math.cube on complex, real part

Percentage Accurate: 82.5% → 96.4%

Time: 6.5s

Alternatives: 7

Speedup: 1.5×

• 43.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

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_re) - (((x_46_re * x_46_im) + (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_46re) - (x_46im * x_46im)) * x_46re) - (((x_46re * x_46im) + (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_re) - (x_46_im * x_46_im)) * x_46_re) - (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im);
}

Python

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

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_re) - Float64(Float64(Float64(x_46_re * x_46_im) + Float64(x_46_im * x_46_re)) * x_46_im))
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_re) - (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im);
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$46$re), $MachinePrecision] - N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision]

Initial Program: 82.5% accurate, 1.0× speedup

Math

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

FPCore

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

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_re) - (((x_46_re * x_46_im) + (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_46re) - (x_46im * x_46im)) * x_46re) - (((x_46re * x_46im) + (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_re) - (x_46_im * x_46_im)) * x_46_re) - (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im);
}

Python

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

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_re) - Float64(Float64(Float64(x_46_re * x_46_im) + Float64(x_46_im * x_46_re)) * x_46_im))
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_re) - (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im);
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$46$re), $MachinePrecision] - N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision]

Alternative 1: 96.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.im \leq 6.5 \cdot 10^{+153}:\\ \;\;\;\;x.re \cdot \mathsf{fma}\left(x.re, x.re, x.im \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im 6.5e+153)
   (* x.re (fma x.re x.re (* x.im (* x.im -3.0))))
   (* x.im (* x.im (* x.re -3.0)))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 6.5e+153) {
		tmp = x_46_re * fma(x_46_re, x_46_re, (x_46_im * (x_46_im * -3.0)));
	} else {
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0));
	}
	return tmp;
}

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= 6.5e+153)
		tmp = Float64(x_46_re * fma(x_46_re, x_46_re, Float64(x_46_im * Float64(x_46_im * -3.0))));
	else
		tmp = Float64(x_46_im * Float64(x_46_im * Float64(x_46_re * -3.0)));
	end
	return tmp
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, 6.5e+153], N[(x$46$re * N[(x$46$re * x$46$re + N[(x$46$im * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(x$46$im * N[(x$46$re * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 6.49999999999999972e153

   1. Initial program 85.9%

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

      1. *-commutative 85.9%

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

      2. distribute-lft-out 85.9%

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

      3. associate-*l* 85.9%

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

      4. *-commutative 85.9%

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

      5. distribute-rgt-out-- 92.1%

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

      6. associate--l- 92.1%

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

      7. associate--l- 92.1%

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

      8. sub-neg 92.1%

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

      9. associate--l+ 92.1%

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

      10. fma-udef 94.8%

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

      11. neg-mul-1 94.8%

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

      12. count-2 94.8%

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

      13. associate-*l* 94.8%

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

      14. distribute-rgt-out-- 94.8%

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

      15. associate-*r* 94.7%

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

      16. metadata-eval 94.7%

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

   3. Simplified 94.7%

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

   if 6.49999999999999972e153 < x.im

   1. Initial program 60.0%

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

      1. *-commutative 60.0%

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

      2. distribute-lft-out 60.0%

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

      3. associate-*l* 60.0%

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

      4. *-commutative 60.0%

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

      5. distribute-rgt-out-- 60.0%

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

      6. associate--l- 60.0%

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

      7. associate--l- 60.0%

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

      8. sub-neg 60.0%

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

      9. associate--l+ 60.0%

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

      10. fma-udef 70.3%

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

      11. neg-mul-1 70.3%

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

      12. count-2 70.3%

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

      13. associate-*l* 70.3%

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

      14. distribute-rgt-out-- 70.3%

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

      15. associate-*r* 70.3%

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

      16. metadata-eval 70.3%

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

   3. Simplified 70.3%

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

   4. Taylor expanded in x.re around 0 70.3%

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

      1. associate-*r* 70.3%

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

      2. unpow2 70.3%

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

   6. Simplified 70.3%

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

      1. add-sqr-sqrt 21.4%

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

      2. pow2 21.4%

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

      3. *-commutative 21.4%

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

      4. sqrt-prod 21.4%

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

      5. sqrt-prod 34.3%

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

      6. add-sqr-sqrt 34.3%

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

      7. *-commutative 34.3%

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

   8. Applied egg-rr 34.3%

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

      1. add-cbrt-cube 27.9%

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

      2. unpow2 27.9%

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

      3. pow1 27.9%

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

      4. pow-prod-up 27.9%

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

      5. metadata-eval 27.9%

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

   10. Applied egg-rr 27.9%

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

       1. rem-cbrt-cube 34.3%

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

       2. unpow-prod-down 21.4%

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

       3. pow2 21.4%

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

       4. pow2 21.4%

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

       5. add-sqr-sqrt 70.3%

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

       6. *-commutative 70.3%

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

       7. associate-*l* 96.5%

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

   12. Applied egg-rr 96.5%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 6.5 \cdot 10^{+153}:\\ \;\;\;\;x.re \cdot \mathsf{fma}\left(x.re, x.re, x.im \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \end{array} \]

