math.cube on complex, imaginary part

Percentage Accurate: 82.6% → 99.8%

Time: 8.8s

Alternatives: 7

Speedup: 1.2×

• 45.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 82.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im));
	double tmp;
	if (((x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + t_0) <= ((double) INFINITY)) {
		tmp = t_0 + ((x_46_re + x_46_im) * (x_46_im * (x_46_re - x_46_im)));
	} else {
		tmp = (x_46_re * x_46_re) * (x_46_im + (x_46_im * (2.0 - ((x_46_im / (x_46_re / x_46_im)) / x_46_re))));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im));
	double tmp;
	if (((x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + t_0) <= Double.POSITIVE_INFINITY) {
		tmp = t_0 + ((x_46_re + x_46_im) * (x_46_im * (x_46_re - x_46_im)));
	} else {
		tmp = (x_46_re * x_46_re) * (x_46_im + (x_46_im * (2.0 - ((x_46_im / (x_46_re / x_46_im)) / x_46_re))));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	t_0 = x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im))
	tmp = 0
	if ((x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + t_0) <= math.inf:
		tmp = t_0 + ((x_46_re + x_46_im) * (x_46_im * (x_46_re - x_46_im)))
	else:
		tmp = (x_46_re * x_46_re) * (x_46_im + (x_46_im * (2.0 - ((x_46_im / (x_46_re / x_46_im)) / x_46_re))))
	return tmp

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(x_46_re * Float64(Float64(x_46_re * x_46_im) + Float64(x_46_re * x_46_im)))
	tmp = 0.0
	if (Float64(Float64(x_46_im * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im))) + t_0) <= Inf)
		tmp = Float64(t_0 + Float64(Float64(x_46_re + x_46_im) * Float64(x_46_im * Float64(x_46_re - x_46_im))));
	else
		tmp = Float64(Float64(x_46_re * x_46_re) * Float64(x_46_im + Float64(x_46_im * Float64(2.0 - Float64(Float64(x_46_im / Float64(x_46_re / x_46_im)) / x_46_re)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	t_0 = x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im));
	tmp = 0.0;
	if (((x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + t_0) <= Inf)
		tmp = t_0 + ((x_46_re + x_46_im) * (x_46_im * (x_46_re - x_46_im)));
	else
		tmp = (x_46_re * x_46_re) * (x_46_im + (x_46_im * (2.0 - ((x_46_im / (x_46_re / x_46_im)) / x_46_re))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(x$46$re * N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(x$46$im * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], Infinity], N[(t$95$0 + N[(N[(x$46$re + x$46$im), $MachinePrecision] * N[(x$46$im * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * x$46$re), $MachinePrecision] * N[(x$46$im + N[(x$46$im * N[(2.0 - N[(N[(x$46$im / N[(x$46$re / x$46$im), $MachinePrecision]), $MachinePrecision] / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 93.9%

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

      1. difference-of-squares N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right) \cdot x.im\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(x.re, x.im\right)}, \mathsf{*.f64}\left(x.im, x.re\right)\right), x.re\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.im\right), \mathsf{*.f64}\left(x.im, x.re\right)\right)}, x.re\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x.re + x.im\right), \left(\left(x.re - x.im\right) \cdot x.im\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.im\right), \mathsf{*.f64}\left(x.im, x.re\right)\right)}, x.re\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x.im + x.re\right), \left(\left(x.re - x.im\right) \cdot x.im\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(x.re, x.im\right)}, \mathsf{*.f64}\left(x.im, x.re\right)\right), x.re\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x.im, x.re\right), \left(\left(x.re - x.im\right) \cdot x.im\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(x.re, x.im\right)}, \mathsf{*.f64}\left(x.im, x.re\right)\right), x.re\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(\left(x.re - x.im\right), x.im\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.im\right), \color{blue}{\mathsf{*.f64}\left(x.im, x.re\right)}\right), x.re\right)\right) \]

      7. --lowering--.f64 99.8%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(x.re, x.im\right), x.im\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.im\right), \mathsf{*.f64}\left(\color{blue}{x.im}, x.re\right)\right), x.re\right)\right) \]

