math.cube on complex, imaginary part

Percentage Accurate: 82.6% → 99.8%

Time: 8.4s

Alternatives: 8

Speedup: 0.4×

• 44.9% 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: 82.6% 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: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m + \left(x.re \cdot x.im\_m + x.im\_m \cdot x.re\right) \cdot x.re\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 5 \cdot 10^{+248}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;x.re \cdot \left(\left(3 \cdot x.re\right) \cdot x.im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im\_m, x.im\_m + x.re, 2\right) \cdot x.im\_m\\ \end{array} \end{array} \end{array} \]

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.re) (* x.im_m x.im_m)) x.im_m)
          (* (+ (* x.re x.im_m) (* x.im_m x.re)) x.re))))
   (*
    x.im_s
    (if (<= t_0 5e+248)
      t_0
      (if (<= t_0 INFINITY)
        (* x.re (* (* 3.0 x.re) x.im_m))
        (* (fma (- x.re x.im_m) (+ x.im_m x.re) 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_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	double tmp;
	if (t_0 <= 5e+248) {
		tmp = t_0;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = x_46_re * ((3.0 * x_46_re) * x_46_im_m);
	} else {
		tmp = fma((x_46_re - x_46_im_m), (x_46_im_m + x_46_re), 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_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m) + Float64(Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_im_m * x_46_re)) * x_46_re))
	tmp = 0.0
	if (t_0 <= 5e+248)
		tmp = t_0;
	elseif (t_0 <= Inf)
		tmp = Float64(x_46_re * Float64(Float64(3.0 * x_46_re) * x_46_im_m));
	else
		tmp = Float64(fma(Float64(x_46_re - x_46_im_m), Float64(x_46_im_m + x_46_re), 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$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] + N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$0, 5e+248], t$95$0, If[LessEqual[t$95$0, Infinity], N[(x$46$re * N[(N[(3.0 * x$46$re), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$46$re - x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m + x$46$re), $MachinePrecision] + 2.0), $MachinePrecision] * x$46$im$95$m), $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)) < 4.9999999999999996e248

   1. Initial program 96.5%

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

   if 4.9999999999999996e248 < (+.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 81.6%

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

   3. Taylor expanded in x.re around inf

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft1-in N/A

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

      7. lower-*.f64 N/A

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

      8. distribute-lft1-in N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 42.7

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

   5. Applied rewrites 42.7%

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

   7. Applied rewrites 60.9%

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

   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. sub-neg N/A

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

      3. flip-+ N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow-prod-down N/A

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

      9. pow-prod-up N/A

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

      10. lower-pow.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

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

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. lift-*.f64 N/A

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

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

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

      19. lower-*.f64 N/A

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

      20. lower-neg.f64 N/A

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

      21. lower--.f64 N/A

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

   4. Applied rewrites 0.0%

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

   6. Applied rewrites 0.0%

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

   8. Applied rewrites 100.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. distribute-rgt-out N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 100.0

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

   10. Applied rewrites 100.0%

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

Alternative 2: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m + \left(x.re \cdot x.im\_m + x.im\_m \cdot x.re\right) \cdot x.re\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 5 \cdot 10^{+233}:\\ \;\;\;\;\left(-\mathsf{fma}\left(x.im\_m, x.im\_m, -3 \cdot \left(x.re \cdot x.re\right)\right)\right) \cdot x.im\_m\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left(\left(x.im\_m \cdot x.re\right) \cdot x.re\right) \cdot 3\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im\_m, x.im\_m + x.re, 2\right) \cdot x.im\_m\\ \end{array} \end{array} \end{array} \]

