math.cube on complex, real part

Percentage Accurate: 82.7% → 96.8%

Time: 8.6s

Alternatives: 15

Speedup: 1.0×

• 44.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 82.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.8% accurate, 0.1× speedup

Math

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

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)
   (fma
    (- 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 = fma((x_46_re - x_46_im), (x_46_re * (x_46_re + x_46_im)), (-x_46_im * (x_46_re * (x_46_im + x_46_im))));
	} else {
		tmp = 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 = fma(Float64(x_46_re - x_46_im), Float64(x_46_re * Float64(x_46_re + x_46_im)), Float64(Float64(-x_46_im) * Float64(x_46_re * Float64(x_46_im + x_46_im))));
	else
		tmp = x_46_re ^ 3.0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[N[(N[(x$46$re * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(x$46$re - x$46$im), $MachinePrecision] * N[(x$46$re * N[(x$46$re + x$46$im), $MachinePrecision]), $MachinePrecision] + N[((-x$46$im) * N[(x$46$re * N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[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 91.9%

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

      1. sqr-neg 91.9%

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

      2. difference-of-squares 91.9%

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

      3. sub-neg 91.9%

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

      4. associate-*l* 99.8%

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

      5. sub-neg 99.8%

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

      6. remove-double-neg 99.8%

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

      7. +-commutative 99.8%

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

      8. *-commutative 99.8%

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

      9. *-commutative 99.8%

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

      10. distribute-rgt-out 99.8%

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

   3. Simplified 99.8%

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

      1. cancel-sign-sub-inv 99.8%

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

      2. fma-def 99.8%

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

      3. *-commutative 99.8%

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

   5. Applied egg-rr 99.8%

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

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

   1. Initial program 0.0%

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

      1. sqr-neg 0.0%

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

      2. difference-of-squares 22.9%

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

      3. sub-neg 22.9%

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

      4. associate-*l* 22.9%

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

      5. sub-neg 22.9%

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

      6. remove-double-neg 22.9%

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

      7. +-commutative 22.9%

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

      8. *-commutative 22.9%

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

      9. *-commutative 22.9%

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

      10. distribute-rgt-out 22.9%

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

   3. Simplified 22.9%

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

   4. Taylor expanded in x.re around inf 77.1%

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

4. Final simplification 96.7%

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

Alternative 2: 96.8% 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 91.9%

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

      1. sqr-neg 91.9%

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

      2. difference-of-squares 91.9%

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

      3. sub-neg 91.9%

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

      4. associate-*l* 99.8%

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

      5. sub-neg 99.8%

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

      6. remove-double-neg 99.8%

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

      7. +-commutative 99.8%

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

      8. *-commutative 99.8%

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

      9. *-commutative 99.8%

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

      10. distribute-rgt-out 99.8%

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

   3. Simplified 99.8%

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

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

   1. Initial program 0.0%

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

      1. sqr-neg 0.0%

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

      2. difference-of-squares 22.9%

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

      3. sub-neg 22.9%

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

      4. associate-*l* 22.9%

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

      5. sub-neg 22.9%

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

      6. remove-double-neg 22.9%

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

      7. +-commutative 22.9%

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

      8. *-commutative 22.9%

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

      9. *-commutative 22.9%

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

      10. distribute-rgt-out 22.9%

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

   3. Simplified 22.9%

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

   4. Taylor expanded in x.re around inf 77.1%

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

4. Final simplification 96.7%

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

Alternative 3: 96.8% accurate, 0.5× speedup

Math

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

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))))

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);
	}
	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);
	}
	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)
	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(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_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);
	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[(x$46$im * x$46$im), $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 91.9%

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

      1. sqr-neg 91.9%

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

      2. difference-of-squares 91.9%

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

      3. sub-neg 91.9%

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

      4. associate-*l* 99.8%

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

      5. sub-neg 99.8%

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

      6. remove-double-neg 99.8%

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

      7. +-commutative 99.8%

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

      8. *-commutative 99.8%

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

      9. *-commutative 99.8%

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

      10. distribute-rgt-out 99.8%

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

   3. Simplified 99.8%

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

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

   1. Initial program 0.0%

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

      1. sqr-neg 0.0%

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

      2. difference-of-squares 22.9%

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

      3. sub-neg 22.9%

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

      4. associate-*l* 22.9%

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

      5. sub-neg 22.9%

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

      6. remove-double-neg 22.9%

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

      7. +-commutative 22.9%

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

      8. *-commutative 22.9%

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

      9. *-commutative 22.9%

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

      10. distribute-rgt-out 22.9%

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

   3. Simplified 22.9%

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

   4. Taylor expanded in x.im around 0 22.9%

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

      1. associate-*r* 22.9%

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

      2. *-commutative 22.9%

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

      3. *-commutative 22.9%

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

      4. unpow2 22.9%

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

      5. associate-*l* 22.9%

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

      6. *-commutative 22.9%

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

      7. count-2 22.9%

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

   6. Simplified 22.9%

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

   7. Taylor expanded in x.re around 0 22.9%

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

      1. associate-*r* 22.9%

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

      2. neg-mul-1 22.9%

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

      3. unpow2 22.9%

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

      4. distribute-rgt-neg-in 22.9%

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

   9. Simplified 22.9%

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

   11. Applied egg-rr 77.1%

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

4. Final simplification 96.7%

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

Alternative 4: 64.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -2.3 \cdot 10^{+144}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.im - x.re\right)\right)\\ \mathbf{elif}\;x.re \leq 2.8 \cdot 10^{+153}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.re - x.im\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.im\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.re -2.3e+144)
   (* x.im (* x.re (- x.im x.re)))
   (if (<= x.re 2.8e+153)
     (- (* (* x.re x.im) (- x.re x.im)) (* x.im (* x.re (+ x.im x.im))))
     (* (* x.re x.re) x.im))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= -2.3e+144) {
		tmp = x_46_im * (x_46_re * (x_46_im - x_46_re));
	} else if (x_46_re <= 2.8e+153) {
		tmp = ((x_46_re * x_46_im) * (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_re) * x_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re <= (-2.3d+144)) then
        tmp = x_46im * (x_46re * (x_46im - x_46re))
    else if (x_46re <= 2.8d+153) then
        tmp = ((x_46re * x_46im) * (x_46re - x_46im)) - (x_46im * (x_46re * (x_46im + x_46im)))
    else
        tmp = (x_46re * x_46re) * x_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= -2.3e+144) {
		tmp = x_46_im * (x_46_re * (x_46_im - x_46_re));
	} else if (x_46_re <= 2.8e+153) {
		tmp = ((x_46_re * x_46_im) * (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_re) * x_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_re <= -2.3e+144:
		tmp = x_46_im * (x_46_re * (x_46_im - x_46_re))
	elif x_46_re <= 2.8e+153:
		tmp = ((x_46_re * x_46_im) * (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_re) * x_46_im
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_re <= -2.3e+144)
		tmp = Float64(x_46_im * Float64(x_46_re * Float64(x_46_im - x_46_re)));
	elseif (x_46_re <= 2.8e+153)
		tmp = Float64(Float64(Float64(x_46_re * x_46_im) * 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(Float64(x_46_re * x_46_re) * x_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_re <= -2.3e+144)
		tmp = x_46_im * (x_46_re * (x_46_im - x_46_re));
	elseif (x_46_re <= 2.8e+153)
		tmp = ((x_46_re * x_46_im) * (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_re) * x_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$re, -2.3e+144], N[(x$46$im * N[(x$46$re * N[(x$46$im - x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 2.8e+153], N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(x$46$re * N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * x$46$re), $MachinePrecision] * x$46$im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -2.3000000000000001e144

   1. Initial program 42.5%

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

      1. sqr-neg 42.5%

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

      2. difference-of-squares 62.5%

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

      3. sub-neg 62.5%

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

      4. associate-*l* 62.5%

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

      5. sub-neg 62.5%

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

      6. remove-double-neg 62.5%

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

      7. +-commutative 62.5%

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

      8. *-commutative 62.5%

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

      9. *-commutative 62.5%

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

      10. distribute-rgt-out 62.5%

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

   3. Simplified 62.5%

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

      1. cancel-sign-sub-inv 62.5%

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

      2. fma-def 62.5%

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

      3. *-commutative 62.5%

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

   5. Applied egg-rr 62.5%

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

   6. Taylor expanded in x.re around 0 30.7%

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

      1. *-commutative 30.7%

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

   8. Simplified 30.7%

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

   10. Applied egg-rr 38.3%

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

       1. +-lft-identity 38.3%

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

       2. *-commutative 38.3%

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

       3. associate-*l* 50.1%

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

   12. Simplified 50.1%

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

   if -2.3000000000000001e144 < x.re < 2.79999999999999985e153

   1. Initial program 90.5%

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

      1. sqr-neg 90.5%

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

      2. difference-of-squares 90.5%

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

      3. sub-neg 90.5%

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

      4. associate-*l* 99.7%

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

      5. sub-neg 99.7%

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

      6. remove-double-neg 99.7%

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

      7. +-commutative 99.7%

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

      8. *-commutative 99.7%

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

      9. *-commutative 99.7%

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

      10. distribute-rgt-out 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x.re around 0 75.8%