   4. Applied egg-rr 99.8%

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

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

   1. Initial program 0.0%

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

      1. difference-of-squares N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right) \cdot x.im\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(x.re, x.im\right)}, \mathsf{*.f64}\left(x.im, x.re\right)\right), x.re\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.im\right), \mathsf{*.f64}\left(x.im, x.re\right)\right)}, x.re\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x.re + x.im\right), \left(\left(x.re - x.im\right) \cdot x.im\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.im\right), \mathsf{*.f64}\left(x.im, x.re\right)\right)}, x.re\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x.im + x.re\right), \left(\left(x.re - x.im\right) \cdot x.im\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(x.re, x.im\right)}, \mathsf{*.f64}\left(x.im, x.re\right)\right), x.re\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x.im, x.re\right), \left(\left(x.re - x.im\right) \cdot x.im\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(x.re, x.im\right)}, \mathsf{*.f64}\left(x.im, x.re\right)\right), x.re\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(\left(x.re - x.im\right), x.im\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.im\right), \color{blue}{\mathsf{*.f64}\left(x.im, x.re\right)}\right), x.re\right)\right) \]

      7. --lowering--.f64 20.6%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(x.re, x.im\right), x.im\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.im\right), \mathsf{*.f64}\left(\color{blue}{x.im}, x.re\right)\right), x.re\right)\right) \]

   4. Applied egg-rr 20.6%

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

   5. Taylor expanded in x.re around inf

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. sum3-define N/A

         \[\leadsto {x.re}^{2} \cdot \mathsf{sum3}\left(\left(-1 \cdot \frac{{x.im}^{3}}{{x.re}^{2}} + \frac{x.im \cdot \left(x.im + -1 \cdot x.im\right)}{x.re}\right), \color{blue}{\left(2 \cdot x.im\right)}, x.im\right) \]

   7. Simplified 79.4%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{+.f64}\left(x.im, \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\left(x.im \cdot \frac{x.im}{x.re}\right), x.re\right)\right)\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{+.f64}\left(x.im, \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\left(x.im \cdot \frac{1}{\frac{x.re}{x.im}}\right), x.re\right)\right)\right)\right)\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{+.f64}\left(x.im, \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{x.im}{\frac{x.re}{x.im}}\right), x.re\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{+.f64}\left(x.im, \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x.im, \left(\frac{x.re}{x.im}\right)\right), x.re\right)\right)\right)\right)\right) \]

      5. /-lowering-/.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{+.f64}\left(x.im, \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x.im, \mathsf{/.f64}\left(x.re, x.im\right)\right), x.re\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 100.0%

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

4. Final simplification 99.8%

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

Alternative 2: 92.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq 7.5 \cdot 10^{+153}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.re \cdot 3\right) - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.re 7.5e+153)
   (* x.im (- (* x.re (* x.re 3.0)) (* x.im x.im)))
   (* 3.0 (* x.re (* x.re x.im)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= 7.5e+153) {
		tmp = x_46_im * ((x_46_re * (x_46_re * 3.0)) - (x_46_im * x_46_im));
	} else {
		tmp = 3.0 * (x_46_re * (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 <= 7.5d+153) then
        tmp = x_46im * ((x_46re * (x_46re * 3.0d0)) - (x_46im * x_46im))
    else
        tmp = 3.0d0 * (x_46re * (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 <= 7.5e+153) {
		tmp = x_46_im * ((x_46_re * (x_46_re * 3.0)) - (x_46_im * x_46_im));
	} else {
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_re <= 7.5e+153:
		tmp = x_46_im * ((x_46_re * (x_46_re * 3.0)) - (x_46_im * x_46_im))
	else:
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im))
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_re <= 7.5e+153)
		tmp = Float64(x_46_im * Float64(Float64(x_46_re * Float64(x_46_re * 3.0)) - Float64(x_46_im * x_46_im)));
	else
		tmp = Float64(3.0 * Float64(x_46_re * 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 <= 7.5e+153)
		tmp = x_46_im * ((x_46_re * (x_46_re * 3.0)) - (x_46_im * x_46_im));
	else
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$re, 7.5e+153], N[(x$46$im * N[(N[(x$46$re * N[(x$46$re * 3.0), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 * N[(x$46$re * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 7.50000000000000065e153

   1. Initial program 86.1%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-out N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \color{blue}{\left(\left(x.re + x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right) \]

      8. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right) - \color{blue}{x.im \cdot x.im}\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right), \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

      10. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(2 \cdot x.re\right) \cdot x.re + x.re \cdot x.re\right), \left(x.im \cdot x.im\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(2 \cdot \left(x.re \cdot x.re\right) + x.re \cdot x.re\right), \left(x.im \cdot x.im\right)\right)\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(2 + 1\right) \cdot \left(x.re \cdot x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(3 \cdot \left(x.re \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re \cdot x.re\right) \cdot 3\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x.re \cdot x.re\right), 3\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3\right), \left(x.im \cdot x.im\right)\right)\right) \]