Alternative 2: 75.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.im \leq 3.6 \cdot 10^{-34} \lor \neg \left(x.im \leq 4.4 \cdot 10^{+100}\right) \land x.im \leq 2.4 \cdot 10^{+126}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \left(x.re \cdot \left(x.im \cdot x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im 3.6e-34) (and (not (<= x.im 4.4e+100)) (<= x.im 2.4e+126)))
   (* x.re (* x.re x.re))
   (* -3.0 (* x.re (* x.im x.im)))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= 3.6e-34) || (!(x_46_im <= 4.4e+100) && (x_46_im <= 2.4e+126))) {
		tmp = x_46_re * (x_46_re * x_46_re);
	} else {
		tmp = -3.0 * (x_46_re * (x_46_im * x_46_im));
	}
	return tmp;
}

Fortran

NOTE: x.im should be positive before calling this function
real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46im <= 3.6d-34) .or. (.not. (x_46im <= 4.4d+100)) .and. (x_46im <= 2.4d+126)) then
        tmp = x_46re * (x_46re * x_46re)
    else
        tmp = (-3.0d0) * (x_46re * (x_46im * x_46im))
    end if
    code = tmp
end function

Java

x.im = Math.abs(x.im);
public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= 3.6e-34) || (!(x_46_im <= 4.4e+100) && (x_46_im <= 2.4e+126))) {
		tmp = x_46_re * (x_46_re * x_46_re);
	} else {
		tmp = -3.0 * (x_46_re * (x_46_im * x_46_im));
	}
	return tmp;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_im <= 3.6e-34) or (not (x_46_im <= 4.4e+100) and (x_46_im <= 2.4e+126)):
		tmp = x_46_re * (x_46_re * x_46_re)
	else:
		tmp = -3.0 * (x_46_re * (x_46_im * x_46_im))
	return tmp

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_im <= 3.6e-34) || (!(x_46_im <= 4.4e+100) && (x_46_im <= 2.4e+126)))
		tmp = Float64(x_46_re * Float64(x_46_re * x_46_re));
	else
		tmp = Float64(-3.0 * Float64(x_46_re * Float64(x_46_im * x_46_im)));
	end
	return tmp
end

MATLAB

x.im = abs(x.im)
function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_im <= 3.6e-34) || (~((x_46_im <= 4.4e+100)) && (x_46_im <= 2.4e+126)))
		tmp = x_46_re * (x_46_re * x_46_re);
	else
		tmp = -3.0 * (x_46_re * (x_46_im * x_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$im, 3.6e-34], And[N[Not[LessEqual[x$46$im, 4.4e+100]], $MachinePrecision], LessEqual[x$46$im, 2.4e+126]]], N[(x$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision], N[(-3.0 * N[(x$46$re * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 3.60000000000000008e-34 or 4.4000000000000001e100 < x.im < 2.40000000000000012e126

   1. Initial program 85.2%

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

      1. *-commutative 85.2%

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

      2. distribute-lft-out 85.2%

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

      3. associate-*l* 85.2%

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

      4. *-commutative 85.2%

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

      5. distribute-rgt-out-- 90.6%

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

      6. associate--l- 90.6%

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

      7. associate--l- 90.6%

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

      8. sub-neg 90.6%

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

      9. associate--l+ 90.6%

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

      10. fma-udef 93.8%

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

      11. neg-mul-1 93.8%

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

      12. count-2 93.8%

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

      13. associate-*l* 93.8%

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

      14. distribute-rgt-out-- 93.8%

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

      15. associate-*r* 93.7%

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

      16. metadata-eval 93.7%

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

   3. Simplified 93.7%

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

   4. Taylor expanded in x.re around inf 65.9%

      \[\leadsto x.re \cdot \color{blue}{{x.re}^{2}} \]
   5. Step-by-step derivation

      1. unpow2 65.9%

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

   6. Simplified 65.9%

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

   if 3.60000000000000008e-34 < x.im < 4.4000000000000001e100 or 2.40000000000000012e126 < x.im