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.re) (* x.im_m x.im_m)) x.im_m)
          (* (+ (* x.re x.im_m) (* x.im_m x.re)) x.re))))
   (*
    x.im_s
    (if (<= t_0 5e+233)
      (* (- (fma x.im_m x.im_m (* -3.0 (* x.re x.re)))) x.im_m)
      (if (<= t_0 INFINITY)
        (* (* (* x.im_m x.re) x.re) 3.0)
        (* (fma (- x.re x.im_m) (+ x.im_m x.re) 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_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	double tmp;
	if (t_0 <= 5e+233) {
		tmp = -fma(x_46_im_m, x_46_im_m, (-3.0 * (x_46_re * x_46_re))) * x_46_im_m;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = ((x_46_im_m * x_46_re) * x_46_re) * 3.0;
	} else {
		tmp = fma((x_46_re - x_46_im_m), (x_46_im_m + x_46_re), 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_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m) + Float64(Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_im_m * x_46_re)) * x_46_re))
	tmp = 0.0
	if (t_0 <= 5e+233)
		tmp = Float64(Float64(-fma(x_46_im_m, x_46_im_m, Float64(-3.0 * Float64(x_46_re * x_46_re)))) * x_46_im_m);
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(Float64(x_46_im_m * x_46_re) * x_46_re) * 3.0);
	else
		tmp = Float64(fma(Float64(x_46_re - x_46_im_m), Float64(x_46_im_m + x_46_re), 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$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] + N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$0, 5e+233], N[((-N[(x$46$im$95$m * x$46$im$95$m + N[(-3.0 * 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[(x$46$im$95$m * x$46$re), $MachinePrecision] * x$46$re), $MachinePrecision] * 3.0), $MachinePrecision], N[(N[(N[(x$46$re - x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m + x$46$re), $MachinePrecision] + 2.0), $MachinePrecision] * x$46$im$95$m), $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)) < 5.00000000000000009e233

   1. Initial program 96.4%

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

   3. Taylor expanded in x.re around 0

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

      1. distribute-rgt-in N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-lft-in N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. distribute-rgt-in N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   5. Applied rewrites 96.4%

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

   if 5.00000000000000009e233 < (+.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 81.9%

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

   3. Taylor expanded in x.re around inf

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft1-in N/A

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

      7. lower-*.f64 N/A

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

      8. distribute-lft1-in N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 43.6

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

   5. Applied rewrites 43.6%

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

   7. Applied rewrites 61.5%

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

   9. Applied rewrites 61.5%

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

   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. sub-neg N/A

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

      3. flip-+ N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow-prod-down N/A

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

      9. pow-prod-up N/A

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

      10. lower-pow.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

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

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. lift-*.f64 N/A

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

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

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

      19. lower-*.f64 N/A

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

      20. lower-neg.f64 N/A

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

      21. lower--.f64 N/A

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

   4. Applied rewrites 0.0%

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

   6. Applied rewrites 0.0%

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

   8. Applied rewrites 100.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. distribute-rgt-out N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 100.0

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

   10. Applied rewrites 100.0%

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

Alternative 3: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m + \left(x.re \cdot x.im\_m + x.im\_m \cdot x.re\right) \cdot x.re\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-317}:\\ \;\;\;\;\left(x.im\_m \cdot x.im\_m\right) \cdot \left(-x.im\_m\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;x.re \cdot \left(\left(3 \cdot x.re\right) \cdot x.im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im\_m, x.im\_m + x.re, 2\right) \cdot x.im\_m\\ \end{array} \end{array} \end{array} \]

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.re) (* x.im_m x.im_m)) x.im_m)
          (* (+ (* x.re x.im_m) (* x.im_m x.re)) x.re))))
   (*
    x.im_s
    (if (<= t_0 -1e-317)
      (* (* x.im_m x.im_m) (- x.im_m))
      (if (<= t_0 INFINITY)
        (* x.re (* (* 3.0 x.re) x.im_m))
        (* (fma (- x.re x.im_m) (+ x.im_m x.re) 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_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	double tmp;
	if (t_0 <= -1e-317) {
		tmp = (x_46_im_m * x_46_im_m) * -x_46_im_m;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = x_46_re * ((3.0 * x_46_re) * x_46_im_m);
	} else {
		tmp = fma((x_46_re - x_46_im_m), (x_46_im_m + x_46_re), 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_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m) + Float64(Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_im_m * x_46_re)) * x_46_re))
	tmp = 0.0
	if (t_0 <= -1e-317)
		tmp = Float64(Float64(x_46_im_m * x_46_im_m) * Float64(-x_46_im_m));
	elseif (t_0 <= Inf)
		tmp = Float64(x_46_re * Float64(Float64(3.0 * x_46_re) * x_46_im_m));
	else
		tmp = Float64(fma(Float64(x_46_re - x_46_im_m), Float64(x_46_im_m + x_46_re), 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$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] + N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$0, -1e-317], 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[(x$46$re * N[(N[(3.0 * x$46$re), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$46$re - x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m + x$46$re), $MachinePrecision] + 2.0), $MachinePrecision] * x$46$im$95$m), $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)) < -1.00000023e-317

