math.cube on complex, real part

Percentage Accurate: 83.5% → 96.6%

Time: 11.2s

Alternatives: 8

Speedup: 1.5×

• 44.4% 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: 83.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.6% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<=
      (-
       (* x.re (- (* x.re x.re) (* x.im x.im)))
       (* x.im (+ (* x.re x.im) (* x.re x.im))))
      INFINITY)
   (fma
    (- x.re x.im)
    (* x.re (+ x.re x.im))
    (* (- x.im) (* x.re (+ x.im x.im))))
   (* x.re (fma x.im (* x.im -3.0) (* x.re x.re)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (((x_46_re * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re * x_46_im) + (x_46_re * x_46_im)))) <= ((double) INFINITY)) {
		tmp = fma((x_46_re - x_46_im), (x_46_re * (x_46_re + x_46_im)), (-x_46_im * (x_46_re * (x_46_im + x_46_im))));
	} else {
		tmp = x_46_re * fma(x_46_im, (x_46_im * -3.0), (x_46_re * x_46_re));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (Float64(Float64(x_46_re * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im))) - Float64(x_46_im * Float64(Float64(x_46_re * x_46_im) + Float64(x_46_re * x_46_im)))) <= Inf)
		tmp = fma(Float64(x_46_re - x_46_im), Float64(x_46_re * Float64(x_46_re + x_46_im)), Float64(Float64(-x_46_im) * Float64(x_46_re * Float64(x_46_im + x_46_im))));
	else
		tmp = Float64(x_46_re * fma(x_46_im, Float64(x_46_im * -3.0), Float64(x_46_re * x_46_re)));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 93.5%

