math.cube on complex, imaginary part

Percentage Accurate: 82.6% → 99.8%

Time: 7.0s

Alternatives: 8

Speedup: 0.9×

• 44.2% 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

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

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im)
 :precision binary64
 (let* ((t_0 (* (- x.re_m x.im) x.im)))
   (if (<=
        (+
         (* (- (* x.re_m x.re_m) (* x.im x.im)) x.im)
         (* (+ (* x.re_m x.im) (* x.im x.re_m)) x.re_m))
        INFINITY)
     (+
      (* (fma (sqrt x.re_m) (sqrt x.re_m) x.im) t_0)
      (* (* (* 2.0 x.re_m) x.im) x.re_m))
     (/ (* t_0 (* (+ x.im x.re_m) (- x.im x.re_m))) (- x.im x.re_m)))))

C

x.re_m = fabs(x_46_re);
double code(double x_46_re_m, double x_46_im) {
	double t_0 = (x_46_re_m - x_46_im) * x_46_im;
	double tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_im) + (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_re_m)) <= ((double) INFINITY)) {
		tmp = (fma(sqrt(x_46_re_m), sqrt(x_46_re_m), x_46_im) * t_0) + (((2.0 * x_46_re_m) * x_46_im) * x_46_re_m);
	} else {
		tmp = (t_0 * ((x_46_im + x_46_re_m) * (x_46_im - x_46_re_m))) / (x_46_im - x_46_re_m);
	}
	return tmp;
}

Julia

x.re_m = abs(x_46_re)
function code(x_46_re_m, x_46_im)
	t_0 = Float64(Float64(x_46_re_m - x_46_im) * x_46_im)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im)) * x_46_im) + Float64(Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_im * x_46_re_m)) * x_46_re_m)) <= Inf)
		tmp = Float64(Float64(fma(sqrt(x_46_re_m), sqrt(x_46_re_m), x_46_im) * t_0) + Float64(Float64(Float64(2.0 * x_46_re_m) * x_46_im) * x_46_re_m));
	else
		tmp = Float64(Float64(t_0 * Float64(Float64(x_46_im + x_46_re_m) * Float64(x_46_im - x_46_re_m))) / Float64(x_46_im - x_46_re_m));
	end
	return tmp
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
code[x$46$re$95$m_, x$46$im_] := Block[{t$95$0 = N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * x$46$im), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision] + N[(N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(N[Sqrt[x$46$re$95$m], $MachinePrecision] * N[Sqrt[x$46$re$95$m], $MachinePrecision] + x$46$im), $MachinePrecision] * t$95$0), $MachinePrecision] + N[(N[(N[(2.0 * x$46$re$95$m), $MachinePrecision] * x$46$im), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(N[(x$46$im + x$46$re$95$m), $MachinePrecision] * N[(x$46$im - x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$46$im - x$46$re$95$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)) < +inf.0

   1. Initial program 94.3%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. difference-of-squares N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow1 N/A

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

      4. metadata-eval N/A

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

      5. sqrt-pow1 N/A

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

      6. pow2 N/A

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

      7. rem-sqrt-square-rev N/A

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

      8. unpow1 N/A

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

      9. metadata-eval N/A

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

      10. sqrt-pow1 N/A

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

      11. pow2 N/A

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

      12. rem-sqrt-square-rev N/A

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

      13. rem-sqrt-square-rev N/A

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

      14. pow2 N/A

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

      15. sqrt-pow1 N/A

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

      16. metadata-eval N/A

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

      17. unpow1 N/A

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

      18. unpow1 N/A

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

      19. metadata-eval N/A

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

      20. sqrt-pow1 N/A

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

      21. pow2 N/A

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

      22. sqrt-prod N/A

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

      23. rem-sqrt-square-rev N/A

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

      24. pow2 N/A

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

      25. sqrt-pow1 N/A

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

      26. metadata-eval N/A

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

      27. unpow1 N/A

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

      28. lower-fma.f64 N/A

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

      29. lower-sqrt.f64 N/A

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

      30. lower-sqrt.f64 49.2

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

   6. Applied rewrites 49.2%

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

   7. Taylor expanded in x.re around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 49.2

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

   9. Applied rewrites 49.2%

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

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

   1. Initial program 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. difference-of-squares N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 21.2