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

      1. *-commutative 75.8%

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

   6. Simplified 75.8%

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

   if 2.79999999999999985e153 < x.re

   1. Initial program 58.6%

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

      1. sqr-neg 58.6%

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

      2. difference-of-squares 58.6%

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

      3. sub-neg 58.6%

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

      4. associate-*l* 58.6%

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

      5. sub-neg 58.6%

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

      6. remove-double-neg 58.6%

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

      7. +-commutative 58.6%

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

      8. *-commutative 58.6%

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

      9. *-commutative 58.6%

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

      10. distribute-rgt-out 58.6%

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

   3. Simplified 58.6%

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

   4. Taylor expanded in x.re around 0 15.4%

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

      1. *-commutative 15.4%

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

   6. Simplified 15.4%

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

   7. Taylor expanded in x.re around inf 69.0%

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

      1. unpow2 69.0%

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

   9. Simplified 69.0%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -2.3 \cdot 10^{+144}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.im - x.re\right)\right)\\ \mathbf{elif}\;x.re \leq 2.8 \cdot 10^{+153}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.re - x.im\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.im\\ \end{array} \]

Alternative 5: 88.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq 1.5 \cdot 10^{+35}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.im \cdot \left(x.im \cdot \left(x.re \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.re - x.im \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im 1.5e+35)
   (+ (* x.re (- (* x.re x.re) (* x.im x.im))) (* x.im (* x.im (* x.re -2.0))))
   (* (* x.re x.im) (- x.re (* x.im 3.0)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 1.5e+35) {
		tmp = (x_46_re * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_im * (x_46_im * (x_46_re * -2.0)));
	} else {
		tmp = (x_46_re * x_46_im) * (x_46_re - (x_46_im * 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46im <= 1.5d+35) then
        tmp = (x_46re * ((x_46re * x_46re) - (x_46im * x_46im))) + (x_46im * (x_46im * (x_46re * (-2.0d0))))
    else
        tmp = (x_46re * x_46im) * (x_46re - (x_46im * 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 1.5e+35) {
		tmp = (x_46_re * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_im * (x_46_im * (x_46_re * -2.0)));
	} else {
		tmp = (x_46_re * x_46_im) * (x_46_re - (x_46_im * 3.0));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= 1.5e+35:
		tmp = (x_46_re * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_im * (x_46_im * (x_46_re * -2.0)))
	else:
		tmp = (x_46_re * x_46_im) * (x_46_re - (x_46_im * 3.0))
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= 1.5e+35)
		tmp = 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(x_46_im * Float64(x_46_re * -2.0))));
	else
		tmp = Float64(Float64(x_46_re * x_46_im) * Float64(x_46_re - 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 <= 1.5e+35)
		tmp = (x_46_re * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_im * (x_46_im * (x_46_re * -2.0)));
	else
		tmp = (x_46_re * x_46_im) * (x_46_re - (x_46_im * 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, 1.5e+35], 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[(x$46$im * N[(x$46$re * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(x$46$re - N[(x$46$im * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.49999999999999995e35

   1. Initial program 89.5%

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

      1. sqr-neg 89.5%

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

      2. *-commutative 89.5%

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

      3. fma-neg 89.5%

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

      4. sqr-neg 89.5%

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

      5. +-commutative 89.5%

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

      6. *-commutative 89.5%

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

      7. *-commutative 89.5%

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

      8. distribute-lft-out 89.5%

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

      9. associate-*r* 89.5%

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

      10. distribute-rgt-neg-in 89.5%

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

      11. distribute-neg-out 89.5%

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

      12. neg-mul-1 89.5%

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

      13. neg-mul-1 89.5%

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

      14. distribute-rgt-out 89.5%

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

      15. metadata-eval 89.5%

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

   3. Simplified 89.5%

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

      1. fma-udef 89.5%

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

      2. associate-*l* 89.5%

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

   5. Applied egg-rr 89.5%

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

   if 1.49999999999999995e35 < x.im

   1. Initial program 52.4%

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

      1. sqr-neg 52.4%

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

      2. difference-of-squares 56.7%

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

      3. sub-neg 56.7%

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

      4. associate-*l* 71.2%

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

      5. sub-neg 71.2%

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

      6. remove-double-neg 71.2%

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

      7. +-commutative 71.2%

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

      8. *-commutative 71.2%

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

      9. *-commutative 71.2%

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

      10. distribute-rgt-out 71.2%

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

   3. Simplified 71.2%

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

      1. cancel-sign-sub-inv 71.2%

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

      2. fma-def 71.3%

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

      3. *-commutative 71.3%

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

   5. Applied egg-rr 71.3%

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

   6. Taylor expanded in x.re around 0 69.8%

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

      1. *-commutative 69.8%

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

   8. Simplified 69.8%

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

   9. Taylor expanded in x.re around 0 52.4%

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

       1. unpow2 52.4%

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

       2. associate-*r* 52.4%

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

       3. *-commutative 52.4%

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

       4. *-commutative 52.4%

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

       5. distribute-rgt-out 58.1%

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

       6. unpow2 58.1%

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

       7. unpow2 58.1%

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

       8. distribute-rgt-out 58.1%

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

       9. metadata-eval 58.1%

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

       10. metadata-eval 58.1%

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

       11. distribute-rgt-neg-in 58.1%

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

       12. associate-*r* 58.1%

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

       13. unsub-neg 58.1%

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

       14. distribute-lft-out-- 52.4%

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

       15. *-commutative 52.4%

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

       16. associate-*r* 66.9%

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

       17. distribute-lft-out-- 82.6%

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

   11. Simplified 82.6%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 1.5 \cdot 10^{+35}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.im \cdot \left(x.im \cdot \left(x.re \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.re - x.im \cdot 3\right)\\ \end{array} \]

Alternative 6: 78.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq 6.5 \cdot 10^{-31}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot \left(x.re - x.im\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.re - x.im \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im 6.5e-31)
   (- (* (* x.re x.re) (- x.re x.im)) (* x.im (* x.re (+ x.im x.im))))
   (* (* x.re x.im) (- x.re (* x.im 3.0)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 6.5e-31) {
		tmp = ((x_46_re * 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_re - (x_46_im * 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46im <= 6.5d-31) then
        tmp = ((x_46re * x_46re) * (x_46re - x_46im)) - (x_46im * (x_46re * (x_46im + x_46im)))
    else
        tmp = (x_46re * x_46im) * (x_46re - (x_46im * 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 6.5e-31) {
		tmp = ((x_46_re * 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_re - (x_46_im * 3.0));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= 6.5e-31:
		tmp = ((x_46_re * 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_re - (x_46_im * 3.0))
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= 6.5e-31)
		tmp = Float64(Float64(Float64(x_46_re * 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(Float64(x_46_re * x_46_im) * Float64(x_46_re - 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 <= 6.5e-31)
		tmp = ((x_46_re * 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_re - (x_46_im * 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, 6.5e-31], N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(x$46$re * N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(x$46$re - N[(x$46$im * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 6.49999999999999967e-31

