math.cube on complex, real part

Percentage Accurate: 83.0% → 96.8%

Time: 5.7s

Alternatives: 8

Speedup: 1.5×

• 45.4% 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: 83.0% 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: 96.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.im \leq 1.95 \cdot 10^{+153}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re + x.im \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im, x.re \cdot \left(x.im + x.re\right), x.im \cdot \left(\left(x.im + x.im\right) \cdot \left(-x.re\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im 1.95e+153)
   (* x.re (+ (* x.re x.re) (* x.im (* x.im -3.0))))
   (fma
    (- x.re x.im)
    (* x.re (+ x.im x.re))
    (* x.im (* (+ x.im x.im) (- x.re))))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 1.95e+153) {
		tmp = x_46_re * ((x_46_re * x_46_re) + (x_46_im * (x_46_im * -3.0)));
	} else {
		tmp = fma((x_46_re - x_46_im), (x_46_re * (x_46_im + x_46_re)), (x_46_im * ((x_46_im + x_46_im) * -x_46_re)));
	}
	return tmp;
}

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= 1.95e+153)
		tmp = Float64(x_46_re * Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * Float64(x_46_im * -3.0))));
	else
		tmp = fma(Float64(x_46_re - x_46_im), Float64(x_46_re * Float64(x_46_im + x_46_re)), Float64(x_46_im * Float64(Float64(x_46_im + x_46_im) * Float64(-x_46_re))));
	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, 1.95e+153], N[(x$46$re * N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re - x$46$im), $MachinePrecision] * N[(x$46$re * N[(x$46$im + x$46$re), $MachinePrecision]), $MachinePrecision] + N[(x$46$im * N[(N[(x$46$im + x$46$im), $MachinePrecision] * (-x$46$re)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.94999999999999992e153

   1. Initial program 87.0%

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

      1. sqr-neg 87.0%

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

      2. difference-of-squares 88.4%

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

      3. sub-neg 88.4%

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

      4. associate-*l* 92.6%

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

      5. sub-neg 92.6%

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

      6. remove-double-neg 92.6%

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

      7. +-commutative 92.6%

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

      8. *-commutative 92.6%

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

      9. *-commutative 92.6%

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

      10. distribute-rgt-out 92.6%

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

   3. Simplified 92.6%

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

      1. add-cube-cbrt 92.3%

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

      2. pow3 92.3%

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

      3. *-commutative 92.3%

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

   5. Applied egg-rr 92.3%

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

      1. cancel-sign-sub-inv 92.3%

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

      2. unpow3 92.3%

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

      3. add-cube-cbrt 92.6%

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

      4. *-commutative 92.6%

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

      5. fma-def 92.6%

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

      6. *-commutative 92.6%

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

   7. Applied egg-rr 92.6%

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

   8. Taylor expanded in x.re around 0 68.1%

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

   10. Simplified 94.3%

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

       1. fma-udef 92.9%

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

       2. associate-*l* 92.9%

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

       3. +-commutative 92.9%

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

       4. associate-*l* 92.9%

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

   12. Applied egg-rr 92.9%

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

   if 1.94999999999999992e153 < x.im

   1. Initial program 65.0%

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

      1. sqr-neg 65.0%

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

      2. difference-of-squares 73.6%

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

      3. sub-neg 73.6%

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

      4. associate-*l* 99.8%

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

      5. sub-neg 99.8%

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

      6. remove-double-neg 99.8%

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

      7. +-commutative 99.8%

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

      8. *-commutative 99.8%

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

      9. *-commutative 99.8%

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

      10. distribute-rgt-out 99.8%

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

   3. Simplified 99.8%

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

      1. add-cube-cbrt 99.8%

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

      2. pow3 99.7%

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

      3. *-commutative 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. cancel-sign-sub-inv 99.7%

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

      2. unpow3 99.8%

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

      3. add-cube-cbrt 99.8%

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

      4. *-commutative 99.8%

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

      5. fma-def 100.0%

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

      6. *-commutative 100.0%

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

   7. Applied egg-rr 100.0%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 1.95 \cdot 10^{+153}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re + x.im \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im, x.re \cdot \left(x.im + x.re\right), x.im \cdot \left(\left(x.im + x.im\right) \cdot \left(-x.re\right)\right)\right)\\ \end{array} \]

