math.cube on complex, real part

Percentage Accurate: 82.2% → 99.8%

Time: 11.7s

Alternatives: 14

Speedup: 0.5×

• 45.0% 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: 82.2% 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: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.re \leq -5.5 \cdot 10^{+102}:\\ \;\;\;\;x.re \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re + -27\right)\right)\\ \mathbf{elif}\;x.re \leq 5 \cdot 10^{+101}:\\ \;\;\;\;\mathsf{fma}\left(x.re \cdot x.im, x.im \cdot -3, {x.re}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right) - x.im \cdot -3\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.re -5.5e+102)
   (* x.re (* (+ x.re x.im) (+ x.re -27.0)))
   (if (<= x.re 5e+101)
     (fma (* x.re x.im) (* x.im -3.0) (pow x.re 3.0))
     (- (* x.re (* (+ x.re x.im) (- x.re x.im))) (* x.im -3.0)))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= -5.5e+102) {
		tmp = x_46_re * ((x_46_re + x_46_im) * (x_46_re + -27.0));
	} else if (x_46_re <= 5e+101) {
		tmp = fma((x_46_re * x_46_im), (x_46_im * -3.0), pow(x_46_re, 3.0));
	} else {
		tmp = (x_46_re * ((x_46_re + x_46_im) * (x_46_re - x_46_im))) - (x_46_im * -3.0);
	}
	return tmp;
}

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_re <= -5.5e+102)
		tmp = Float64(x_46_re * Float64(Float64(x_46_re + x_46_im) * Float64(x_46_re + -27.0)));
	elseif (x_46_re <= 5e+101)
		tmp = fma(Float64(x_46_re * x_46_im), Float64(x_46_im * -3.0), (x_46_re ^ 3.0));
	else
		tmp = Float64(Float64(x_46_re * Float64(Float64(x_46_re + x_46_im) * Float64(x_46_re - x_46_im))) - Float64(x_46_im * -3.0));
	end
	return tmp
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := If[LessEqual[x$46$re, -5.5e+102], N[(x$46$re * N[(N[(x$46$re + x$46$im), $MachinePrecision] * N[(x$46$re + -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 5e+101], N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(x$46$im * -3.0), $MachinePrecision] + N[Power[x$46$re, 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * N[(N[(x$46$re + x$46$im), $MachinePrecision] * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -5.49999999999999981e102

   1. Initial program 53.3%

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

      1. difference-of-squares 64.4%

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

   3. Applied egg-rr 64.4%

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

   5. Simplified 51.1%

      \[\leadsto \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   7. Applied egg-rr 86.7%

      \[\leadsto \color{blue}{\left(\left(x.re + x.im\right) \cdot \left(x.re + -27\right)\right) \cdot x.re} \]

   if -5.49999999999999981e102 < x.re < 4.99999999999999989e101

   1. Initial program 89.8%

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

   3. Simplified 89.8%

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

      1. +-commutative 89.8%

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

      2. associate-*r* 99.8%

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

      3. fma-def 99.8%

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

   5. Applied egg-rr 99.8%

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

   if 4.99999999999999989e101 < x.re

   1. Initial program 62.8%

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

      1. difference-of-squares 72.1%

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

      2. *-commutative 72.1%

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

   3. Applied egg-rr 72.1%

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

      1. *-commutative 72.1%

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

      2. flip-+ 0.0%

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

      3. +-inverses 0.0%

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

      4. metadata-eval 0.0%

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

      5. +-inverses 0.0%

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

      6. distribute-neg-frac 0.0%

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

      7. flip-+ 69.8%

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

      8. neg-sub0 69.8%

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

      9. flip-+ 0.0%

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

      10. +-inverses 0.0%

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

      11. +-inverses 0.0%

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

   5. Applied egg-rr 0.0%

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

   7. Simplified 100.0%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -5.5 \cdot 10^{+102}:\\ \;\;\;\;x.re \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re + -27\right)\right)\\ \mathbf{elif}\;x.re \leq 5 \cdot 10^{+101}:\\ \;\;\;\;\mathsf{fma}\left(x.re \cdot x.im, x.im \cdot -3, {x.re}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right) - x.im \cdot -3\\ \end{array} \]

Alternative 2: 93.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} t_0 := x.re \cdot x.re - x.im \cdot x.im\\ \mathbf{if}\;x.re \cdot t_0 - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq 5 \cdot 10^{+256}:\\ \;\;\;\;\mathsf{fma}\left(x.re, t_0, x.im \cdot \left(\left(x.re \cdot x.im\right) \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right) - x.im \cdot -3\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (- (* x.re x.re) (* x.im x.im))))
   (if (<= (- (* x.re t_0) (* x.im (+ (* x.re x.im) (* x.re x.im)))) 5e+256)
     (fma x.re t_0 (* x.im (* (* x.re x.im) -2.0)))
     (- (* x.re (* (+ x.re x.im) (- x.re x.im))) (* x.im -3.0)))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double t_0 = (x_46_re * x_46_re) - (x_46_im * x_46_im);
	double tmp;
	if (((x_46_re * t_0) - (x_46_im * ((x_46_re * x_46_im) + (x_46_re * x_46_im)))) <= 5e+256) {
		tmp = fma(x_46_re, t_0, (x_46_im * ((x_46_re * x_46_im) * -2.0)));
	} else {
		tmp = (x_46_re * ((x_46_re + x_46_im) * (x_46_re - x_46_im))) - (x_46_im * -3.0);
	}
	return tmp;
}

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	t_0 = Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im))
	tmp = 0.0
	if (Float64(Float64(x_46_re * t_0) - Float64(x_46_im * Float64(Float64(x_46_re * x_46_im) + Float64(x_46_re * x_46_im)))) <= 5e+256)
		tmp = fma(x_46_re, t_0, Float64(x_46_im * Float64(Float64(x_46_re * x_46_im) * -2.0)));
	else
		tmp = Float64(Float64(x_46_re * Float64(Float64(x_46_re + x_46_im) * Float64(x_46_re - x_46_im))) - Float64(x_46_im * -3.0));
	end
	return tmp
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(x$46$re * t$95$0), $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], 5e+256], N[(x$46$re * t$95$0 + N[(x$46$im * N[(N[(x$46$re * x$46$im), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * N[(N[(x$46$re + x$46$im), $MachinePrecision] * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < 5.00000000000000015e256

   1. Initial program 95.5%

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

   3. Simplified 95.5%

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

   if 5.00000000000000015e256 < (-.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 44.8%

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

      1. difference-of-squares 55.5%

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

      2. *-commutative 55.5%

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

   3. Applied egg-rr 55.5%

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

      1. *-commutative 55.5%

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

      2. flip-+ 0.0%

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

      3. +-inverses 0.0%

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

      4. metadata-eval 0.0%

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

      5. +-inverses 0.0%

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

      6. distribute-neg-frac 0.0%

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

      7. flip-+ 59.1%

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

      8. neg-sub0 59.1%

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

      9. flip-+ 0.0%

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

      10. +-inverses 0.0%

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

      11. +-inverses 0.0%

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

   5. Applied egg-rr 0.0%

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

   7. Simplified 88.9%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq 5 \cdot 10^{+256}:\\ \;\;\;\;\mathsf{fma}\left(x.re, x.re \cdot x.re - x.im \cdot x.im, x.im \cdot \left(\left(x.re \cdot x.im\right) \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right) - x.im \cdot -3\\ \end{array} \]