   1. Initial program 89.1%

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

      1. sqr-neg 89.1%

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

      2. difference-of-squares 91.9%

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

      3. sub-neg 91.9%

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

      4. associate-*l* 95.9%

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

      5. sub-neg 95.9%

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

      6. remove-double-neg 95.9%

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

      7. +-commutative 95.9%

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

      8. *-commutative 95.9%

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

      9. *-commutative 95.9%

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

      10. distribute-rgt-out 95.9%

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

   3. Simplified 95.9%

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

   4. Taylor expanded in x.re around inf 76.8%

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

      1. unpow2 76.8%

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

   6. Simplified 76.8%

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

   if 6.49999999999999967e-31 < x.im

   1. Initial program 56.7%

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

      1. sqr-neg 56.7%

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

      2. difference-of-squares 60.6%

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

      3. sub-neg 60.6%

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

      4. associate-*l* 73.8%

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

      5. sub-neg 73.8%

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

      6. remove-double-neg 73.8%

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

      7. +-commutative 73.8%

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

      8. *-commutative 73.8%

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

      9. *-commutative 73.8%

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

      10. distribute-rgt-out 73.8%

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

   3. Simplified 73.8%

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

      1. cancel-sign-sub-inv 73.8%

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

      2. fma-def 73.9%

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

      3. *-commutative 73.9%

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

   5. Applied egg-rr 73.9%

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

   6. Taylor expanded in x.re around 0 70.2%

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

      1. *-commutative 70.2%

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

   8. Simplified 70.2%

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

   9. Taylor expanded in x.re around 0 54.3%

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

       1. unpow2 54.3%

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

       2. associate-*r* 54.3%

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

       3. *-commutative 54.3%

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

       4. *-commutative 54.3%

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

       5. distribute-rgt-out 59.5%

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

       6. unpow2 59.5%

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

       7. unpow2 59.5%

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

       8. distribute-rgt-out 59.5%

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

       9. metadata-eval 59.5%

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

       10. metadata-eval 59.5%

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

       11. distribute-rgt-neg-in 59.5%

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

       12. associate-*r* 59.4%

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

       13. unsub-neg 59.4%

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

       14. distribute-lft-out-- 54.3%

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

       15. *-commutative 54.3%

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

       16. associate-*r* 67.5%

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

       17. distribute-lft-out-- 81.8%

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

   11. Simplified 81.8%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 6.5 \cdot 10^{-31}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot \left(x.re - x.im\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.re - x.im \cdot 3\right)\\ \end{array} \]

Alternative 7: 64.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -6.6 \cdot 10^{+145}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.im - x.re\right)\right)\\ \mathbf{elif}\;x.re \leq 1.5 \cdot 10^{+154}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.re - x.im \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.im\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.re -6.6e+145)
   (* x.im (* x.re (- x.im x.re)))
   (if (<= x.re 1.5e+154)
     (* (* x.re x.im) (- x.re (* x.im 3.0)))
     (* (* x.re x.re) x.im))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= -6.6e+145) {
		tmp = x_46_im * (x_46_re * (x_46_im - x_46_re));
	} else if (x_46_re <= 1.5e+154) {
		tmp = (x_46_re * x_46_im) * (x_46_re - (x_46_im * 3.0));
	} else {
		tmp = (x_46_re * x_46_re) * x_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re <= (-6.6d+145)) then
        tmp = x_46im * (x_46re * (x_46im - x_46re))
    else if (x_46re <= 1.5d+154) then
        tmp = (x_46re * x_46im) * (x_46re - (x_46im * 3.0d0))
    else
        tmp = (x_46re * x_46re) * x_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= -6.6e+145) {
		tmp = x_46_im * (x_46_re * (x_46_im - x_46_re));
	} else if (x_46_re <= 1.5e+154) {
		tmp = (x_46_re * x_46_im) * (x_46_re - (x_46_im * 3.0));
	} else {
		tmp = (x_46_re * x_46_re) * x_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_re <= -6.6e+145:
		tmp = x_46_im * (x_46_re * (x_46_im - x_46_re))
	elif x_46_re <= 1.5e+154:
		tmp = (x_46_re * x_46_im) * (x_46_re - (x_46_im * 3.0))
	else:
		tmp = (x_46_re * x_46_re) * x_46_im
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_re <= -6.6e+145)
		tmp = Float64(x_46_im * Float64(x_46_re * Float64(x_46_im - x_46_re)));
	elseif (x_46_re <= 1.5e+154)
		tmp = Float64(Float64(x_46_re * x_46_im) * Float64(x_46_re - Float64(x_46_im * 3.0)));
	else
		tmp = Float64(Float64(x_46_re * x_46_re) * x_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_re <= -6.6e+145)
		tmp = x_46_im * (x_46_re * (x_46_im - x_46_re));
	elseif (x_46_re <= 1.5e+154)
		tmp = (x_46_re * x_46_im) * (x_46_re - (x_46_im * 3.0));
	else
		tmp = (x_46_re * x_46_re) * x_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$re, -6.6e+145], N[(x$46$im * N[(x$46$re * N[(x$46$im - x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 1.5e+154], N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(x$46$re - N[(x$46$im * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * x$46$re), $MachinePrecision] * x$46$im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -6.60000000000000054e145

   1. Initial program 42.5%

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

      1. sqr-neg 42.5%

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

      2. difference-of-squares 62.5%

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

      3. sub-neg 62.5%

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

      4. associate-*l* 62.5%

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

      5. sub-neg 62.5%

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

      6. remove-double-neg 62.5%

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

      7. +-commutative 62.5%

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

      8. *-commutative 62.5%

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

      9. *-commutative 62.5%

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

      10. distribute-rgt-out 62.5%

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

   3. Simplified 62.5%

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

      1. cancel-sign-sub-inv 62.5%

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

      2. fma-def 62.5%

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

      3. *-commutative 62.5%

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

   5. Applied egg-rr 62.5%

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

   6. Taylor expanded in x.re around 0 30.7%

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

      1. *-commutative 30.7%

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

   8. Simplified 30.7%

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

   10. Applied egg-rr 38.3%

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

       1. +-lft-identity 38.3%

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

       2. *-commutative 38.3%

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

       3. associate-*l* 50.1%

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

   12. Simplified 50.1%

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

   if -6.60000000000000054e145 < x.re < 1.50000000000000013e154

   1. Initial program 90.5%

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

      1. sqr-neg 90.5%

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

      2. difference-of-squares 90.5%

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

      3. sub-neg 90.5%

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

      4. associate-*l* 99.7%

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

      5. sub-neg 99.7%

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

      6. remove-double-neg 99.7%

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

      7. +-commutative 99.7%

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

      8. *-commutative 99.7%

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

      9. *-commutative 99.7%

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

      10. distribute-rgt-out 99.7%

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

   3. Simplified 99.7%

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

      1. cancel-sign-sub-inv 99.7%

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

      2. fma-def 99.8%

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

      3. *-commutative 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in x.re around 0 75.9%

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

      1. *-commutative 75.9%

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

   8. Simplified 75.9%

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

   9. Taylor expanded in x.re around 0 63.3%

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

       1. unpow2 63.3%

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

       2. associate-*r* 63.3%

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

       3. *-commutative 63.3%

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

       4. *-commutative 63.3%

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

       5. distribute-rgt-out 64.4%

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

       6. unpow2 64.4%

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

       7. unpow2 64.4%

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

       8. distribute-rgt-out 64.4%

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

       9. metadata-eval 64.4%

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

       10. metadata-eval 64.4%

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

       11. distribute-rgt-neg-in 64.4%

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

       12. associate-*r* 64.3%

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

       13. unsub-neg 64.3%

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

       14. distribute-lft-out-- 63.3%

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

       15. *-commutative 63.3%

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

       16. associate-*r* 72.6%

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

       17. distribute-lft-out-- 75.8%

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

   11. Simplified 75.8%

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

   if 1.50000000000000013e154 < x.re

   1. Initial program 58.6%

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

      1. sqr-neg 58.6%

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

      2. difference-of-squares 58.6%

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

      3. sub-neg 58.6%

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

      4. associate-*l* 58.6%

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

      5. sub-neg 58.6%

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

      6. remove-double-neg 58.6%

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

      7. +-commutative 58.6%

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

      8. *-commutative 58.6%

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

      9. *-commutative 58.6%

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

      10. distribute-rgt-out 58.6%

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

   3. Simplified 58.6%

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

   4. Taylor expanded in x.re around 0 15.4%

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

      1. *-commutative 15.4%

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

   6. Simplified 15.4%

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

   7. Taylor expanded in x.re around inf 69.0%

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

      1. unpow2 69.0%

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

   9. Simplified 69.0%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -6.6 \cdot 10^{+145}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.im - x.re\right)\right)\\ \mathbf{elif}\;x.re \leq 1.5 \cdot 10^{+154}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.re - x.im \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.im\\ \end{array} \]

