math.cube on complex, real part

Percentage Accurate: 82.8% → 96.5%

Time: 6.9s

Alternatives: 9

Speedup: 1.3×

• 43.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 82.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.5% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (((x_46_re * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re * x_46_im) + (x_46_re * x_46_im)))) <= Double.POSITIVE_INFINITY) {
		tmp = ((x_46_re + x_46_im) * (x_46_re * (x_46_re - x_46_im))) - (x_46_im * (x_46_re * (x_46_im + x_46_im)));
	} else {
		tmp = Math.pow(x_46_re, 3.0);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if ((x_46_re * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re * x_46_im) + (x_46_re * x_46_im)))) <= math.inf:
		tmp = ((x_46_re + x_46_im) * (x_46_re * (x_46_re - x_46_im))) - (x_46_im * (x_46_re * (x_46_im + x_46_im)))
	else:
		tmp = math.pow(x_46_re, 3.0)
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 92.1%

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

      1. *-commutative 92.1%

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

      2. *-commutative 92.1%

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

      3. *-commutative 92.1%

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

      4. distribute-lft-out 92.1%

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

   3. Simplified 92.1%

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

      1. add-log-exp 52.3%

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

      2. exp-prod 54.2%

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

   5. Applied egg-rr 54.2%

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

      1. log-pow 46.0%

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

      2. difference-of-squares 46.0%

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

      3. add-log-exp 92.1%

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

      4. associate-*l* 99.7%

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

   7. Applied egg-rr 99.7%

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

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

   1. Initial program 0.0%

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

      1. *-commutative 0.0%

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

      2. distribute-lft-out 0.0%

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

      3. associate-*l* 0.0%

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

      4. *-commutative 0.0%

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

      5. distribute-rgt-out-- 41.7%

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

      6. associate--l- 41.7%

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

      7. associate--l- 41.7%

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

      8. sub-neg 41.7%

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

      9. associate--l+ 41.7%

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

      10. fma-udef 69.4%

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

      11. neg-mul-1 69.4%

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

      12. count-2 69.4%

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

      13. associate-*l* 69.4%

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

      14. distribute-rgt-out-- 69.4%

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

      15. associate-*r* 69.4%

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

      16. metadata-eval 69.4%

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

   3. Simplified 69.4%

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

   4. Taylor expanded in x.re around inf 72.2%

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

4. Final simplification 95.8%

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

Alternative 2: 96.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (((x_46_re * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re * x_46_im) + (x_46_re * x_46_im)))) <= Double.POSITIVE_INFINITY) {
		tmp = ((x_46_re + x_46_im) * (x_46_re * (x_46_re - x_46_im))) - (x_46_im * (x_46_re * (x_46_im + x_46_im)));
	} else {
		tmp = x_46_re * ((x_46_im * x_46_im) * 3.0);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if ((x_46_re * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re * x_46_im) + (x_46_re * x_46_im)))) <= math.inf:
		tmp = ((x_46_re + x_46_im) * (x_46_re * (x_46_re - x_46_im))) - (x_46_im * (x_46_re * (x_46_im + x_46_im)))
	else:
		tmp = x_46_re * ((x_46_im * x_46_im) * 3.0)
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 92.1%