Alternative 3: 93.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} t_0 := x.re \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)\\ \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq 5 \cdot 10^{+256}:\\ \;\;\;\;t_0 - x.im \cdot \left(\left(x.re \cdot x.im\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 - x.im \cdot -3\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (* x.re (* (+ x.re x.im) (- x.re x.im)))))
   (if (<=
        (-
         (* x.re (- (* x.re x.re) (* x.im x.im)))
         (* x.im (+ (* x.re x.im) (* x.re x.im))))
        5e+256)
     (- t_0 (* x.im (* (* x.re x.im) 2.0)))
     (- t_0 (* x.im -3.0)))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_re * ((x_46_re + x_46_im) * (x_46_re - 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)))) <= 5e+256) {
		tmp = t_0 - (x_46_im * ((x_46_re * x_46_im) * 2.0));
	} else {
		tmp = t_0 - (x_46_im * -3.0);
	}
	return tmp;
}

Fortran

NOTE: x.im should be positive before calling this function
real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x_46re * ((x_46re + x_46im) * (x_46re - x_46im))
    if (((x_46re * ((x_46re * x_46re) - (x_46im * x_46im))) - (x_46im * ((x_46re * x_46im) + (x_46re * x_46im)))) <= 5d+256) then
        tmp = t_0 - (x_46im * ((x_46re * x_46im) * 2.0d0))
    else
        tmp = t_0 - (x_46im * (-3.0d0))
    end if
    code = tmp
end function

Java

x.im = Math.abs(x.im);
public static double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_re * ((x_46_re + x_46_im) * (x_46_re - 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)))) <= 5e+256) {
		tmp = t_0 - (x_46_im * ((x_46_re * x_46_im) * 2.0));
	} else {
		tmp = t_0 - (x_46_im * -3.0);
	}
	return tmp;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	t_0 = x_46_re * ((x_46_re + x_46_im) * (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)))) <= 5e+256:
		tmp = t_0 - (x_46_im * ((x_46_re * x_46_im) * 2.0))
	else:
		tmp = t_0 - (x_46_im * -3.0)
	return tmp

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	t_0 = Float64(x_46_re * Float64(Float64(x_46_re + x_46_im) * Float64(x_46_re - x_46_im)))
	tmp = 0.0
	if (Float64(Float64(x_46_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)))) <= 5e+256)
		tmp = Float64(t_0 - Float64(x_46_im * Float64(Float64(x_46_re * x_46_im) * 2.0)));
	else
		tmp = Float64(t_0 - Float64(x_46_im * -3.0));
	end
	return tmp
end

MATLAB

x.im = abs(x.im)
function tmp_2 = code(x_46_re, x_46_im)
	t_0 = x_46_re * ((x_46_re + x_46_im) * (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)))) <= 5e+256)
		tmp = t_0 - (x_46_im * ((x_46_re * x_46_im) * 2.0));
	else
		tmp = t_0 - (x_46_im * -3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(x$46$re * N[(N[(x$46$re + x$46$im), $MachinePrecision] * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, 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], 5e+256], N[(t$95$0 - N[(x$46$im * N[(N[(x$46$re * x$46$im), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < 5.00000000000000015e256

   1. Initial program 95.5%

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

      1. difference-of-squares 95.5%

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

      2. *-commutative 95.5%

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

   3. Applied egg-rr 95.5%

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

      1. *-commutative 29.5%

         \[\leadsto x.im \cdot \left(x.re \cdot -27\right) - \left(x.re \cdot x.im + \color{blue}{x.re \cdot x.im}\right) \cdot x.im \]

      2. *-un-lft-identity 29.5%

         \[\leadsto x.im \cdot \left(x.re \cdot -27\right) - \left(x.re \cdot x.im + \color{blue}{1 \cdot \left(x.re \cdot x.im\right)}\right) \cdot x.im \]

      3. *-un-lft-identity 29.5%

         \[\leadsto x.im \cdot \left(x.re \cdot -27\right) - \left(\color{blue}{1 \cdot \left(x.re \cdot x.im\right)} + 1 \cdot \left(x.re \cdot x.im\right)\right) \cdot x.im \]

      4. distribute-rgt-out 29.5%

         \[\leadsto x.im \cdot \left(x.re \cdot -27\right) - \color{blue}{\left(\left(x.re \cdot x.im\right) \cdot \left(1 + 1\right)\right)} \cdot x.im \]

      5. metadata-eval 29.5%

         \[\leadsto x.im \cdot \left(x.re \cdot -27\right) - \left(\left(x.re \cdot x.im\right) \cdot \color{blue}{2}\right) \cdot x.im \]

   5. Applied egg-rr 95.5%

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

   if 5.00000000000000015e256 < (-.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 44.8%

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

      1. difference-of-squares 55.5%

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

      2. *-commutative 55.5%

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

   3. Applied egg-rr 55.5%

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

      1. *-commutative 55.5%

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

      2. flip-+ 0.0%

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

      3. +-inverses 0.0%

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

      4. metadata-eval 0.0%

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

      5. +-inverses 0.0%

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

      6. distribute-neg-frac 0.0%

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

      7. flip-+ 59.1%

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

      8. neg-sub0 59.1%

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

      9. flip-+ 0.0%

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

      10. +-inverses 0.0%

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

      11. +-inverses 0.0%

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

   5. Applied egg-rr 0.0%

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

   7. Simplified 88.9%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq 5 \cdot 10^{+256}:\\ \;\;\;\;x.re \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right) - x.im \cdot \left(\left(x.re \cdot x.im\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right) - x.im \cdot -3\\ \end{array} \]

Alternative 4: 72.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} t_0 := x.re \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)\\ \mathbf{if}\;x.im \leq 1.85 \cdot 10^{-156}:\\ \;\;\;\;x.re \cdot x.im + t_0\\ \mathbf{elif}\;x.im \leq 5.1 \cdot 10^{+49}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.re - 27\right)\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 - x.im \cdot -3\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (* x.re (* (+ x.re x.im) (- x.re x.im)))))
   (if (<= x.im 1.85e-156)
     (+ (* x.re x.im) t_0)
     (if (<= x.im 5.1e+49)
       (-
        (* x.re (* x.re (- x.re 27.0)))
        (* x.im (+ (* x.re x.im) (* x.re x.im))))
       (- t_0 (* x.im -3.0))))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_re * ((x_46_re + x_46_im) * (x_46_re - x_46_im));
	double tmp;
	if (x_46_im <= 1.85e-156) {
		tmp = (x_46_re * x_46_im) + t_0;
	} else if (x_46_im <= 5.1e+49) {
		tmp = (x_46_re * (x_46_re * (x_46_re - 27.0))) - (x_46_im * ((x_46_re * x_46_im) + (x_46_re * x_46_im)));
	} else {
		tmp = t_0 - (x_46_im * -3.0);
	}
	return tmp;
}