Alternative 8: 62.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -1.05 \cdot 10^{+202}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.im\right)\\ \mathbf{elif}\;x.re \leq 6.2 \cdot 10^{+141}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.im\right) \cdot -3\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.im\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.re -1.05e+202)
   (* x.im (* x.re x.im))
   (if (<= x.re 6.2e+141)
     (* x.im (* (* x.re x.im) -3.0))
     (* (* x.re x.re) x.im))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= -1.05e+202) {
		tmp = x_46_im * (x_46_re * x_46_im);
	} else if (x_46_re <= 6.2e+141) {
		tmp = x_46_im * ((x_46_re * x_46_im) * -3.0);
	} else {
		tmp = (x_46_re * x_46_re) * x_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re <= (-1.05d+202)) then
        tmp = x_46im * (x_46re * x_46im)
    else if (x_46re <= 6.2d+141) then
        tmp = x_46im * ((x_46re * x_46im) * (-3.0d0))
    else
        tmp = (x_46re * x_46re) * x_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= -1.05e+202) {
		tmp = x_46_im * (x_46_re * x_46_im);
	} else if (x_46_re <= 6.2e+141) {
		tmp = x_46_im * ((x_46_re * x_46_im) * -3.0);
	} else {
		tmp = (x_46_re * x_46_re) * x_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_re <= -1.05e+202:
		tmp = x_46_im * (x_46_re * x_46_im)
	elif x_46_re <= 6.2e+141:
		tmp = x_46_im * ((x_46_re * x_46_im) * -3.0)
	else:
		tmp = (x_46_re * x_46_re) * x_46_im
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_re <= -1.05e+202)
		tmp = Float64(x_46_im * Float64(x_46_re * x_46_im));
	elseif (x_46_re <= 6.2e+141)
		tmp = Float64(x_46_im * Float64(Float64(x_46_re * x_46_im) * -3.0));
	else
		tmp = Float64(Float64(x_46_re * x_46_re) * x_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_re <= -1.05e+202)
		tmp = x_46_im * (x_46_re * x_46_im);
	elseif (x_46_re <= 6.2e+141)
		tmp = x_46_im * ((x_46_re * x_46_im) * -3.0);
	else
		tmp = (x_46_re * x_46_re) * x_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$re, -1.05e+202], N[(x$46$im * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 6.2e+141], N[(x$46$im * N[(N[(x$46$re * x$46$im), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * x$46$re), $MachinePrecision] * x$46$im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -1.05e202

   1. Initial program 36.7%

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

      1. sqr-neg 36.7%

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

      2. difference-of-squares 53.3%

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

      3. sub-neg 53.3%

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

      4. associate-*l* 53.3%

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

      5. sub-neg 53.3%

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

      6. remove-double-neg 53.3%

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

      7. +-commutative 53.3%

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

      8. *-commutative 53.3%

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

      9. *-commutative 53.3%

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

      10. distribute-rgt-out 53.3%

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

   3. Simplified 53.3%

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

   4. Taylor expanded in x.im around 0 53.3%

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

      1. associate-*r* 53.3%

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

      2. *-commutative 53.3%

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

      3. *-commutative 53.3%

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

      4. unpow2 53.3%

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

      5. associate-*l* 53.3%

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

      6. *-commutative 53.3%

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

      7. count-2 53.3%

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

   6. Simplified 53.3%

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

   7. Taylor expanded in x.re around 0 17.0%

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

      1. associate-*r* 17.0%

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

      2. neg-mul-1 17.0%

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

      3. unpow2 17.0%

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

      4. distribute-rgt-neg-in 17.0%

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

   9. Simplified 17.0%

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

   11. Applied egg-rr 47.9%

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

   if -1.05e202 < x.re < 6.20000000000000007e141

   1. Initial program 88.9%

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

      1. sqr-neg 88.9%

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

      2. difference-of-squares 90.4%

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

      3. sub-neg 90.4%

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

      4. associate-*l* 99.3%

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

      5. sub-neg 99.3%

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

      6. remove-double-neg 99.3%

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

      7. +-commutative 99.3%

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

      8. *-commutative 99.3%

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

      9. *-commutative 99.3%

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

      10. distribute-rgt-out 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in x.re around 0 64.4%

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

      1. distribute-rgt-out-- 64.4%

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

      2. unpow2 64.4%

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

      3. metadata-eval 64.4%

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

      4. associate-*r* 64.4%

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

      5. associate-*r* 73.3%

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

      6. *-commutative 73.3%

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

      7. associate-*l* 73.2%

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

   6. Simplified 73.2%

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

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

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

   if 6.20000000000000007e141 < x.re

   1. Initial program 58.6%

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

      1. sqr-neg 58.6%

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

      2. difference-of-squares 58.6%

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

      3. sub-neg 58.6%

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

      4. associate-*l* 58.6%

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

      5. sub-neg 58.6%

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

      6. remove-double-neg 58.6%

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

      7. +-commutative 58.6%

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

      8. *-commutative 58.6%

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

      9. *-commutative 58.6%

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

      10. distribute-rgt-out 58.6%

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

   3. Simplified 58.6%

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

   4. Taylor expanded in x.re around 0 15.4%

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

      1. *-commutative 15.4%

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

   6. Simplified 15.4%

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

   7. Taylor expanded in x.re around inf 69.0%

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

      1. unpow2 69.0%

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

   9. Simplified 69.0%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -1.05 \cdot 10^{+202}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.im\right)\\ \mathbf{elif}\;x.re \leq 6.2 \cdot 10^{+141}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.im\right) \cdot -3\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.im\\ \end{array} \]

Alternative 9: 62.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -3.6 \cdot 10^{+201}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.im\right)\\ \mathbf{elif}\;x.re \leq 7.5 \cdot 10^{+140}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.im\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.re -3.6e+201)
   (* x.im (* x.re x.im))
   (if (<= x.re 7.5e+140)
     (* x.im (* x.re (* x.im -3.0)))
     (* (* x.re x.re) x.im))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= -3.6e+201) {
		tmp = x_46_im * (x_46_re * x_46_im);
	} else if (x_46_re <= 7.5e+140) {
		tmp = x_46_im * (x_46_re * (x_46_im * -3.0));
	} else {
		tmp = (x_46_re * x_46_re) * x_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re <= (-3.6d+201)) then
        tmp = x_46im * (x_46re * x_46im)
    else if (x_46re <= 7.5d+140) then
        tmp = x_46im * (x_46re * (x_46im * (-3.0d0)))
    else
        tmp = (x_46re * x_46re) * x_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= -3.6e+201) {
		tmp = x_46_im * (x_46_re * x_46_im);
	} else if (x_46_re <= 7.5e+140) {
		tmp = x_46_im * (x_46_re * (x_46_im * -3.0));
	} else {
		tmp = (x_46_re * x_46_re) * x_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_re <= -3.6e+201:
		tmp = x_46_im * (x_46_re * x_46_im)
	elif x_46_re <= 7.5e+140:
		tmp = x_46_im * (x_46_re * (x_46_im * -3.0))
	else:
		tmp = (x_46_re * x_46_re) * x_46_im
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_re <= -3.6e+201)
		tmp = Float64(x_46_im * Float64(x_46_re * x_46_im));
	elseif (x_46_re <= 7.5e+140)
		tmp = Float64(x_46_im * Float64(x_46_re * Float64(x_46_im * -3.0)));
	else
		tmp = Float64(Float64(x_46_re * x_46_re) * x_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_re <= -3.6e+201)
		tmp = x_46_im * (x_46_re * x_46_im);
	elseif (x_46_re <= 7.5e+140)
		tmp = x_46_im * (x_46_re * (x_46_im * -3.0));
	else
		tmp = (x_46_re * x_46_re) * x_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$re, -3.6e+201], N[(x$46$im * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 7.5e+140], N[(x$46$im * N[(x$46$re * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * x$46$re), $MachinePrecision] * x$46$im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -3.59999999999999976e201

   1. Initial program 36.7%

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

      1. sqr-neg 36.7%

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

      2. difference-of-squares 53.3%

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

      3. sub-neg 53.3%

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

      4. associate-*l* 53.3%

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

      5. sub-neg 53.3%

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

      6. remove-double-neg 53.3%

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

      7. +-commutative 53.3%

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

      8. *-commutative 53.3%

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

      9. *-commutative 53.3%

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

      10. distribute-rgt-out 53.3%

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

   3. Simplified 53.3%

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

   4. Taylor expanded in x.im around 0 53.3%

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

      1. associate-*r* 53.3%

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

      2. *-commutative 53.3%

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

      3. *-commutative 53.3%

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

      4. unpow2 53.3%

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

      5. associate-*l* 53.3%

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

      6. *-commutative 53.3%

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

      7. count-2 53.3%

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

   6. Simplified 53.3%

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

   7. Taylor expanded in x.re around 0 17.0%

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

      1. associate-*r* 17.0%

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

      2. neg-mul-1 17.0%

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

      3. unpow2 17.0%

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

      4. distribute-rgt-neg-in 17.0%

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

   9. Simplified 17.0%

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

   11. Applied egg-rr 47.9%

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

   if -3.59999999999999976e201 < x.re < 7.4999999999999997e140

   1. Initial program 88.9%

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

      1. sqr-neg 88.9%

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

      2. difference-of-squares 90.4%

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

      3. sub-neg 90.4%

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

      4. associate-*l* 99.3%

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

      5. sub-neg 99.3%

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

      6. remove-double-neg 99.3%

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

      7. +-commutative 99.3%

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

      8. *-commutative 99.3%

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

      9. *-commutative 99.3%

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

      10. distribute-rgt-out 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in x.re around 0 64.4%