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

   4. Applied rewrites 21.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. difference-of-squares N/A

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

      9. lift-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 21.2

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

   6. Applied rewrites 21.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. unpow1 N/A

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

      8. metadata-eval N/A

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

      9. sqrt-pow1 N/A

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

      10. pow2 N/A

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

      11. rem-sqrt-square-rev N/A

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

      12. rem-square-sqrt N/A

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

      13. sqrt-prod N/A

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

      14. sqr-neg-rev N/A

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

      15. sqrt-unprod N/A

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

      16. rem-square-sqrt N/A

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

      17. rem-sqrt-square-rev N/A

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

      18. pow2 N/A

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

      19. sqrt-pow1 N/A

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

      20. metadata-eval N/A

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

      21. unpow1 N/A

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

      22. *-commutative N/A

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

      23. lift-*.f64 N/A

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

      24. +-inverses N/A

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

      25. +-inverses N/A

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

      26. lift-*.f64 N/A

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

      27. *-commutative N/A

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

      28. lift-*.f64 N/A

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

   8. Applied rewrites 100.0%

      \[\leadsto \frac{\left(\left(x.re - x.im\right) \cdot x.im\right) \cdot \left(\left(x.im + x.re\right) \cdot \left(x.im - x.re\right)\right)}{x.im - x.re} + \color{blue}{x.re \cdot 0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.8%

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

Alternative 2: 59.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ \begin{array}{l} t_0 := \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) \cdot x.im + \left(x.re\_m \cdot x.im + x.im \cdot x.re\_m\right) \cdot x.re\_m\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-304} \lor \neg \left(t\_0 \leq \infty\right):\\ \;\;\;\;\left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\left(x.im \cdot x.re\_m\right) \cdot \left(3 \cdot x.re\_m\right)\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* x.re_m x.re_m) (* x.im x.im)) x.im)
          (* (+ (* x.re_m x.im) (* x.im x.re_m)) x.re_m))))
   (if (or (<= t_0 -1e-304) (not (<= t_0 INFINITY)))
     (* (* (- x.im) x.im) x.im)
     (* (* x.im x.re_m) (* 3.0 x.re_m)))))

C

x.re_m = fabs(x_46_re);
double code(double x_46_re_m, double x_46_im) {
	double t_0 = (((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_im) + (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_re_m);
	double tmp;
	if ((t_0 <= -1e-304) || !(t_0 <= ((double) INFINITY))) {
		tmp = (-x_46_im * x_46_im) * x_46_im;
	} else {
		tmp = (x_46_im * x_46_re_m) * (3.0 * x_46_re_m);
	}
	return tmp;
}

Java

x.re_m = Math.abs(x_46_re);
public static double code(double x_46_re_m, double x_46_im) {
	double t_0 = (((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_im) + (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_re_m);
	double tmp;
	if ((t_0 <= -1e-304) || !(t_0 <= Double.POSITIVE_INFINITY)) {
		tmp = (-x_46_im * x_46_im) * x_46_im;
	} else {
		tmp = (x_46_im * x_46_re_m) * (3.0 * x_46_re_m);
	}
	return tmp;
}

Python

x.re_m = math.fabs(x_46_re)
def code(x_46_re_m, x_46_im):
	t_0 = (((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_im) + (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_re_m)
	tmp = 0
	if (t_0 <= -1e-304) or not (t_0 <= math.inf):
		tmp = (-x_46_im * x_46_im) * x_46_im
	else:
		tmp = (x_46_im * x_46_re_m) * (3.0 * x_46_re_m)
	return tmp

