$46$re - x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re + x$46$im$95$m), $MachinePrecision] * N[(N[(x$46$re - x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] + N[(2.0 * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   1. Initial program 91.1%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 99.7

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

   4. Applied rewrites 99.7%

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

   if +inf.0 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re))

   1. Initial program 0.0%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 25.0

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

   4. Applied rewrites 25.0%

      \[\leadsto \color{blue}{\left(x.im \cdot \left(x.im + x.re\right)\right) \cdot \left(x.re - x.im\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-+.f64 N/A

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

      9. flip-+ N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. div-inv N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 25.0%

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

   8. Applied rewrites 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.im + x.re \cdot x.im\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im \leq \infty:\\ \;\;\;\;\left(x.re \cdot x.im + x.re \cdot x.im\right) \cdot x.re + \left(\left(x.re + x.im\right) \cdot x.im\right) \cdot \left(x.re - x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re + x.im, \left(x.re - x.im\right) \cdot x.im, 2 \cdot x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.im\_m + x.re \cdot x.im\_m\right) \cdot x.re + \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m \leq \infty:\\ \;\;\;\;\left(\left(x.re + x.im\_m\right) \cdot x.im\_m\right) \cdot \left(x.re - x.im\_m\right) + \left(\left(x.im\_m + x.im\_m\right) \cdot x.re\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re + x.im\_m, \left(x.re - x.im\_m\right) \cdot x.im\_m, 2 \cdot x.im\_m\right)\\ \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<=
       (+
        (* (+ (* x.re x.im_m) (* x.re x.im_m)) x.re)
        (* (- (* x.re x.re) (* x.im_m x.im_m)) x.im_m))
       INFINITY)
    (+
     (* (* (+ x.re x.im_m) x.im_m) (- x.re x.im_m))
     (* (* (+ x.im_m x.im_m) x.re) x.re))
    (fma (+ x.re x.im_m) (* (- x.re x.im_m) x.im_m) (* 2.0 x.im_m)))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double tmp;
	if (((((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)) * x_46_re) + (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m)) <= ((double) INFINITY)) {
		tmp = (((x_46_re + x_46_im_m) * x_46_im_m) * (x_46_re - x_46_im_m)) + (((x_46_im_m + x_46_im_m) * x_46_re) * x_46_re);
	} else {
		tmp = fma((x_46_re + x_46_im_m), ((x_46_re - x_46_im_m) * x_46_im_m), (2.0 * x_46_im_m));
	}
	return x_46_im_s * tmp;
}

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_re * x_46_im_m)) * x_46_re) + Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m)) <= Inf)
		tmp = Float64(Float64(Float64(Float64(x_46_re + x_46_im_m) * x_46_im_m) * Float64(x_46_re - x_46_im_m)) + Float64(Float64(Float64(x_46_im_m + x_46_im_m) * x_46_re) * x_46_re));
	else
		tmp = fma(Float64(x_46_re + x_46_im_m), Float64(Float64(x_46_re - x_46_im_m) * x_46_im_m), Float64(2.0 * x_46_im_m));
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[N[(N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$re * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision] + N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(N[(x$46$re + x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] * N[(x$46$re - x$46$im$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x$46$im$95$m + x$46$im$95$m), $MachinePrecision] * x$46$re), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re + x$46$im$95$m), $MachinePrecision] * N[(N[(x$46$re - x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] + N[(2.0 * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   1. Initial program 91.1%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 99.7

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

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\left(x.im \cdot \left(x.im + x.re\right)\right) \cdot \left(x.re - x.im\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 99.3

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

   6. Applied rewrites 99.3%

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

   if +inf.0 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re))

   1. Initial program 0.0%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 25.0

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

   4. Applied rewrites 25.0%

      \[\leadsto \color{blue}{\left(x.im \cdot \left(x.im + x.re\right)\right) \cdot \left(x.re - x.im\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-+.f64 N/A

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

      9. flip-+ N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. div-inv N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 25.0%

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

   8. Applied rewrites 100.0%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.im + x.re \cdot x.im\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im \leq \infty:\\ \;\;\;\;\left(\left(x.re + x.im\right) \cdot x.im\right) \cdot \left(x.re - x.im\right) + \left(\left(x.im + x.im\right) \cdot x.re\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re + x.im, \left(x.re - x.im\right) \cdot x.im, 2 \cdot x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.1% accurate, 3.1× speedup

Math

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

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (* x.im_s (* (* (- x.im_m) x.im_m) x.im_m)))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	return x_46_im_s * ((-x_46_im_m * x_46_im_m) * x_46_im_m);
}

Fortran

x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    code = x_46im_s * ((-x_46im_m * x_46im_m) * x_46im_m)
end function

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	return x_46_im_s * ((-x_46_im_m * x_46_im_m) * x_46_im_m);
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	return x_46_im_s * ((-x_46_im_m * x_46_im_m) * x_46_im_m)

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	return Float64(x_46_im_s * Float64(Float64(Float64(-x_46_im_m) * x_46_im_m) * x_46_im_m))
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp = code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = x_46_im_s * ((-x_46_im_m * x_46_im_m) * x_46_im_m);
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := N[(x$46$im$95$s * N[(N[((-x$46$im$95$m) * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.9%

   \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
2. Add Preprocessing

3. Taylor expanded in x.im around inf

   \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left({x.im}^{3}\right)} \]

   2. lower-neg.f64 N/A

      \[\leadsto \color{blue}{-{x.im}^{3}} \]

   3. lower-pow.f64 56.1

      \[\leadsto -\color{blue}{{x.im}^{3}} \]

5. Applied rewrites 56.1%

   \[\leadsto \color{blue}{-{x.im}^{3}} \]
6. Step-by-step derivation

7. Applied rewrites 56.0%

   \[\leadsto \left(\left(-x.im\right) \cdot x.im\right) \cdot \color{blue}{x.im} \]
8. Add Preprocessing

Developer Target 1: 91.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(x.re \cdot x.im\right) \cdot \left(2 \cdot x.re\right) + \left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.re + x.im\right) \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (+ (* (* x.re x.im) (* 2.0 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im))))

C

double code(double x_46_re, double x_46_im) {
	return ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = ((x_46re * x_46im) * (2.0d0 * x_46re)) + ((x_46im * (x_46re - x_46im)) * (x_46re + x_46im))
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
}

Python

def code(x_46_re, x_46_im):
	return ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im))

Julia

function code(x_46_re, x_46_im)
	return Float64(Float64(Float64(x_46_re * x_46_im) * Float64(2.0 * x_46_re)) + Float64(Float64(x_46_im * Float64(x_46_re - x_46_im)) * Float64(x_46_re + x_46_im)))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
end

Wolfram

code[x$46$re_, x$46$im_] := N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(2.0 * x$46$re), $MachinePrecision]), $MachinePrecision] + N[(N[(x$46$im * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision] * N[(x$46$re + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024264 
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (* (* x.re x.im) (* 2 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im))))

  (+ (* (- (* x.re x.re) (* x.im x.im)) x.im) (* (+ (* x.re x.im) (* x.im x.re)) x.re)))