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

      1. *-commutative 92.1%

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

      2. *-commutative 92.1%

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

      3. *-commutative 92.1%

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

      4. distribute-lft-out 92.1%

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

   3. Simplified 92.1%

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

      1. add-log-exp 52.3%

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

      2. exp-prod 54.2%

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

   5. Applied egg-rr 54.2%

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

      1. log-pow 46.0%

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

      2. difference-of-squares 46.0%

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

      3. add-log-exp 92.1%

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

      4. associate-*l* 99.7%

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

   7. Applied egg-rr 99.7%

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

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

   1. Initial program 0.0%

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

      1. *-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. *-commutative 0.0%

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

      4. distribute-lft-out 0.0%

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

   3. Simplified 0.0%

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

      1. add-log-exp 0.0%

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

      2. exp-prod 0.0%

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

   5. Applied egg-rr 0.0%

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

   6. Taylor expanded in x.re around 0 27.8%

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

      1. associate-*r* 27.8%

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

      2. neg-mul-1 27.8%

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

      3. unpow2 27.8%

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

   8. Simplified 27.8%

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

      1. sub-neg 27.8%

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

      2. distribute-rgt-neg-in 27.8%

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

      3. distribute-lft-neg-in 27.8%

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

      4. add-sqr-sqrt 13.9%

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

      5. sqrt-unprod 13.9%

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

      6. sqr-neg 13.9%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. associate-*r* 0.0%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 38.9%

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

      12. sqr-neg 38.9%

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

      13. sqrt-unprod 38.9%

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

      14. add-sqr-sqrt 72.2%

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

      15. associate-*r* 72.2%

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

   10. Applied egg-rr 72.2%

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

       1. associate-*l* 72.2%

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

       2. *-commutative 72.2%

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

       3. *-commutative 72.2%

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

       4. associate-*r* 72.2%

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

       5. distribute-rgt-out 72.2%

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

       6. count-2 72.2%

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

       7. associate-*r* 72.2%

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

       8. distribute-rgt1-in 72.2%

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

       9. metadata-eval 72.2%

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

   12. Simplified 72.2%

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

4. Final simplification 95.8%

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

Alternative 3: 96.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -8 \cdot 10^{+153}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \mathbf{elif}\;x.im \leq 4.8 \cdot 10^{+139}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re + x.im \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(-x.im\right)\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im -8e+153)
   (* x.im (* x.im (* x.re -3.0)))
   (if (<= x.im 4.8e+139)
     (* x.re (+ (* x.re x.re) (* x.im (* x.im -3.0))))
     (- (* x.im (* x.re (- x.im))) (* x.im (* x.re (+ x.im x.im)))))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= -8e+153) {
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0));
	} else if (x_46_im <= 4.8e+139) {
		tmp = x_46_re * ((x_46_re * x_46_re) + (x_46_im * (x_46_im * -3.0)));
	} else {
		tmp = (x_46_im * (x_46_re * -x_46_im)) - (x_46_im * (x_46_re * (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 <= (-8d+153)) then
        tmp = x_46im * (x_46im * (x_46re * (-3.0d0)))
    else if (x_46im <= 4.8d+139) then
        tmp = x_46re * ((x_46re * x_46re) + (x_46im * (x_46im * (-3.0d0))))
    else
        tmp = (x_46im * (x_46re * -x_46im)) - (x_46im * (x_46re * (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 <= -8e+153) {
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0));
	} else if (x_46_im <= 4.8e+139) {
		tmp = x_46_re * ((x_46_re * x_46_re) + (x_46_im * (x_46_im * -3.0)));
	} else {
		tmp = (x_46_im * (x_46_re * -x_46_im)) - (x_46_im * (x_46_re * (x_46_im + x_46_im)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= -8e+153:
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0))
	elif x_46_im <= 4.8e+139:
		tmp = x_46_re * ((x_46_re * x_46_re) + (x_46_im * (x_46_im * -3.0)))
	else:
		tmp = (x_46_im * (x_46_re * -x_46_im)) - (x_46_im * (x_46_re * (x_46_im + x_46_im)))
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= -8e+153)
		tmp = Float64(x_46_im * Float64(x_46_im * Float64(x_46_re * -3.0)));
	elseif (x_46_im <= 4.8e+139)
		tmp = Float64(x_46_re * Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * Float64(x_46_im * -3.0))));
	else
		tmp = Float64(Float64(x_46_im * Float64(x_46_re * Float64(-x_46_im))) - Float64(x_46_im * Float64(x_46_re * 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 <= -8e+153)
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0));
	elseif (x_46_im <= 4.8e+139)
		tmp = x_46_re * ((x_46_re * x_46_re) + (x_46_im * (x_46_im * -3.0)));
	else
		tmp = (x_46_im * (x_46_re * -x_46_im)) - (x_46_im * (x_46_re * (x_46_im + x_46_im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, -8e+153], N[(x$46$im * N[(x$46$im * N[(x$46$re * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 4.8e+139], N[(x$46$re * N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im * N[(x$46$re * (-x$46$im)), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(x$46$re * N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -8e153