Julia

x.re_m = abs(x_46_re)
function code(x_46_re_m, x_46_im)
	t_0 = Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im)) * x_46_im) + Float64(Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_im * x_46_re_m)) * x_46_re_m))
	tmp = 0.0
	if ((t_0 <= -1e-304) || !(t_0 <= Inf))
		tmp = Float64(Float64(Float64(-x_46_im) * x_46_im) * x_46_im);
	else
		tmp = Float64(Float64(x_46_im * x_46_re_m) * Float64(3.0 * x_46_re_m));
	end
	return tmp
end

MATLAB

x.re_m = abs(x_46_re);
function tmp_2 = code(x_46_re_m, x_46_im)
	t_0 = (((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_im) + (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_re_m);
	tmp = 0.0;
	if ((t_0 <= -1e-304) || ~((t_0 <= Inf)))
		tmp = (-x_46_im * x_46_im) * x_46_im;
	else
		tmp = (x_46_im * x_46_re_m) * (3.0 * x_46_re_m);
	end
	tmp_2 = tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
code[x$46$re$95$m_, x$46$im_] := Block[{t$95$0 = N[(N[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision] + N[(N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -1e-304], N[Not[LessEqual[t$95$0, Infinity]], $MachinePrecision]], N[(N[((-x$46$im) * x$46$im), $MachinePrecision] * x$46$im), $MachinePrecision], N[(N[(x$46$im * x$46$re$95$m), $MachinePrecision] * N[(3.0 * x$46$re$95$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)) < -9.99999999999999971e-305 or +inf.0 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re))

   1. Initial program 68.8%

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

   3. Taylor expanded in x.re around 0

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

      1. mul-1-neg N/A

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

      2. cube-neg-rev N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-neg.f64 54.0

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

   5. Applied rewrites 54.0%

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

   7. Applied rewrites 53.9%

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

   if -9.99999999999999971e-305 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   1. Initial program 96.7%

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

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt1-in N/A

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

      7. metadata-eval N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 66.9

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

   5. Applied rewrites 66.9%

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

   7. Applied rewrites 66.9%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq -1 \cdot 10^{-304} \lor \neg \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq \infty\right):\\ \;\;\;\;\left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\left(x.im \cdot x.re\right) \cdot \left(3 \cdot x.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ \begin{array}{l} t_0 := \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) \cdot x.im + \left(x.re\_m \cdot x.im + x.im \cdot x.re\_m\right) \cdot x.re\_m\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-304} \lor \neg \left(t\_0 \leq \infty\right):\\ \;\;\;\;\left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\left(x.im \cdot x.re\_m\right) \cdot x.re\_m\right)\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* x.re_m x.re_m) (* x.im x.im)) x.im)
          (* (+ (* x.re_m x.im) (* x.im x.re_m)) x.re_m))))
   (if (or (<= t_0 -1e-304) (not (<= t_0 INFINITY)))
     (* (* (- x.im) x.im) x.im)
     (* 3.0 (* (* x.im x.re_m) x.re_m)))))

C

x.re_m = fabs(x_46_re);
double code(double x_46_re_m, double x_46_im) {
	double t_0 = (((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_im) + (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_re_m);
	double tmp;
	if ((t_0 <= -1e-304) || !(t_0 <= ((double) INFINITY))) {
		tmp = (-x_46_im * x_46_im) * x_46_im;
	} else {
		tmp = 3.0 * ((x_46_im * x_46_re_m) * x_46_re_m);
	}
	return tmp;
}

Java

x.re_m = Math.abs(x_46_re);
public static double code(double x_46_re_m, double x_46_im) {
	double t_0 = (((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_im) + (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_re_m);
	double tmp;
	if ((t_0 <= -1e-304) || !(t_0 <= Double.POSITIVE_INFINITY)) {
		tmp = (-x_46_im * x_46_im) * x_46_im;
	} else {
		tmp = 3.0 * ((x_46_im * x_46_re_m) * x_46_re_m);
	}
	return tmp;
}

Python

x.re_m = math.fabs(x_46_re)
def code(x_46_re_m, x_46_im):
	t_0 = (((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_im) + (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_re_m)
	tmp = 0
	if (t_0 <= -1e-304) or not (t_0 <= math.inf):
		tmp = (-x_46_im * x_46_im) * x_46_im
	else:
		tmp = 3.0 * ((x_46_im * x_46_re_m) * x_46_re_m)
	return tmp