Fortran

NOTE: x.im should be positive before calling this function
real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x_46re * ((x_46re + x_46im) * (x_46re - x_46im))
    if (x_46im <= 1.85d-156) then
        tmp = (x_46re * x_46im) + t_0
    else if (x_46im <= 5.1d+49) then
        tmp = (x_46re * (x_46re * (x_46re - 27.0d0))) - (x_46im * ((x_46re * x_46im) + (x_46re * x_46im)))
    else
        tmp = t_0 - (x_46im * (-3.0d0))
    end if
    code = tmp
end function

Java

x.im = Math.abs(x.im);
public static double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_re * ((x_46_re + x_46_im) * (x_46_re - x_46_im));
	double tmp;
	if (x_46_im <= 1.85e-156) {
		tmp = (x_46_re * x_46_im) + t_0;
	} else if (x_46_im <= 5.1e+49) {
		tmp = (x_46_re * (x_46_re * (x_46_re - 27.0))) - (x_46_im * ((x_46_re * x_46_im) + (x_46_re * x_46_im)));
	} else {
		tmp = t_0 - (x_46_im * -3.0);
	}
	return tmp;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	t_0 = x_46_re * ((x_46_re + x_46_im) * (x_46_re - x_46_im))
	tmp = 0
	if x_46_im <= 1.85e-156:
		tmp = (x_46_re * x_46_im) + t_0
	elif x_46_im <= 5.1e+49:
		tmp = (x_46_re * (x_46_re * (x_46_re - 27.0))) - (x_46_im * ((x_46_re * x_46_im) + (x_46_re * x_46_im)))
	else:
		tmp = t_0 - (x_46_im * -3.0)
	return tmp

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	t_0 = Float64(x_46_re * Float64(Float64(x_46_re + x_46_im) * Float64(x_46_re - x_46_im)))
	tmp = 0.0
	if (x_46_im <= 1.85e-156)
		tmp = Float64(Float64(x_46_re * x_46_im) + t_0);
	elseif (x_46_im <= 5.1e+49)
		tmp = Float64(Float64(x_46_re * Float64(x_46_re * Float64(x_46_re - 27.0))) - Float64(x_46_im * Float64(Float64(x_46_re * x_46_im) + Float64(x_46_re * x_46_im))));
	else
		tmp = Float64(t_0 - Float64(x_46_im * -3.0));
	end
	return tmp
end

MATLAB

x.im = abs(x.im)
function tmp_2 = code(x_46_re, x_46_im)
	t_0 = x_46_re * ((x_46_re + x_46_im) * (x_46_re - x_46_im));
	tmp = 0.0;
	if (x_46_im <= 1.85e-156)
		tmp = (x_46_re * x_46_im) + t_0;
	elseif (x_46_im <= 5.1e+49)
		tmp = (x_46_re * (x_46_re * (x_46_re - 27.0))) - (x_46_im * ((x_46_re * x_46_im) + (x_46_re * x_46_im)));
	else
		tmp = t_0 - (x_46_im * -3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(x$46$re * N[(N[(x$46$re + x$46$im), $MachinePrecision] * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$im, 1.85e-156], N[(N[(x$46$re * x$46$im), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[x$46$im, 5.1e+49], N[(N[(x$46$re * N[(x$46$re * N[(x$46$re - 27.0), $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], N[(t$95$0 - N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x.im < 1.85e-156

   1. Initial program 82.2%

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

      1. difference-of-squares 85.8%

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

      2. *-commutative 85.8%

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

   3. Applied egg-rr 85.8%

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

   4. Taylor expanded in x.re around 0 85.8%

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

   6. Simplified 67.7%

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

   if 1.85e-156 < x.im < 5.09999999999999956e49

   1. Initial program 100.0%

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

      1. difference-of-squares 100.0%

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

   3. Applied egg-rr 100.0%

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

   5. Simplified 46.4%

      \[\leadsto \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   6. Taylor expanded in x.im around 0 55.3%

      \[\leadsto \color{blue}{\left(x.re \cdot \left(x.re - 27\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   if 5.09999999999999956e49 < x.im

   1. Initial program 55.2%

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

      1. difference-of-squares 60.7%

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

      2. *-commutative 60.7%

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

   3. Applied egg-rr 60.7%

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

      1. *-commutative 60.7%

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

      2. flip-+ 0.0%

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

      3. +-inverses 0.0%

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

      4. metadata-eval 0.0%

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

      5. +-inverses 0.0%

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

      6. distribute-neg-frac 0.0%

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

      7. flip-+ 24.4%

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

      8. neg-sub0 24.4%

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

      9. flip-+ 0.0%

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

      10. +-inverses 0.0%

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

      11. +-inverses 0.0%

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

   5. Applied egg-rr 0.0%

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

   7. Simplified 61.7%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 1.85 \cdot 10^{-156}:\\ \;\;\;\;x.re \cdot x.im + x.re \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)\\ \mathbf{elif}\;x.im \leq 5.1 \cdot 10^{+49}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.re - 27\right)\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right) - x.im \cdot -3\\ \end{array} \]

Alternative 5: 72.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.re \leq -2.2 \cdot 10^{-58} \lor \neg \left(x.re \leq 6.1 \cdot 10^{-61}\right):\\ \;\;\;\;x.re \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right) - x.im \cdot -3\\ \mathbf{else}:\\ \;\;\;\;-27 \cdot \left(x.re \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.re -2.2e-58) (not (<= x.re 6.1e-61)))
   (- (* x.re (* (+ x.re x.im) (- x.re x.im))) (* x.im -3.0))
   (- (* -27.0 (* x.re x.im)) (* x.im (+ (* x.re x.im) (* x.re x.im))))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -2.2e-58) || !(x_46_re <= 6.1e-61)) {
		tmp = (x_46_re * ((x_46_re + x_46_im) * (x_46_re - x_46_im))) - (x_46_im * -3.0);
	} else {
		tmp = (-27.0 * (x_46_re * x_46_im)) - (x_46_im * ((x_46_re * x_46_im) + (x_46_re * x_46_im)));
	}
	return tmp;
}

Fortran

NOTE: x.im should be positive before calling this function
real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46re <= (-2.2d-58)) .or. (.not. (x_46re <= 6.1d-61))) then
        tmp = (x_46re * ((x_46re + x_46im) * (x_46re - x_46im))) - (x_46im * (-3.0d0))
    else
        tmp = ((-27.0d0) * (x_46re * x_46im)) - (x_46im * ((x_46re * x_46im) + (x_46re * x_46im)))
    end if
    code = tmp
end function