      17. *-lowering-*.f64 93.2%

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3\right), \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

   3. Simplified 93.2%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(x.re \cdot x.re\right) \cdot 3 - x.im \cdot x.im\right), \color{blue}{x.im}\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\left(x.re \cdot x.re\right) \cdot 3\right), \left(x.im \cdot x.im\right)\right), x.im\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(x.re \cdot \left(x.re \cdot 3\right)\right), \left(x.im \cdot x.im\right)\right), x.im\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(x.re \cdot 3\right)\right), \left(x.im \cdot x.im\right)\right), x.im\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \left(x.im \cdot x.im\right)\right), x.im\right) \]

      7. *-lowering-*.f64 93.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), x.im\right) \]

   6. Applied egg-rr 93.3%

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

   if 7.50000000000000065e153 < x.re

   1. Initial program 49.8%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-out N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \color{blue}{\left(\left(x.re + x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right) \]

      8. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right) - \color{blue}{x.im \cdot x.im}\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right), \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

      10. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(2 \cdot x.re\right) \cdot x.re + x.re \cdot x.re\right), \left(x.im \cdot x.im\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(2 \cdot \left(x.re \cdot x.re\right) + x.re \cdot x.re\right), \left(x.im \cdot x.im\right)\right)\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(2 + 1\right) \cdot \left(x.re \cdot x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(3 \cdot \left(x.re \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re \cdot x.re\right) \cdot 3\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x.re \cdot x.re\right), 3\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3\right), \left(x.im \cdot x.im\right)\right)\right) \]

      17. *-lowering-*.f64 49.8%

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3\right), \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

   3. Simplified 49.8%

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

   5. Taylor expanded in x.im around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(3, \color{blue}{\left(x.im \cdot {x.re}^{2}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(x.im, \color{blue}{\left({x.re}^{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(x.im, \left(x.re \cdot \color{blue}{x.re}\right)\right)\right) \]

      4. *-lowering-*.f64 58.9%

         \[\leadsto \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.re, \color{blue}{x.re}\right)\right)\right) \]

   7. Simplified 58.9%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(3, \left(\left(x.im \cdot x.re\right) \cdot \color{blue}{x.re}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(\left(x.im \cdot x.re\right), \color{blue}{x.re}\right)\right) \]

      3. *-lowering-*.f64 81.8%

         \[\leadsto \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.re\right), x.re\right)\right) \]

   9. Applied egg-rr 81.8%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 7.5 \cdot 10^{+153}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.re \cdot 3\right) - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq 7.5 \cdot 10^{+153}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.re\right) \cdot 3 - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.re 7.5e+153)
   (* x.im (- (* (* x.re x.re) 3.0) (* x.im x.im)))
   (* 3.0 (* x.re (* x.re x.im)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= 7.5e+153) {
		tmp = x_46_im * (((x_46_re * x_46_re) * 3.0) - (x_46_im * x_46_im));
	} else {
		tmp = 3.0 * (x_46_re * (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 <= 7.5d+153) then
        tmp = x_46im * (((x_46re * x_46re) * 3.0d0) - (x_46im * x_46im))
    else
        tmp = 3.0d0 * (x_46re * (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 <= 7.5e+153) {
		tmp = x_46_im * (((x_46_re * x_46_re) * 3.0) - (x_46_im * x_46_im));
	} else {
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_re <= 7.5e+153:
		tmp = x_46_im * (((x_46_re * x_46_re) * 3.0) - (x_46_im * x_46_im))
	else:
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im))
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_re <= 7.5e+153)
		tmp = Float64(x_46_im * Float64(Float64(Float64(x_46_re * x_46_re) * 3.0) - Float64(x_46_im * x_46_im)));
	else
		tmp = Float64(3.0 * Float64(x_46_re * 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 <= 7.5e+153)
		tmp = x_46_im * (((x_46_re * x_46_re) * 3.0) - (x_46_im * x_46_im));
	else
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$re, 7.5e+153], N[(x$46$im * N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] * 3.0), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 * N[(x$46$re * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 7.50000000000000065e153

   1. Initial program 86.1%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-out N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \color{blue}{\left(\left(x.re + x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right) \]

      8. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right) - \color{blue}{x.im \cdot x.im}\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right), \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