Julia

x.re_m = abs(x_46_re)
function code(x_46_re_m, x_46_im)
	t_0 = Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im)) * x_46_im) + Float64(Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_im * x_46_re_m)) * x_46_re_m))
	tmp = 0.0
	if ((t_0 <= -1e-304) || !(t_0 <= Inf))
		tmp = Float64(Float64(Float64(-x_46_im) * x_46_im) * x_46_im);
	else
		tmp = Float64(3.0 * Float64(Float64(x_46_im * x_46_re_m) * x_46_re_m));
	end
	return tmp
end

MATLAB

x.re_m = abs(x_46_re);
function tmp_2 = code(x_46_re_m, x_46_im)
	t_0 = (((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_im) + (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_re_m);
	tmp = 0.0;
	if ((t_0 <= -1e-304) || ~((t_0 <= Inf)))
		tmp = (-x_46_im * x_46_im) * x_46_im;
	else
		tmp = 3.0 * ((x_46_im * x_46_re_m) * x_46_re_m);
	end
	tmp_2 = tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
code[x$46$re$95$m_, x$46$im_] := Block[{t$95$0 = N[(N[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision] + N[(N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -1e-304], N[Not[LessEqual[t$95$0, Infinity]], $MachinePrecision]], N[(N[((-x$46$im) * x$46$im), $MachinePrecision] * x$46$im), $MachinePrecision], N[(3.0 * N[(N[(x$46$im * x$46$re$95$m), $MachinePrecision] * x$46$re$95$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)) < -9.99999999999999971e-305 or +inf.0 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re))

   1. Initial program 68.8%

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

   3. Taylor expanded in x.re around 0

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

      1. mul-1-neg N/A

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

      2. cube-neg-rev N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-neg.f64 54.0

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

   5. Applied rewrites 54.0%

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

   7. Applied rewrites 53.9%

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

   if -9.99999999999999971e-305 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   1. Initial program 96.7%

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

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt1-in N/A

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

      7. metadata-eval N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 66.9

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

   5. Applied rewrites 66.9%

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

   7. Applied rewrites 66.9%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq -1 \cdot 10^{-304} \lor \neg \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq \infty\right):\\ \;\;\;\;\left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\left(x.im \cdot x.re\right) \cdot x.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ \begin{array}{l} t_0 := \left(x.re\_m - x.im\right) \cdot x.im\\ t_1 := \left(x.re\_m \cdot x.im + x.im \cdot x.re\_m\right) \cdot x.re\_m\\ \mathbf{if}\;\left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) \cdot x.im + t\_1 \leq \infty:\\ \;\;\;\;\left(x.im + x.re\_m\right) \cdot t\_0 + t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot \left(\left(x.im + x.re\_m\right) \cdot \left(x.im - x.re\_m\right)\right)}{x.im - x.re\_m}\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im)
 :precision binary64
 (let* ((t_0 (* (- x.re_m x.im) x.im))
        (t_1 (* (+ (* x.re_m x.im) (* x.im x.re_m)) x.re_m)))
   (if (<= (+ (* (- (* x.re_m x.re_m) (* x.im x.im)) x.im) t_1) INFINITY)
     (+ (* (+ x.im x.re_m) t_0) t_1)
     (/ (* t_0 (* (+ x.im x.re_m) (- x.im x.re_m))) (- x.im x.re_m)))))

C

x.re_m = fabs(x_46_re);
double code(double x_46_re_m, double x_46_im) {
	double t_0 = (x_46_re_m - x_46_im) * x_46_im;
	double t_1 = ((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_re_m;
	double tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_im) + t_1) <= ((double) INFINITY)) {
		tmp = ((x_46_im + x_46_re_m) * t_0) + t_1;
	} else {
		tmp = (t_0 * ((x_46_im + x_46_re_m) * (x_46_im - x_46_re_m))) / (x_46_im - x_46_re_m);
	}
	return tmp;
}