   1. Initial program 76.8%

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

      1. *-commutative 76.8%

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

      2. distribute-lft-out 76.8%

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

      3. associate-*l* 76.8%

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

      4. *-commutative 76.8%

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

      5. distribute-rgt-out-- 82.7%

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

      6. associate--l- 82.7%

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

      7. associate--l- 82.7%

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

      8. sub-neg 82.7%

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

      9. associate--l+ 82.7%

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

      10. fma-udef 87.1%

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

      11. neg-mul-1 87.1%

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

      12. count-2 87.1%

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

      13. associate-*l* 87.1%

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

      14. distribute-rgt-out-- 87.1%

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

      15. associate-*r* 87.1%

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

      16. metadata-eval 87.1%

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

   3. Simplified 87.1%

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

      1. fma-udef 82.7%

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

   5. Applied egg-rr 82.7%

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

   6. Taylor expanded in x.re around 0 69.5%

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

      1. unpow2 69.5%

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

   8. Simplified 69.5%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 3.6 \cdot 10^{-34} \lor \neg \left(x.im \leq 4.4 \cdot 10^{+100}\right) \land x.im \leq 2.4 \cdot 10^{+126}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \left(x.re \cdot \left(x.im \cdot x.im\right)\right)\\ \end{array} \]

Alternative 3: 80.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.im \leq 9 \cdot 10^{-34} \lor \neg \left(x.im \leq 4.4 \cdot 10^{+100}\right) \land x.im \leq 1.3 \cdot 10^{+126}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im 9e-34) (and (not (<= x.im 4.4e+100)) (<= x.im 1.3e+126)))
   (* x.re (* x.re x.re))
   (* x.im (* x.im (* x.re -3.0)))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= 9e-34) || (!(x_46_im <= 4.4e+100) && (x_46_im <= 1.3e+126))) {
		tmp = x_46_re * (x_46_re * x_46_re);
	} else {
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0));
	}
	return tmp;
}

Fortran

NOTE: x.im should be positive before calling this function
real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46im <= 9d-34) .or. (.not. (x_46im <= 4.4d+100)) .and. (x_46im <= 1.3d+126)) then
        tmp = x_46re * (x_46re * x_46re)
    else
        tmp = x_46im * (x_46im * (x_46re * (-3.0d0)))
    end if
    code = tmp
end function

Java

x.im = Math.abs(x.im);
public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= 9e-34) || (!(x_46_im <= 4.4e+100) && (x_46_im <= 1.3e+126))) {
		tmp = x_46_re * (x_46_re * x_46_re);
	} else {
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0));
	}
	return tmp;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_im <= 9e-34) or (not (x_46_im <= 4.4e+100) and (x_46_im <= 1.3e+126)):
		tmp = x_46_re * (x_46_re * x_46_re)
	else:
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0))
	return tmp

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_im <= 9e-34) || (!(x_46_im <= 4.4e+100) && (x_46_im <= 1.3e+126)))
		tmp = Float64(x_46_re * Float64(x_46_re * x_46_re));
	else
		tmp = Float64(x_46_im * Float64(x_46_im * Float64(x_46_re * -3.0)));
	end
	return tmp
end

MATLAB

x.im = abs(x.im)
function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_im <= 9e-34) || (~((x_46_im <= 4.4e+100)) && (x_46_im <= 1.3e+126)))
		tmp = x_46_re * (x_46_re * x_46_re);
	else
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$im, 9e-34], And[N[Not[LessEqual[x$46$im, 4.4e+100]], $MachinePrecision], LessEqual[x$46$im, 1.3e+126]]], N[(x$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(x$46$im * N[(x$46$re * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 9.00000000000000085e-34 or 4.4000000000000001e100 < x.im < 1.3e126