   1. Initial program 93.7%

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

   3. Taylor expanded in x.re around 0

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

      1. distribute-rgt-in N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-lft-in N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. distribute-rgt-in N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   5. Applied rewrites 93.6%

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

   6. Taylor expanded in x.re around 0

      \[\leadsto \left(-{x.im}^{2}\right) \cdot x.im \]
   7. Step-by-step derivation

   8. Applied rewrites 44.4%

      \[\leadsto \left(-x.im \cdot x.im\right) \cdot x.im \]

   if -1.00000023e-317 < (+.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 91.7%

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

   3. Taylor expanded in x.re around inf

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft1-in N/A

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

      7. lower-*.f64 N/A

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

      8. distribute-lft1-in N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 65.7

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

   5. Applied rewrites 65.7%

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

   7. Applied rewrites 73.8%

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

   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. sub-neg N/A

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

      3. flip-+ N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow-prod-down N/A

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

      9. pow-prod-up N/A

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

      10. lower-pow.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

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

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. lift-*.f64 N/A

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

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

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

      19. lower-*.f64 N/A

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

      20. lower-neg.f64 N/A

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

      21. lower--.f64 N/A

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

   4. Applied rewrites 0.0%

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

   6. Applied rewrites 0.0%

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

   8. Applied rewrites 100.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. distribute-rgt-out N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 100.0

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

   10. Applied rewrites 100.0%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq -1 \cdot 10^{-317}:\\ \;\;\;\;\left(x.im \cdot x.im\right) \cdot \left(-x.im\right)\\ \mathbf{elif}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq \infty:\\ \;\;\;\;x.re \cdot \left(\left(3 \cdot x.re\right) \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im, x.im + x.re, 2\right) \cdot x.im\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m + \left(x.re \cdot x.im\_m + x.im\_m \cdot x.re\right) \cdot x.re\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-317} \lor \neg \left(t\_0 \leq \infty\right):\\ \;\;\;\;\left(x.im\_m \cdot x.im\_m\right) \cdot \left(-x.im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(3 \cdot x.re\right) \cdot x.im\_m\right)\\ \end{array} \end{array} \end{array} \]

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.re) (* x.im_m x.im_m)) x.im_m)
          (* (+ (* x.re x.im_m) (* x.im_m x.re)) x.re))))
   (*
    x.im_s
    (if (or (<= t_0 -1e-317) (not (<= t_0 INFINITY)))
      (* (* x.im_m x.im_m) (- x.im_m))
      (* x.re (* (* 3.0 x.re) 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_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	double tmp;
	if ((t_0 <= -1e-317) || !(t_0 <= ((double) INFINITY))) {
		tmp = (x_46_im_m * x_46_im_m) * -x_46_im_m;
	} else {
		tmp = x_46_re * ((3.0 * x_46_re) * x_46_im_m);
	}
	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_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	double tmp;
	if ((t_0 <= -1e-317) || !(t_0 <= Double.POSITIVE_INFINITY)) {
		tmp = (x_46_im_m * x_46_im_m) * -x_46_im_m;
	} else {
		tmp = x_46_re * ((3.0 * x_46_re) * x_46_im_m);
	}
	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_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re)
	tmp = 0
	if (t_0 <= -1e-317) or not (t_0 <= math.inf):
		tmp = (x_46_im_m * x_46_im_m) * -x_46_im_m
	else:
		tmp = x_46_re * ((3.0 * x_46_re) * 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_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m) + Float64(Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_im_m * x_46_re)) * x_46_re))
	tmp = 0.0
	if ((t_0 <= -1e-317) || !(t_0 <= Inf))
		tmp = Float64(Float64(x_46_im_m * x_46_im_m) * Float64(-x_46_im_m));
	else
		tmp = Float64(x_46_re * Float64(Float64(3.0 * x_46_re) * x_46_im_m));
	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_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	tmp = 0.0;
	if ((t_0 <= -1e-317) || ~((t_0 <= Inf)))
		tmp = (x_46_im_m * x_46_im_m) * -x_46_im_m;
	else
		tmp = x_46_re * ((3.0 * x_46_re) * x_46_im_m);
	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[(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] + N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[Or[LessEqual[t$95$0, -1e-317], N[Not[LessEqual[t$95$0, Infinity]], $MachinePrecision]], N[(N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision] * (-x$46$im$95$m)), $MachinePrecision], N[(x$46$re * N[(N[(3.0 * x$46$re), $MachinePrecision] * 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)) < -1.00000023e-317 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 71.6%