      10. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(2 \cdot x.re\right) \cdot x.re + x.re \cdot x.re\right), \left(x.im \cdot x.im\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(2 \cdot \left(x.re \cdot x.re\right) + x.re \cdot x.re\right), \left(x.im \cdot x.im\right)\right)\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(2 + 1\right) \cdot \left(x.re \cdot x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(3 \cdot \left(x.re \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re \cdot x.re\right) \cdot 3\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x.re \cdot x.re\right), 3\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3\right), \left(x.im \cdot x.im\right)\right)\right) \]

      17. *-lowering-*.f64 93.2%

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3\right), \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

   3. Simplified 93.2%

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

   if 7.50000000000000065e153 < x.re

   1. Initial program 49.8%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-out N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \color{blue}{\left(\left(x.re + x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right) \]

      8. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right) - \color{blue}{x.im \cdot x.im}\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right), \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

      10. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(2 \cdot x.re\right) \cdot x.re + x.re \cdot x.re\right), \left(x.im \cdot x.im\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(2 \cdot \left(x.re \cdot x.re\right) + x.re \cdot x.re\right), \left(x.im \cdot x.im\right)\right)\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(2 + 1\right) \cdot \left(x.re \cdot x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(3 \cdot \left(x.re \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re \cdot x.re\right) \cdot 3\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x.re \cdot x.re\right), 3\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3\right), \left(x.im \cdot x.im\right)\right)\right) \]

      17. *-lowering-*.f64 49.8%

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3\right), \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

   3. Simplified 49.8%

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

   5. Taylor expanded in x.im around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(3, \color{blue}{\left(x.im \cdot {x.re}^{2}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(x.im, \color{blue}{\left({x.re}^{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(x.im, \left(x.re \cdot \color{blue}{x.re}\right)\right)\right) \]

      4. *-lowering-*.f64 58.9%

         \[\leadsto \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.re, \color{blue}{x.re}\right)\right)\right) \]

   7. Simplified 58.9%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(3, \left(\left(x.im \cdot x.re\right) \cdot \color{blue}{x.re}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(\left(x.im \cdot x.re\right), \color{blue}{x.re}\right)\right) \]

      3. *-lowering-*.f64 81.8%

         \[\leadsto \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.re\right), x.re\right)\right) \]

   9. Applied egg-rr 81.8%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 7.5 \cdot 10^{+153}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.re\right) \cdot 3 - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq 1.65 \cdot 10^{+83}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;0 - x.im \cdot \left(x.im \cdot x.im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im 1.65e+83)
   (* (* x.re x.im) (* x.re 3.0))
   (- 0.0 (* x.im (* x.im x.im)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 1.65e+83) {
		tmp = (x_46_re * x_46_im) * (x_46_re * 3.0);
	} else {
		tmp = 0.0 - (x_46_im * (x_46_im * 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_46im <= 1.65d+83) then
        tmp = (x_46re * x_46im) * (x_46re * 3.0d0)
    else
        tmp = 0.0d0 - (x_46im * (x_46im * 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_im <= 1.65e+83) {
		tmp = (x_46_re * x_46_im) * (x_46_re * 3.0);
	} else {
		tmp = 0.0 - (x_46_im * (x_46_im * x_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= 1.65e+83:
		tmp = (x_46_re * x_46_im) * (x_46_re * 3.0)
	else:
		tmp = 0.0 - (x_46_im * (x_46_im * x_46_im))
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= 1.65e+83)
		tmp = Float64(Float64(x_46_re * x_46_im) * Float64(x_46_re * 3.0));
	else
		tmp = Float64(0.0 - Float64(x_46_im * Float64(x_46_im * x_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_im <= 1.65e+83)
		tmp = (x_46_re * x_46_im) * (x_46_re * 3.0);
	else
		tmp = 0.0 - (x_46_im * (x_46_im * x_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, 1.65e+83], N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(x$46$re * 3.0), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(x$46$im * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.64999999999999992e83

   1. Initial program 84.3%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-out N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \color{blue}{\left(\left(x.re + x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right) \]

      8. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right) - \color{blue}{x.im \cdot x.im}\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right), \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

      10. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(2 \cdot x.re\right) \cdot x.re + x.re \cdot x.re\right), \left(x.im \cdot x.im\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(2 \cdot \left(x.re \cdot x.re\right) + x.re \cdot x.re\right), \left(x.im \cdot x.im\right)\right)\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(2 + 1\right) \cdot \left(x.re \cdot x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(3 \cdot \left(x.re \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re \cdot x.re\right) \cdot 3\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x.re \cdot x.re\right), 3\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3\right), \left(x.im \cdot x.im\right)\right)\right) \]