Alternative 2: 82.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.im \leq 1300 \lor \neg \left(x.im \leq 2.8 \cdot 10^{+60}\right) \land x.im \leq 2.05 \cdot 10^{+96}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.im \cdot x.re\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im 1300.0) (and (not (<= x.im 2.8e+60)) (<= x.im 2.05e+96)))
   (* x.re (* x.re x.re))
   (* -3.0 (* x.im (* x.im x.re)))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= 1300.0) || (!(x_46_im <= 2.8e+60) && (x_46_im <= 2.05e+96))) {
		tmp = x_46_re * (x_46_re * x_46_re);
	} else {
		tmp = -3.0 * (x_46_im * (x_46_im * x_46_re));
	}
	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 <= 1300.0d0) .or. (.not. (x_46im <= 2.8d+60)) .and. (x_46im <= 2.05d+96)) then
        tmp = x_46re * (x_46re * x_46re)
    else
        tmp = (-3.0d0) * (x_46im * (x_46im * x_46re))
    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 <= 1300.0) || (!(x_46_im <= 2.8e+60) && (x_46_im <= 2.05e+96))) {
		tmp = x_46_re * (x_46_re * x_46_re);
	} else {
		tmp = -3.0 * (x_46_im * (x_46_im * x_46_re));
	}
	return tmp;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_im <= 1300.0) or (not (x_46_im <= 2.8e+60) and (x_46_im <= 2.05e+96)):
		tmp = x_46_re * (x_46_re * x_46_re)
	else:
		tmp = -3.0 * (x_46_im * (x_46_im * x_46_re))
	return tmp

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_im <= 1300.0) || (!(x_46_im <= 2.8e+60) && (x_46_im <= 2.05e+96)))
		tmp = Float64(x_46_re * Float64(x_46_re * x_46_re));
	else
		tmp = Float64(-3.0 * Float64(x_46_im * Float64(x_46_im * x_46_re)));
	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 <= 1300.0) || (~((x_46_im <= 2.8e+60)) && (x_46_im <= 2.05e+96)))
		tmp = x_46_re * (x_46_re * x_46_re);
	else
		tmp = -3.0 * (x_46_im * (x_46_im * x_46_re));
	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, 1300.0], And[N[Not[LessEqual[x$46$im, 2.8e+60]], $MachinePrecision], LessEqual[x$46$im, 2.05e+96]]], N[(x$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision], N[(-3.0 * N[(x$46$im * N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 1300 or 2.8e60 < x.im < 2.04999999999999999e96