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

      1. sqr-neg 93.5%

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

      2. difference-of-squares 93.5%

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

      3. sub-neg 93.5%

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

      4. associate-*l* 99.7%

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

      5. sub-neg 99.7%

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

      6. remove-double-neg 99.7%

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

      7. +-commutative 99.7%

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

      8. *-commutative 99.7%

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

      9. *-commutative 99.7%

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

      10. distribute-rgt-out 99.7%

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

   3. Simplified 99.7%

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

      1. cancel-sign-sub-inv 99.7%

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

      2. fma-def 99.8%

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

      3. *-commutative 99.8%

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

   5. Applied egg-rr 99.8%

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

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

   1. Initial program 0.0%

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

      1. sqr-neg 0.0%

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

      2. difference-of-squares 22.9%

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

      3. sub-neg 22.9%

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

      4. associate-*l* 22.9%

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

      5. sub-neg 22.9%

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

      6. remove-double-neg 22.9%

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

      7. +-commutative 22.9%

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

      8. *-commutative 22.9%

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

      9. *-commutative 22.9%

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

      10. distribute-rgt-out 22.9%

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

   3. Simplified 22.9%

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

      1. sub-neg 22.9%

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

      2. *-commutative 22.9%

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

      3. associate-*r* 22.9%

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

      4. *-commutative 22.9%

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

   5. Applied egg-rr 22.9%

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

      1. unsub-neg 22.9%

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

      2. *-commutative 22.9%

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

      3. associate-*r* 22.9%

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

      4. *-commutative 22.9%

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

      5. difference-of-squares 0.0%

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

      6. *-commutative 0.0%

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

      7. associate-*l* 0.0%

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

      8. distribute-lft-out-- 45.7%

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

   7. Simplified 45.7%

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

   8. Taylor expanded in x.re around 0 45.7%

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

      1. unpow2 45.7%

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

      2. mul-1-neg 45.7%

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

      3. distribute-rgt1-in 45.7%

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

      4. metadata-eval 45.7%

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

      5. unpow2 45.7%

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

   10. Simplified 45.7%

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

       1. +-commutative 45.7%

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

       2. distribute-lft-neg-in 45.7%

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

       3. metadata-eval 45.7%

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

       4. *-commutative 45.7%

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

       5. associate-*l* 45.7%

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

       6. fma-def 74.3%

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

   12. Applied egg-rr 74.3%

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

4. Final simplification 96.3%

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

Alternative 2: 93.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq 3.2 \cdot 10^{+164}:\\ \;\;\;\;x.re \cdot \mathsf{fma}\left(x.im, x.im \cdot -3, x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im 3.2e+164)
   (* x.re (fma x.im (* x.im -3.0) (* x.re x.re)))
   (* x.im (* x.im (* x.re -3.0)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 3.2e+164) {
		tmp = x_46_re * fma(x_46_im, (x_46_im * -3.0), (x_46_re * x_46_re));
	} else {
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= 3.2e+164)
		tmp = Float64(x_46_re * fma(x_46_im, Float64(x_46_im * -3.0), Float64(x_46_re * x_46_re)));
	else
		tmp = Float64(x_46_im * Float64(x_46_im * Float64(x_46_re * -3.0)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, 3.2e+164], N[(x$46$re * N[(x$46$im * N[(x$46$im * -3.0), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(x$46$im * N[(x$46$re * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 3.1999999999999998e164

   1. Initial program 83.8%

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

      1. sqr-neg 83.8%

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

      2. difference-of-squares 85.6%

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

      3. sub-neg 85.6%

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

      4. associate-*l* 89.6%

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

      5. sub-neg 89.6%

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

      6. remove-double-neg 89.6%

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

      7. +-commutative 89.6%

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

      8. *-commutative 89.6%

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

      9. *-commutative 89.6%

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

      10. distribute-rgt-out 89.6%

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

   3. Simplified 89.6%

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

      1. sub-neg 89.6%

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

      2. *-commutative 89.6%

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

      3. associate-*r* 89.6%

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

      4. *-commutative 89.6%

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

   5. Applied egg-rr 89.6%

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

      1. unsub-neg 89.6%

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

      2. *-commutative 89.6%

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

      3. associate-*r* 85.6%

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

      4. *-commutative 85.6%

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

      5. difference-of-squares 83.8%

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

      6. *-commutative 83.8%

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

      7. associate-*l* 83.8%

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

      8. distribute-lft-out-- 90.8%

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

   7. Simplified 90.8%

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

   8. Taylor expanded in x.re around 0 90.8%

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

      1. unpow2 90.8%

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

      2. mul-1-neg 90.8%

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

      3. distribute-rgt1-in 90.8%

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

      4. metadata-eval 90.8%

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

      5. unpow2 90.8%

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

   10. Simplified 90.8%

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

       1. +-commutative 90.8%

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

       2. distribute-lft-neg-in 90.8%

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

       3. metadata-eval 90.8%

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

       4. *-commutative 90.8%

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

       5. associate-*l* 90.8%

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

       6. fma-def 93.5%

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

   12. Applied egg-rr 93.5%

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

   if 3.1999999999999998e164 < x.im

   1. Initial program 56.1%

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

      1. sqr-neg 56.1%

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

      2. difference-of-squares 69.9%

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

      3. sub-neg 69.9%

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

      4. associate-*l* 86.1%

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

      5. sub-neg 86.1%

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

      6. remove-double-neg 86.1%

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

      7. +-commutative 86.1%

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

      8. *-commutative 86.1%

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

      9. *-commutative 86.1%

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

      10. distribute-rgt-out 86.1%

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

   3. Simplified 86.1%

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

      1. cancel-sign-sub-inv 86.1%

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

      2. fma-def 86.2%

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

      3. *-commutative 86.2%

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

   5. Applied egg-rr 86.2%

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

   6. Taylor expanded in x.re around 0 69.9%

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

      1. distribute-rgt-out 69.9%

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

      2. metadata-eval 69.9%

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

      3. associate-*r* 69.9%

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

      4. *-commutative 69.9%

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

      5. unpow2 69.9%

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

      6. associate-*l* 86.2%

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

      7. associate-*r* 86.1%

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

   8. Simplified 86.1%

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

      1. pow1 86.1%

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

      2. associate-*l* 86.2%

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

   10. Applied egg-rr 86.2%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 3.2 \cdot 10^{+164}:\\ \;\;\;\;x.re \cdot \mathsf{fma}\left(x.im, x.im \cdot -3, x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \end{array} \]