Java

x.im = Math.abs(x.im);
public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -2.2e-58) || !(x_46_re <= 6.1e-61)) {
		tmp = (x_46_re * ((x_46_re + x_46_im) * (x_46_re - x_46_im))) - (x_46_im * -3.0);
	} else {
		tmp = (-27.0 * (x_46_re * x_46_im)) - (x_46_im * ((x_46_re * x_46_im) + (x_46_re * x_46_im)));
	}
	return tmp;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_re <= -2.2e-58) or not (x_46_re <= 6.1e-61):
		tmp = (x_46_re * ((x_46_re + x_46_im) * (x_46_re - x_46_im))) - (x_46_im * -3.0)
	else:
		tmp = (-27.0 * (x_46_re * x_46_im)) - (x_46_im * ((x_46_re * x_46_im) + (x_46_re * x_46_im)))
	return tmp

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_re <= -2.2e-58) || !(x_46_re <= 6.1e-61))
		tmp = Float64(Float64(x_46_re * Float64(Float64(x_46_re + x_46_im) * Float64(x_46_re - x_46_im))) - Float64(x_46_im * -3.0));
	else
		tmp = Float64(Float64(-27.0 * Float64(x_46_re * x_46_im)) - Float64(x_46_im * Float64(Float64(x_46_re * x_46_im) + Float64(x_46_re * x_46_im))));
	end
	return tmp
end

MATLAB

x.im = abs(x.im)
function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_re <= -2.2e-58) || ~((x_46_re <= 6.1e-61)))
		tmp = (x_46_re * ((x_46_re + x_46_im) * (x_46_re - x_46_im))) - (x_46_im * -3.0);
	else
		tmp = (-27.0 * (x_46_re * x_46_im)) - (x_46_im * ((x_46_re * x_46_im) + (x_46_re * x_46_im)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$re, -2.2e-58], N[Not[LessEqual[x$46$re, 6.1e-61]], $MachinePrecision]], N[(N[(x$46$re * N[(N[(x$46$re + x$46$im), $MachinePrecision] * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision], N[(N[(-27.0 * N[(x$46$re * x$46$im), $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]]

Derivation

1. Split input into 2 regimes

2. if x.re < -2.20000000000000006e-58 or 6.1000000000000001e-61 < x.re

   1. Initial program 74.7%

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

      1. difference-of-squares 80.6%

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

      2. *-commutative 80.6%

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

   3. Applied egg-rr 80.6%

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

      1. *-commutative 80.6%

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

      2. flip-+ 0.0%

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

      3. +-inverses 0.0%

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

      4. metadata-eval 0.0%

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

      5. +-inverses 0.0%

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

      6. distribute-neg-frac 0.0%

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

      7. flip-+ 65.5%

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

      8. neg-sub0 65.5%

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

      9. flip-+ 0.0%

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

      10. +-inverses 0.0%

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

      11. +-inverses 0.0%

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

   5. Applied egg-rr 0.0%

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

   7. Simplified 90.4%

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

   if -2.20000000000000006e-58 < x.re < 6.1000000000000001e-61

   1. Initial program 85.1%

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

      1. difference-of-squares 85.1%

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

   3. Applied egg-rr 85.1%

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

   5. Simplified 35.1%

      \[\leadsto \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   6. Taylor expanded in x.re around 0 39.1%

      \[\leadsto \color{blue}{-27 \cdot \left(x.im \cdot x.re\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -2.2 \cdot 10^{-58} \lor \neg \left(x.re \leq 6.1 \cdot 10^{-61}\right):\\ \;\;\;\;x.re \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right) - x.im \cdot -3\\ \mathbf{else}:\\ \;\;\;\;-27 \cdot \left(x.re \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\\ \end{array} \]

Alternative 6: 72.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.re \leq -3.6 \cdot 10^{-58} \lor \neg \left(x.re \leq 6.1 \cdot 10^{-61}\right):\\ \;\;\;\;x.re \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right) - x.im \cdot -3\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot -27\right) - x.im \cdot \left(\left(x.re \cdot x.im\right) \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.re -3.6e-58) (not (<= x.re 6.1e-61)))
   (- (* x.re (* (+ x.re x.im) (- x.re x.im))) (* x.im -3.0))
   (- (* x.im (* x.re -27.0)) (* x.im (* (* x.re x.im) 2.0)))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -3.6e-58) || !(x_46_re <= 6.1e-61)) {
		tmp = (x_46_re * ((x_46_re + x_46_im) * (x_46_re - x_46_im))) - (x_46_im * -3.0);
	} else {
		tmp = (x_46_im * (x_46_re * -27.0)) - (x_46_im * ((x_46_re * x_46_im) * 2.0));
	}
	return tmp;
}

Fortran

NOTE: x.im should be positive before calling this function
real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46re <= (-3.6d-58)) .or. (.not. (x_46re <= 6.1d-61))) then
        tmp = (x_46re * ((x_46re + x_46im) * (x_46re - x_46im))) - (x_46im * (-3.0d0))
    else
        tmp = (x_46im * (x_46re * (-27.0d0))) - (x_46im * ((x_46re * x_46im) * 2.0d0))
    end if
    code = tmp
end function

Java

x.im = Math.abs(x.im);
public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -3.6e-58) || !(x_46_re <= 6.1e-61)) {
		tmp = (x_46_re * ((x_46_re + x_46_im) * (x_46_re - x_46_im))) - (x_46_im * -3.0);
	} else {
		tmp = (x_46_im * (x_46_re * -27.0)) - (x_46_im * ((x_46_re * x_46_im) * 2.0));
	}
	return tmp;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_re <= -3.6e-58) or not (x_46_re <= 6.1e-61):
		tmp = (x_46_re * ((x_46_re + x_46_im) * (x_46_re - x_46_im))) - (x_46_im * -3.0)
	else:
		tmp = (x_46_im * (x_46_re * -27.0)) - (x_46_im * ((x_46_re * x_46_im) * 2.0))
	return tmp

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_re <= -3.6e-58) || !(x_46_re <= 6.1e-61))
		tmp = Float64(Float64(x_46_re * Float64(Float64(x_46_re + x_46_im) * Float64(x_46_re - x_46_im))) - Float64(x_46_im * -3.0));
	else
		tmp = Float64(Float64(x_46_im * Float64(x_46_re * -27.0)) - Float64(x_46_im * Float64(Float64(x_46_re * x_46_im) * 2.0)));
	end
	return tmp
end

MATLAB

x.im = abs(x.im)
function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_re <= -3.6e-58) || ~((x_46_re <= 6.1e-61)))
		tmp = (x_46_re * ((x_46_re + x_46_im) * (x_46_re - x_46_im))) - (x_46_im * -3.0);
	else
		tmp = (x_46_im * (x_46_re * -27.0)) - (x_46_im * ((x_46_re * x_46_im) * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$re, -3.6e-58], N[Not[LessEqual[x$46$re, 6.1e-61]], $MachinePrecision]], N[(N[(x$46$re * N[(N[(x$46$re + x$46$im), $MachinePrecision] * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im * N[(x$46$re * -27.0), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(N[(x$46$re * x$46$im), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < -3.60000000000000009e-58 or 6.1000000000000001e-61 < x.re