      17. *-lowering-*.f64 89.5%

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3\right), \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

   3. Simplified 89.5%

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

   5. Taylor expanded in x.im around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(3, \color{blue}{\left(x.im \cdot {x.re}^{2}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(x.im, \color{blue}{\left({x.re}^{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(x.im, \left(x.re \cdot \color{blue}{x.re}\right)\right)\right) \]

      4. *-lowering-*.f64 54.1%

         \[\leadsto \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.re, \color{blue}{x.re}\right)\right)\right) \]

   7. Simplified 54.1%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

         \[\leadsto \left(x.im \cdot \left(x.re \cdot x.re\right)\right) \cdot 3 \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(x.im \cdot x.re\right) \cdot x.re\right) \cdot 3 \]

      5. associate-*l* N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x.im \cdot x.re\right), \color{blue}{\left(x.re \cdot 3\right)}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.re\right), \left(\color{blue}{x.re} \cdot 3\right)\right) \]

      8. *-lowering-*.f64 60.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(x.re, \color{blue}{3}\right)\right) \]

   9. Applied egg-rr 60.4%

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

   if 1.64999999999999992e83 < x.im

   1. Initial program 69.4%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-out N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \color{blue}{\left(\left(x.re + x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right) \]

      8. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right) - \color{blue}{x.im \cdot x.im}\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right), \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

      10. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(2 \cdot x.re\right) \cdot x.re + x.re \cdot x.re\right), \left(x.im \cdot x.im\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(2 \cdot \left(x.re \cdot x.re\right) + x.re \cdot x.re\right), \left(x.im \cdot x.im\right)\right)\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(2 + 1\right) \cdot \left(x.re \cdot x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(3 \cdot \left(x.re \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re \cdot x.re\right) \cdot 3\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x.re \cdot x.re\right), 3\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3\right), \left(x.im \cdot x.im\right)\right)\right) \]

      17. *-lowering-*.f64 79.6%

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3\right), \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

   3. Simplified 79.6%

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

   5. Taylor expanded in x.im around inf

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

      1. mul-1-neg N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

         \[\leadsto \mathsf{neg}\left({x.im}^{2} \cdot x.im\right) \]

      4. distribute-rgt-neg-in N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({x.im}^{2}\right), \color{blue}{\left(\mathsf{neg}\left(x.im\right)\right)}\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x.im \cdot x.im\right), \left(\mathsf{neg}\left(\color{blue}{x.im}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(\mathsf{neg}\left(\color{blue}{x.im}\right)\right)\right) \]

      8. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(0 - \color{blue}{x.im}\right)\right) \]

      9. --lowering--.f64 85.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{\_.f64}\left(0, \color{blue}{x.im}\right)\right) \]

   7. Simplified 85.7%

      \[\leadsto \color{blue}{\left(x.im \cdot x.im\right) \cdot \left(0 - x.im\right)} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(\mathsf{neg}\left(x.im\right)\right)\right) \]

      2. neg-lowering-neg.f64 85.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{neg.f64}\left(x.im\right)\right) \]

   9. Applied egg-rr 85.7%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 1.65 \cdot 10^{+83}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;0 - x.im \cdot \left(x.im \cdot x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq 1.65 \cdot 10^{+83}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0 - x.im \cdot \left(x.im \cdot x.im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im 1.65e+83)
   (* 3.0 (* x.re (* x.re x.im)))
   (- 0.0 (* x.im (* x.im x.im)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 1.65e+83) {
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im));
	} else {
		tmp = 0.0 - (x_46_im * (x_46_im * 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_46im <= 1.65d+83) then
        tmp = 3.0d0 * (x_46re * (x_46re * x_46im))
    else
        tmp = 0.0d0 - (x_46im * (x_46im * 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_im <= 1.65e+83) {
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im));
	} else {
		tmp = 0.0 - (x_46_im * (x_46_im * x_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= 1.65e+83:
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im))
	else:
		tmp = 0.0 - (x_46_im * (x_46_im * x_46_im))
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= 1.65e+83)
		tmp = Float64(3.0 * Float64(x_46_re * Float64(x_46_re * x_46_im)));
	else
		tmp = Float64(0.0 - Float64(x_46_im * Float64(x_46_im * x_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_im <= 1.65e+83)
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im));
	else
		tmp = 0.0 - (x_46_im * (x_46_im * x_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, 1.65e+83], N[(3.0 * N[(x$46$re * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(x$46$im * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.64999999999999992e83