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

   3. Taylor expanded in x.re around 0

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

      1. distribute-rgt-in N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-lft-in N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. distribute-rgt-in N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   5. Applied rewrites 90.9%

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

   6. Taylor expanded in x.re around 0

      \[\leadsto \left(-{x.im}^{2}\right) \cdot x.im \]
   7. Step-by-step derivation

   8. Applied rewrites 51.6%

      \[\leadsto \left(-x.im \cdot x.im\right) \cdot x.im \]

   if -1.00000023e-317 < (+.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 91.7%

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

   3. Taylor expanded in x.re around inf

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft1-in N/A

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

      7. lower-*.f64 N/A

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

      8. distribute-lft1-in N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 65.7

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

   5. Applied rewrites 65.7%

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

   7. Applied rewrites 73.8%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq -1 \cdot 10^{-317} \lor \neg \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq \infty\right):\\ \;\;\;\;\left(x.im \cdot x.im\right) \cdot \left(-x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(3 \cdot x.re\right) \cdot x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m + \left(x.re \cdot x.im\_m + x.im\_m \cdot x.re\right) \cdot x.re\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-317} \lor \neg \left(t\_0 \leq \infty\right):\\ \;\;\;\;\left(x.im\_m \cdot x.im\_m\right) \cdot \left(-x.im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(3 \cdot x.im\_m\right)\right)\\ \end{array} \end{array} \end{array} \]

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.re) (* x.im_m x.im_m)) x.im_m)
          (* (+ (* x.re x.im_m) (* x.im_m x.re)) x.re))))
   (*
    x.im_s
    (if (or (<= t_0 -1e-317) (not (<= t_0 INFINITY)))
      (* (* x.im_m x.im_m) (- x.im_m))
      (* x.re (* x.re (* 3.0 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_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	double tmp;
	if ((t_0 <= -1e-317) || !(t_0 <= ((double) INFINITY))) {
		tmp = (x_46_im_m * x_46_im_m) * -x_46_im_m;
	} else {
		tmp = x_46_re * (x_46_re * (3.0 * x_46_im_m));
	}
	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_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	double tmp;
	if ((t_0 <= -1e-317) || !(t_0 <= Double.POSITIVE_INFINITY)) {
		tmp = (x_46_im_m * x_46_im_m) * -x_46_im_m;
	} else {
		tmp = x_46_re * (x_46_re * (3.0 * x_46_im_m));
	}
	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_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re)
	tmp = 0
	if (t_0 <= -1e-317) or not (t_0 <= math.inf):
		tmp = (x_46_im_m * x_46_im_m) * -x_46_im_m
	else:
		tmp = x_46_re * (x_46_re * (3.0 * 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_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m) + Float64(Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_im_m * x_46_re)) * x_46_re))
	tmp = 0.0
	if ((t_0 <= -1e-317) || !(t_0 <= Inf))
		tmp = Float64(Float64(x_46_im_m * x_46_im_m) * Float64(-x_46_im_m));
	else
		tmp = Float64(x_46_re * Float64(x_46_re * Float64(3.0 * x_46_im_m)));
	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_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	tmp = 0.0;
	if ((t_0 <= -1e-317) || ~((t_0 <= Inf)))
		tmp = (x_46_im_m * x_46_im_m) * -x_46_im_m;
	else
		tmp = x_46_re * (x_46_re * (3.0 * x_46_im_m));
	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[(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] + N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[Or[LessEqual[t$95$0, -1e-317], N[Not[LessEqual[t$95$0, Infinity]], $MachinePrecision]], N[(N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision] * (-x$46$im$95$m)), $MachinePrecision], N[(x$46$re * N[(x$46$re * N[(3.0 * x$46$im$95$m), $MachinePrecision]), $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)) < -1.00000023e-317 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 71.6%