   1. Initial program 50.1%

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

      1. *-commutative 50.1%

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

      2. *-commutative 50.1%

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

      3. *-commutative 50.1%

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

      4. distribute-lft-out 50.1%

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

   3. Simplified 50.1%

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

      1. add-log-exp 50.1%

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

      2. exp-prod 53.1%

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

   5. Applied egg-rr 53.1%

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

      1. log-pow 31.4%

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

      2. difference-of-squares 42.9%

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

      3. add-log-exp 61.5%

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

      4. associate-*l* 82.6%

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

   7. Applied egg-rr 82.6%

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

   8. Taylor expanded in x.re around 0 61.5%

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

      1. unpow2 61.5%

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

      2. unpow2 61.5%

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

      3. distribute-rgt-out-- 61.5%

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

      4. metadata-eval 61.5%

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

      5. *-commutative 61.5%

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

      6. associate-*l* 61.5%

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

      7. *-commutative 61.5%

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

      8. associate-*l* 82.8%

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

   10. Simplified 82.8%

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

   if -8e153 < x.im < 4.80000000000000016e139

   1. Initial program 91.3%

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

      1. *-commutative 91.3%

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

      2. distribute-lft-out 91.3%

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

      3. associate-*l* 91.3%

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

      4. *-commutative 91.3%

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

      5. distribute-rgt-out-- 99.7%

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

      6. associate--l- 99.7%

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

      7. associate--l- 99.7%

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

      8. sub-neg 99.7%

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

      9. associate--l+ 99.7%

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

      10. fma-udef 99.7%

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

      11. neg-mul-1 99.7%

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

      12. count-2 99.7%

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

      13. associate-*l* 99.7%

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

      14. distribute-rgt-out-- 99.7%

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

      15. associate-*r* 99.7%

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

      16. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. fma-udef 99.7%

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

   5. Applied egg-rr 99.7%

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

   if 4.80000000000000016e139 < x.im

   1. Initial program 53.3%

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

      1. *-commutative 53.3%

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

      2. *-commutative 53.3%

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

      3. *-commutative 53.3%

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

      4. distribute-lft-out 53.3%

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

   3. Simplified 53.3%

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

      1. add-log-exp 48.3%

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

      2. exp-prod 51.4%

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

   5. Applied egg-rr 51.4%

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

      1. log-pow 21.8%

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

      2. difference-of-squares 35.5%

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

      3. add-log-exp 67.0%

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

      4. associate-*l* 88.5%

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

   7. Applied egg-rr 88.5%

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

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

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

      1. mul-1-neg 67.0%

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

      2. unpow2 67.0%

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

      3. associate-*l* 88.5%

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

      4. distribute-rgt-neg-in 88.5%

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

   10. Simplified 88.5%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -8 \cdot 10^{+153}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \mathbf{elif}\;x.im \leq 4.8 \cdot 10^{+139}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re + x.im \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(-x.im\right)\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\\ \end{array} \]