   1. Initial program 87.1%

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

      1. sqr-neg 87.1%

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

      2. difference-of-squares 88.6%

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

      3. sub-neg 88.6%

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

      4. associate-*l* 93.3%

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

      5. sub-neg 93.3%

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

      6. remove-double-neg 93.3%

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

      7. +-commutative 93.3%

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

      8. *-commutative 93.3%

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

      9. *-commutative 93.3%

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

      10. distribute-rgt-out 93.3%

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

   3. Simplified 93.3%

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

      1. add-cube-cbrt 93.0%

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

      2. pow3 93.0%

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

      3. *-commutative 93.0%

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

   5. Applied egg-rr 93.0%

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

      1. cancel-sign-sub-inv 93.0%

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

      2. unpow3 93.0%

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

      3. add-cube-cbrt 93.3%

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

      4. *-commutative 93.3%

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

      5. fma-def 93.3%

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

      6. *-commutative 93.3%

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

   7. Applied egg-rr 93.3%

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

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

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

   10. Simplified 93.7%

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

   11. Taylor expanded in x.im around 0 71.4%

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

       1. unpow2 71.4%

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

   13. Simplified 71.4%

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

   if 1300 < x.im < 2.8e60 or 2.04999999999999999e96 < x.im

   1. Initial program 73.6%

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

      1. sqr-neg 73.6%

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

      2. difference-of-squares 78.7%

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

      3. sub-neg 78.7%

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

      4. associate-*l* 94.5%

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

      5. sub-neg 94.5%

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

      6. remove-double-neg 94.5%

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

      7. +-commutative 94.5%

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

      8. *-commutative 94.5%

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

      9. *-commutative 94.5%

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

      10. distribute-rgt-out 94.5%

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

   3. Simplified 94.5%

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

      1. add-cube-cbrt 94.3%

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

      2. pow3 94.3%

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

      3. *-commutative 94.3%

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

   5. Applied egg-rr 94.3%

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

      1. cancel-sign-sub-inv 94.3%

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

      2. unpow3 94.3%

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

      3. add-cube-cbrt 94.5%

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

      4. *-commutative 94.5%

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

      5. fma-def 94.6%

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

      6. *-commutative 94.6%

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

   7. Applied egg-rr 94.6%

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

   8. Taylor expanded in x.re around 0 59.7%

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

   10. Simplified 78.6%

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

   11. Taylor expanded in x.re around 0 73.5%

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

       1. unpow2 73.5%

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

       2. associate-*r* 89.4%

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

   13. Simplified 89.4%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 1300 \lor \neg \left(x.im \leq 2.8 \cdot 10^{+60}\right) \land x.im \leq 2.05 \cdot 10^{+96}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.im \cdot x.re\right)\right)\\ \end{array} \]

Alternative 3: 82.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} t_0 := x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{if}\;x.im \leq 420:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.im \leq 2.45 \cdot 10^{+61}:\\ \;\;\;\;x.re \cdot \left(-3 \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{elif}\;x.im \leq 2.02 \cdot 10^{+96}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.im \cdot x.re\right)\right)\\ \end{array} \end{array} \]

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 420.0)
     t_0
     (if (<= x.im 2.45e+61)
       (* x.re (* -3.0 (* x.im x.im)))
       (if (<= x.im 2.02e+96) t_0 (* -3.0 (* x.im (* x.im x.re))))))))

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 <= 420.0) {
		tmp = t_0;
	} else if (x_46_im <= 2.45e+61) {
		tmp = x_46_re * (-3.0 * (x_46_im * x_46_im));
	} else if (x_46_im <= 2.02e+96) {
		tmp = t_0;
	} else {
		tmp = -3.0 * (x_46_im * (x_46_im * x_46_re));
	}
	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 <= 420.0d0) then
        tmp = t_0
    else if (x_46im <= 2.45d+61) then
        tmp = x_46re * ((-3.0d0) * (x_46im * x_46im))
    else if (x_46im <= 2.02d+96) then
        tmp = t_0
    else
        tmp = (-3.0d0) * (x_46im * (x_46im * x_46re))
    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 <= 420.0) {
		tmp = t_0;
	} else if (x_46_im <= 2.45e+61) {
		tmp = x_46_re * (-3.0 * (x_46_im * x_46_im));
	} else if (x_46_im <= 2.02e+96) {
		tmp = t_0;
	} else {
		tmp = -3.0 * (x_46_im * (x_46_im * x_46_re));
	}
	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 <= 420.0:
		tmp = t_0
	elif x_46_im <= 2.45e+61:
		tmp = x_46_re * (-3.0 * (x_46_im * x_46_im))
	elif x_46_im <= 2.02e+96:
		tmp = t_0
	else:
		tmp = -3.0 * (x_46_im * (x_46_im * x_46_re))
	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 <= 420.0)
		tmp = t_0;
	elseif (x_46_im <= 2.45e+61)
		tmp = Float64(x_46_re * Float64(-3.0 * Float64(x_46_im * x_46_im)));
	elseif (x_46_im <= 2.02e+96)
		tmp = t_0;
	else
		tmp = Float64(-3.0 * Float64(x_46_im * Float64(x_46_im * x_46_re)));
	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 <= 420.0)
		tmp = t_0;
	elseif (x_46_im <= 2.45e+61)
		tmp = x_46_re * (-3.0 * (x_46_im * x_46_im));
	elseif (x_46_im <= 2.02e+96)
		tmp = t_0;
	else
		tmp = -3.0 * (x_46_im * (x_46_im * x_46_re));
	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, 420.0], t$95$0, If[LessEqual[x$46$im, 2.45e+61], N[(x$46$re * N[(-3.0 * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 2.02e+96], t$95$0, N[(-3.0 * N[(x$46$im * N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x.im < 420 or 2.45000000000000013e61 < x.im < 2.02000000000000006e96