Alternative 3: 92.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq 4.5 \cdot 10^{+152}:\\ \;\;\;\;x.re \cdot \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.im + x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.im\right) \cdot -3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im 4.5e+152)
   (* x.re (- (- (* x.re x.re) (* x.im x.im)) (* x.im (+ x.im x.im))))
   (* x.im (* (* x.re x.im) -3.0))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 4.5e+152) {
		tmp = x_46_re * (((x_46_re * x_46_re) - (x_46_im * x_46_im)) - (x_46_im * (x_46_im + x_46_im)));
	} else {
		tmp = x_46_im * ((x_46_re * x_46_im) * -3.0);
	}
	return tmp;
}

Fortran

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 <= 4.5d+152) then
        tmp = x_46re * (((x_46re * x_46re) - (x_46im * x_46im)) - (x_46im * (x_46im + x_46im)))
    else
        tmp = x_46im * ((x_46re * x_46im) * (-3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 4.5e+152) {
		tmp = x_46_re * (((x_46_re * x_46_re) - (x_46_im * x_46_im)) - (x_46_im * (x_46_im + x_46_im)));
	} else {
		tmp = x_46_im * ((x_46_re * x_46_im) * -3.0);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= 4.5e+152:
		tmp = x_46_re * (((x_46_re * x_46_re) - (x_46_im * x_46_im)) - (x_46_im * (x_46_im + x_46_im)))
	else:
		tmp = x_46_im * ((x_46_re * x_46_im) * -3.0)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= 4.5e+152)
		tmp = Float64(x_46_re * Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im)) - Float64(x_46_im * Float64(x_46_im + x_46_im))));
	else
		tmp = Float64(x_46_im * Float64(Float64(x_46_re * x_46_im) * -3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_im <= 4.5e+152)
		tmp = x_46_re * (((x_46_re * x_46_re) - (x_46_im * x_46_im)) - (x_46_im * (x_46_im + x_46_im)));
	else
		tmp = x_46_im * ((x_46_re * x_46_im) * -3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, 4.5e+152], N[(x$46$re * N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(N[(x$46$re * x$46$im), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 4.5000000000000001e152

   1. Initial program 83.8%

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

      1. sqr-neg 83.8%

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

      2. difference-of-squares 85.6%

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

      3. sub-neg 85.6%

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

      4. associate-*l* 89.6%

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

      5. sub-neg 89.6%

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

      6. remove-double-neg 89.6%

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

      7. +-commutative 89.6%

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

      8. *-commutative 89.6%

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

      9. *-commutative 89.6%

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

      10. distribute-rgt-out 89.6%

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

   3. Simplified 89.6%

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

      1. sub-neg 89.6%

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

      2. *-commutative 89.6%

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

      3. associate-*r* 89.6%

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

      4. *-commutative 89.6%

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

   5. Applied egg-rr 89.6%

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

      1. unsub-neg 89.6%

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

      2. *-commutative 89.6%

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

      3. associate-*r* 85.6%

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

      4. *-commutative 85.6%

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

      5. difference-of-squares 83.8%

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

      6. *-commutative 83.8%

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

      7. associate-*l* 83.8%

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

      8. distribute-lft-out-- 90.8%

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

   7. Simplified 90.8%

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

   if 4.5000000000000001e152 < x.im

   1. Initial program 56.1%

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

      1. sqr-neg 56.1%

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

      2. difference-of-squares 69.9%

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

      3. sub-neg 69.9%

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

      4. associate-*l* 86.1%

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

      5. sub-neg 86.1%

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

      6. remove-double-neg 86.1%

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

      7. +-commutative 86.1%

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

      8. *-commutative 86.1%

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

      9. *-commutative 86.1%

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

      10. distribute-rgt-out 86.1%

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

   3. Simplified 86.1%

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

      1. cancel-sign-sub-inv 86.1%

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

      2. fma-def 86.2%

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

      3. *-commutative 86.2%

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

   5. Applied egg-rr 86.2%

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

   6. Taylor expanded in x.re around 0 69.9%

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

      1. distribute-rgt-out 69.9%

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

      2. metadata-eval 69.9%

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

      3. associate-*r* 69.9%

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

      4. *-commutative 69.9%

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

      5. unpow2 69.9%

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

      6. associate-*l* 86.2%

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

      7. associate-*r* 86.1%

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

   8. Simplified 86.1%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 4.5 \cdot 10^{+152}:\\ \;\;\;\;x.re \cdot \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.im + x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.im\right) \cdot -3\right)\\ \end{array} \]