Java

x.re_m = Math.abs(x_46_re);
public static double code(double x_46_re_m, double x_46_im) {
	double t_0 = (x_46_re_m - x_46_im) * x_46_im;
	double t_1 = ((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_re_m;
	double tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_im) + t_1) <= Double.POSITIVE_INFINITY) {
		tmp = ((x_46_im + x_46_re_m) * t_0) + t_1;
	} else {
		tmp = (t_0 * ((x_46_im + x_46_re_m) * (x_46_im - x_46_re_m))) / (x_46_im - x_46_re_m);
	}
	return tmp;
}

Python

x.re_m = math.fabs(x_46_re)
def code(x_46_re_m, x_46_im):
	t_0 = (x_46_re_m - x_46_im) * x_46_im
	t_1 = ((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_re_m
	tmp = 0
	if ((((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_im) + t_1) <= math.inf:
		tmp = ((x_46_im + x_46_re_m) * t_0) + t_1
	else:
		tmp = (t_0 * ((x_46_im + x_46_re_m) * (x_46_im - x_46_re_m))) / (x_46_im - x_46_re_m)
	return tmp

Julia

x.re_m = abs(x_46_re)
function code(x_46_re_m, x_46_im)
	t_0 = Float64(Float64(x_46_re_m - x_46_im) * x_46_im)
	t_1 = Float64(Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_im * x_46_re_m)) * x_46_re_m)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im)) * x_46_im) + t_1) <= Inf)
		tmp = Float64(Float64(Float64(x_46_im + x_46_re_m) * t_0) + t_1);
	else
		tmp = Float64(Float64(t_0 * Float64(Float64(x_46_im + x_46_re_m) * Float64(x_46_im - x_46_re_m))) / Float64(x_46_im - x_46_re_m));
	end
	return tmp
end

MATLAB

x.re_m = abs(x_46_re);
function tmp_2 = code(x_46_re_m, x_46_im)
	t_0 = (x_46_re_m - x_46_im) * x_46_im;
	t_1 = ((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_re_m;
	tmp = 0.0;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_im) + t_1) <= Inf)
		tmp = ((x_46_im + x_46_re_m) * t_0) + t_1;
	else
		tmp = (t_0 * ((x_46_im + x_46_re_m) * (x_46_im - x_46_re_m))) / (x_46_im - x_46_re_m);
	end
	tmp_2 = tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
code[x$46$re$95$m_, x$46$im_] := Block[{t$95$0 = N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * x$46$im), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision] + t$95$1), $MachinePrecision], Infinity], N[(N[(N[(x$46$im + x$46$re$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(t$95$0 * N[(N[(x$46$im + x$46$re$95$m), $MachinePrecision] * N[(x$46$im - x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$46$im - x$46$re$95$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)) < +inf.0

   1. Initial program 94.3%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. difference-of-squares N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 99.7

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

   4. Applied rewrites 99.7%

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

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

   1. Initial program 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. difference-of-squares N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 21.2

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

   4. Applied rewrites 21.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. difference-of-squares N/A

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

      9. lift-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 21.2

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

   6. Applied rewrites 21.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. unpow1 N/A

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

      8. metadata-eval N/A

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

      9. sqrt-pow1 N/A

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

      10. pow2 N/A

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

      11. rem-sqrt-square-rev N/A

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

      12. rem-square-sqrt N/A

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

      13. sqrt-prod N/A

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

      14. sqr-neg-rev N/A

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

      15. sqrt-unprod N/A

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

      16. rem-square-sqrt N/A

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

      17. rem-sqrt-square-rev N/A

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

      18. pow2 N/A

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

      19. sqrt-pow1 N/A

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

      20. metadata-eval N/A

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

      21. unpow1 N/A

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

      22. *-commutative N/A

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

      23. lift-*.f64 N/A

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

      24. +-inverses N/A

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

      25. +-inverses N/A

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

      26. lift-*.f64 N/A

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

      27. *-commutative N/A

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

      28. lift-*.f64 N/A

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

   8. Applied rewrites 100.0%

      \[\leadsto \frac{\left(\left(x.re - x.im\right) \cdot x.im\right) \cdot \left(\left(x.im + x.re\right) \cdot \left(x.im - x.re\right)\right)}{x.im - x.re} + \color{blue}{x.re \cdot 0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