   1. Initial program 87.1%

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

      1. sqr-neg 87.1%

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

      2. difference-of-squares 88.6%

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

      3. sub-neg 88.6%

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

      4. associate-*l* 93.3%

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

      5. sub-neg 93.3%

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

      6. remove-double-neg 93.3%

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

      7. +-commutative 93.3%

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

      8. *-commutative 93.3%

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

      9. *-commutative 93.3%

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

      10. distribute-rgt-out 93.3%

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

   3. Simplified 93.3%

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

      1. add-cube-cbrt 93.0%

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

      2. pow3 93.0%

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

      3. *-commutative 93.0%

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

   5. Applied egg-rr 93.0%

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

      1. cancel-sign-sub-inv 93.0%

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

      2. unpow3 93.0%

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

      3. add-cube-cbrt 93.3%

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

      4. *-commutative 93.3%

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

      5. fma-def 93.3%

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

      6. *-commutative 93.3%

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

   7. Applied egg-rr 93.3%

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

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

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

   10. Simplified 93.7%

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

   11. Taylor expanded in x.im around 0 71.4%

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

       1. unpow2 71.4%

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

   13. Simplified 71.4%

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

   if 420 < x.im < 2.45000000000000013e61

   1. Initial program 92.5%

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

      1. sqr-neg 92.5%

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

      2. difference-of-squares 92.5%

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

      3. sub-neg 92.5%

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

      4. associate-*l* 92.5%

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

      5. sub-neg 92.5%

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

      6. remove-double-neg 92.5%

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

      7. +-commutative 92.5%

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

      8. *-commutative 92.5%

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

      9. *-commutative 92.5%

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

      10. distribute-rgt-out 92.5%

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

   3. Simplified 92.5%

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

      1. add-cube-cbrt 91.8%

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

      2. pow3 91.9%

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

      3. *-commutative 91.9%

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

   5. Applied egg-rr 91.9%

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

      1. cancel-sign-sub-inv 91.9%

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

      2. unpow3 91.8%

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

      3. add-cube-cbrt 92.5%

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

      4. *-commutative 92.5%

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

      5. fma-def 92.4%

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

      6. *-commutative 92.4%

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

   7. Applied egg-rr 92.4%

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

   8. Taylor expanded in x.re around 0 70.9%

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

      1. distribute-rgt-out 70.9%

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

      2. unpow2 70.9%

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

      3. metadata-eval 70.9%

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

   10. Simplified 70.9%

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

   if 2.02000000000000006e96 < x.im

   1. Initial program 67.5%

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

      1. sqr-neg 67.5%

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

      2. difference-of-squares 74.3%

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

      3. sub-neg 74.3%

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

      4. associate-*l* 95.2%

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

      5. sub-neg 95.2%

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

      6. remove-double-neg 95.2%

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

      7. +-commutative 95.2%

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

      8. *-commutative 95.2%

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

      9. *-commutative 95.2%

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

      10. distribute-rgt-out 95.2%

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

   3. Simplified 95.2%

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

      1. add-cube-cbrt 95.1%

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

      2. pow3 95.1%

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

      3. *-commutative 95.1%

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

   5. Applied egg-rr 95.1%

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

      1. cancel-sign-sub-inv 95.1%

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

      2. unpow3 95.1%

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

      3. add-cube-cbrt 95.2%

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

      4. *-commutative 95.2%

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

      5. fma-def 95.3%

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

      6. *-commutative 95.3%

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

   7. Applied egg-rr 95.3%

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

   8. Taylor expanded in x.re around 0 53.8%

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

   10. Simplified 72.0%

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

   11. Taylor expanded in x.re around 0 74.3%

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

       1. unpow2 74.3%

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

       2. associate-*r* 95.2%

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

   13. Simplified 95.2%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 420:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{elif}\;x.im \leq 2.45 \cdot 10^{+61}:\\ \;\;\;\;x.re \cdot \left(-3 \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{elif}\;x.im \leq 2.02 \cdot 10^{+96}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.im \cdot x.re\right)\right)\\ \end{array} \]