Alternative 4: 92.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq 4.5 \cdot 10^{+152}:\\ \;\;\;\;x.re \cdot \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.im + x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im 4.5e+152)
   (* x.re (- (- (* x.re x.re) (* x.im x.im)) (* x.im (+ x.im x.im))))
   (* x.im (* x.im (* x.re -3.0)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 4.5e+152) {
		tmp = x_46_re * (((x_46_re * x_46_re) - (x_46_im * x_46_im)) - (x_46_im * (x_46_im + x_46_im)));
	} else {
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0));
	}
	return tmp;
}

Fortran

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 <= 4.5d+152) then
        tmp = x_46re * (((x_46re * x_46re) - (x_46im * x_46im)) - (x_46im * (x_46im + x_46im)))
    else
        tmp = x_46im * (x_46im * (x_46re * (-3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 4.5e+152) {
		tmp = x_46_re * (((x_46_re * x_46_re) - (x_46_im * x_46_im)) - (x_46_im * (x_46_im + x_46_im)));
	} else {
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= 4.5e+152:
		tmp = x_46_re * (((x_46_re * x_46_re) - (x_46_im * x_46_im)) - (x_46_im * (x_46_im + x_46_im)))
	else:
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0))
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= 4.5e+152)
		tmp = Float64(x_46_re * Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im)) - Float64(x_46_im * Float64(x_46_im + x_46_im))));
	else
		tmp = Float64(x_46_im * Float64(x_46_im * Float64(x_46_re * -3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_im <= 4.5e+152)
		tmp = x_46_re * (((x_46_re * x_46_re) - (x_46_im * x_46_im)) - (x_46_im * (x_46_im + x_46_im)));
	else
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, 4.5e+152], N[(x$46$re * N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(x$46$im * N[(x$46$re * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 4.5000000000000001e152

   1. Initial program 83.8%

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

      1. sqr-neg 83.8%

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

      2. difference-of-squares 85.6%

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

      3. sub-neg 85.6%

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

      4. associate-*l* 89.6%

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

      5. sub-neg 89.6%

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

      6. remove-double-neg 89.6%

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

      7. +-commutative 89.6%

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

      8. *-commutative 89.6%

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

      9. *-commutative 89.6%

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

      10. distribute-rgt-out 89.6%

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

   3. Simplified 89.6%

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

      1. sub-neg 89.6%

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

      2. *-commutative 89.6%

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

      3. associate-*r* 89.6%

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

      4. *-commutative 89.6%

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

   5. Applied egg-rr 89.6%

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

      1. unsub-neg 89.6%

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

      2. *-commutative 89.6%

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

      3. associate-*r* 85.6%

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

      4. *-commutative 85.6%

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

      5. difference-of-squares 83.8%

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

      6. *-commutative 83.8%

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

      7. associate-*l* 83.8%

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

      8. distribute-lft-out-- 90.8%

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

   7. Simplified 90.8%

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

   if 4.5000000000000001e152 < x.im

   1. Initial program 56.1%

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

      1. sqr-neg 56.1%

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

      2. difference-of-squares 69.9%

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

      3. sub-neg 69.9%

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

      4. associate-*l* 86.1%

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

      5. sub-neg 86.1%

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

      6. remove-double-neg 86.1%

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

      7. +-commutative 86.1%

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

      8. *-commutative 86.1%

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

      9. *-commutative 86.1%

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

      10. distribute-rgt-out 86.1%

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

   3. Simplified 86.1%

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

      1. cancel-sign-sub-inv 86.1%

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

      2. fma-def 86.2%

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

      3. *-commutative 86.2%

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

   5. Applied egg-rr 86.2%

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

   6. Taylor expanded in x.re around 0 69.9%

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

      1. distribute-rgt-out 69.9%

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

      2. metadata-eval 69.9%

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

      3. associate-*r* 69.9%

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

      4. *-commutative 69.9%

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

      5. unpow2 69.9%

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

      6. associate-*l* 86.2%

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

      7. associate-*r* 86.1%

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

   8. Simplified 86.1%

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

      1. pow1 86.1%

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

      2. associate-*l* 86.2%

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

   10. Applied egg-rr 86.2%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 4.5 \cdot 10^{+152}:\\ \;\;\;\;x.re \cdot \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.im + x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \end{array} \]