Alternative 4: 96.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -8 \cdot 10^{+153} \lor \neg \left(x.im \leq 4.8 \cdot 10^{+139}\right):\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re + x.im \cdot \left(x.im \cdot -3\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -8e+153) (not (<= x.im 4.8e+139)))
   (* x.im (* x.im (* x.re -3.0)))
   (* x.re (+ (* x.re x.re) (* x.im (* x.im -3.0))))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -8e+153) || !(x_46_im <= 4.8e+139)) {
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0));
	} else {
		tmp = x_46_re * ((x_46_re * x_46_re) + (x_46_im * (x_46_im * -3.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46im <= (-8d+153)) .or. (.not. (x_46im <= 4.8d+139))) then
        tmp = x_46im * (x_46im * (x_46re * (-3.0d0)))
    else
        tmp = x_46re * ((x_46re * x_46re) + (x_46im * (x_46im * (-3.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -8e+153) || !(x_46_im <= 4.8e+139)) {
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0));
	} else {
		tmp = x_46_re * ((x_46_re * x_46_re) + (x_46_im * (x_46_im * -3.0)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_im <= -8e+153) or not (x_46_im <= 4.8e+139):
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0))
	else:
		tmp = x_46_re * ((x_46_re * x_46_re) + (x_46_im * (x_46_im * -3.0)))
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_im <= -8e+153) || !(x_46_im <= 4.8e+139))
		tmp = Float64(x_46_im * Float64(x_46_im * Float64(x_46_re * -3.0)));
	else
		tmp = Float64(x_46_re * Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * Float64(x_46_im * -3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_im <= -8e+153) || ~((x_46_im <= 4.8e+139)))
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0));
	else
		tmp = x_46_re * ((x_46_re * x_46_re) + (x_46_im * (x_46_im * -3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$im, -8e+153], N[Not[LessEqual[x$46$im, 4.8e+139]], $MachinePrecision]], N[(x$46$im * N[(x$46$im * N[(x$46$re * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re * N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -8e153 or 4.80000000000000016e139 < x.im

   1. Initial program 51.9%

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

      1. *-commutative 51.9%

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

      2. *-commutative 51.9%

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

      3. *-commutative 51.9%

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

      4. distribute-lft-out 51.9%

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

   3. Simplified 51.9%

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

      1. add-log-exp 49.1%

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

      2. exp-prod 52.1%

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

   5. Applied egg-rr 52.1%

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

      1. log-pow 26.1%

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

      2. difference-of-squares 38.7%

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

      3. add-log-exp 64.6%

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

      4. associate-*l* 85.9%

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

   7. Applied egg-rr 85.9%

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

   8. Taylor expanded in x.re around 0 63.4%

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

      1. unpow2 63.4%

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

      2. unpow2 63.4%

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

      3. distribute-rgt-out-- 63.4%

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

      4. metadata-eval 63.4%

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

      5. *-commutative 63.4%

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

      6. associate-*l* 64.6%

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

      7. *-commutative 64.6%

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

      8. associate-*l* 86.0%

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

   10. Simplified 86.0%

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

   if -8e153 < x.im < 4.80000000000000016e139

   1. Initial program 91.3%

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

      1. *-commutative 91.3%

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

      2. distribute-lft-out 91.3%

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

      3. associate-*l* 91.3%

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

      4. *-commutative 91.3%

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

      5. distribute-rgt-out-- 99.7%

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

      6. associate--l- 99.7%

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

      7. associate--l- 99.7%

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

      8. sub-neg 99.7%

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

      9. associate--l+ 99.7%

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

      10. fma-udef 99.7%

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

      11. neg-mul-1 99.7%

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

      12. count-2 99.7%

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

      13. associate-*l* 99.7%

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

      14. distribute-rgt-out-- 99.7%

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

      15. associate-*r* 99.7%

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

      16. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. fma-udef 99.7%

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

   5. Applied egg-rr 99.7%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -8 \cdot 10^{+153} \lor \neg \left(x.im \leq 4.8 \cdot 10^{+139}\right):\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re + x.im \cdot \left(x.im \cdot -3\right)\right)\\ \end{array} \]