   1. Initial program 74.7%

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

      1. difference-of-squares 80.6%

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

      2. *-commutative 80.6%

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

   3. Applied egg-rr 80.6%

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

      1. *-commutative 80.6%

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

      2. flip-+ 0.0%

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

      3. +-inverses 0.0%

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

      4. metadata-eval 0.0%

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

      5. +-inverses 0.0%

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

      6. distribute-neg-frac 0.0%

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

      7. flip-+ 65.5%

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

      8. neg-sub0 65.5%

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

      9. flip-+ 0.0%

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

      10. +-inverses 0.0%

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

      11. +-inverses 0.0%

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

   5. Applied egg-rr 0.0%

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

   7. Simplified 90.4%

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

   if -3.60000000000000009e-58 < x.re < 6.1000000000000001e-61

   1. Initial program 85.1%

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

      1. difference-of-squares 85.1%

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

   3. Applied egg-rr 85.1%

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

   5. Simplified 35.1%

      \[\leadsto \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   6. Taylor expanded in x.re around 0 39.1%

      \[\leadsto \color{blue}{-27 \cdot \left(x.im \cdot x.re\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   7. Step-by-step derivation

      1. associate-*r* 39.1%

         \[\leadsto \color{blue}{\left(-27 \cdot x.im\right) \cdot x.re} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      2. *-commutative 39.1%

         \[\leadsto \color{blue}{\left(x.im \cdot -27\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      3. associate-*l* 39.1%

         \[\leadsto \color{blue}{x.im \cdot \left(-27 \cdot x.re\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      4. *-commutative 39.1%

         \[\leadsto x.im \cdot \color{blue}{\left(x.re \cdot -27\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   8. Simplified 39.1%

      \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot -27\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   9. Step-by-step derivation

      1. *-commutative 39.1%

         \[\leadsto x.im \cdot \left(x.re \cdot -27\right) - \left(x.re \cdot x.im + \color{blue}{x.re \cdot x.im}\right) \cdot x.im \]

      2. *-un-lft-identity 39.1%

         \[\leadsto x.im \cdot \left(x.re \cdot -27\right) - \left(x.re \cdot x.im + \color{blue}{1 \cdot \left(x.re \cdot x.im\right)}\right) \cdot x.im \]

      3. *-un-lft-identity 39.1%

         \[\leadsto x.im \cdot \left(x.re \cdot -27\right) - \left(\color{blue}{1 \cdot \left(x.re \cdot x.im\right)} + 1 \cdot \left(x.re \cdot x.im\right)\right) \cdot x.im \]

      4. distribute-rgt-out 39.1%

         \[\leadsto x.im \cdot \left(x.re \cdot -27\right) - \color{blue}{\left(\left(x.re \cdot x.im\right) \cdot \left(1 + 1\right)\right)} \cdot x.im \]

      5. metadata-eval 39.1%

         \[\leadsto x.im \cdot \left(x.re \cdot -27\right) - \left(\left(x.re \cdot x.im\right) \cdot \color{blue}{2}\right) \cdot x.im \]

   10. Applied egg-rr 39.1%

       \[\leadsto x.im \cdot \left(x.re \cdot -27\right) - \color{blue}{\left(\left(x.re \cdot x.im\right) \cdot 2\right)} \cdot x.im \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -3.6 \cdot 10^{-58} \lor \neg \left(x.re \leq 6.1 \cdot 10^{-61}\right):\\ \;\;\;\;x.re \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right) - x.im \cdot -3\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot -27\right) - x.im \cdot \left(\left(x.re \cdot x.im\right) \cdot 2\right)\\ \end{array} \]

Alternative 7: 40.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} t_0 := x.re \cdot \left(x.re \cdot -27\right)\\ t_1 := x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right)\\ \mathbf{if}\;x.re \leq -3.5 \cdot 10^{+149}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.re \leq -6 \cdot 10^{-160}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x.re \leq 1.9 \cdot 10^{-180}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.re \leq 1.1 \cdot 10^{+157}:\\ \;\;\;\;-27 \cdot \left(x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (* x.re (* x.re -27.0))) (t_1 (* x.im (* x.re (- x.re 27.0)))))
   (if (<= x.re -3.5e+149)
     t_0
     (if (<= x.re -6e-160)
       t_1
       (if (<= x.re 1.9e-180)
         t_0
         (if (<= x.re 1.1e+157) (* -27.0 (* x.re x.im)) t_1))))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_re * (x_46_re * -27.0);
	double t_1 = x_46_im * (x_46_re * (x_46_re - 27.0));
	double tmp;
	if (x_46_re <= -3.5e+149) {
		tmp = t_0;
	} else if (x_46_re <= -6e-160) {
		tmp = t_1;
	} else if (x_46_re <= 1.9e-180) {
		tmp = t_0;
	} else if (x_46_re <= 1.1e+157) {
		tmp = -27.0 * (x_46_re * x_46_im);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: x.im should be positive before calling this function
real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x_46re * (x_46re * (-27.0d0))
    t_1 = x_46im * (x_46re * (x_46re - 27.0d0))
    if (x_46re <= (-3.5d+149)) then
        tmp = t_0
    else if (x_46re <= (-6d-160)) then
        tmp = t_1
    else if (x_46re <= 1.9d-180) then
        tmp = t_0
    else if (x_46re <= 1.1d+157) then
        tmp = (-27.0d0) * (x_46re * x_46im)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

x.im = Math.abs(x.im);
public static double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_re * (x_46_re * -27.0);
	double t_1 = x_46_im * (x_46_re * (x_46_re - 27.0));
	double tmp;
	if (x_46_re <= -3.5e+149) {
		tmp = t_0;
	} else if (x_46_re <= -6e-160) {
		tmp = t_1;
	} else if (x_46_re <= 1.9e-180) {
		tmp = t_0;
	} else if (x_46_re <= 1.1e+157) {
		tmp = -27.0 * (x_46_re * x_46_im);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	t_0 = x_46_re * (x_46_re * -27.0)
	t_1 = x_46_im * (x_46_re * (x_46_re - 27.0))
	tmp = 0
	if x_46_re <= -3.5e+149:
		tmp = t_0
	elif x_46_re <= -6e-160:
		tmp = t_1
	elif x_46_re <= 1.9e-180:
		tmp = t_0
	elif x_46_re <= 1.1e+157:
		tmp = -27.0 * (x_46_re * x_46_im)
	else:
		tmp = t_1
	return tmp