Alternative 5: 92.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq 4.5 \cdot 10^{+152}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re + x.im \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.im\right) \cdot -3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im 4.5e+152)
   (* x.re (+ (* x.re x.re) (* x.im (* x.im -3.0))))
   (* x.im (* (* x.re x.im) -3.0))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 4.5e+152) {
		tmp = x_46_re * ((x_46_re * x_46_re) + (x_46_im * (x_46_im * -3.0)));
	} else {
		tmp = x_46_im * ((x_46_re * x_46_im) * -3.0);
	}
	return tmp;
}

Fortran

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 <= 4.5d+152) then
        tmp = x_46re * ((x_46re * x_46re) + (x_46im * (x_46im * (-3.0d0))))
    else
        tmp = x_46im * ((x_46re * x_46im) * (-3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 4.5e+152) {
		tmp = x_46_re * ((x_46_re * x_46_re) + (x_46_im * (x_46_im * -3.0)));
	} else {
		tmp = x_46_im * ((x_46_re * x_46_im) * -3.0);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= 4.5e+152:
		tmp = x_46_re * ((x_46_re * x_46_re) + (x_46_im * (x_46_im * -3.0)))
	else:
		tmp = x_46_im * ((x_46_re * x_46_im) * -3.0)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= 4.5e+152)
		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(x_46_im * Float64(Float64(x_46_re * x_46_im) * -3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_im <= 4.5e+152)
		tmp = x_46_re * ((x_46_re * x_46_re) + (x_46_im * (x_46_im * -3.0)));
	else
		tmp = x_46_im * ((x_46_re * x_46_im) * -3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, 4.5e+152], 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[(x$46$im * N[(N[(x$46$re * x$46$im), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 4.5000000000000001e152