   1. Initial program 84.3%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-out N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \color{blue}{\left(\left(x.re + x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right) \]

      8. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right) - \color{blue}{x.im \cdot x.im}\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right), \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

      10. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(2 \cdot x.re\right) \cdot x.re + x.re \cdot x.re\right), \left(x.im \cdot x.im\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(2 \cdot \left(x.re \cdot x.re\right) + x.re \cdot x.re\right), \left(x.im \cdot x.im\right)\right)\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(2 + 1\right) \cdot \left(x.re \cdot x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(3 \cdot \left(x.re \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re \cdot x.re\right) \cdot 3\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x.re \cdot x.re\right), 3\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3\right), \left(x.im \cdot x.im\right)\right)\right) \]

      17. *-lowering-*.f64 89.5%

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3\right), \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

   3. Simplified 89.5%

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

   5. Taylor expanded in x.im around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(3, \color{blue}{\left(x.im \cdot {x.re}^{2}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(x.im, \color{blue}{\left({x.re}^{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(x.im, \left(x.re \cdot \color{blue}{x.re}\right)\right)\right) \]

      4. *-lowering-*.f64 54.1%

         \[\leadsto \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.re, \color{blue}{x.re}\right)\right)\right) \]

   7. Simplified 54.1%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(3, \left(\left(x.im \cdot x.re\right) \cdot \color{blue}{x.re}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(\left(x.im \cdot x.re\right), \color{blue}{x.re}\right)\right) \]

      3. *-lowering-*.f64 60.4%

         \[\leadsto \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.re\right), x.re\right)\right) \]

   9. Applied egg-rr 60.4%

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

   if 1.64999999999999992e83 < x.im

   1. Initial program 69.4%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-out N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \color{blue}{\left(\left(x.re + x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right) \]

      8. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right) - \color{blue}{x.im \cdot x.im}\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right), \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

      10. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(2 \cdot x.re\right) \cdot x.re + x.re \cdot x.re\right), \left(x.im \cdot x.im\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(2 \cdot \left(x.re \cdot x.re\right) + x.re \cdot x.re\right), \left(x.im \cdot x.im\right)\right)\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(2 + 1\right) \cdot \left(x.re \cdot x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(3 \cdot \left(x.re \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re \cdot x.re\right) \cdot 3\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x.re \cdot x.re\right), 3\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3\right), \left(x.im \cdot x.im\right)\right)\right) \]

      17. *-lowering-*.f64 79.6%

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3\right), \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

   3. Simplified 79.6%

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

   5. Taylor expanded in x.im around inf

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

      1. mul-1-neg N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

         \[\leadsto \mathsf{neg}\left({x.im}^{2} \cdot x.im\right) \]

      4. distribute-rgt-neg-in N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({x.im}^{2}\right), \color{blue}{\left(\mathsf{neg}\left(x.im\right)\right)}\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x.im \cdot x.im\right), \left(\mathsf{neg}\left(\color{blue}{x.im}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(\mathsf{neg}\left(\color{blue}{x.im}\right)\right)\right) \]

      8. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(0 - \color{blue}{x.im}\right)\right) \]

      9. --lowering--.f64 85.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{\_.f64}\left(0, \color{blue}{x.im}\right)\right) \]

   7. Simplified 85.7%

      \[\leadsto \color{blue}{\left(x.im \cdot x.im\right) \cdot \left(0 - x.im\right)} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(\mathsf{neg}\left(x.im\right)\right)\right) \]

      2. neg-lowering-neg.f64 85.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{neg.f64}\left(x.im\right)\right) \]