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	t_0 = Float64(x_46_re * Float64(x_46_re * -27.0))
	t_1 = Float64(x_46_im * Float64(x_46_re * Float64(x_46_re - 27.0)))
	tmp = 0.0
	if (x_46_re <= -3.5e+149)
		tmp = t_0;
	elseif (x_46_re <= -6e-160)
		tmp = t_1;
	elseif (x_46_re <= 1.9e-180)
		tmp = t_0;
	elseif (x_46_re <= 1.1e+157)
		tmp = Float64(-27.0 * Float64(x_46_re * x_46_im));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x.im = abs(x.im)
function tmp_2 = code(x_46_re, x_46_im)
	t_0 = x_46_re * (x_46_re * -27.0);
	t_1 = x_46_im * (x_46_re * (x_46_re - 27.0));
	tmp = 0.0;
	if (x_46_re <= -3.5e+149)
		tmp = t_0;
	elseif (x_46_re <= -6e-160)
		tmp = t_1;
	elseif (x_46_re <= 1.9e-180)
		tmp = t_0;
	elseif (x_46_re <= 1.1e+157)
		tmp = -27.0 * (x_46_re * x_46_im);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(x$46$re * N[(x$46$re * -27.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x$46$im * N[(x$46$re * N[(x$46$re - 27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$re, -3.5e+149], t$95$0, If[LessEqual[x$46$re, -6e-160], t$95$1, If[LessEqual[x$46$re, 1.9e-180], t$95$0, If[LessEqual[x$46$re, 1.1e+157], N[(-27.0 * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -3.50000000000000011e149 or -5.99999999999999993e-160 < x.re < 1.9e-180

   1. Initial program 65.2%

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

      1. difference-of-squares 70.6%

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

   3. Applied egg-rr 70.6%

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

   5. Simplified 49.2%

      \[\leadsto \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   7. Applied egg-rr 61.3%

      \[\leadsto \color{blue}{\left(\left(x.re + x.im\right) \cdot \left(x.re + -27\right)\right) \cdot x.re} \]

   8. Taylor expanded in x.im around 0 64.8%

      \[\leadsto \color{blue}{\left(x.re \cdot \left(x.re - 27\right)\right)} \cdot x.re \]

   9. Taylor expanded in x.re around 0 63.9%

      \[\leadsto \color{blue}{\left(-27 \cdot x.re\right)} \cdot x.re \]
   10. Step-by-step derivation

       1. *-commutative 63.9%

          \[\leadsto \color{blue}{\left(x.re \cdot -27\right)} \cdot x.re \]

   11. Simplified 63.9%

       \[\leadsto \color{blue}{\left(x.re \cdot -27\right)} \cdot x.re \]

   if -3.50000000000000011e149 < x.re < -5.99999999999999993e-160 or 1.1000000000000001e157 < x.re

   1. Initial program 82.2%

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

      1. difference-of-squares 85.2%

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

   3. Applied egg-rr 85.2%

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

   5. Simplified 50.1%

      \[\leadsto \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   7. Applied egg-rr 50.1%

      \[\leadsto \color{blue}{\left(\left(x.re + x.im\right) \cdot \left(x.re + -27\right)\right) \cdot x.re} \]

   8. Taylor expanded in x.im around inf 23.4%

      \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right)} \]

   if 1.9e-180 < x.re < 1.1000000000000001e157

   1. Initial program 93.9%

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

      1. difference-of-squares 95.5%

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

   3. Applied egg-rr 95.5%

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

   5. Simplified 43.0%

      \[\leadsto \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   7. Applied egg-rr 30.2%

      \[\leadsto \color{blue}{\left(\left(x.re + x.im\right) \cdot \left(x.re + -27\right)\right) \cdot x.re} \]

   8. Taylor expanded in x.re around 0 18.2%

      \[\leadsto \color{blue}{-27 \cdot \left(x.im \cdot x.re\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 36.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -3.5 \cdot 10^{+149}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot -27\right)\\ \mathbf{elif}\;x.re \leq -6 \cdot 10^{-160}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right)\\ \mathbf{elif}\;x.re \leq 1.9 \cdot 10^{-180}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot -27\right)\\ \mathbf{elif}\;x.re \leq 1.1 \cdot 10^{+157}:\\ \;\;\;\;-27 \cdot \left(x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right)\\ \end{array} \]

Alternative 8: 62.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.im \leq 2.2 \cdot 10^{+129}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.re - 27\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot -27\right) - x.im \cdot \left(\left(x.re \cdot x.im\right) \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im 2.2e+129)
   (* x.re (* x.re (- x.re 27.0)))
   (- (* x.im (* x.re -27.0)) (* x.im (* (* x.re x.im) 2.0)))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 2.2e+129) {
		tmp = x_46_re * (x_46_re * (x_46_re - 27.0));
	} else {
		tmp = (x_46_im * (x_46_re * -27.0)) - (x_46_im * ((x_46_re * x_46_im) * 2.0));
	}
	return tmp;
}

Fortran

NOTE: x.im should be positive before calling this function
real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46im <= 2.2d+129) then
        tmp = x_46re * (x_46re * (x_46re - 27.0d0))
    else
        tmp = (x_46im * (x_46re * (-27.0d0))) - (x_46im * ((x_46re * x_46im) * 2.0d0))
    end if
    code = tmp
end function

Java

x.im = Math.abs(x.im);
public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 2.2e+129) {
		tmp = x_46_re * (x_46_re * (x_46_re - 27.0));
	} else {
		tmp = (x_46_im * (x_46_re * -27.0)) - (x_46_im * ((x_46_re * x_46_im) * 2.0));
	}
	return tmp;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= 2.2e+129:
		tmp = x_46_re * (x_46_re * (x_46_re - 27.0))
	else:
		tmp = (x_46_im * (x_46_re * -27.0)) - (x_46_im * ((x_46_re * x_46_im) * 2.0))
	return tmp

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= 2.2e+129)
		tmp = Float64(x_46_re * Float64(x_46_re * Float64(x_46_re - 27.0)));
	else
		tmp = Float64(Float64(x_46_im * Float64(x_46_re * -27.0)) - Float64(x_46_im * Float64(Float64(x_46_re * x_46_im) * 2.0)));
	end
	return tmp
end

MATLAB

x.im = abs(x.im)
function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_im <= 2.2e+129)
		tmp = x_46_re * (x_46_re * (x_46_re - 27.0));
	else
		tmp = (x_46_im * (x_46_re * -27.0)) - (x_46_im * ((x_46_re * x_46_im) * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, 2.2e+129], N[(x$46$re * N[(x$46$re * N[(x$46$re - 27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im * N[(x$46$re * -27.0), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(N[(x$46$re * x$46$im), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 2.1999999999999999e129

   1. Initial program 83.6%

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

      1. difference-of-squares 86.3%

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

   3. Applied egg-rr 86.3%

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

   5. Simplified 49.9%

      \[\leadsto \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   7. Applied egg-rr 53.6%

      \[\leadsto \color{blue}{\left(\left(x.re + x.im\right) \cdot \left(x.re + -27\right)\right) \cdot x.re} \]

   8. Taylor expanded in x.im around 0 52.5%

      \[\leadsto \color{blue}{\left(x.re \cdot \left(x.re - 27\right)\right)} \cdot x.re \]

   if 2.1999999999999999e129 < x.im

   1. Initial program 50.9%

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

      1. difference-of-squares 59.1%

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

   3. Applied egg-rr 59.1%

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

   5. Simplified 37.3%

      \[\leadsto \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   6. Taylor expanded in x.re around 0 58.9%

      \[\leadsto \color{blue}{-27 \cdot \left(x.im \cdot x.re\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   7. Step-by-step derivation