Alternative 5: 61.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -1.75 \cdot 10^{+177} \lor \neg \left(x.re \leq 5.5 \cdot 10^{+149}\right):\\ \;\;\;\;x.re \cdot \left(\left(x.im \cdot x.im\right) \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.re -1.75e+177) (not (<= x.re 5.5e+149)))
   (* x.re (* (* x.im x.im) 3.0))
   (* x.im (* x.im (* x.re -3.0)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -1.75e+177) || !(x_46_re <= 5.5e+149)) {
		tmp = x_46_re * ((x_46_im * x_46_im) * 3.0);
	} else {
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0));
	}
	return tmp;
}

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 <= (-1.75d+177)) .or. (.not. (x_46re <= 5.5d+149))) then
        tmp = x_46re * ((x_46im * x_46im) * 3.0d0)
    else
        tmp = x_46im * (x_46im * (x_46re * (-3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -1.75e+177) || !(x_46_re <= 5.5e+149)) {
		tmp = x_46_re * ((x_46_im * x_46_im) * 3.0);
	} else {
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_re <= -1.75e+177) or not (x_46_re <= 5.5e+149):
		tmp = x_46_re * ((x_46_im * x_46_im) * 3.0)
	else:
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0))
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_re <= -1.75e+177) || !(x_46_re <= 5.5e+149))
		tmp = Float64(x_46_re * Float64(Float64(x_46_im * x_46_im) * 3.0));
	else
		tmp = Float64(x_46_im * Float64(x_46_im * Float64(x_46_re * -3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_re <= -1.75e+177) || ~((x_46_re <= 5.5e+149)))
		tmp = x_46_re * ((x_46_im * x_46_im) * 3.0);
	else
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$re, -1.75e+177], N[Not[LessEqual[x$46$re, 5.5e+149]], $MachinePrecision]], N[(x$46$re * N[(N[(x$46$im * x$46$im), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(x$46$im * N[(x$46$re * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < -1.74999999999999996e177 or 5.49999999999999999e149 < x.re