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

   3. Taylor expanded in x.re around 0

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

      1. distribute-rgt-in N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-lft-in N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. distribute-rgt-in N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   5. Applied rewrites 90.9%

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

   6. Taylor expanded in x.re around 0

      \[\leadsto \left(-{x.im}^{2}\right) \cdot x.im \]
   7. Step-by-step derivation

   8. Applied rewrites 51.6%

      \[\leadsto \left(-x.im \cdot x.im\right) \cdot x.im \]

   if -1.00000023e-317 < (+.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 91.7%

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

   3. Taylor expanded in x.re around inf

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft1-in N/A

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

      7. lower-*.f64 N/A

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

      8. distribute-lft1-in N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 65.7

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

   5. Applied rewrites 65.7%

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

   7. Applied rewrites 73.8%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq -1 \cdot 10^{-317} \lor \neg \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq \infty\right):\\ \;\;\;\;\left(x.im \cdot x.im\right) \cdot \left(-x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(3 \cdot x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m + \left(x.re \cdot x.im\_m + x.im\_m \cdot x.re\right) \cdot x.re\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 10^{-275} \lor \neg \left(t\_0 \leq \infty\right):\\ \;\;\;\;\left(x.im\_m \cdot x.im\_m\right) \cdot \left(-x.im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im\_m \cdot x.re\right) \cdot \left(2 + x.re\right)\\ \end{array} \end{array} \end{array} \]

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.re) (* x.im_m x.im_m)) x.im_m)
          (* (+ (* x.re x.im_m) (* x.im_m x.re)) x.re))))
   (*
    x.im_s
    (if (or (<= t_0 1e-275) (not (<= t_0 INFINITY)))
      (* (* x.im_m x.im_m) (- x.im_m))
      (* (* x.im_m x.re) (+ 2.0 x.re))))))

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_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	double tmp;
	if ((t_0 <= 1e-275) || !(t_0 <= ((double) INFINITY))) {
		tmp = (x_46_im_m * x_46_im_m) * -x_46_im_m;
	} else {
		tmp = (x_46_im_m * x_46_re) * (2.0 + x_46_re);
	}
	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_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	double tmp;
	if ((t_0 <= 1e-275) || !(t_0 <= Double.POSITIVE_INFINITY)) {
		tmp = (x_46_im_m * x_46_im_m) * -x_46_im_m;
	} else {
		tmp = (x_46_im_m * x_46_re) * (2.0 + x_46_re);
	}
	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_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re)
	tmp = 0
	if (t_0 <= 1e-275) or not (t_0 <= math.inf):
		tmp = (x_46_im_m * x_46_im_m) * -x_46_im_m
	else:
		tmp = (x_46_im_m * x_46_re) * (2.0 + x_46_re)
	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_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m) + Float64(Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_im_m * x_46_re)) * x_46_re))
	tmp = 0.0
	if ((t_0 <= 1e-275) || !(t_0 <= Inf))
		tmp = Float64(Float64(x_46_im_m * x_46_im_m) * Float64(-x_46_im_m));
	else
		tmp = Float64(Float64(x_46_im_m * x_46_re) * Float64(2.0 + x_46_re));
	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_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	tmp = 0.0;
	if ((t_0 <= 1e-275) || ~((t_0 <= Inf)))
		tmp = (x_46_im_m * x_46_im_m) * -x_46_im_m;
	else
		tmp = (x_46_im_m * x_46_re) * (2.0 + x_46_re);
	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[(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] + N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[Or[LessEqual[t$95$0, 1e-275], N[Not[LessEqual[t$95$0, Infinity]], $MachinePrecision]], N[(N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision] * (-x$46$im$95$m)), $MachinePrecision], N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] * N[(2.0 + x$46$re), $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)) < 9.99999999999999934e-276 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 79.4%