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

Alternative 5: 94.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ \begin{array}{l} \mathbf{if}\;x.im \leq 1.65 \cdot 10^{+199}:\\ \;\;\;\;\left(x.im + x.re\_m\right) \cdot \left(\left(x.re\_m - x.im\right) \cdot x.im\right) + \left(x.re\_m \cdot x.im + x.im \cdot x.re\_m\right) \cdot x.re\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im)
 :precision binary64
 (if (<= x.im 1.65e+199)
   (+
    (* (+ x.im x.re_m) (* (- x.re_m x.im) x.im))
    (* (+ (* x.re_m x.im) (* x.im x.re_m)) x.re_m))
   (* (* (- x.im) x.im) x.im)))

C

x.re_m = fabs(x_46_re);
double code(double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_im <= 1.65e+199) {
		tmp = ((x_46_im + x_46_re_m) * ((x_46_re_m - x_46_im) * x_46_im)) + (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_re_m);
	} else {
		tmp = (-x_46_im * x_46_im) * x_46_im;
	}
	return tmp;
}

Fortran

x.re_m = abs(x_46re)
real(8) function code(x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46im <= 1.65d+199) then
        tmp = ((x_46im + x_46re_m) * ((x_46re_m - x_46im) * x_46im)) + (((x_46re_m * x_46im) + (x_46im * x_46re_m)) * x_46re_m)
    else
        tmp = (-x_46im * x_46im) * x_46im
    end if
    code = tmp
end function

Java

x.re_m = Math.abs(x_46_re);
public static double code(double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_im <= 1.65e+199) {
		tmp = ((x_46_im + x_46_re_m) * ((x_46_re_m - x_46_im) * x_46_im)) + (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_re_m);
	} else {
		tmp = (-x_46_im * x_46_im) * x_46_im;
	}
	return tmp;
}

Python

x.re_m = math.fabs(x_46_re)
def code(x_46_re_m, x_46_im):
	tmp = 0
	if x_46_im <= 1.65e+199:
		tmp = ((x_46_im + x_46_re_m) * ((x_46_re_m - x_46_im) * x_46_im)) + (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_re_m)
	else:
		tmp = (-x_46_im * x_46_im) * x_46_im
	return tmp

Julia

x.re_m = abs(x_46_re)
function code(x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_im <= 1.65e+199)
		tmp = Float64(Float64(Float64(x_46_im + x_46_re_m) * Float64(Float64(x_46_re_m - x_46_im) * x_46_im)) + Float64(Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_im * x_46_re_m)) * x_46_re_m));
	else
		tmp = Float64(Float64(Float64(-x_46_im) * x_46_im) * x_46_im);
	end
	return tmp
end

MATLAB

x.re_m = abs(x_46_re);
function tmp_2 = code(x_46_re_m, x_46_im)
	tmp = 0.0;
	if (x_46_im <= 1.65e+199)
		tmp = ((x_46_im + x_46_re_m) * ((x_46_re_m - x_46_im) * x_46_im)) + (((x_46_re_m * x_46_im) + (x_46_im * x_46_re_m)) * x_46_re_m);
	else
		tmp = (-x_46_im * x_46_im) * x_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
code[x$46$re$95$m_, x$46$im_] := If[LessEqual[x$46$im, 1.65e+199], N[(N[(N[(x$46$im + x$46$re$95$m), $MachinePrecision] * N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]), $MachinePrecision], N[(N[((-x$46$im) * x$46$im), $MachinePrecision] * x$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.6499999999999999e199

   1. Initial program 86.4%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. difference-of-squares N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 94.6