   1. Initial program 85.2%

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

      1. *-commutative 85.2%

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

      2. distribute-lft-out 85.2%

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

      3. associate-*l* 85.2%

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

      4. *-commutative 85.2%

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

      5. distribute-rgt-out-- 90.6%

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

      6. associate--l- 90.6%

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

      7. associate--l- 90.6%

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

      8. sub-neg 90.6%

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

      9. associate--l+ 90.6%

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

      10. fma-udef 93.8%

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

      11. neg-mul-1 93.8%

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

      12. count-2 93.8%

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

      13. associate-*l* 93.8%

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

      14. distribute-rgt-out-- 93.8%

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

      15. associate-*r* 93.7%

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

      16. metadata-eval 93.7%

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

   3. Simplified 93.7%

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

   4. Taylor expanded in x.re around inf 65.9%

      \[\leadsto x.re \cdot \color{blue}{{x.re}^{2}} \]
   5. Step-by-step derivation

      1. unpow2 65.9%

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

   6. Simplified 65.9%

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

   if 9.00000000000000085e-34 < x.im < 4.4000000000000001e100 or 1.3e126 < x.im

   1. Initial program 76.8%

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

      1. *-commutative 76.8%

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

      2. distribute-lft-out 76.8%

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

      3. associate-*l* 76.8%

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

      4. *-commutative 76.8%

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

      5. distribute-rgt-out-- 82.7%

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

      6. associate--l- 82.7%

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

      7. associate--l- 82.7%

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

      8. sub-neg 82.7%

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

      9. associate--l+ 82.7%

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

      10. fma-udef 87.1%

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

      11. neg-mul-1 87.1%

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

      12. count-2 87.1%

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

      13. associate-*l* 87.1%

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

      14. distribute-rgt-out-- 87.1%

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

      15. associate-*r* 87.1%

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

      16. metadata-eval 87.1%

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

   3. Simplified 87.1%

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

   4. Taylor expanded in x.re around 0 69.5%

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

      1. associate-*r* 69.6%

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

      2. unpow2 69.6%

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

   6. Simplified 69.6%

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

      1. add-sqr-sqrt 26.7%

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

      2. pow2 26.7%

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

      3. *-commutative 26.7%

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

      4. sqrt-prod 26.7%

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

      5. sqrt-prod 32.1%

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

      6. add-sqr-sqrt 32.2%

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

      7. *-commutative 32.2%

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

   8. Applied egg-rr 32.2%

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

      1. add-cbrt-cube 26.7%

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

      2. unpow2 26.7%

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

      3. pow1 26.7%

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

      4. pow-prod-up 26.8%

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

      5. metadata-eval 26.8%

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

   10. Applied egg-rr 26.8%

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

       1. rem-cbrt-cube 32.2%

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

       2. unpow-prod-down 26.6%

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

       3. pow2 26.6%

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

       4. pow2 26.6%

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

       5. add-sqr-sqrt 69.6%

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

       6. *-commutative 69.6%

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

       7. associate-*l* 80.7%

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

   12. Applied egg-rr 80.7%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 9 \cdot 10^{-34} \lor \neg \left(x.im \leq 4.4 \cdot 10^{+100}\right) \land x.im \leq 1.3 \cdot 10^{+126}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \end{array} \]

Alternative 4: 80.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} t_0 := x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{if}\;x.im \leq 7.6 \cdot 10^{-34}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.im \leq 4.4 \cdot 10^{+100}:\\ \;\;\;\;x.re \cdot \left(-3 \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{elif}\;x.im \leq 1.3 \cdot 10^{+126}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (* x.re (* x.re x.re))))
   (if (<= x.im 7.6e-34)
     t_0
     (if (<= x.im 4.4e+100)
       (* x.re (* -3.0 (* x.im x.im)))
       (if (<= x.im 1.3e+126) t_0 (* x.im (* x.im (* x.re -3.0))))))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_re * (x_46_re * x_46_re);
	double tmp;
	if (x_46_im <= 7.6e-34) {
		tmp = t_0;
	} else if (x_46_im <= 4.4e+100) {
		tmp = x_46_re * (-3.0 * (x_46_im * x_46_im));
	} else if (x_46_im <= 1.3e+126) {
		tmp = t_0;
	} else {
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0));
	}
	return tmp;
}

Fortran

NOTE: x.im should be positive before calling this function
real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x_46re * (x_46re * x_46re)
    if (x_46im <= 7.6d-34) then
        tmp = t_0
    else if (x_46im <= 4.4d+100) then
        tmp = x_46re * ((-3.0d0) * (x_46im * x_46im))
    else if (x_46im <= 1.3d+126) then
        tmp = t_0
    else
        tmp = x_46im * (x_46im * (x_46re * (-3.0d0)))
    end if
    code = tmp
end function

Java

x.im = Math.abs(x.im);
public static double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_re * (x_46_re * x_46_re);
	double tmp;
	if (x_46_im <= 7.6e-34) {
		tmp = t_0;
	} else if (x_46_im <= 4.4e+100) {
		tmp = x_46_re * (-3.0 * (x_46_im * x_46_im));
	} else if (x_46_im <= 1.3e+126) {
		tmp = t_0;
	} else {
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0));
	}
	return tmp;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	t_0 = x_46_re * (x_46_re * x_46_re)
	tmp = 0
	if x_46_im <= 7.6e-34:
		tmp = t_0
	elif x_46_im <= 4.4e+100:
		tmp = x_46_re * (-3.0 * (x_46_im * x_46_im))
	elif x_46_im <= 1.3e+126:
		tmp = t_0
	else:
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0))
	return tmp