Alternative 4: 96.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.im \leq 1.95 \cdot 10^{+153}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re + x.im \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.im \cdot x.re\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im 1.95e+153)
   (* x.re (+ (* x.re x.re) (* x.im (* x.im -3.0))))
   (* -3.0 (* x.im (* x.im x.re)))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 1.95e+153) {
		tmp = x_46_re * ((x_46_re * x_46_re) + (x_46_im * (x_46_im * -3.0)));
	} else {
		tmp = -3.0 * (x_46_im * (x_46_im * x_46_re));
	}
	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 <= 1.95d+153) then
        tmp = x_46re * ((x_46re * x_46re) + (x_46im * (x_46im * (-3.0d0))))
    else
        tmp = (-3.0d0) * (x_46im * (x_46im * x_46re))
    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 <= 1.95e+153) {
		tmp = x_46_re * ((x_46_re * x_46_re) + (x_46_im * (x_46_im * -3.0)));
	} else {
		tmp = -3.0 * (x_46_im * (x_46_im * x_46_re));
	}
	return tmp;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= 1.95e+153:
		tmp = x_46_re * ((x_46_re * x_46_re) + (x_46_im * (x_46_im * -3.0)))
	else:
		tmp = -3.0 * (x_46_im * (x_46_im * x_46_re))
	return tmp

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= 1.95e+153)
		tmp = Float64(x_46_re * Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * Float64(x_46_im * -3.0))));
	else
		tmp = Float64(-3.0 * Float64(x_46_im * Float64(x_46_im * x_46_re)));
	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 <= 1.95e+153)
		tmp = x_46_re * ((x_46_re * x_46_re) + (x_46_im * (x_46_im * -3.0)));
	else
		tmp = -3.0 * (x_46_im * (x_46_im * x_46_re));
	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, 1.95e+153], N[(x$46$re * N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-3.0 * N[(x$46$im * N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.94999999999999992e153

   1. Initial program 87.0%

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

      1. sqr-neg 87.0%

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

      2. difference-of-squares 88.4%

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

      3. sub-neg 88.4%

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

      4. associate-*l* 92.6%

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

      5. sub-neg 92.6%

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

      6. remove-double-neg 92.6%

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

      7. +-commutative 92.6%

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

      8. *-commutative 92.6%

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

      9. *-commutative 92.6%

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

      10. distribute-rgt-out 92.6%

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

   3. Simplified 92.6%

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

      1. add-cube-cbrt 92.3%

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

      2. pow3 92.3%

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

      3. *-commutative 92.3%

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

   5. Applied egg-rr 92.3%

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

      1. cancel-sign-sub-inv 92.3%

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

      2. unpow3 92.3%

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

      3. add-cube-cbrt 92.6%

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

      4. *-commutative 92.6%

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

      5. fma-def 92.6%

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

      6. *-commutative 92.6%

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

   7. Applied egg-rr 92.6%

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

   8. Taylor expanded in x.re around 0 68.1%

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

   10. Simplified 94.3%

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

       1. fma-udef 92.9%

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

       2. associate-*l* 92.9%

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

       3. +-commutative 92.9%

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

       4. associate-*l* 92.9%

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

   12. Applied egg-rr 92.9%

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

   if 1.94999999999999992e153 < x.im

   1. Initial program 65.0%

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

      1. sqr-neg 65.0%

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

      2. difference-of-squares 73.6%

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

      3. sub-neg 73.6%

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

      4. associate-*l* 99.8%

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

      5. sub-neg 99.8%

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

      6. remove-double-neg 99.8%

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

      7. +-commutative 99.8%

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

      8. *-commutative 99.8%

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

      9. *-commutative 99.8%

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

      10. distribute-rgt-out 99.8%

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

   3. Simplified 99.8%

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

      1. add-cube-cbrt 99.8%

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

      2. pow3 99.7%

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

      3. *-commutative 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. cancel-sign-sub-inv 99.7%

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

      2. unpow3 99.8%

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

      3. add-cube-cbrt 99.8%

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

      4. *-commutative 99.8%

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

      5. fma-def 100.0%

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

      6. *-commutative 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in x.re around 0 47.8%

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

   10. Simplified 65.0%

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

   11. Taylor expanded in x.re around 0 73.6%

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

       1. unpow2 73.6%

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

       2. associate-*r* 99.8%

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

   13. Simplified 99.8%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 1.95 \cdot 10^{+153}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re + x.im \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.im \cdot x.re\right)\right)\\ \end{array} \]