   1. Initial program 47.5%

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

      1. *-commutative 47.5%

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

      2. *-commutative 47.5%

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

      3. *-commutative 47.5%

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

      4. distribute-lft-out 47.5%

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

   3. Simplified 47.5%

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

      1. add-log-exp 47.5%

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

      2. exp-prod 47.5%

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

   5. Applied egg-rr 47.5%

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

   6. Taylor expanded in x.re around 0 11.9%

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

      1. associate-*r* 11.9%

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

      2. neg-mul-1 11.9%

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

      3. unpow2 11.9%

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

   8. Simplified 11.9%

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

      1. sub-neg 11.9%

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

      2. distribute-rgt-neg-in 11.9%

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

      3. distribute-lft-neg-in 11.9%

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

      4. add-sqr-sqrt 3.5%

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

      5. sqrt-unprod 29.6%

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

      6. sqr-neg 29.6%

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

      7. sqrt-unprod 0.9%

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

      8. add-sqr-sqrt 1.5%

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

      9. associate-*r* 1.5%

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

      10. add-sqr-sqrt 0.6%

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

      11. sqrt-unprod 49.3%

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

      12. sqr-neg 49.3%

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

      13. sqrt-unprod 23.8%

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

      14. add-sqr-sqrt 42.4%

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

      15. associate-*r* 42.4%

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

   10. Applied egg-rr 42.4%

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

       1. associate-*l* 42.4%

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

       2. *-commutative 42.4%

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

       3. *-commutative 42.4%

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

       4. associate-*r* 42.3%

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

       5. distribute-rgt-out 42.3%

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

       6. count-2 42.3%

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

       7. associate-*r* 42.3%

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

       8. distribute-rgt1-in 42.3%

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

       9. metadata-eval 42.3%

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

   12. Simplified 42.3%

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

   if -1.74999999999999996e177 < x.re < 5.49999999999999999e149

   1. Initial program 89.0%

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

      1. *-commutative 89.0%

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

      2. *-commutative 89.0%

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

      3. *-commutative 89.0%

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

      4. distribute-lft-out 89.0%

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

   3. Simplified 89.0%

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

      1. add-log-exp 44.1%

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

      2. exp-prod 46.2%

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

   5. Applied egg-rr 46.2%

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

      1. log-pow 37.0%

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

      2. difference-of-squares 38.5%

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

      3. add-log-exp 90.5%

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

      4. associate-*l* 99.1%

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

   7. Applied egg-rr 99.1%

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

   8. Taylor expanded in x.re around 0 57.9%

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

      1. unpow2 57.9%

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

      2. unpow2 57.9%

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

      3. distribute-rgt-out-- 57.9%

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

      4. metadata-eval 57.9%

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

      5. *-commutative 57.9%

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

      6. associate-*l* 58.5%

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

      7. *-commutative 58.5%

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

      8. associate-*l* 67.1%

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

   10. Simplified 67.1%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -1.75 \cdot 10^{+177} \lor \neg \left(x.re \leq 5.5 \cdot 10^{+149}\right):\\ \;\;\;\;x.re \cdot \left(\left(x.im \cdot x.im\right) \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \end{array} \]

Alternative 6: 61.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -1.55 \cdot 10^{+174}:\\ \;\;\;\;x.re \cdot \left(\left(x.im \cdot x.im\right) \cdot 3\right)\\ \mathbf{elif}\;x.re \leq 6.2 \cdot 10^{+148}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im \cdot x.im\right) \cdot \left(x.re \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.re -1.55e+174)
   (* x.re (* (* x.im x.im) 3.0))
   (if (<= x.re 6.2e+148)
     (* x.im (* x.im (* x.re -3.0)))
     (* (* x.im x.im) (* x.re 3.0)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= -1.55e+174) {
		tmp = x_46_re * ((x_46_im * x_46_im) * 3.0);
	} else if (x_46_re <= 6.2e+148) {
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0));
	} else {
		tmp = (x_46_im * x_46_im) * (x_46_re * 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re <= (-1.55d+174)) then
        tmp = x_46re * ((x_46im * x_46im) * 3.0d0)
    else if (x_46re <= 6.2d+148) then
        tmp = x_46im * (x_46im * (x_46re * (-3.0d0)))
    else
        tmp = (x_46im * x_46im) * (x_46re * 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= -1.55e+174) {
		tmp = x_46_re * ((x_46_im * x_46_im) * 3.0);
	} else if (x_46_re <= 6.2e+148) {
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0));
	} else {
		tmp = (x_46_im * x_46_im) * (x_46_re * 3.0);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_re <= -1.55e+174:
		tmp = x_46_re * ((x_46_im * x_46_im) * 3.0)
	elif x_46_re <= 6.2e+148:
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0))
	else:
		tmp = (x_46_im * x_46_im) * (x_46_re * 3.0)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_re <= -1.55e+174)
		tmp = Float64(x_46_re * Float64(Float64(x_46_im * x_46_im) * 3.0));
	elseif (x_46_re <= 6.2e+148)
		tmp = Float64(x_46_im * Float64(x_46_im * Float64(x_46_re * -3.0)));
	else
		tmp = Float64(Float64(x_46_im * x_46_im) * Float64(x_46_re * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_re <= -1.55e+174)
		tmp = x_46_re * ((x_46_im * x_46_im) * 3.0);
	elseif (x_46_re <= 6.2e+148)
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0));
	else
		tmp = (x_46_im * x_46_im) * (x_46_re * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$re, -1.55e+174], N[(x$46$re * N[(N[(x$46$im * x$46$im), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 6.2e+148], N[(x$46$im * N[(x$46$im * N[(x$46$re * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im * x$46$im), $MachinePrecision] * N[(x$46$re * 3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -1.55e174