      1. associate-*r* 58.9%

         \[\leadsto \color{blue}{\left(-27 \cdot x.im\right) \cdot x.re} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      2. *-commutative 58.9%

         \[\leadsto \color{blue}{\left(x.im \cdot -27\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      3. associate-*l* 58.9%

         \[\leadsto \color{blue}{x.im \cdot \left(-27 \cdot x.re\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

      4. *-commutative 58.9%

         \[\leadsto x.im \cdot \color{blue}{\left(x.re \cdot -27\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   8. Simplified 58.9%

      \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot -27\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   9. Step-by-step derivation

      1. *-commutative 58.9%

         \[\leadsto x.im \cdot \left(x.re \cdot -27\right) - \left(x.re \cdot x.im + \color{blue}{x.re \cdot x.im}\right) \cdot x.im \]

      2. *-un-lft-identity 58.9%

         \[\leadsto x.im \cdot \left(x.re \cdot -27\right) - \left(x.re \cdot x.im + \color{blue}{1 \cdot \left(x.re \cdot x.im\right)}\right) \cdot x.im \]

      3. *-un-lft-identity 58.9%

         \[\leadsto x.im \cdot \left(x.re \cdot -27\right) - \left(\color{blue}{1 \cdot \left(x.re \cdot x.im\right)} + 1 \cdot \left(x.re \cdot x.im\right)\right) \cdot x.im \]

      4. distribute-rgt-out 58.9%

         \[\leadsto x.im \cdot \left(x.re \cdot -27\right) - \color{blue}{\left(\left(x.re \cdot x.im\right) \cdot \left(1 + 1\right)\right)} \cdot x.im \]

      5. metadata-eval 58.9%

         \[\leadsto x.im \cdot \left(x.re \cdot -27\right) - \left(\left(x.re \cdot x.im\right) \cdot \color{blue}{2}\right) \cdot x.im \]

   10. Applied egg-rr 58.9%

       \[\leadsto x.im \cdot \left(x.re \cdot -27\right) - \color{blue}{\left(\left(x.re \cdot x.im\right) \cdot 2\right)} \cdot x.im \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 2.2 \cdot 10^{+129}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.re - 27\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot -27\right) - x.im \cdot \left(\left(x.re \cdot x.im\right) \cdot 2\right)\\ \end{array} \]

Alternative 9: 55.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.im \leq 7 \cdot 10^{-14}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.re - 27\right)\right)\\ \mathbf{elif}\;x.im \leq 1.78 \cdot 10^{+227}:\\ \;\;\;\;x.re \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re + -27\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re + x.im\right) \cdot \left(x.re \cdot -27\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im 7e-14)
   (* x.re (* x.re (- x.re 27.0)))
   (if (<= x.im 1.78e+227)
     (* x.re (* (+ x.re x.im) (+ x.re -27.0)))
     (* (+ x.re x.im) (* x.re -27.0)))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 7e-14) {
		tmp = x_46_re * (x_46_re * (x_46_re - 27.0));
	} else if (x_46_im <= 1.78e+227) {
		tmp = x_46_re * ((x_46_re + x_46_im) * (x_46_re + -27.0));
	} else {
		tmp = (x_46_re + x_46_im) * (x_46_re * -27.0);
	}
	return tmp;
}

Fortran

NOTE: x.im should be positive before calling this function
real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46im <= 7d-14) then
        tmp = x_46re * (x_46re * (x_46re - 27.0d0))
    else if (x_46im <= 1.78d+227) then
        tmp = x_46re * ((x_46re + x_46im) * (x_46re + (-27.0d0)))
    else
        tmp = (x_46re + x_46im) * (x_46re * (-27.0d0))
    end if
    code = tmp
end function

Java

x.im = Math.abs(x.im);
public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 7e-14) {
		tmp = x_46_re * (x_46_re * (x_46_re - 27.0));
	} else if (x_46_im <= 1.78e+227) {
		tmp = x_46_re * ((x_46_re + x_46_im) * (x_46_re + -27.0));
	} else {
		tmp = (x_46_re + x_46_im) * (x_46_re * -27.0);
	}
	return tmp;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= 7e-14:
		tmp = x_46_re * (x_46_re * (x_46_re - 27.0))
	elif x_46_im <= 1.78e+227:
		tmp = x_46_re * ((x_46_re + x_46_im) * (x_46_re + -27.0))
	else:
		tmp = (x_46_re + x_46_im) * (x_46_re * -27.0)
	return tmp

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= 7e-14)
		tmp = Float64(x_46_re * Float64(x_46_re * Float64(x_46_re - 27.0)));
	elseif (x_46_im <= 1.78e+227)
		tmp = Float64(x_46_re * Float64(Float64(x_46_re + x_46_im) * Float64(x_46_re + -27.0)));
	else
		tmp = Float64(Float64(x_46_re + x_46_im) * Float64(x_46_re * -27.0));
	end
	return tmp
end

MATLAB

x.im = abs(x.im)
function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_im <= 7e-14)
		tmp = x_46_re * (x_46_re * (x_46_re - 27.0));
	elseif (x_46_im <= 1.78e+227)
		tmp = x_46_re * ((x_46_re + x_46_im) * (x_46_re + -27.0));
	else
		tmp = (x_46_re + x_46_im) * (x_46_re * -27.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, 7e-14], N[(x$46$re * N[(x$46$re * N[(x$46$re - 27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 1.78e+227], N[(x$46$re * N[(N[(x$46$re + x$46$im), $MachinePrecision] * N[(x$46$re + -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re + x$46$im), $MachinePrecision] * N[(x$46$re * -27.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.im < 7.0000000000000005e-14

   1. Initial program 84.3%

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

      1. difference-of-squares 87.4%

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

   3. Applied egg-rr 87.4%

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

   5. Simplified 53.2%

      \[\leadsto \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   7. Applied egg-rr 55.3%

      \[\leadsto \color{blue}{\left(\left(x.re + x.im\right) \cdot \left(x.re + -27\right)\right) \cdot x.re} \]

   8. Taylor expanded in x.im around 0 54.2%

      \[\leadsto \color{blue}{\left(x.re \cdot \left(x.re - 27\right)\right)} \cdot x.re \]

   if 7.0000000000000005e-14 < x.im < 1.77999999999999995e227

   1. Initial program 65.8%

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

      1. difference-of-squares 67.8%

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

   3. Applied egg-rr 67.8%

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

   5. Simplified 32.2%

      \[\leadsto \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   7. Applied egg-rr 33.3%

      \[\leadsto \color{blue}{\left(\left(x.re + x.im\right) \cdot \left(x.re + -27\right)\right) \cdot x.re} \]

   if 1.77999999999999995e227 < x.im

   1. Initial program 54.9%

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

      1. difference-of-squares 68.2%

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

   3. Applied egg-rr 68.2%

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

   5. Simplified 37.3%

      \[\leadsto \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   7. Applied egg-rr 28.5%

      \[\leadsto \color{blue}{\left(x.re + x.im\right) \cdot \left(x.re \cdot \left(x.re + -27\right)\right)} \]