Alternative 5: 59.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.im \leq 4.2 \cdot 10^{+169}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot x.re\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im 4.2e+169) (* x.re (* x.re x.re)) (* x.re x.re)))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 4.2e+169) {
		tmp = x_46_re * (x_46_re * x_46_re);
	} else {
		tmp = x_46_re * x_46_re;
	}
	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 <= 4.2d+169) then
        tmp = x_46re * (x_46re * x_46re)
    else
        tmp = x_46re * x_46re
    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 <= 4.2e+169) {
		tmp = x_46_re * (x_46_re * x_46_re);
	} else {
		tmp = x_46_re * x_46_re;
	}
	return tmp;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= 4.2e+169:
		tmp = x_46_re * (x_46_re * x_46_re)
	else:
		tmp = x_46_re * x_46_re
	return tmp

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= 4.2e+169)
		tmp = Float64(x_46_re * Float64(x_46_re * x_46_re));
	else
		tmp = Float64(x_46_re * x_46_re);
	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 <= 4.2e+169)
		tmp = x_46_re * (x_46_re * x_46_re);
	else
		tmp = x_46_re * x_46_re;
	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, 4.2e+169], N[(x$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision], N[(x$46$re * x$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 4.2000000000000002e169

   1. Initial program 86.4%

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

      1. sqr-neg 86.4%

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

      2. difference-of-squares 87.8%

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

      3. sub-neg 87.8%

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

      4. associate-*l* 92.7%

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

      5. sub-neg 92.7%

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

      6. remove-double-neg 92.7%

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

      7. +-commutative 92.7%

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

      8. *-commutative 92.7%

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

      9. *-commutative 92.7%

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

      10. distribute-rgt-out 92.7%

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

   3. Simplified 92.7%

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

      1. add-cube-cbrt 92.4%

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

      2. pow3 92.4%

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

      3. *-commutative 92.4%

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

   5. Applied egg-rr 92.4%

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

      1. cancel-sign-sub-inv 92.4%

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

      2. unpow3 92.4%

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

      3. add-cube-cbrt 92.7%

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

      4. *-commutative 92.7%

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

      5. fma-def 92.7%

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

      6. *-commutative 92.7%

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

   7. Applied egg-rr 92.7%

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

   8. Taylor expanded in x.re around 0 67.8%

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

   10. Simplified 93.5%

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

   11. Taylor expanded in x.im around 0 65.8%

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

       1. unpow2 65.8%

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

   13. Simplified 65.8%

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

   if 4.2000000000000002e169 < x.im

   1. Initial program 66.7%

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

   3. Simplified 47.3%

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

   4. Taylor expanded in x.re around inf 0.8%

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

   6. Simplified 8.9%

      \[\leadsto \color{blue}{x.re \cdot x.re} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 4.2 \cdot 10^{+169}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot x.re\\ \end{array} \]

Alternative 6: 65.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.im \leq 1.2 \cdot 10^{+170}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im \cdot x.im\right) \cdot -2\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im 1.2e+170) (* x.re (* x.re x.re)) (* (* x.im x.im) -2.0)))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 1.2e+170) {
		tmp = x_46_re * (x_46_re * x_46_re);
	} else {
		tmp = (x_46_im * x_46_im) * -2.0;
	}
	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 <= 1.2d+170) then
        tmp = x_46re * (x_46re * x_46re)
    else
        tmp = (x_46im * x_46im) * (-2.0d0)
    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 <= 1.2e+170) {
		tmp = x_46_re * (x_46_re * x_46_re);
	} else {
		tmp = (x_46_im * x_46_im) * -2.0;
	}
	return tmp;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= 1.2e+170:
		tmp = x_46_re * (x_46_re * x_46_re)
	else:
		tmp = (x_46_im * x_46_im) * -2.0
	return tmp

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= 1.2e+170)
		tmp = Float64(x_46_re * Float64(x_46_re * x_46_re));
	else
		tmp = Float64(Float64(x_46_im * x_46_im) * -2.0);
	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 <= 1.2e+170)
		tmp = x_46_re * (x_46_re * x_46_re);
	else
		tmp = (x_46_im * x_46_im) * -2.0;
	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, 1.2e+170], N[(x$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im * x$46$im), $MachinePrecision] * -2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.2e170