   9. Applied egg-rr 85.7%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 1.65 \cdot 10^{+83}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0 - x.im \cdot \left(x.im \cdot x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq 1.65 \cdot 10^{+83}:\\ \;\;\;\;3 \cdot \left(\left(x.re \cdot x.re\right) \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;0 - x.im \cdot \left(x.im \cdot x.im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im 1.65e+83)
   (* 3.0 (* (* x.re x.re) x.im))
   (- 0.0 (* x.im (* x.im x.im)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 1.65e+83) {
		tmp = 3.0 * ((x_46_re * x_46_re) * x_46_im);
	} else {
		tmp = 0.0 - (x_46_im * (x_46_im * 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_46im <= 1.65d+83) then
        tmp = 3.0d0 * ((x_46re * x_46re) * x_46im)
    else
        tmp = 0.0d0 - (x_46im * (x_46im * 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_im <= 1.65e+83) {
		tmp = 3.0 * ((x_46_re * x_46_re) * x_46_im);
	} else {
		tmp = 0.0 - (x_46_im * (x_46_im * x_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= 1.65e+83:
		tmp = 3.0 * ((x_46_re * x_46_re) * x_46_im)
	else:
		tmp = 0.0 - (x_46_im * (x_46_im * x_46_im))
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= 1.65e+83)
		tmp = Float64(3.0 * Float64(Float64(x_46_re * x_46_re) * x_46_im));
	else
		tmp = Float64(0.0 - Float64(x_46_im * Float64(x_46_im * x_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_im <= 1.65e+83)
		tmp = 3.0 * ((x_46_re * x_46_re) * x_46_im);
	else
		tmp = 0.0 - (x_46_im * (x_46_im * x_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, 1.65e+83], N[(3.0 * N[(N[(x$46$re * x$46$re), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(x$46$im * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.64999999999999992e83

   1. Initial program 84.3%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-out N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \color{blue}{\left(\left(x.re + x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right) \]

      8. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right) - \color{blue}{x.im \cdot x.im}\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right), \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

      10. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(2 \cdot x.re\right) \cdot x.re + x.re \cdot x.re\right), \left(x.im \cdot x.im\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(2 \cdot \left(x.re \cdot x.re\right) + x.re \cdot x.re\right), \left(x.im \cdot x.im\right)\right)\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(2 + 1\right) \cdot \left(x.re \cdot x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(3 \cdot \left(x.re \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re \cdot x.re\right) \cdot 3\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x.re \cdot x.re\right), 3\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3\right), \left(x.im \cdot x.im\right)\right)\right) \]

      17. *-lowering-*.f64 89.5%

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3\right), \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

   3. Simplified 89.5%

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

   5. Taylor expanded in x.im around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(3, \color{blue}{\left(x.im \cdot {x.re}^{2}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(x.im, \color{blue}{\left({x.re}^{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(x.im, \left(x.re \cdot \color{blue}{x.re}\right)\right)\right) \]

      4. *-lowering-*.f64 54.1%

         \[\leadsto \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.re, \color{blue}{x.re}\right)\right)\right) \]

   7. Simplified 54.1%

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

   if 1.64999999999999992e83 < x.im

   1. Initial program 69.4%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-out N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \color{blue}{\left(\left(x.re + x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right) \]

      8. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right) - \color{blue}{x.im \cdot x.im}\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right), \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

      10. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(2 \cdot x.re\right) \cdot x.re + x.re \cdot x.re\right), \left(x.im \cdot x.im\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(2 \cdot \left(x.re \cdot x.re\right) + x.re \cdot x.re\right), \left(x.im \cdot x.im\right)\right)\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(2 + 1\right) \cdot \left(x.re \cdot x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(3 \cdot \left(x.re \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re \cdot x.re\right) \cdot 3\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x.re \cdot x.re\right), 3\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3\right), \left(x.im \cdot x.im\right)\right)\right) \]

      17. *-lowering-*.f64 79.6%

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3\right), \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

   3. Simplified 79.6%

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

   5. Taylor expanded in x.im around inf

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

      1. mul-1-neg N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

         \[\leadsto \mathsf{neg}\left({x.im}^{2} \cdot x.im\right) \]

      4. distribute-rgt-neg-in N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({x.im}^{2}\right), \color{blue}{\left(\mathsf{neg}\left(x.im\right)\right)}\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x.im \cdot x.im\right), \left(\mathsf{neg}\left(\color{blue}{x.im}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(\mathsf{neg}\left(\color{blue}{x.im}\right)\right)\right) \]

      8. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(0 - \color{blue}{x.im}\right)\right) \]

      9. --lowering--.f64 85.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{\_.f64}\left(0, \color{blue}{x.im}\right)\right) \]

   7. Simplified 85.7%

      \[\leadsto \color{blue}{\left(x.im \cdot x.im\right) \cdot \left(0 - x.im\right)} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(\mathsf{neg}\left(x.im\right)\right)\right) \]

      2. neg-lowering-neg.f64 85.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{neg.f64}\left(x.im\right)\right) \]

   9. Applied egg-rr 85.7%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 1.65 \cdot 10^{+83}:\\ \;\;\;\;3 \cdot \left(\left(x.re \cdot x.re\right) \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;0 - x.im \cdot \left(x.im \cdot x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 58.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ 0 - x.im \cdot \left(x.im \cdot x.im\right) \end{array} \]