   1. Initial program 83.8%

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

      1. sqr-neg 83.8%

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

      2. difference-of-squares 85.6%

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

      3. sub-neg 85.6%

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

      4. associate-*l* 89.6%

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

      5. sub-neg 89.6%

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

      6. remove-double-neg 89.6%

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

      7. +-commutative 89.6%

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

      8. *-commutative 89.6%

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

      9. *-commutative 89.6%

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

      10. distribute-rgt-out 89.6%

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

   3. Simplified 89.6%

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

      1. sub-neg 89.6%

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

      2. *-commutative 89.6%

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

      3. associate-*r* 89.6%

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

      4. *-commutative 89.6%

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

   5. Applied egg-rr 89.6%

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

      1. unsub-neg 89.6%

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

      2. *-commutative 89.6%

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

      3. associate-*r* 85.6%

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

      4. *-commutative 85.6%

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

      5. difference-of-squares 83.8%

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

      6. *-commutative 83.8%

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

      7. associate-*l* 83.8%

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

      8. distribute-lft-out-- 90.8%

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

   7. Simplified 90.8%

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

   8. Taylor expanded in x.re around 0 90.8%

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

      1. unpow2 90.8%

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

      2. mul-1-neg 90.8%

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

      3. distribute-rgt1-in 90.8%

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

      4. metadata-eval 90.8%

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

      5. unpow2 90.8%

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

   10. Simplified 90.8%

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

       1. +-commutative 90.8%

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

       2. distribute-lft-neg-in 90.8%

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

       3. metadata-eval 90.8%

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

       4. *-commutative 90.8%

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

       5. associate-*l* 90.8%

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

       6. fma-def 93.5%

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

   12. Applied egg-rr 93.5%

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

       1. fma-udef 90.8%

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

   14. Applied egg-rr 90.8%

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

   if 4.5000000000000001e152 < x.im

   1. Initial program 56.1%

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

      1. sqr-neg 56.1%

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

      2. difference-of-squares 69.9%

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

      3. sub-neg 69.9%

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

      4. associate-*l* 86.1%

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

      5. sub-neg 86.1%

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

      6. remove-double-neg 86.1%

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

      7. +-commutative 86.1%

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

      8. *-commutative 86.1%

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

      9. *-commutative 86.1%

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

      10. distribute-rgt-out 86.1%

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

   3. Simplified 86.1%

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

      1. cancel-sign-sub-inv 86.1%

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

      2. fma-def 86.2%

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

      3. *-commutative 86.2%

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

   5. Applied egg-rr 86.2%

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

   6. Taylor expanded in x.re around 0 69.9%

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

      1. distribute-rgt-out 69.9%

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

      2. metadata-eval 69.9%

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

      3. associate-*r* 69.9%

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

      4. *-commutative 69.9%

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

      5. unpow2 69.9%

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

      6. associate-*l* 86.2%

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

      7. associate-*r* 86.1%

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

   8. Simplified 86.1%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 4.5 \cdot 10^{+152}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re + x.im \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.im\right) \cdot -3\right)\\ \end{array} \]

Alternative 6: 92.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq 4.5 \cdot 10^{+152}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - \left(x.im \cdot x.im\right) \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.im\right) \cdot -3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im 4.5e+152)
   (* x.re (- (* x.re x.re) (* (* x.im x.im) 3.0)))
   (* x.im (* (* x.re x.im) -3.0))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 4.5e+152) {
		tmp = x_46_re * ((x_46_re * x_46_re) - ((x_46_im * x_46_im) * 3.0));
	} else {
		tmp = x_46_im * ((x_46_re * x_46_im) * -3.0);
	}
	return tmp;
}

Fortran

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 <= 4.5d+152) then
        tmp = x_46re * ((x_46re * x_46re) - ((x_46im * x_46im) * 3.0d0))
    else
        tmp = x_46im * ((x_46re * x_46im) * (-3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 4.5e+152) {
		tmp = x_46_re * ((x_46_re * x_46_re) - ((x_46_im * x_46_im) * 3.0));
	} else {
		tmp = x_46_im * ((x_46_re * x_46_im) * -3.0);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= 4.5e+152:
		tmp = x_46_re * ((x_46_re * x_46_re) - ((x_46_im * x_46_im) * 3.0))
	else:
		tmp = x_46_im * ((x_46_re * x_46_im) * -3.0)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= 4.5e+152)
		tmp = Float64(x_46_re * Float64(Float64(x_46_re * x_46_re) - Float64(Float64(x_46_im * x_46_im) * 3.0)));
	else
		tmp = Float64(x_46_im * Float64(Float64(x_46_re * x_46_im) * -3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_im <= 4.5e+152)
		tmp = x_46_re * ((x_46_re * x_46_re) - ((x_46_im * x_46_im) * 3.0));
	else
		tmp = x_46_im * ((x_46_re * x_46_im) * -3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, 4.5e+152], N[(x$46$re * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(N[(x$46$im * x$46$im), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(N[(x$46$re * x$46$im), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 4.5000000000000001e152