   1. Initial program 86.1%

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

      1. sqr-neg 86.1%

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

      2. difference-of-squares 87.4%

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

      3. sub-neg 87.4%

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

      4. associate-*l* 92.7%

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

      5. sub-neg 92.7%

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

      6. remove-double-neg 92.7%

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

      7. +-commutative 92.7%

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

      8. *-commutative 92.7%

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

      9. *-commutative 92.7%

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

      10. distribute-rgt-out 92.7%

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

   3. Simplified 92.7%

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

      1. add-cube-cbrt 92.4%

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

      2. pow3 92.5%

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

      3. *-commutative 92.5%

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

   5. Applied egg-rr 92.5%

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

      1. cancel-sign-sub-inv 92.5%

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

      2. unpow3 92.4%

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

      3. add-cube-cbrt 92.7%

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

      4. *-commutative 92.7%

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

      5. fma-def 92.8%

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

      6. *-commutative 92.8%

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

   7. Applied egg-rr 92.8%

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

   8. Taylor expanded in x.re around 0 67.5%

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

   10. Simplified 93.1%

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

   11. Taylor expanded in x.im around 0 65.5%

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

       1. unpow2 65.5%

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

   13. Simplified 65.5%

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

   if 1.2e170 < x.im

   1. Initial program 68.7%

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

      1. sqr-neg 68.7%

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

      2. difference-of-squares 78.7%

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

      3. sub-neg 78.7%

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

      4. associate-*l* 99.8%

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

      5. sub-neg 99.8%

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

      6. remove-double-neg 99.8%

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

      7. +-commutative 99.8%

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

      8. *-commutative 99.8%

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

      9. *-commutative 99.8%

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

      10. distribute-rgt-out 99.8%

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

   3. Simplified 99.8%

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

      1. distribute-lft-in 99.8%

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

      2. flip-+ 0.0%

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

      3. +-inverses 0.0%

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

      4. +-inverses 0.0%

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

      5. +-inverses 0.0%

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

      6. +-inverses 0.0%

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

      7. flip-+ 31.8%

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

   5. Applied egg-rr 31.8%

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

   6. Taylor expanded in x.re around 0 31.8%

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

      1. *-commutative 31.8%

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

      2. unpow2 31.8%

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

   8. Simplified 31.8%

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

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 1.2 \cdot 10^{+170}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im \cdot x.im\right) \cdot -2\\ \end{array} \]

Alternative 7: 26.1% accurate, 6.3× speedup

Math

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

FPCore

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

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
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;
}

Python

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

Wolfram

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

Derivation

1. Initial program 84.0%

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

3. Simplified 80.5%

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

4. Taylor expanded in x.re around inf 57.9%

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

6. Simplified 27.9%

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

7. Final simplification 27.9%

   \[\leadsto x.re \cdot x.re \]

Alternative 8: 2.7% accurate, 19.0× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ 1 \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im) :precision binary64 1.0)

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	return 1.0;
}

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 = 1.0d0
end function

Java

x.im = Math.abs(x.im);
public static double code(double x_46_re, double x_46_im) {
	return 1.0;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	return 1.0

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	return 1.0
end

MATLAB

x.im = abs(x.im)
function tmp = code(x_46_re, x_46_im)
	tmp = 1.0;
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := 1.0

Derivation

1. Initial program 84.0%

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

3. Simplified 80.5%

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

   1. associate-*r* 80.5%

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

   2. associate-*l* 80.5%

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

   3. +-commutative 80.5%

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

   4. flip-+ 17.7%

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

   5. pow2 17.7%

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

   6. associate-*l* 17.6%

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

   7. associate-*r* 17.6%

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

   8. associate-*r* 17.6%

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

   9. pow-prod-up 17.6%

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

   10. metadata-eval 17.6%

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

   11. associate-*l* 17.6%

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

   12. associate-*r* 17.6%

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

   13. associate-*r* 19.7%

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

5. Applied egg-rr 19.7%

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

7. Simplified 2.2%

   \[\leadsto \color{blue}{\frac{-27 - x.re \cdot x.re}{-27 - x.re \cdot x.re}} \]

8. Taylor expanded in x.re around 0 2.6%

   \[\leadsto \color{blue}{1} \]

9. Final simplification 2.6%

   \[\leadsto 1 \]

Developer target: 87.6% 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 2023292 
(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)))