FPCore

(FPCore (x.re x.im) :precision binary64 (- 0.0 (* x.im (* x.im x.im))))

C

double code(double x_46_re, double x_46_im) {
	return 0.0 - (x_46_im * (x_46_im * 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 = 0.0d0 - (x_46im * (x_46im * x_46im))
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return 0.0 - (x_46_im * (x_46_im * x_46_im));
}

Python

def code(x_46_re, x_46_im):
	return 0.0 - (x_46_im * (x_46_im * x_46_im))

Julia

function code(x_46_re, x_46_im)
	return Float64(0.0 - Float64(x_46_im * Float64(x_46_im * x_46_im)))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = 0.0 - (x_46_im * (x_46_im * x_46_im));
end

Wolfram

code[x$46$re_, x$46$im_] := N[(0.0 - N[(x$46$im * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.5%

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. distribute-lft-out N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. distribute-lft-out N/A

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

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x.im, \color{blue}{\left(\left(x.re + x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right) \]

   8. associate-+r- N/A

      \[\leadsto \mathsf{*.f64}\left(x.im, \left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right) - \color{blue}{x.im \cdot x.im}\right)\right) \]

   9. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right), \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

   10. count-2 N/A

       \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(2 \cdot x.re\right) \cdot x.re + x.re \cdot x.re\right), \left(x.im \cdot x.im\right)\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(2 \cdot \left(x.re \cdot x.re\right) + x.re \cdot x.re\right), \left(x.im \cdot x.im\right)\right)\right) \]

   12. distribute-lft1-in N/A

       \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(2 + 1\right) \cdot \left(x.re \cdot x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(3 \cdot \left(x.re \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re \cdot x.re\right) \cdot 3\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x.re \cdot x.re\right), 3\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

   16. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3\right), \left(x.im \cdot x.im\right)\right)\right) \]

   17. *-lowering-*.f64 87.6%

       \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3\right), \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

3. Simplified 87.6%

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

5. Taylor expanded in x.im around inf

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

   1. mul-1-neg N/A

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

   2. unpow3 N/A

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

   3. unpow2 N/A

      \[\leadsto \mathsf{neg}\left({x.im}^{2} \cdot x.im\right) \]

   4. distribute-rgt-neg-in N/A

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

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left({x.im}^{2}\right), \color{blue}{\left(\mathsf{neg}\left(x.im\right)\right)}\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(x.im \cdot x.im\right), \left(\mathsf{neg}\left(\color{blue}{x.im}\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(\mathsf{neg}\left(\color{blue}{x.im}\right)\right)\right) \]

   8. neg-sub0 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(0 - \color{blue}{x.im}\right)\right) \]

   9. --lowering--.f64 60.7%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{\_.f64}\left(0, \color{blue}{x.im}\right)\right) \]

7. Simplified 60.7%

   \[\leadsto \color{blue}{\left(x.im \cdot x.im\right) \cdot \left(0 - x.im\right)} \]
8. Step-by-step derivation

   1. sub0-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(\mathsf{neg}\left(x.im\right)\right)\right) \]

   2. neg-lowering-neg.f64 60.7%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{neg.f64}\left(x.im\right)\right) \]

9. Applied egg-rr 60.7%

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

10. Final simplification 60.7%

    \[\leadsto 0 - x.im \cdot \left(x.im \cdot x.im\right) \]
11. Add Preprocessing

Developer Target 1: 91.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(x.re \cdot x.im\right) \cdot \left(2 \cdot x.re\right) + \left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.re + x.im\right) \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (+ (* (* x.re x.im) (* 2.0 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im))))

C

double code(double x_46_re, double x_46_im) {
	return ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = ((x_46re * x_46im) * (2.0d0 * x_46re)) + ((x_46im * (x_46re - x_46im)) * (x_46re + x_46im))
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
}

Python

def code(x_46_re, x_46_im):
	return ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im))

Julia

function code(x_46_re, x_46_im)
	return Float64(Float64(Float64(x_46_re * x_46_im) * Float64(2.0 * x_46_re)) + Float64(Float64(x_46_im * Float64(x_46_re - x_46_im)) * Float64(x_46_re + x_46_im)))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
end

Wolfram

code[x$46$re_, x$46$im_] := N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(2.0 * x$46$re), $MachinePrecision]), $MachinePrecision] + N[(N[(x$46$im * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision] * N[(x$46$re + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024144 
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (* (* x.re x.im) (* 2 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im))))

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