   1. Initial program 45.8%

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

      1. *-commutative 45.8%

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

      2. *-commutative 45.8%

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

      3. *-commutative 45.8%

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

      4. distribute-lft-out 45.8%

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

   3. Simplified 45.8%

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

      1. add-log-exp 45.8%

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

      2. exp-prod 45.8%

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

   5. Applied egg-rr 45.8%

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

   6. Taylor expanded in x.re around 0 8.8%

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

      1. associate-*r* 8.8%

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

      2. neg-mul-1 8.8%

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

      3. unpow2 8.8%

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

   8. Simplified 8.8%

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

      1. sub-neg 8.8%

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

      2. distribute-rgt-neg-in 8.8%

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

      3. distribute-lft-neg-in 8.8%

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

      4. add-sqr-sqrt 8.8%

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

      5. sqrt-unprod 8.3%

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

      6. sqr-neg 8.3%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 1.4%

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

      9. associate-*r* 1.4%

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

      10. add-sqr-sqrt 1.4%

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

      11. sqrt-unprod 0.0%

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

      12. sqr-neg 0.0%

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

      13. sqrt-unprod 0.0%

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

      14. add-sqr-sqrt 47.3%

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

      15. associate-*r* 47.3%

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

   10. Applied egg-rr 47.3%

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

       1. associate-*l* 47.3%

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

       2. *-commutative 47.3%

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

       3. *-commutative 47.3%

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

       4. associate-*r* 47.2%

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

       5. distribute-rgt-out 47.2%

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

       6. count-2 47.2%

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

       7. associate-*r* 47.2%

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

       8. distribute-rgt1-in 47.2%

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

       9. metadata-eval 47.2%

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

   12. Simplified 47.2%

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

   if -1.55e174 < x.re < 6.19999999999999951e148

   1. Initial program 89.0%

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

      1. *-commutative 89.0%

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

      2. *-commutative 89.0%

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

      3. *-commutative 89.0%

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

      4. distribute-lft-out 89.0%

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

   3. Simplified 89.0%

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

      1. add-log-exp 44.1%

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

      2. exp-prod 46.2%

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

   5. Applied egg-rr 46.2%

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

      1. log-pow 37.0%

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

      2. difference-of-squares 38.5%

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

      3. add-log-exp 90.5%

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

      4. associate-*l* 99.1%

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

   7. Applied egg-rr 99.1%

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

   8. Taylor expanded in x.re around 0 57.9%

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

      1. unpow2 57.9%

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

      2. unpow2 57.9%

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

      3. distribute-rgt-out-- 57.9%

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

      4. metadata-eval 57.9%

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

      5. *-commutative 57.9%

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

      6. associate-*l* 58.5%

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

      7. *-commutative 58.5%

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

      8. associate-*l* 67.1%

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

   10. Simplified 67.1%

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

   if 6.19999999999999951e148 < x.re

   1. Initial program 48.6%

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

      1. *-commutative 48.6%

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

      2. *-commutative 48.6%

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

      3. *-commutative 48.6%

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

      4. distribute-lft-out 48.6%

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

   3. Simplified 48.6%

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

      1. add-log-exp 48.6%

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

      2. exp-prod 48.6%

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

   5. Applied egg-rr 48.6%

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

   6. Taylor expanded in x.re around 0 13.9%

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

      1. associate-*r* 13.9%

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

      2. neg-mul-1 13.9%

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

      3. unpow2 13.9%

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

   8. Simplified 13.9%

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

      1. sub-neg 13.9%

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

      2. distribute-rgt-neg-in 13.9%

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

      3. distribute-lft-neg-in 13.9%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 43.4%