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

      1. distribute-rgt-out-- 64.4%

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

      2. unpow2 64.4%

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

      3. metadata-eval 64.4%

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

      4. associate-*r* 64.4%

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

      5. associate-*r* 73.3%

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

      6. *-commutative 73.3%

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

      7. associate-*l* 73.2%

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

   6. Simplified 73.2%

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

   if 7.4999999999999997e140 < x.re

   1. Initial program 58.6%

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

      1. sqr-neg 58.6%

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

      2. difference-of-squares 58.6%

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

      3. sub-neg 58.6%

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

      4. associate-*l* 58.6%

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

      5. sub-neg 58.6%

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

      6. remove-double-neg 58.6%

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

      7. +-commutative 58.6%

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

      8. *-commutative 58.6%

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

      9. *-commutative 58.6%

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

      10. distribute-rgt-out 58.6%

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

   3. Simplified 58.6%

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

   4. Taylor expanded in x.re around 0 15.4%

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

      1. *-commutative 15.4%

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

   6. Simplified 15.4%

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

   7. Taylor expanded in x.re around inf 69.0%

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

      1. unpow2 69.0%

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

   9. Simplified 69.0%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -3.6 \cdot 10^{+201}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.im\right)\\ \mathbf{elif}\;x.re \leq 7.5 \cdot 10^{+140}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.im\\ \end{array} \]

Alternative 10: 63.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -1.9 \cdot 10^{+141}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.im - x.re\right)\right)\\ \mathbf{elif}\;x.re \leq 5.2 \cdot 10^{+150}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.im\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.re -1.9e+141)
   (* x.im (* x.re (- x.im x.re)))
   (if (<= x.re 5.2e+150)
     (* x.im (* x.re (* x.im -3.0)))
     (* (* x.re x.re) x.im))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= -1.9e+141) {
		tmp = x_46_im * (x_46_re * (x_46_im - x_46_re));
	} else if (x_46_re <= 5.2e+150) {
		tmp = x_46_im * (x_46_re * (x_46_im * -3.0));
	} else {
		tmp = (x_46_re * x_46_re) * x_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re <= (-1.9d+141)) then
        tmp = x_46im * (x_46re * (x_46im - x_46re))
    else if (x_46re <= 5.2d+150) then
        tmp = x_46im * (x_46re * (x_46im * (-3.0d0)))
    else
        tmp = (x_46re * x_46re) * x_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= -1.9e+141) {
		tmp = x_46_im * (x_46_re * (x_46_im - x_46_re));
	} else if (x_46_re <= 5.2e+150) {
		tmp = x_46_im * (x_46_re * (x_46_im * -3.0));
	} else {
		tmp = (x_46_re * x_46_re) * x_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_re <= -1.9e+141:
		tmp = x_46_im * (x_46_re * (x_46_im - x_46_re))
	elif x_46_re <= 5.2e+150:
		tmp = x_46_im * (x_46_re * (x_46_im * -3.0))
	else:
		tmp = (x_46_re * x_46_re) * x_46_im
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_re <= -1.9e+141)
		tmp = Float64(x_46_im * Float64(x_46_re * Float64(x_46_im - x_46_re)));
	elseif (x_46_re <= 5.2e+150)
		tmp = Float64(x_46_im * Float64(x_46_re * Float64(x_46_im * -3.0)));
	else
		tmp = Float64(Float64(x_46_re * x_46_re) * x_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_re <= -1.9e+141)
		tmp = x_46_im * (x_46_re * (x_46_im - x_46_re));
	elseif (x_46_re <= 5.2e+150)
		tmp = x_46_im * (x_46_re * (x_46_im * -3.0));
	else
		tmp = (x_46_re * x_46_re) * x_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$re, -1.9e+141], N[(x$46$im * N[(x$46$re * N[(x$46$im - x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 5.2e+150], N[(x$46$im * N[(x$46$re * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * x$46$re), $MachinePrecision] * x$46$im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -1.89999999999999988e141

   1. Initial program 42.5%

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

      1. sqr-neg 42.5%

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

      2. difference-of-squares 62.5%

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

      3. sub-neg 62.5%

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

      4. associate-*l* 62.5%

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

      5. sub-neg 62.5%

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

      6. remove-double-neg 62.5%

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

      7. +-commutative 62.5%

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

      8. *-commutative 62.5%

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

      9. *-commutative 62.5%

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

      10. distribute-rgt-out 62.5%

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

   3. Simplified 62.5%

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

      1. cancel-sign-sub-inv 62.5%

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

      2. fma-def 62.5%

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

      3. *-commutative 62.5%

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

   5. Applied egg-rr 62.5%

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

   6. Taylor expanded in x.re around 0 30.7%

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

      1. *-commutative 30.7%

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

   8. Simplified 30.7%

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

   10. Applied egg-rr 38.3%

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

       1. +-lft-identity 38.3%

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

       2. *-commutative 38.3%

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

       3. associate-*l* 50.1%

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

   12. Simplified 50.1%

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

   if -1.89999999999999988e141 < x.re < 5.20000000000000012e150

   1. Initial program 90.5%

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

      1. sqr-neg 90.5%

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

      2. difference-of-squares 90.5%

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

      3. sub-neg 90.5%

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

      4. associate-*l* 99.7%

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

      5. sub-neg 99.7%

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

      6. remove-double-neg 99.7%

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

      7. +-commutative 99.7%

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

      8. *-commutative 99.7%

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

      9. *-commutative 99.7%

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

      10. distribute-rgt-out 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x.re around 0 66.2%

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

      1. distribute-rgt-out-- 66.2%

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

      2. unpow2 66.2%

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

      3. metadata-eval 66.2%

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

      4. associate-*r* 66.2%

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

      5. associate-*r* 75.5%

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

      6. *-commutative 75.5%

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

      7. associate-*l* 75.5%

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

   6. Simplified 75.5%

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

   if 5.20000000000000012e150 < x.re

   1. Initial program 58.6%

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

      1. sqr-neg 58.6%

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

      2. difference-of-squares 58.6%

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

      3. sub-neg 58.6%

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

      4. associate-*l* 58.6%

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

      5. sub-neg 58.6%

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

      6. remove-double-neg 58.6%

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

      7. +-commutative 58.6%

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

      8. *-commutative 58.6%

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

      9. *-commutative 58.6%

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

      10. distribute-rgt-out 58.6%

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

   3. Simplified 58.6%

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

   4. Taylor expanded in x.re around 0 15.4%

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

      1. *-commutative 15.4%

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

   6. Simplified 15.4%

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

   7. Taylor expanded in x.re around inf 69.0%

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

      1. unpow2 69.0%

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

   9. Simplified 69.0%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -1.9 \cdot 10^{+141}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.im - x.re\right)\right)\\ \mathbf{elif}\;x.re \leq 5.2 \cdot 10^{+150}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.im\\ \end{array} \]

Alternative 11: 63.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -1.45 \cdot 10^{+143}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.im - x.re\right)\right)\\ \mathbf{elif}\;x.re \leq 5.2 \cdot 10^{+146}:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.im\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.re -1.45e+143)
   (* x.im (* x.re (- x.im x.re)))
   (if (<= x.re 5.2e+146)
     (* -3.0 (* x.im (* x.re x.im)))
     (* (* x.re x.re) x.im))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= -1.45e+143) {
		tmp = x_46_im * (x_46_re * (x_46_im - x_46_re));
	} else if (x_46_re <= 5.2e+146) {
		tmp = -3.0 * (x_46_im * (x_46_re * x_46_im));
	} else {
		tmp = (x_46_re * x_46_re) * x_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re <= (-1.45d+143)) then
        tmp = x_46im * (x_46re * (x_46im - x_46re))
    else if (x_46re <= 5.2d+146) then
        tmp = (-3.0d0) * (x_46im * (x_46re * x_46im))
    else
        tmp = (x_46re * x_46re) * x_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= -1.45e+143) {
		tmp = x_46_im * (x_46_re * (x_46_im - x_46_re));
	} else if (x_46_re <= 5.2e+146) {
		tmp = -3.0 * (x_46_im * (x_46_re * x_46_im));
	} else {
		tmp = (x_46_re * x_46_re) * x_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_re <= -1.45e+143:
		tmp = x_46_im * (x_46_re * (x_46_im - x_46_re))
	elif x_46_re <= 5.2e+146:
		tmp = -3.0 * (x_46_im * (x_46_re * x_46_im))
	else:
		tmp = (x_46_re * x_46_re) * x_46_im
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_re <= -1.45e+143)
		tmp = Float64(x_46_im * Float64(x_46_re * Float64(x_46_im - x_46_re)));
	elseif (x_46_re <= 5.2e+146)
		tmp = Float64(-3.0 * Float64(x_46_im * Float64(x_46_re * x_46_im)));
	else
		tmp = Float64(Float64(x_46_re * x_46_re) * x_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_re <= -1.45e+143)
		tmp = x_46_im * (x_46_re * (x_46_im - x_46_re));
	elseif (x_46_re <= 5.2e+146)
		tmp = -3.0 * (x_46_im * (x_46_re * x_46_im));
	else
		tmp = (x_46_re * x_46_re) * x_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$re, -1.45e+143], N[(x$46$im * N[(x$46$re * N[(x$46$im - x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 5.2e+146], N[(-3.0 * N[(x$46$im * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * x$46$re), $MachinePrecision] * x$46$im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -1.4499999999999999e143