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

   4. Applied rewrites 94.6%

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

   if 1.6499999999999999e199 < x.im

   1. Initial program 39.1%

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

   3. Taylor expanded in x.re around 0

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

      1. mul-1-neg N/A

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

      2. cube-neg-rev N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

Alternative 6: 76.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ \begin{array}{l} \mathbf{if}\;x.im \leq 6.6 \cdot 10^{-118}:\\ \;\;\;\;\left(x.im \cdot x.re\_m\right) \cdot \left(3 \cdot x.re\_m\right)\\ \mathbf{elif}\;x.im \leq 5 \cdot 10^{+200}:\\ \;\;\;\;\left(-x.im\right) \cdot \mathsf{fma}\left(x.im, x.im, -3 \cdot \left(x.re\_m \cdot x.re\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im)
 :precision binary64
 (if (<= x.im 6.6e-118)
   (* (* x.im x.re_m) (* 3.0 x.re_m))
   (if (<= x.im 5e+200)
     (* (- x.im) (fma x.im x.im (* -3.0 (* x.re_m x.re_m))))
     (* (* (- x.im) x.im) x.im))))

C

x.re_m = fabs(x_46_re);
double code(double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_im <= 6.6e-118) {
		tmp = (x_46_im * x_46_re_m) * (3.0 * x_46_re_m);
	} else if (x_46_im <= 5e+200) {
		tmp = -x_46_im * fma(x_46_im, x_46_im, (-3.0 * (x_46_re_m * x_46_re_m)));
	} else {
		tmp = (-x_46_im * x_46_im) * x_46_im;
	}
	return tmp;
}

Julia

x.re_m = abs(x_46_re)
function code(x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_im <= 6.6e-118)
		tmp = Float64(Float64(x_46_im * x_46_re_m) * Float64(3.0 * x_46_re_m));
	elseif (x_46_im <= 5e+200)
		tmp = Float64(Float64(-x_46_im) * fma(x_46_im, x_46_im, Float64(-3.0 * Float64(x_46_re_m * x_46_re_m))));
	else
		tmp = Float64(Float64(Float64(-x_46_im) * x_46_im) * x_46_im);
	end
	return tmp
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
code[x$46$re$95$m_, x$46$im_] := If[LessEqual[x$46$im, 6.6e-118], N[(N[(x$46$im * x$46$re$95$m), $MachinePrecision] * N[(3.0 * x$46$re$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 5e+200], N[((-x$46$im) * N[(x$46$im * x$46$im + N[(-3.0 * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-x$46$im) * x$46$im), $MachinePrecision] * x$46$im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.im < 6.5999999999999999e-118

   1. Initial program 84.8%

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

   3. Taylor expanded in x.re around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt1-in N/A

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

      7. metadata-eval N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 70.3

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

   5. Applied rewrites 70.3%

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

   7. Applied rewrites 70.3%

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

   if 6.5999999999999999e-118 < x.im < 5.00000000000000019e200

   1. Initial program 90.4%

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

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-in N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. mul-1-neg N/A

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

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

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

      11. distribute-lft-out-- N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. distribute-lft1-in N/A

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

      15. metadata-eval N/A

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

      16. *-commutative N/A

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

      17. remove-double-neg N/A

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

   5. Applied rewrites 98.1%

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

   if 5.00000000000000019e200 < x.im

   1. Initial program 39.1%

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

   3. Taylor expanded in x.re around 0

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

      1. mul-1-neg N/A

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

      2. cube-neg-rev N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

Alternative 7: 62.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ \begin{array}{l} \mathbf{if}\;x.re\_m \leq 8.6 \cdot 10^{+200}:\\ \;\;\;\;\left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\left(x.im \cdot x.im\right) \cdot x.im\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im)
 :precision binary64
 (if (<= x.re_m 8.6e+200) (* (* (- x.im) x.im) x.im) (* (* x.im x.im) x.im)))

C

x.re_m = fabs(x_46_re);
double code(double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 8.6e+200) {
		tmp = (-x_46_im * x_46_im) * x_46_im;
	} else {
		tmp = (x_46_im * x_46_im) * x_46_im;
	}
	return tmp;
}