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	t_0 = Float64(x_46_re * Float64(x_46_re * x_46_re))
	tmp = 0.0
	if (x_46_im <= 7.6e-34)
		tmp = t_0;
	elseif (x_46_im <= 4.4e+100)
		tmp = Float64(x_46_re * Float64(-3.0 * Float64(x_46_im * x_46_im)));
	elseif (x_46_im <= 1.3e+126)
		tmp = t_0;
	else
		tmp = Float64(x_46_im * Float64(x_46_im * Float64(x_46_re * -3.0)));
	end
	return tmp
end

MATLAB

x.im = abs(x.im)
function tmp_2 = code(x_46_re, x_46_im)
	t_0 = x_46_re * (x_46_re * x_46_re);
	tmp = 0.0;
	if (x_46_im <= 7.6e-34)
		tmp = t_0;
	elseif (x_46_im <= 4.4e+100)
		tmp = x_46_re * (-3.0 * (x_46_im * x_46_im));
	elseif (x_46_im <= 1.3e+126)
		tmp = t_0;
	else
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(x$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$im, 7.6e-34], t$95$0, If[LessEqual[x$46$im, 4.4e+100], N[(x$46$re * N[(-3.0 * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 1.3e+126], t$95$0, N[(x$46$im * N[(x$46$im * N[(x$46$re * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x.im < 7.6000000000000002e-34 or 4.4000000000000001e100 < x.im < 1.3e126