   1. Initial program 83.8%

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

      1. sqr-neg 83.8%

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

      2. difference-of-squares 85.6%

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

      3. sub-neg 85.6%

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

      4. associate-*l* 89.6%

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

      5. sub-neg 89.6%

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

      6. remove-double-neg 89.6%

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

      7. +-commutative 89.6%

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

      8. *-commutative 89.6%

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

      9. *-commutative 89.6%

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

      10. distribute-rgt-out 89.6%

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

   3. Simplified 89.6%

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

      1. sub-neg 89.6%

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

      2. *-commutative 89.6%

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

      3. associate-*r* 89.6%

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

      4. *-commutative 89.6%

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

   5. Applied egg-rr 89.6%

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

      1. unsub-neg 89.6%

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

      2. *-commutative 89.6%

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

      3. associate-*r* 85.6%

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

      4. *-commutative 85.6%

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

      5. difference-of-squares 83.8%

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

      6. *-commutative 83.8%

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

      7. associate-*l* 83.8%

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

      8. distribute-lft-out-- 90.8%

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

   7. Simplified 90.8%

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

   8. Taylor expanded in x.re around 0 90.8%

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

      1. unpow2 90.8%

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

      2. mul-1-neg 90.8%

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

      3. distribute-rgt1-in 90.8%

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

      4. metadata-eval 90.8%

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

      5. unpow2 90.8%

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

   10. Simplified 90.8%

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

       1. unsub-neg 90.8%

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

   12. Applied egg-rr 90.8%

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

   if 4.5000000000000001e152 < x.im

   1. Initial program 56.1%

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

      1. sqr-neg 56.1%

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

      2. difference-of-squares 69.9%

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

      3. sub-neg 69.9%

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

      4. associate-*l* 86.1%

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

      5. sub-neg 86.1%

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

      6. remove-double-neg 86.1%

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

      7. +-commutative 86.1%

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

      8. *-commutative 86.1%

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

      9. *-commutative 86.1%

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

      10. distribute-rgt-out 86.1%

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

   3. Simplified 86.1%

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

      1. cancel-sign-sub-inv 86.1%

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

      2. fma-def 86.2%

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

      3. *-commutative 86.2%

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

   5. Applied egg-rr 86.2%

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

   6. Taylor expanded in x.re around 0 69.9%

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

      1. distribute-rgt-out 69.9%

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

      2. metadata-eval 69.9%

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

      3. associate-*r* 69.9%

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

      4. *-commutative 69.9%

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

      5. unpow2 69.9%

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

      6. associate-*l* 86.2%

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

      7. associate-*r* 86.1%

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

   8. Simplified 86.1%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 4.5 \cdot 10^{+152}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - \left(x.im \cdot x.im\right) \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.im\right) \cdot -3\right)\\ \end{array} \]

Alternative 7: 80.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -8.8 \cdot 10^{+16} \lor \neg \left(x.re \leq 6 \cdot 10^{-28}\right):\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.re -8.8e+16) (not (<= x.re 6e-28)))
   (* x.re (* x.re x.re))
   (* -3.0 (* x.im (* x.re x.im)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -8.8e+16) || !(x_46_re <= 6e-28)) {
		tmp = x_46_re * (x_46_re * x_46_re);
	} else {
		tmp = -3.0 * (x_46_im * (x_46_re * x_46_im));
	}
	return tmp;
}

Fortran

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_46re <= (-8.8d+16)) .or. (.not. (x_46re <= 6d-28))) then
        tmp = x_46re * (x_46re * x_46re)
    else
        tmp = (-3.0d0) * (x_46im * (x_46re * x_46im))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -8.8e+16) || !(x_46_re <= 6e-28)) {
		tmp = x_46_re * (x_46_re * x_46_re);
	} else {
		tmp = -3.0 * (x_46_im * (x_46_re * x_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_re <= -8.8e+16) or not (x_46_re <= 6e-28):
		tmp = x_46_re * (x_46_re * x_46_re)
	else:
		tmp = -3.0 * (x_46_im * (x_46_re * x_46_im))
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_re <= -8.8e+16) || !(x_46_re <= 6e-28))
		tmp = Float64(x_46_re * Float64(x_46_re * x_46_re));
	else
		tmp = Float64(-3.0 * Float64(x_46_im * Float64(x_46_re * x_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_re <= -8.8e+16) || ~((x_46_re <= 6e-28)))
		tmp = x_46_re * (x_46_re * x_46_re);
	else
		tmp = -3.0 * (x_46_im * (x_46_re * x_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$re, -8.8e+16], N[Not[LessEqual[x$46$re, 6e-28]], $MachinePrecision]], N[(x$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision], N[(-3.0 * N[(x$46$im * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < -8.8e16 or 6.00000000000000005e-28 < x.re