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

   3. Taylor expanded in x.re around 0

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

      1. distribute-rgt-in N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-lft-in N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. distribute-rgt-in N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   5. Applied rewrites 93.4%

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

   6. Taylor expanded in x.re around 0

      \[\leadsto \left(-{x.im}^{2}\right) \cdot x.im \]
   7. Step-by-step derivation

   8. Applied rewrites 63.6%

      \[\leadsto \left(-x.im \cdot x.im\right) \cdot x.im \]

   if 9.99999999999999934e-276 < (+.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 87.7%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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. sub-neg N/A

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

      3. flip-+ N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow-prod-down N/A

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

      9. pow-prod-up N/A

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

      10. lower-pow.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

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

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. lift-*.f64 N/A

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

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

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

      19. lower-*.f64 N/A

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

      20. lower-neg.f64 N/A

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

      21. lower--.f64 N/A

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

   4. Applied rewrites 30.0%

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

   6. Applied rewrites 43.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. flip-+ N/A

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

      6. +-inverses N/A

         \[\leadsto \left(\frac{x.im}{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)} \cdot \mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right) \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right) + \frac{\left(x.im \cdot x.re\right) \cdot \left(x.im \cdot x.re\right) - \left(x.im \cdot x.re\right) \cdot \left(x.im \cdot x.re\right)}{\color{blue}{0}} \cdot x.re \]

      7. +-inverses N/A

         \[\leadsto \left(\frac{x.im}{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)} \cdot \mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right) \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right) + \frac{\color{blue}{0}}{0} \cdot x.re \]

      8. +-inverses N/A

         \[\leadsto \left(\frac{x.im}{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)} \cdot \mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right) \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right) + \frac{\color{blue}{x.im \cdot x.im - x.im \cdot x.im}}{0} \cdot x.re \]

      9. +-inverses N/A

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

      10. flip-+ N/A

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

      11. lift-+.f64 23.8

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

   8. Applied rewrites 23.8%

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

   9. Taylor expanded in x.im around 0

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

       1. distribute-lft-in N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. unpow2 N/A

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

       5. associate-*r* N/A

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

       6. distribute-lft-out N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. lower-+.f64 31.8

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

   11. Applied rewrites 31.8%

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

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq 10^{-275} \lor \neg \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq \infty\right):\\ \;\;\;\;\left(x.im \cdot x.im\right) \cdot \left(-x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im \cdot x.re\right) \cdot \left(2 + x.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m + \left(x.re \cdot x.im\_m + x.im\_m \cdot x.re\right) \cdot x.re\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-317} \lor \neg \left(t\_0 \leq \infty\right):\\ \;\;\;\;\left(x.im\_m \cdot x.im\_m\right) \cdot \left(-x.im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.im\_m\\ \end{array} \end{array} \end{array} \]