   1. Initial program 85.2%

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

      1. *-commutative 85.2%

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

      2. distribute-lft-out 85.2%

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

      3. associate-*l* 85.2%

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

      4. *-commutative 85.2%

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

      5. distribute-rgt-out-- 90.6%

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

      6. associate--l- 90.6%

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

      7. associate--l- 90.6%

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

      8. sub-neg 90.6%

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

      9. associate--l+ 90.6%

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

      10. fma-udef 93.8%

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

      11. neg-mul-1 93.8%

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

      12. count-2 93.8%

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

      13. associate-*l* 93.8%

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

      14. distribute-rgt-out-- 93.8%

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

      15. associate-*r* 93.7%

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

      16. metadata-eval 93.7%

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

   3. Simplified 93.7%

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

   4. Taylor expanded in x.re around inf 65.9%

      \[\leadsto x.re \cdot \color{blue}{{x.re}^{2}} \]
   5. Step-by-step derivation

      1. unpow2 65.9%

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

   6. Simplified 65.9%

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

   if 7.6000000000000002e-34 < x.im < 4.4000000000000001e100

   1. Initial program 90.2%

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

      1. *-commutative 90.2%

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

      2. distribute-lft-out 90.2%

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

      3. associate-*l* 90.2%

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

      4. *-commutative 90.2%

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

      5. distribute-rgt-out-- 99.5%

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

      6. associate--l- 99.5%

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

      7. associate--l- 99.5%

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

      8. sub-neg 99.5%

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

      9. associate--l+ 99.5%

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

      10. fma-udef 99.5%

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

      11. neg-mul-1 99.5%

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

      12. count-2 99.5%

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

      13. associate-*l* 99.5%

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

      14. distribute-rgt-out-- 99.5%

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

      15. associate-*r* 99.6%

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

      16. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x.re around 0 65.4%

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

      1. associate-*r* 65.5%

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

      2. *-commutative 65.5%

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

      3. metadata-eval 65.5%

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

      4. distribute-rgt-out 65.5%

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

      5. metadata-eval 65.5%

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

      6. cancel-sign-sub-inv 65.5%

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

      7. *-commutative 65.5%

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

      8. cancel-sign-sub-inv 65.5%

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

      9. metadata-eval 65.5%

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

      10. +-commutative 65.5%

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

      11. distribute-lft-in 65.4%

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

   6. Simplified 65.4%

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

   if 1.3e126 < x.im

   1. Initial program 64.9%

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

      1. *-commutative 64.9%

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

      2. distribute-lft-out 64.9%

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

      3. associate-*l* 64.9%

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

      4. *-commutative 64.9%

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

      5. distribute-rgt-out-- 67.7%

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

      6. associate--l- 67.7%

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

      7. associate--l- 67.7%

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

      8. sub-neg 67.7%

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

      9. associate--l+ 67.7%

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

      10. fma-udef 76.0%

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

      11. neg-mul-1 76.0%

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

      12. count-2 76.0%

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

      13. associate-*l* 76.0%

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

      14. distribute-rgt-out-- 76.0%

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

      15. associate-*r* 76.0%

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

      16. metadata-eval 76.0%

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

   3. Simplified 76.0%

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

   4. Taylor expanded in x.re around 0 73.2%

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

      1. associate-*r* 73.3%

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

      2. unpow2 73.3%

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

   6. Simplified 73.3%

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

      1. add-sqr-sqrt 28.4%

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

      2. pow2 28.4%

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

      3. *-commutative 28.4%

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

      4. sqrt-prod 28.4%

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

      5. sqrt-prod 38.7%

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

      6. add-sqr-sqrt 38.8%

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

      7. *-commutative 38.8%

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

   8. Applied egg-rr 38.8%

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

      1. add-cbrt-cube 31.1%

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

      2. unpow2 31.1%

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

      3. pow1 31.1%

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

      4. pow-prod-up 31.1%

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

      5. metadata-eval 31.1%

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

   10. Applied egg-rr 31.1%

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

       1. rem-cbrt-cube 38.8%

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

       2. unpow-prod-down 28.3%

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

       3. pow2 28.3%

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

       4. pow2 28.3%

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

       5. add-sqr-sqrt 73.3%

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

       6. *-commutative 73.3%

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

       7. associate-*l* 94.3%

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

   12. Applied egg-rr 94.3%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 7.6 \cdot 10^{-34}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{elif}\;x.im \leq 4.4 \cdot 10^{+100}:\\ \;\;\;\;x.re \cdot \left(-3 \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{elif}\;x.im \leq 1.3 \cdot 10^{+126}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \end{array} \]

Alternative 5: 96.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.im \leq 6.5 \cdot 10^{+153}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.im \cdot -3\right) + x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im 6.5e+153)
   (* x.re (+ (* x.im (* x.im -3.0)) (* x.re x.re)))
   (* x.im (* x.im (* x.re -3.0)))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 6.5e+153) {
		tmp = x_46_re * ((x_46_im * (x_46_im * -3.0)) + (x_46_re * x_46_re));
	} else {
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0));
	}
	return tmp;
}

Fortran

NOTE: x.im should be positive before calling this function
real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46im <= 6.5d+153) then
        tmp = x_46re * ((x_46im * (x_46im * (-3.0d0))) + (x_46re * x_46re))
    else
        tmp = x_46im * (x_46im * (x_46re * (-3.0d0)))
    end if
    code = tmp
end function

Java

x.im = Math.abs(x.im);
public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 6.5e+153) {
		tmp = x_46_re * ((x_46_im * (x_46_im * -3.0)) + (x_46_re * x_46_re));
	} else {
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0));
	}
	return tmp;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= 6.5e+153:
		tmp = x_46_re * ((x_46_im * (x_46_im * -3.0)) + (x_46_re * x_46_re))
	else:
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0))
	return tmp

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= 6.5e+153)
		tmp = Float64(x_46_re * Float64(Float64(x_46_im * Float64(x_46_im * -3.0)) + Float64(x_46_re * x_46_re)));
	else
		tmp = Float64(x_46_im * Float64(x_46_im * Float64(x_46_re * -3.0)));
	end
	return tmp
end

MATLAB

x.im = abs(x.im)
function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_im <= 6.5e+153)
		tmp = x_46_re * ((x_46_im * (x_46_im * -3.0)) + (x_46_re * x_46_re));
	else
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, 6.5e+153], N[(x$46$re * N[(N[(x$46$im * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(x$46$im * N[(x$46$re * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 6.49999999999999972e153