   1. Initial program 42.5%

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

      1. sqr-neg 42.5%

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

      2. difference-of-squares 62.5%

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

      3. sub-neg 62.5%

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

      4. associate-*l* 62.5%

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

      5. sub-neg 62.5%

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

      6. remove-double-neg 62.5%

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

      7. +-commutative 62.5%

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

      8. *-commutative 62.5%

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

      9. *-commutative 62.5%

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

      10. distribute-rgt-out 62.5%

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

   3. Simplified 62.5%

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

      1. cancel-sign-sub-inv 62.5%

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

      2. fma-def 62.5%

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

      3. *-commutative 62.5%

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

   5. Applied egg-rr 62.5%

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

   6. Taylor expanded in x.re around 0 30.7%

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

      1. *-commutative 30.7%

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

   8. Simplified 30.7%

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

   10. Applied egg-rr 38.3%

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

       1. +-lft-identity 38.3%

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

       2. *-commutative 38.3%

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

       3. associate-*l* 50.1%

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

   12. Simplified 50.1%

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

   if -1.4499999999999999e143 < x.re < 5.20000000000000028e146

   1. Initial program 90.5%

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

      1. sqr-neg 90.5%

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

      2. difference-of-squares 90.5%

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

      3. sub-neg 90.5%

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

      4. associate-*l* 99.7%

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

      5. sub-neg 99.7%

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

      6. remove-double-neg 99.7%

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

      7. +-commutative 99.7%

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

      8. *-commutative 99.7%

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

      9. *-commutative 99.7%

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

      10. distribute-rgt-out 99.7%

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

   3. Simplified 99.7%

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

      1. cancel-sign-sub-inv 99.7%

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

      2. fma-def 99.8%

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

      3. *-commutative 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in x.re around 0 75.9%

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

      1. *-commutative 75.9%

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

   8. Simplified 75.9%

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

   9. Taylor expanded in x.re around 0 66.2%

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

       1. unpow2 66.2%

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

       2. unpow2 66.2%

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

       3. distribute-rgt-out 66.2%

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

       4. associate-*r* 66.2%

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

       5. associate-*r* 75.6%

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

       6. *-commutative 75.6%

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

       7. metadata-eval 75.6%

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

   11. Simplified 75.6%

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

   if 5.20000000000000028e146 < x.re

   1. Initial program 58.6%

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

      1. sqr-neg 58.6%

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

      2. difference-of-squares 58.6%

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

      3. sub-neg 58.6%

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

      4. associate-*l* 58.6%

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

      5. sub-neg 58.6%

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

      6. remove-double-neg 58.6%

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

      7. +-commutative 58.6%

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

      8. *-commutative 58.6%

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

      9. *-commutative 58.6%

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

      10. distribute-rgt-out 58.6%

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

   3. Simplified 58.6%

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

   4. Taylor expanded in x.re around 0 15.4%

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

      1. *-commutative 15.4%

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

   6. Simplified 15.4%

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

   7. Taylor expanded in x.re around inf 69.0%

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

      1. unpow2 69.0%

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

   9. Simplified 69.0%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -1.45 \cdot 10^{+143}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.im - x.re\right)\right)\\ \mathbf{elif}\;x.re \leq 5.2 \cdot 10^{+146}:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.im\\ \end{array} \]

Alternative 12: 42.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -1.05 \cdot 10^{+202}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.im\right)\\ \mathbf{elif}\;x.re \leq 1.2 \cdot 10^{+151}:\\ \;\;\;\;x.re \cdot \left(-x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.im\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.re -1.05e+202)
   (* x.im (* x.re x.im))
   (if (<= x.re 1.2e+151) (* x.re (- (* x.im x.im))) (* (* x.re x.re) x.im))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= -1.05e+202) {
		tmp = x_46_im * (x_46_re * x_46_im);
	} else if (x_46_re <= 1.2e+151) {
		tmp = x_46_re * -(x_46_im * x_46_im);
	} else {
		tmp = (x_46_re * x_46_re) * x_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re <= (-1.05d+202)) then
        tmp = x_46im * (x_46re * x_46im)
    else if (x_46re <= 1.2d+151) then
        tmp = x_46re * -(x_46im * x_46im)
    else
        tmp = (x_46re * x_46re) * x_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= -1.05e+202) {
		tmp = x_46_im * (x_46_re * x_46_im);
	} else if (x_46_re <= 1.2e+151) {
		tmp = x_46_re * -(x_46_im * x_46_im);
	} else {
		tmp = (x_46_re * x_46_re) * x_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_re <= -1.05e+202:
		tmp = x_46_im * (x_46_re * x_46_im)
	elif x_46_re <= 1.2e+151:
		tmp = x_46_re * -(x_46_im * x_46_im)
	else:
		tmp = (x_46_re * x_46_re) * x_46_im
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_re <= -1.05e+202)
		tmp = Float64(x_46_im * Float64(x_46_re * x_46_im));
	elseif (x_46_re <= 1.2e+151)
		tmp = Float64(x_46_re * Float64(-Float64(x_46_im * x_46_im)));
	else
		tmp = Float64(Float64(x_46_re * x_46_re) * x_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_re <= -1.05e+202)
		tmp = x_46_im * (x_46_re * x_46_im);
	elseif (x_46_re <= 1.2e+151)
		tmp = x_46_re * -(x_46_im * x_46_im);
	else
		tmp = (x_46_re * x_46_re) * x_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$re, -1.05e+202], N[(x$46$im * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 1.2e+151], N[(x$46$re * (-N[(x$46$im * x$46$im), $MachinePrecision])), $MachinePrecision], N[(N[(x$46$re * x$46$re), $MachinePrecision] * x$46$im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -1.05e202

   1. Initial program 36.7%

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

      1. sqr-neg 36.7%

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

      2. difference-of-squares 53.3%

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

      3. sub-neg 53.3%

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

      4. associate-*l* 53.3%

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

      5. sub-neg 53.3%

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

      6. remove-double-neg 53.3%

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

      7. +-commutative 53.3%

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

      8. *-commutative 53.3%

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

      9. *-commutative 53.3%

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

      10. distribute-rgt-out 53.3%

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

   3. Simplified 53.3%

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

   4. Taylor expanded in x.im around 0 53.3%

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

      1. associate-*r* 53.3%

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

      2. *-commutative 53.3%

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

      3. *-commutative 53.3%

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

      4. unpow2 53.3%

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

      5. associate-*l* 53.3%

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

      6. *-commutative 53.3%

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

      7. count-2 53.3%

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

   6. Simplified 53.3%

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

   7. Taylor expanded in x.re around 0 17.0%

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

      1. associate-*r* 17.0%

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

      2. neg-mul-1 17.0%

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

      3. unpow2 17.0%

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

      4. distribute-rgt-neg-in 17.0%

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

   9. Simplified 17.0%

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

   11. Applied egg-rr 47.9%

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

   if -1.05e202 < x.re < 1.20000000000000005e151

   1. Initial program 88.9%

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

      1. sqr-neg 88.9%

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

      2. difference-of-squares 90.4%

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

      3. sub-neg 90.4%

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

      4. associate-*l* 99.3%

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

      5. sub-neg 99.3%

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

      6. remove-double-neg 99.3%

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

      7. +-commutative 99.3%

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

      8. *-commutative 99.3%

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

      9. *-commutative 99.3%

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

      10. distribute-rgt-out 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in x.im around 0 90.4%