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.re) (* x.im_m x.im_m)) x.im_m)
          (* (+ (* x.re x.im_m) (* x.im_m x.re)) x.re))))
   (*
    x.im_s
    (if (or (<= t_0 -1e-317) (not (<= t_0 INFINITY)))
      (* (* x.im_m x.im_m) (- x.im_m))
      (* (* x.re x.re) 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_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	double tmp;
	if ((t_0 <= -1e-317) || !(t_0 <= ((double) INFINITY))) {
		tmp = (x_46_im_m * x_46_im_m) * -x_46_im_m;
	} else {
		tmp = (x_46_re * x_46_re) * x_46_im_m;
	}
	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_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	double tmp;
	if ((t_0 <= -1e-317) || !(t_0 <= Double.POSITIVE_INFINITY)) {
		tmp = (x_46_im_m * x_46_im_m) * -x_46_im_m;
	} else {
		tmp = (x_46_re * x_46_re) * x_46_im_m;
	}
	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_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re)
	tmp = 0
	if (t_0 <= -1e-317) or not (t_0 <= math.inf):
		tmp = (x_46_im_m * x_46_im_m) * -x_46_im_m
	else:
		tmp = (x_46_re * x_46_re) * 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_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m) + Float64(Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_im_m * x_46_re)) * x_46_re))
	tmp = 0.0
	if ((t_0 <= -1e-317) || !(t_0 <= Inf))
		tmp = Float64(Float64(x_46_im_m * x_46_im_m) * Float64(-x_46_im_m));
	else
		tmp = Float64(Float64(x_46_re * x_46_re) * x_46_im_m);
	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_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	tmp = 0.0;
	if ((t_0 <= -1e-317) || ~((t_0 <= Inf)))
		tmp = (x_46_im_m * x_46_im_m) * -x_46_im_m;
	else
		tmp = (x_46_re * x_46_re) * x_46_im_m;
	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[(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] + N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[Or[LessEqual[t$95$0, -1e-317], N[Not[LessEqual[t$95$0, Infinity]], $MachinePrecision]], N[(N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision] * (-x$46$im$95$m)), $MachinePrecision], N[(N[(x$46$re * x$46$re), $MachinePrecision] * x$46$im$95$m), $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)) < -1.00000023e-317 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 71.6%

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

   3. Taylor expanded in x.re around 0

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

      1. distribute-rgt-in N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-lft-in N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. distribute-rgt-in N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   5. Applied rewrites 90.9%

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

   6. Taylor expanded in x.re around 0

      \[\leadsto \left(-{x.im}^{2}\right) \cdot x.im \]
   7. Step-by-step derivation

   8. Applied rewrites 51.6%

      \[\leadsto \left(-x.im \cdot x.im\right) \cdot x.im \]

   if -1.00000023e-317 < (+.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 91.7%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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. sub-neg N/A

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

      3. flip-+ N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow-prod-down N/A

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

      9. pow-prod-up N/A

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

      10. lower-pow.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

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

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. lift-*.f64 N/A

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

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

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

      19. lower-*.f64 N/A

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

      20. lower-neg.f64 N/A

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

      21. lower--.f64 N/A

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

   4. Applied rewrites 44.7%

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

   6. Applied rewrites 54.5%

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

   8. Applied rewrites 42.8%

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

   9. Taylor expanded in x.re around inf

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

       1. distribute-rgt-in N/A

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

       2. associate-*l/ N/A

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

       3. *-commutative N/A

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

       4. distribute-rgt1-in N/A

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

       5. metadata-eval N/A

          \[\leadsto x.im \cdot {x.re}^{2} + \frac{\left(\left(\color{blue}{0} \cdot x.im\right) \cdot x.im\right) \cdot {x.re}^{2}}{x.re} \]

       6. mul0-lft N/A

          \[\leadsto x.im \cdot {x.re}^{2} + \frac{\left(\color{blue}{0} \cdot x.im\right) \cdot {x.re}^{2}}{x.re} \]

       7. mul0-lft N/A

          \[\leadsto x.im \cdot {x.re}^{2} + \frac{\color{blue}{0} \cdot {x.re}^{2}}{x.re} \]

       8. unpow2 N/A

          \[\leadsto x.im \cdot {x.re}^{2} + \frac{0 \cdot \color{blue}{\left(x.re \cdot x.re\right)}}{x.re} \]

       9. associate-*r* N/A

          \[\leadsto x.im \cdot {x.re}^{2} + \frac{\color{blue}{\left(0 \cdot x.re\right) \cdot x.re}}{x.re} \]

       10. mul0-lft N/A

           \[\leadsto x.im \cdot {x.re}^{2} + \frac{\color{blue}{0} \cdot x.re}{x.re} \]

       11. mul0-lft N/A

           \[\leadsto x.im \cdot {x.re}^{2} + \frac{\color{blue}{0}}{x.re} \]

       12. mul0-lft N/A

           \[\leadsto x.im \cdot {x.re}^{2} + \frac{\color{blue}{0 \cdot x.im}}{x.re} \]