   1. Initial program 85.9%

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

      1. *-commutative 85.9%

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

      2. distribute-lft-out 85.9%

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

      3. associate-*l* 85.9%

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

      4. *-commutative 85.9%

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

      5. distribute-rgt-out-- 92.1%

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

      6. associate--l- 92.1%

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

      7. associate--l- 92.1%

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

      8. sub-neg 92.1%

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

      9. associate--l+ 92.1%

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

      10. fma-udef 94.8%

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

      11. neg-mul-1 94.8%

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

      12. count-2 94.8%

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

      13. associate-*l* 94.8%

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

      14. distribute-rgt-out-- 94.8%

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

      15. associate-*r* 94.7%

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

      16. metadata-eval 94.7%

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

   3. Simplified 94.7%

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

      1. fma-udef 92.1%

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

   5. Applied egg-rr 92.1%

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

   if 6.49999999999999972e153 < x.im

   1. Initial program 60.0%

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

      1. *-commutative 60.0%

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

      2. distribute-lft-out 60.0%

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

      3. associate-*l* 60.0%

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

      4. *-commutative 60.0%

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

      5. distribute-rgt-out-- 60.0%

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

      6. associate--l- 60.0%

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

      7. associate--l- 60.0%

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

      8. sub-neg 60.0%

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

      9. associate--l+ 60.0%

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

      10. fma-udef 70.3%

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

      11. neg-mul-1 70.3%

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

      12. count-2 70.3%

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

      13. associate-*l* 70.3%

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

      14. distribute-rgt-out-- 70.3%

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

      15. associate-*r* 70.3%

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

      16. metadata-eval 70.3%

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

   3. Simplified 70.3%

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

   4. Taylor expanded in x.re around 0 70.3%

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

      1. associate-*r* 70.3%

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

      2. unpow2 70.3%

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

   6. Simplified 70.3%

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

      1. add-sqr-sqrt 21.4%

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

      2. pow2 21.4%

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

      3. *-commutative 21.4%

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

      4. sqrt-prod 21.4%

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

      5. sqrt-prod 34.3%

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

      6. add-sqr-sqrt 34.3%

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

      7. *-commutative 34.3%

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

   8. Applied egg-rr 34.3%

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

      1. add-cbrt-cube 27.9%

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

      2. unpow2 27.9%

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

      3. pow1 27.9%

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

      4. pow-prod-up 27.9%

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

      5. metadata-eval 27.9%

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

   10. Applied egg-rr 27.9%

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

       1. rem-cbrt-cube 34.3%

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

       2. unpow-prod-down 21.4%

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

       3. pow2 21.4%

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

       4. pow2 21.4%

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

       5. add-sqr-sqrt 70.3%

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

       6. *-commutative 70.3%

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

       7. associate-*l* 96.5%

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

   12. Applied egg-rr 96.5%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 6.5 \cdot 10^{+153}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.im \cdot -3\right) + x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \end{array} \]

Alternative 6: 71.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.im \leq 2.3 \cdot 10^{+126}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im 2.3e+126) (* x.re (* x.re x.re)) (* x.re (* x.im (- x.im)))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 2.3e+126) {
		tmp = x_46_re * (x_46_re * x_46_re);
	} else {
		tmp = x_46_re * (x_46_im * -x_46_im);
	}
	return tmp;
}

Fortran

NOTE: x.im should be positive before calling this function
real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46im <= 2.3d+126) then
        tmp = x_46re * (x_46re * x_46re)
    else
        tmp = x_46re * (x_46im * -x_46im)
    end if
    code = tmp
end function

Java

x.im = Math.abs(x.im);
public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 2.3e+126) {
		tmp = x_46_re * (x_46_re * x_46_re);
	} else {
		tmp = x_46_re * (x_46_im * -x_46_im);
	}
	return tmp;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= 2.3e+126:
		tmp = x_46_re * (x_46_re * x_46_re)
	else:
		tmp = x_46_re * (x_46_im * -x_46_im)
	return tmp

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= 2.3e+126)
		tmp = Float64(x_46_re * Float64(x_46_re * x_46_re));
	else
		tmp = Float64(x_46_re * Float64(x_46_im * Float64(-x_46_im)));
	end
	return tmp
end

MATLAB

x.im = abs(x.im)
function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_im <= 2.3e+126)
		tmp = x_46_re * (x_46_re * x_46_re);
	else
		tmp = x_46_re * (x_46_im * -x_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, 2.3e+126], N[(x$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision], N[(x$46$re * N[(x$46$im * (-x$46$im)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 2.3000000000000001e126

   1. Initial program 86.0%

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

      1. *-commutative 86.0%

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

      2. distribute-lft-out 86.0%

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

      3. associate-*l* 85.9%

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

      4. *-commutative 85.9%

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

      5. distribute-rgt-out-- 91.9%

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

      6. associate--l- 91.9%

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

      7. associate--l- 91.9%

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

      8. sub-neg 91.9%

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

      9. associate--l+ 91.9%

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

      10. fma-udef 94.6%

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

      11. neg-mul-1 94.6%

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

      12. count-2 94.6%

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

      13. associate-*l* 94.6%

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

      14. distribute-rgt-out-- 94.6%

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

      15. associate-*r* 94.6%

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

      16. metadata-eval 94.6%

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

   3. Simplified 94.6%

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

   4. Taylor expanded in x.re around inf 61.9%

      \[\leadsto x.re \cdot \color{blue}{{x.re}^{2}} \]
   5. Step-by-step derivation

      1. unpow2 61.9%

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

   6. Simplified 61.9%

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

   if 2.3000000000000001e126 < x.im

   1. Initial program 64.9%

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

      1. *-commutative 64.9%

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

      2. fma-neg 64.9%

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

      3. distribute-lft-neg-in 64.9%

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

      4. *-commutative 64.9%

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

      5. *-commutative 64.9%

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

      6. count-2 64.9%

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

      7. distribute-lft-neg-in 64.9%

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

      8. metadata-eval 64.9%

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

      9. *-commutative 64.9%

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

   3. Simplified 64.9%

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

   4. Taylor expanded in x.re around 0 73.2%

      \[\leadsto \mathsf{fma}\left(x.re, \color{blue}{-1 \cdot {x.im}^{2}}, x.im \cdot \left(-2 \cdot \left(x.re \cdot x.im\right)\right)\right) \]
   5. Step-by-step derivation