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

      1. associate-*r* 90.4%

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

      2. *-commutative 90.4%

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

      3. *-commutative 90.4%

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

      4. unpow2 90.4%

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

      5. associate-*l* 90.4%

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

      6. *-commutative 90.4%

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

      7. count-2 90.4%

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

   6. Simplified 90.4%

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

   7. Taylor expanded in x.re around 0 64.4%

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

      1. associate-*r* 64.4%

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

      2. neg-mul-1 64.4%

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

      3. unpow2 64.4%

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

      4. distribute-rgt-neg-in 64.4%

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

   9. Simplified 64.4%

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

       1. *-commutative 64.4%

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

       2. cancel-sign-sub-inv 64.4%

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

       3. distribute-lft-neg-in 64.4%

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

       4. associate-*r* 64.5%

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

       5. add-sqr-sqrt 31.6%

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

       6. sqrt-unprod 42.4%

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

       7. sqr-neg 42.4%

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

       8. sqrt-prod 11.1%

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

       9. add-sqr-sqrt 21.4%

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

       10. associate-*r* 20.6%

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

       11. *-commutative 20.6%

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

       12. *-commutative 20.6%

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

       13. distribute-lft-out 20.6%

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

       14. add-sqr-sqrt 9.9%

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

       15. sqrt-unprod 21.1%

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

       16. sqr-neg 21.1%

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

       17. sqrt-prod 11.2%

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

       18. add-sqr-sqrt 21.1%

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

       19. flip-+ 0.0%

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

   11. Applied egg-rr 42.8%

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

       1. distribute-lft-neg-in 42.8%

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

   13. Simplified 42.8%

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

   if 1.20000000000000005e151 < x.re

   1. Initial program 58.6%

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

      1. sqr-neg 58.6%

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

      2. difference-of-squares 58.6%

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

      3. sub-neg 58.6%

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

      4. associate-*l* 58.6%

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

      5. sub-neg 58.6%

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

      6. remove-double-neg 58.6%

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

      7. +-commutative 58.6%

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

      8. *-commutative 58.6%

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

      9. *-commutative 58.6%

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

      10. distribute-rgt-out 58.6%

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

   3. Simplified 58.6%

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

   4. Taylor expanded in x.re around 0 15.4%

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

      1. *-commutative 15.4%

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

   6. Simplified 15.4%

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

   7. Taylor expanded in x.re around inf 69.0%

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

      1. unpow2 69.0%

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

   9. Simplified 69.0%

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

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -1.05 \cdot 10^{+202}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.im\right)\\ \mathbf{elif}\;x.re \leq 1.2 \cdot 10^{+151}:\\ \;\;\;\;x.re \cdot \left(-x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.im\\ \end{array} \]

Alternative 13: 31.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -3.6 \cdot 10^{+201}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.im\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.re -3.6e+201) (* x.re (* x.im x.im)) (* (* x.re x.re) x.im)))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= -3.6e+201) {
		tmp = x_46_re * (x_46_im * x_46_im);
	} else {
		tmp = (x_46_re * x_46_re) * x_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re <= (-3.6d+201)) then
        tmp = x_46re * (x_46im * x_46im)
    else
        tmp = (x_46re * x_46re) * x_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= -3.6e+201) {
		tmp = x_46_re * (x_46_im * x_46_im);
	} else {
		tmp = (x_46_re * x_46_re) * x_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_re <= -3.6e+201:
		tmp = x_46_re * (x_46_im * x_46_im)
	else:
		tmp = (x_46_re * x_46_re) * x_46_im
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_re <= -3.6e+201)
		tmp = Float64(x_46_re * Float64(x_46_im * x_46_im));
	else
		tmp = Float64(Float64(x_46_re * x_46_re) * x_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_re <= -3.6e+201)
		tmp = x_46_re * (x_46_im * x_46_im);
	else
		tmp = (x_46_re * x_46_re) * x_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$re, -3.6e+201], N[(x$46$re * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * x$46$re), $MachinePrecision] * x$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < -3.59999999999999976e201

   1. Initial program 36.7%

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

      1. sqr-neg 36.7%

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

      2. difference-of-squares 53.3%

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

      3. sub-neg 53.3%

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

      4. associate-*l* 53.3%

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

      5. sub-neg 53.3%

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

      6. remove-double-neg 53.3%

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

      7. +-commutative 53.3%

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

      8. *-commutative 53.3%

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

      9. *-commutative 53.3%

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

      10. distribute-rgt-out 53.3%

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

   3. Simplified 53.3%

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

   4. Taylor expanded in x.im around 0 53.3%

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

      1. associate-*r* 53.3%

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

      2. *-commutative 53.3%

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

      3. *-commutative 53.3%

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

      4. unpow2 53.3%

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

      5. associate-*l* 53.3%

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

      6. *-commutative 53.3%

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

      7. count-2 53.3%

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

   6. Simplified 53.3%

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

   7. Taylor expanded in x.re around 0 17.0%

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

      1. associate-*r* 17.0%

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

      2. neg-mul-1 17.0%

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

      3. unpow2 17.0%

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

      4. distribute-rgt-neg-in 17.0%

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

   9. Simplified 17.0%

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

   11. Applied egg-rr 47.8%

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

   if -3.59999999999999976e201 < x.re

   1. Initial program 85.0%

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

      1. sqr-neg 85.0%

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

      2. difference-of-squares 86.4%

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

      3. sub-neg 86.4%

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

      4. associate-*l* 94.0%

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

      5. sub-neg 94.0%

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

      6. remove-double-neg 94.0%

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

      7. +-commutative 94.0%

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

      8. *-commutative 94.0%

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

      9. *-commutative 94.0%

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

      10. distribute-rgt-out 94.0%

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

   3. Simplified 94.0%

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

   4. Taylor expanded in x.re around 0 66.1%

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

      1. *-commutative 66.1%

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

   6. Simplified 66.1%

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

   7. Taylor expanded in x.re around inf 30.6%

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

      1. unpow2 30.6%

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

   9. Simplified 30.6%

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

4. Final simplification 32.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -3.6 \cdot 10^{+201}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.im\\ \end{array} \]

Alternative 14: 31.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -4.8 \cdot 10^{+203}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.im\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.re -4.8e+203) (* x.im (* x.re x.im)) (* (* x.re x.re) x.im)))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= -4.8e+203) {
		tmp = x_46_im * (x_46_re * x_46_im);
	} else {
		tmp = (x_46_re * x_46_re) * x_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re <= (-4.8d+203)) then
        tmp = x_46im * (x_46re * x_46im)
    else
        tmp = (x_46re * x_46re) * x_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= -4.8e+203) {
		tmp = x_46_im * (x_46_re * x_46_im);
	} else {
		tmp = (x_46_re * x_46_re) * x_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_re <= -4.8e+203:
		tmp = x_46_im * (x_46_re * x_46_im)
	else:
		tmp = (x_46_re * x_46_re) * x_46_im
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_re <= -4.8e+203)
		tmp = Float64(x_46_im * Float64(x_46_re * x_46_im));
	else
		tmp = Float64(Float64(x_46_re * x_46_re) * x_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_re <= -4.8e+203)
		tmp = x_46_im * (x_46_re * x_46_im);
	else
		tmp = (x_46_re * x_46_re) * x_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$re, -4.8e+203], N[(x$46$im * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * x$46$re), $MachinePrecision] * x$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < -4.8000000000000002e203

   1. Initial program 36.7%

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

      1. sqr-neg 36.7%

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

      2. difference-of-squares 53.3%

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

      3. sub-neg 53.3%

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

      4. associate-*l* 53.3%

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

      5. sub-neg 53.3%

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

      6. remove-double-neg 53.3%

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

      7. +-commutative 53.3%

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

      8. *-commutative 53.3%

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

      9. *-commutative 53.3%

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

      10. distribute-rgt-out 53.3%

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

   3. Simplified 53.3%

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

   4. Taylor expanded in x.im around 0 53.3%

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

      1. associate-*r* 53.3%

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

      2. *-commutative 53.3%

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

      3. *-commutative 53.3%

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

      4. unpow2 53.3%

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

      5. associate-*l* 53.3%

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

      6. *-commutative 53.3%

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

      7. count-2 53.3%

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

   6. Simplified 53.3%

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

   7. Taylor expanded in x.re around 0 17.0%

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

      1. associate-*r* 17.0%

         \[\leadsto \color{blue}{\left(-1 \cdot {x.im}^{2}\right) \cdot x.re} - x.re \cdot \left(x.im \cdot \left(x.im + x.im\right)\right) \]