   1. Initial program 74.4%

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

      1. sqr-neg 74.4%

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

      2. difference-of-squares 80.2%

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

      3. sub-neg 80.2%

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

      4. associate-*l* 80.2%

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

      5. sub-neg 80.2%

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

      6. remove-double-neg 80.2%

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

      7. +-commutative 80.2%

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

      8. *-commutative 80.2%

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

      9. *-commutative 80.2%

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

      10. distribute-rgt-out 80.2%

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

   3. Simplified 80.2%

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

      1. sub-neg 80.2%

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

      2. *-commutative 80.2%

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

      3. associate-*r* 80.2%

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

      4. *-commutative 80.2%

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

   5. Applied egg-rr 80.2%

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

      1. unsub-neg 80.2%

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

      2. *-commutative 80.2%

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

      3. associate-*r* 80.2%

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

      4. *-commutative 80.2%

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

      5. difference-of-squares 74.4%

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

      6. *-commutative 74.4%

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

      7. associate-*l* 74.4%

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

      8. distribute-lft-out-- 86.0%

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

   7. Simplified 86.0%

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

   8. Taylor expanded in x.re around 0 86.0%

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

      1. unpow2 86.0%

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

      2. mul-1-neg 86.0%

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

      3. distribute-rgt1-in 86.0%

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

      4. metadata-eval 86.0%

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

      5. unpow2 86.0%

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

   10. Simplified 86.0%

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

       1. +-commutative 86.0%

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

       2. distribute-lft-neg-in 86.0%

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

       3. metadata-eval 86.0%

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

       4. *-commutative 86.0%

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

       5. associate-*l* 86.0%

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

       6. fma-def 93.3%

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

   12. Applied egg-rr 93.3%

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

   13. Taylor expanded in x.im around 0 76.6%

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

       1. unpow2 76.6%

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

   15. Simplified 76.6%

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

   if -8.8e16 < x.re < 6.00000000000000005e-28

   1. Initial program 88.0%

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

   3. Simplified 88.0%

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

   4. Taylor expanded in x.re around 0 77.8%

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

      1. unpow2 77.8%

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

   6. Simplified 77.8%

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

   7. Taylor expanded in x.re around 0 77.8%

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

      1. *-commutative 77.8%

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

      2. unpow2 77.8%

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

      3. associate-*l* 89.5%

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

   9. Simplified 89.5%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -8.8 \cdot 10^{+16} \lor \neg \left(x.re \leq 6 \cdot 10^{-28}\right):\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \]

Alternative 8: 58.6% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.7%

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

   1. sqr-neg 80.7%

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

   2. difference-of-squares 83.8%

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

   3. sub-neg 83.8%

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

   4. associate-*l* 89.2%

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

   5. sub-neg 89.2%

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

   6. remove-double-neg 89.2%

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

   7. +-commutative 89.2%

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

   8. *-commutative 89.2%

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

   9. *-commutative 89.2%

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

   10. distribute-rgt-out 89.2%

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

3. Simplified 89.2%

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

   1. sub-neg 89.2%

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

   2. *-commutative 89.2%

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

   3. associate-*r* 89.2%

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

   4. *-commutative 89.2%

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

5. Applied egg-rr 89.2%

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

   1. unsub-neg 89.2%

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

   2. *-commutative 89.2%

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

   3. associate-*r* 83.8%

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

   4. *-commutative 83.8%

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

   5. difference-of-squares 80.7%

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

   6. *-commutative 80.7%

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

   7. associate-*l* 80.7%

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

   8. distribute-lft-out-- 86.9%

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

7. Simplified 86.9%

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

8. Taylor expanded in x.re around 0 86.9%

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

   1. unpow2 86.9%

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

   2. mul-1-neg 86.9%

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

   3. distribute-rgt1-in 86.9%

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

   4. metadata-eval 86.9%

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

   5. unpow2 86.9%

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

10. Simplified 86.9%

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

    1. +-commutative 86.9%

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

    2. distribute-lft-neg-in 86.9%

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

    3. metadata-eval 86.9%

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

    4. *-commutative 86.9%

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

    5. associate-*l* 86.9%

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

    6. fma-def 90.8%

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

12. Applied egg-rr 90.8%

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

13. Taylor expanded in x.im around 0 59.4%

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

    1. unpow2 59.4%

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

15. Simplified 59.4%

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

16. Final simplification 59.4%

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

Developer target: 87.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = ((x_46re * x_46re) * (x_46re - x_46im)) + ((x_46re * x_46im) * (x_46re - (3.0d0 * x_46im)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023274 
(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)))