      1. mul-1-neg 73.2%

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

      2. unpow2 73.2%

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

      3. distribute-rgt-neg-out 73.2%

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

   6. Simplified 73.2%

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

      1. fma-udef 73.2%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 0.1%

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

      4. distribute-rgt-neg-out 0.1%

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

      5. distribute-rgt-neg-out 0.1%

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

      6. sqr-neg 0.1%

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

      7. sqrt-unprod 2.0%

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

      8. add-sqr-sqrt 2.0%

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

      9. associate-*r* 2.0%

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

   8. Applied egg-rr 2.0%

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

      1. associate-*r* 5.9%

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

      2. *-commutative 5.9%

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

      3. distribute-lft-out 67.0%

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

   10. Simplified 67.0%

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

   11. Taylor expanded in x.re around 0 67.0%

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

       1. distribute-lft1-in 67.0%

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

       2. metadata-eval 67.0%

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

       3. neg-mul-1 67.0%

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

       4. *-commutative 67.0%

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

       5. associate-*r* 64.2%

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

   13. Simplified 64.2%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 2.3 \cdot 10^{+126}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \end{array} \]

Alternative 7: 58.6% accurate, 3.8× speedup

Math

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

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im) :precision binary64 (* x.re (* x.re x.re)))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	return x_46_re * (x_46_re * x_46_re);
}

Fortran

NOTE: x.im should be positive before calling this function
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_46re)
end function

Java

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

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	return x_46_re * (x_46_re * x_46_re)

Julia

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

MATLAB

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

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := N[(x$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.0%

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

   1. *-commutative 83.0%

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

   2. distribute-lft-out 83.0%

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

   3. associate-*l* 83.0%

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

   4. *-commutative 83.0%

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

   5. distribute-rgt-out-- 88.5%

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

   6. associate--l- 88.5%

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

   7. associate--l- 88.5%

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

   8. sub-neg 88.5%

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

   9. associate--l+ 88.5%

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

   10. fma-udef 92.0%

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

   11. neg-mul-1 92.0%

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

   12. count-2 92.0%

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

   13. associate-*l* 92.0%

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

   14. distribute-rgt-out-- 92.0%

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

   15. associate-*r* 92.0%

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

   16. metadata-eval 92.0%

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

3. Simplified 92.0%

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

4. Taylor expanded in x.re around inf 54.1%

   \[\leadsto x.re \cdot \color{blue}{{x.re}^{2}} \]
5. Step-by-step derivation

   1. unpow2 54.1%

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

6. Simplified 54.1%

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

7. Final simplification 54.1%

   \[\leadsto x.re \cdot \left(x.re \cdot x.re\right) \]

Developer target: 87.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(x.re \cdot x.re\right) \cdot \left(x.re - x.im\right) + \left(x.re \cdot x.im\right) \cdot \left(x.re - 3 \cdot x.im\right) \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (+ (* (* x.re x.re) (- x.re x.im)) (* (* x.re x.im) (- x.re (* 3.0 x.im)))))

C

double code(double x_46_re, double x_46_im) {
	return ((x_46_re * x_46_re) * (x_46_re - x_46_im)) + ((x_46_re * x_46_im) * (x_46_re - (3.0 * 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_46re) * (x_46re - x_46im)) + ((x_46re * x_46im) * (x_46re - (3.0d0 * x_46im)))
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return ((x_46_re * x_46_re) * (x_46_re - x_46_im)) + ((x_46_re * x_46_im) * (x_46_re - (3.0 * x_46_im)));
}

Python

def code(x_46_re, x_46_im):
	return ((x_46_re * x_46_re) * (x_46_re - x_46_im)) + ((x_46_re * x_46_im) * (x_46_re - (3.0 * x_46_im)))

Julia

function code(x_46_re, x_46_im)
	return Float64(Float64(Float64(x_46_re * x_46_re) * Float64(x_46_re - x_46_im)) + Float64(Float64(x_46_re * x_46_im) * Float64(x_46_re - Float64(3.0 * x_46_im))))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = ((x_46_re * x_46_re) * (x_46_re - x_46_im)) + ((x_46_re * x_46_im) * (x_46_re - (3.0 * x_46_im)));
end

Wolfram

code[x$46$re_, x$46$im_] := N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision] + N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(x$46$re - N[(3.0 * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023200 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64

  :herbie-target
  (+ (* (* x.re x.re) (- x.re x.im)) (* (* x.re x.im) (- x.re (* 3.0 x.im))))

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