      2. neg-mul-1 17.0%

         \[\leadsto \color{blue}{\left(-{x.im}^{2}\right)} \cdot x.re - x.re \cdot \left(x.im \cdot \left(x.im + x.im\right)\right) \]

      3. unpow2 17.0%

         \[\leadsto \left(-\color{blue}{x.im \cdot x.im}\right) \cdot x.re - x.re \cdot \left(x.im \cdot \left(x.im + x.im\right)\right) \]

      4. distribute-rgt-neg-in 17.0%

         \[\leadsto \color{blue}{\left(x.im \cdot \left(-x.im\right)\right)} \cdot x.re - x.re \cdot \left(x.im \cdot \left(x.im + x.im\right)\right) \]

   9. Simplified 17.0%

      \[\leadsto \color{blue}{\left(x.im \cdot \left(-x.im\right)\right) \cdot x.re} - x.re \cdot \left(x.im \cdot \left(x.im + x.im\right)\right) \]

   11. Applied egg-rr 47.9%

       \[\leadsto \color{blue}{\left(x.re \cdot x.im\right) \cdot x.im} \]

   if -4.8000000000000002e203 < x.re

   1. Initial program 85.0%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   2. Step-by-step derivation

      1. sqr-neg 85.0%

         \[\leadsto \left(x.re \cdot x.re - \color{blue}{\left(-x.im\right) \cdot \left(-x.im\right)}\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      2. difference-of-squares 86.4%

         \[\leadsto \color{blue}{\left(\left(x.re + \left(-x.im\right)\right) \cdot \left(x.re - \left(-x.im\right)\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      3. sub-neg 86.4%

         \[\leadsto \left(\color{blue}{\left(x.re - x.im\right)} \cdot \left(x.re - \left(-x.im\right)\right)\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      4. associate-*l* 94.0%

         \[\leadsto \color{blue}{\left(x.re - x.im\right) \cdot \left(\left(x.re - \left(-x.im\right)\right) \cdot x.re\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      5. sub-neg 94.0%

         \[\leadsto \left(x.re - x.im\right) \cdot \left(\color{blue}{\left(x.re + \left(-\left(-x.im\right)\right)\right)} \cdot x.re\right) - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      6. remove-double-neg 94.0%

         \[\leadsto \left(x.re - x.im\right) \cdot \left(\left(x.re + \color{blue}{x.im}\right) \cdot x.re\right) - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      7. +-commutative 94.0%

         \[\leadsto \left(x.re - x.im\right) \cdot \left(\left(x.re + x.im\right) \cdot x.re\right) - \color{blue}{\left(x.im \cdot x.re + x.re \cdot x.im\right)} \cdot x.im \]

      8. *-commutative 94.0%

         \[\leadsto \left(x.re - x.im\right) \cdot \left(\left(x.re + x.im\right) \cdot x.re\right) - \color{blue}{x.im \cdot \left(x.im \cdot x.re + x.re \cdot x.im\right)} \]

      9. *-commutative 94.0%

         \[\leadsto \left(x.re - x.im\right) \cdot \left(\left(x.re + x.im\right) \cdot x.re\right) - x.im \cdot \left(x.im \cdot x.re + \color{blue}{x.im \cdot x.re}\right) \]

      10. distribute-rgt-out 94.0%

          \[\leadsto \left(x.re - x.im\right) \cdot \left(\left(x.re + x.im\right) \cdot x.re\right) - x.im \cdot \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right)} \]

   3. Simplified 94.0%

      \[\leadsto \color{blue}{\left(x.re - x.im\right) \cdot \left(\left(x.re + x.im\right) \cdot x.re\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)} \]

   4. Taylor expanded in x.re around 0 66.1%

      \[\leadsto \left(x.re - x.im\right) \cdot \color{blue}{\left(x.im \cdot x.re\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]
   5. Step-by-step derivation

      1. *-commutative 66.1%

         \[\leadsto \left(x.re - x.im\right) \cdot \color{blue}{\left(x.re \cdot x.im\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

   6. Simplified 66.1%

      \[\leadsto \left(x.re - x.im\right) \cdot \color{blue}{\left(x.re \cdot x.im\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

   7. Taylor expanded in x.re around inf 30.6%

      \[\leadsto \color{blue}{x.im \cdot {x.re}^{2}} \]
   8. Step-by-step derivation

      1. unpow2 30.6%

         \[\leadsto x.im \cdot \color{blue}{\left(x.re \cdot x.re\right)} \]

   9. Simplified 30.6%

      \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot x.re\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -4.8 \cdot 10^{+203}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.im\\ \end{array} \]

Alternative 15: 31.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \left(x.re \cdot x.re\right) \cdot x.im \end{array} \]

FPCore

(FPCore (x.re x.im) :precision binary64 (* (* x.re 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;
}

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
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;
}

Python

def code(x_46_re, x_46_im):
	return (x_46_re * x_46_re) * x_46_im

Julia

function code(x_46_re, x_46_im)
	return Float64(Float64(x_46_re * 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;
end

Wolfram

code[x$46$re_, x$46$im_] := N[(N[(x$46$re * x$46$re), $MachinePrecision] * x$46$im), $MachinePrecision]

Derivation

1. Initial program 79.4%

   \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
2. Step-by-step derivation

   1. sqr-neg 79.4%

      \[\leadsto \left(x.re \cdot x.re - \color{blue}{\left(-x.im\right) \cdot \left(-x.im\right)}\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   2. difference-of-squares 82.5%

      \[\leadsto \color{blue}{\left(\left(x.re + \left(-x.im\right)\right) \cdot \left(x.re - \left(-x.im\right)\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   3. sub-neg 82.5%

      \[\leadsto \left(\color{blue}{\left(x.re - x.im\right)} \cdot \left(x.re - \left(-x.im\right)\right)\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   4. associate-*l* 89.3%

      \[\leadsto \color{blue}{\left(x.re - x.im\right) \cdot \left(\left(x.re - \left(-x.im\right)\right) \cdot x.re\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   5. sub-neg 89.3%

      \[\leadsto \left(x.re - x.im\right) \cdot \left(\color{blue}{\left(x.re + \left(-\left(-x.im\right)\right)\right)} \cdot x.re\right) - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   6. remove-double-neg 89.3%

      \[\leadsto \left(x.re - x.im\right) \cdot \left(\left(x.re + \color{blue}{x.im}\right) \cdot x.re\right) - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   7. +-commutative 89.3%

      \[\leadsto \left(x.re - x.im\right) \cdot \left(\left(x.re + x.im\right) \cdot x.re\right) - \color{blue}{\left(x.im \cdot x.re + x.re \cdot x.im\right)} \cdot x.im \]

   8. *-commutative 89.3%

      \[\leadsto \left(x.re - x.im\right) \cdot \left(\left(x.re + x.im\right) \cdot x.re\right) - \color{blue}{x.im \cdot \left(x.im \cdot x.re + x.re \cdot x.im\right)} \]

   9. *-commutative 89.3%

      \[\leadsto \left(x.re - x.im\right) \cdot \left(\left(x.re + x.im\right) \cdot x.re\right) - x.im \cdot \left(x.im \cdot x.re + \color{blue}{x.im \cdot x.re}\right) \]

   10. distribute-rgt-out 89.3%

       \[\leadsto \left(x.re - x.im\right) \cdot \left(\left(x.re + x.im\right) \cdot x.re\right) - x.im \cdot \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right)} \]

3. Simplified 89.3%

   \[\leadsto \color{blue}{\left(x.re - x.im\right) \cdot \left(\left(x.re + x.im\right) \cdot x.re\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)} \]

4. Taylor expanded in x.re around 0 61.9%

   \[\leadsto \left(x.re - x.im\right) \cdot \color{blue}{\left(x.im \cdot x.re\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]
5. Step-by-step derivation

   1. *-commutative 61.9%

      \[\leadsto \left(x.re - x.im\right) \cdot \color{blue}{\left(x.re \cdot x.im\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

6. Simplified 61.9%

   \[\leadsto \left(x.re - x.im\right) \cdot \color{blue}{\left(x.re \cdot x.im\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

7. Taylor expanded in x.re around inf 32.1%

   \[\leadsto \color{blue}{x.im \cdot {x.re}^{2}} \]
8. Step-by-step derivation

   1. unpow2 32.1%

      \[\leadsto x.im \cdot \color{blue}{\left(x.re \cdot x.re\right)} \]

9. Simplified 32.1%

   \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot x.re\right)} \]

10. Final simplification 32.1%

    \[\leadsto \left(x.re \cdot x.re\right) \cdot x.im \]

Developer target: 87.1% 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 2023291 
(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)))