       13. associate-*r/ N/A

           \[\leadsto x.im \cdot {x.re}^{2} + \color{blue}{0 \cdot \frac{x.im}{x.re}} \]

       14. mul0-lft N/A

           \[\leadsto x.im \cdot {x.re}^{2} + \color{blue}{0} \]

       15. +-rgt-identity N/A

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

       16. *-commutative N/A

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

       17. lower-*.f64 N/A

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

       18. unpow2 N/A

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

       19. lower-*.f64 51.1

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

   11. Applied rewrites 51.1%

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

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq -1 \cdot 10^{-317} \lor \neg \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq \infty\right):\\ \;\;\;\;\left(x.im \cdot x.im\right) \cdot \left(-x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.im\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 35.2% accurate, 3.6× speedup

Math

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

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.re x.re) 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_re * x_46_re) * 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_46re * x_46re) * 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_re * x_46_re) * 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_re * x_46_re) * 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(x_46_re * x_46_re) * 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_re * x_46_re) * 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$46$re * x$46$re), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.4%

   \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
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. sub-neg N/A

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

   3. flip-+ N/A

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

   4. lower-/.f64 N/A

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

   5. lower--.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 N/A

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

   8. pow-prod-down N/A

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

   9. pow-prod-up N/A

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

   10. lower-pow.f64 N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f64 N/A

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

   13. lift-*.f64 N/A

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

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

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

   15. lower-*.f64 N/A

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

   16. lower-neg.f64 N/A

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

   17. lift-*.f64 N/A

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

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

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

   19. lower-*.f64 N/A

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

   20. lower-neg.f64 N/A

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

   21. lower--.f64 N/A

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

4. Applied rewrites 37.2%

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

6. Applied rewrites 46.0%

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

8. Applied rewrites 57.1%

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

9. Taylor expanded in x.re around inf

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

    1. distribute-rgt-in N/A

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

    2. associate-*l/ N/A

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

    3. *-commutative N/A

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

    4. distribute-rgt1-in N/A

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

    5. metadata-eval N/A

       \[\leadsto x.im \cdot {x.re}^{2} + \frac{\left(\left(\color{blue}{0} \cdot x.im\right) \cdot x.im\right) \cdot {x.re}^{2}}{x.re} \]

    6. mul0-lft N/A

       \[\leadsto x.im \cdot {x.re}^{2} + \frac{\left(\color{blue}{0} \cdot x.im\right) \cdot {x.re}^{2}}{x.re} \]

    7. mul0-lft N/A

       \[\leadsto x.im \cdot {x.re}^{2} + \frac{\color{blue}{0} \cdot {x.re}^{2}}{x.re} \]

    8. unpow2 N/A

       \[\leadsto x.im \cdot {x.re}^{2} + \frac{0 \cdot \color{blue}{\left(x.re \cdot x.re\right)}}{x.re} \]

    9. associate-*r* N/A

       \[\leadsto x.im \cdot {x.re}^{2} + \frac{\color{blue}{\left(0 \cdot x.re\right) \cdot x.re}}{x.re} \]

    10. mul0-lft N/A

        \[\leadsto x.im \cdot {x.re}^{2} + \frac{\color{blue}{0} \cdot x.re}{x.re} \]

    11. mul0-lft N/A

        \[\leadsto x.im \cdot {x.re}^{2} + \frac{\color{blue}{0}}{x.re} \]

    12. mul0-lft N/A

        \[\leadsto x.im \cdot {x.re}^{2} + \frac{\color{blue}{0 \cdot x.im}}{x.re} \]

    13. associate-*r/ N/A

        \[\leadsto x.im \cdot {x.re}^{2} + \color{blue}{0 \cdot \frac{x.im}{x.re}} \]

    14. mul0-lft N/A

        \[\leadsto x.im \cdot {x.re}^{2} + \color{blue}{0} \]

    15. +-rgt-identity N/A

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

    16. *-commutative N/A

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

    17. lower-*.f64 N/A

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

    18. unpow2 N/A

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

    19. lower-*.f64 40.8

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

11. Applied rewrites 40.8%

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

Developer Target 1: 91.4% 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 2024315 
(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)))