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

      6. sqr-neg 43.4%

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

      7. sqrt-unprod 1.5%

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

      8. add-sqr-sqrt 1.5%

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

      9. associate-*r* 1.5%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 81.2%

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

      12. sqr-neg 81.2%

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

      13. sqrt-unprod 39.3%

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

      14. add-sqr-sqrt 39.3%

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

      15. associate-*r* 39.3%

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

   10. Applied egg-rr 39.3%

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

       1. associate-*l* 39.3%

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

       2. *-commutative 39.3%

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

       3. *-commutative 39.3%

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

       4. associate-*r* 39.2%

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

       5. distribute-rgt-out 39.2%

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

       6. count-2 39.2%

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

       7. associate-*r* 39.2%

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

       8. distribute-rgt1-in 39.2%

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

       9. metadata-eval 39.2%

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

   12. Simplified 39.2%

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

   13. Taylor expanded in x.re around 0 39.2%

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

       1. associate-*r* 41.8%

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

       2. unpow2 41.8%

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

   15. Simplified 41.8%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -1.55 \cdot 10^{+174}:\\ \;\;\;\;x.re \cdot \left(\left(x.im \cdot x.im\right) \cdot 3\right)\\ \mathbf{elif}\;x.re \leq 6.2 \cdot 10^{+148}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im \cdot x.im\right) \cdot \left(x.re \cdot 3\right)\\ \end{array} \]

Alternative 7: 50.6% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

   1. *-commutative 79.1%

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

   2. distribute-lft-out 79.1%

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

   3. associate-*l* 79.1%

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

   4. *-commutative 79.1%

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

   5. distribute-rgt-out-- 84.6%

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

   6. associate--l- 84.6%

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

   7. associate--l- 84.6%

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

   8. sub-neg 84.6%

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

   9. associate--l+ 84.6%

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

   10. fma-udef 88.5%

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

   11. neg-mul-1 88.5%

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

   12. count-2 88.5%

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

   13. associate-*l* 88.5%

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

   14. distribute-rgt-out-- 88.5%

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

   15. associate-*r* 88.5%

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

   16. metadata-eval 88.5%

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

3. Simplified 88.5%

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

4. Taylor expanded in x.re around 0 47.4%

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

6. Simplified 47.4%

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

7. Final simplification 47.4%

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

Alternative 8: 56.4% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

   1. *-commutative 79.1%

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

   2. *-commutative 79.1%

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

   3. *-commutative 79.1%

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

   4. distribute-lft-out 79.1%

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

3. Simplified 79.1%

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

   1. add-log-exp 44.9%

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

   2. exp-prod 46.5%

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

5. Applied egg-rr 46.5%

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

   1. log-pow 39.5%

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

   2. difference-of-squares 43.4%

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

   3. add-log-exp 83.0%

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

   4. associate-*l* 89.6%

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

7. Applied egg-rr 89.6%

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

8. Taylor expanded in x.re around 0 47.0%

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

   1. unpow2 47.0%

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

   2. unpow2 47.0%

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

   3. distribute-rgt-out-- 47.0%

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

   4. metadata-eval 47.0%

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

   5. *-commutative 47.0%

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

   6. associate-*l* 47.4%

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

   7. *-commutative 47.4%

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

   8. associate-*l* 53.9%

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

10. Simplified 53.9%

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

11. Taylor expanded in x.im around 0 53.9%

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

12. Final simplification 53.9%

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

Alternative 9: 56.4% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

   1. *-commutative 79.1%

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

   2. *-commutative 79.1%

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

   3. *-commutative 79.1%

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

   4. distribute-lft-out 79.1%

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

3. Simplified 79.1%

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

   1. add-log-exp 44.9%

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

   2. exp-prod 46.5%

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

5. Applied egg-rr 46.5%

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

   1. log-pow 39.5%

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

   2. difference-of-squares 43.4%

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

   3. add-log-exp 83.0%

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

   4. associate-*l* 89.6%

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

7. Applied egg-rr 89.6%

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

8. Taylor expanded in x.re around 0 47.0%

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

   1. unpow2 47.0%

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

   2. unpow2 47.0%

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

   3. distribute-rgt-out-- 47.0%

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

   4. metadata-eval 47.0%

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

   5. *-commutative 47.0%

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

   6. associate-*l* 47.4%

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

   7. *-commutative 47.4%

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

   8. associate-*l* 53.9%

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

10. Simplified 53.9%

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

11. Final simplification 53.9%

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

Developer target: 88.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023196 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64

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

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