Fortran

x.re_m = abs(x_46re)
real(8) function code(x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re_m <= 8.6d+200) then
        tmp = (-x_46im * x_46im) * x_46im
    else
        tmp = (x_46im * x_46im) * x_46im
    end if
    code = tmp
end function

Java

x.re_m = Math.abs(x_46_re);
public static double code(double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 8.6e+200) {
		tmp = (-x_46_im * x_46_im) * x_46_im;
	} else {
		tmp = (x_46_im * x_46_im) * x_46_im;
	}
	return tmp;
}

Python

x.re_m = math.fabs(x_46_re)
def code(x_46_re_m, x_46_im):
	tmp = 0
	if x_46_re_m <= 8.6e+200:
		tmp = (-x_46_im * x_46_im) * x_46_im
	else:
		tmp = (x_46_im * x_46_im) * x_46_im
	return tmp

Julia

x.re_m = abs(x_46_re)
function code(x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_re_m <= 8.6e+200)
		tmp = Float64(Float64(Float64(-x_46_im) * x_46_im) * x_46_im);
	else
		tmp = Float64(Float64(x_46_im * x_46_im) * x_46_im);
	end
	return tmp
end

MATLAB

x.re_m = abs(x_46_re);
function tmp_2 = code(x_46_re_m, x_46_im)
	tmp = 0.0;
	if (x_46_re_m <= 8.6e+200)
		tmp = (-x_46_im * x_46_im) * x_46_im;
	else
		tmp = (x_46_im * x_46_im) * x_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
code[x$46$re$95$m_, x$46$im_] := If[LessEqual[x$46$re$95$m, 8.6e+200], N[(N[((-x$46$im) * x$46$im), $MachinePrecision] * x$46$im), $MachinePrecision], N[(N[(x$46$im * x$46$im), $MachinePrecision] * x$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 8.60000000000000062e200

   1. Initial program 82.8%

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

   3. Taylor expanded in x.re around 0

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

      1. mul-1-neg N/A

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

      2. cube-neg-rev N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-neg.f64 59.5

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

   5. Applied rewrites 59.5%

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

   7. Applied rewrites 59.4%

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

   if 8.60000000000000062e200 < x.re

   1. Initial program 72.9%

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

   3. Taylor expanded in x.re around 0

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

      1. mul-1-neg N/A

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

      2. cube-neg-rev N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-neg.f64 11.9

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

   5. Applied rewrites 11.9%

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

   7. Applied rewrites 18.6%

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

Alternative 8: 21.2% accurate, 3.6× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ \left(x.im \cdot x.im\right) \cdot x.im \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im) :precision binary64 (* (* x.im x.im) x.im))

C

x.re_m = fabs(x_46_re);
double code(double x_46_re_m, double x_46_im) {
	return (x_46_im * x_46_im) * x_46_im;
}

Fortran

x.re_m = abs(x_46re)
real(8) function code(x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    code = (x_46im * x_46im) * x_46im
end function

Java

x.re_m = Math.abs(x_46_re);
public static double code(double x_46_re_m, double x_46_im) {
	return (x_46_im * x_46_im) * x_46_im;
}

Python

x.re_m = math.fabs(x_46_re)
def code(x_46_re_m, x_46_im):
	return (x_46_im * x_46_im) * x_46_im

Julia

x.re_m = abs(x_46_re)
function code(x_46_re_m, x_46_im)
	return Float64(Float64(x_46_im * x_46_im) * x_46_im)
end

MATLAB

x.re_m = abs(x_46_re);
function tmp = code(x_46_re_m, x_46_im)
	tmp = (x_46_im * x_46_im) * x_46_im;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
code[x$46$re$95$m_, x$46$im_] := N[(N[(x$46$im * x$46$im), $MachinePrecision] * x$46$im), $MachinePrecision]

Derivation

1. Initial program 82.1%

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

3. Taylor expanded in x.re around 0

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

   1. mul-1-neg N/A

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

   2. cube-neg-rev N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-neg.f64 56.2

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

5. Applied rewrites 56.2%

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

7. Applied rewrites 17.2%

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

Developer Target 1: 91.5% 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 2024344 
(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)))