   8. Taylor expanded in x.re around 0 48.5%

      \[\leadsto \left(x.re + x.im\right) \cdot \color{blue}{\left(-27 \cdot x.re\right)} \]

   10. Simplified 48.5%

       \[\leadsto \left(x.re + x.im\right) \cdot \color{blue}{\left(x.re \cdot -27\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 7 \cdot 10^{-14}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.re - 27\right)\right)\\ \mathbf{elif}\;x.im \leq 1.78 \cdot 10^{+227}:\\ \;\;\;\;x.re \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re + -27\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re + x.im\right) \cdot \left(x.re \cdot -27\right)\\ \end{array} \]

Alternative 10: 52.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.im \leq 9.5 \cdot 10^{+234}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.re - 27\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-27 \cdot \left(x.re \cdot x.im\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im 9.5e+234)
   (* x.re (* x.re (- x.re 27.0)))
   (* -27.0 (* x.re x.im))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 9.5e+234) {
		tmp = x_46_re * (x_46_re * (x_46_re - 27.0));
	} else {
		tmp = -27.0 * (x_46_re * x_46_im);
	}
	return tmp;
}

Fortran

NOTE: x.im should be positive before calling this function
real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46im <= 9.5d+234) then
        tmp = x_46re * (x_46re * (x_46re - 27.0d0))
    else
        tmp = (-27.0d0) * (x_46re * x_46im)
    end if
    code = tmp
end function

Java

x.im = Math.abs(x.im);
public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 9.5e+234) {
		tmp = x_46_re * (x_46_re * (x_46_re - 27.0));
	} else {
		tmp = -27.0 * (x_46_re * x_46_im);
	}
	return tmp;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= 9.5e+234:
		tmp = x_46_re * (x_46_re * (x_46_re - 27.0))
	else:
		tmp = -27.0 * (x_46_re * x_46_im)
	return tmp

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= 9.5e+234)
		tmp = Float64(x_46_re * Float64(x_46_re * Float64(x_46_re - 27.0)));
	else
		tmp = Float64(-27.0 * Float64(x_46_re * x_46_im));
	end
	return tmp
end

MATLAB

x.im = abs(x.im)
function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_im <= 9.5e+234)
		tmp = x_46_re * (x_46_re * (x_46_re - 27.0));
	else
		tmp = -27.0 * (x_46_re * x_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, 9.5e+234], N[(x$46$re * N[(x$46$re * N[(x$46$re - 27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-27.0 * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 9.49999999999999938e234

   1. Initial program 80.1%

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

      1. difference-of-squares 83.0%

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

   3. Applied egg-rr 83.0%

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

   5. Simplified 48.6%

      \[\leadsto \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   7. Applied egg-rr 50.4%

      \[\leadsto \color{blue}{\left(\left(x.re + x.im\right) \cdot \left(x.re + -27\right)\right) \cdot x.re} \]

   8. Taylor expanded in x.im around 0 48.4%

      \[\leadsto \color{blue}{\left(x.re \cdot \left(x.re - 27\right)\right)} \cdot x.re \]

   if 9.49999999999999938e234 < x.im

   1. Initial program 58.1%

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

      1. difference-of-squares 72.4%

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

   3. Applied egg-rr 72.4%

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

   5. Simplified 38.6%

      \[\leadsto \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   7. Applied egg-rr 30.2%

      \[\leadsto \color{blue}{\left(\left(x.re + x.im\right) \cdot \left(x.re + -27\right)\right) \cdot x.re} \]

   8. Taylor expanded in x.re around 0 51.6%

      \[\leadsto \color{blue}{-27 \cdot \left(x.im \cdot x.re\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 9.5 \cdot 10^{+234}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.re - 27\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-27 \cdot \left(x.re \cdot x.im\right)\\ \end{array} \]

Alternative 11: 53.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.im \leq 2.9 \cdot 10^{+190}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.re - 27\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re + x.im\right) \cdot \left(x.re \cdot -27\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im 2.9e+190)
   (* x.re (* x.re (- x.re 27.0)))
   (* (+ x.re x.im) (* x.re -27.0))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 2.9e+190) {
		tmp = x_46_re * (x_46_re * (x_46_re - 27.0));
	} else {
		tmp = (x_46_re + x_46_im) * (x_46_re * -27.0);
	}
	return tmp;
}

Fortran

NOTE: x.im should be positive before calling this function
real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46im <= 2.9d+190) then
        tmp = x_46re * (x_46re * (x_46re - 27.0d0))
    else
        tmp = (x_46re + x_46im) * (x_46re * (-27.0d0))
    end if
    code = tmp
end function

Java

x.im = Math.abs(x.im);
public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 2.9e+190) {
		tmp = x_46_re * (x_46_re * (x_46_re - 27.0));
	} else {
		tmp = (x_46_re + x_46_im) * (x_46_re * -27.0);
	}
	return tmp;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= 2.9e+190:
		tmp = x_46_re * (x_46_re * (x_46_re - 27.0))
	else:
		tmp = (x_46_re + x_46_im) * (x_46_re * -27.0)
	return tmp

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= 2.9e+190)
		tmp = Float64(x_46_re * Float64(x_46_re * Float64(x_46_re - 27.0)));
	else
		tmp = Float64(Float64(x_46_re + x_46_im) * Float64(x_46_re * -27.0));
	end
	return tmp
end

MATLAB

x.im = abs(x.im)
function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_im <= 2.9e+190)
		tmp = x_46_re * (x_46_re * (x_46_re - 27.0));
	else
		tmp = (x_46_re + x_46_im) * (x_46_re * -27.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, 2.9e+190], N[(x$46$re * N[(x$46$re * N[(x$46$re - 27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re + x$46$im), $MachinePrecision] * N[(x$46$re * -27.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 2.89999999999999989e190

   1. Initial program 81.4%

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

      1. difference-of-squares 84.4%

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

   3. Applied egg-rr 84.4%

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

   5. Simplified 48.6%

      \[\leadsto \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   7. Applied egg-rr 51.2%

      \[\leadsto \color{blue}{\left(\left(x.re + x.im\right) \cdot \left(x.re + -27\right)\right) \cdot x.re} \]

   8. Taylor expanded in x.im around 0 49.6%

      \[\leadsto \color{blue}{\left(x.re \cdot \left(x.re - 27\right)\right)} \cdot x.re \]

   if 2.89999999999999989e190 < 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. difference-of-squares 61.4%

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

   3. Applied egg-rr 61.4%

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

   5. Simplified 41.8%

      \[\leadsto \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   7. Applied egg-rr 29.5%

      \[\leadsto \color{blue}{\left(x.re + x.im\right) \cdot \left(x.re \cdot \left(x.re + -27\right)\right)} \]

   8. Taylor expanded in x.re around 0 34.5%

      \[\leadsto \left(x.re + x.im\right) \cdot \color{blue}{\left(-27 \cdot x.re\right)} \]

   10. Simplified 34.5%

       \[\leadsto \left(x.re + x.im\right) \cdot \color{blue}{\left(x.re \cdot -27\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 2.9 \cdot 10^{+190}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.re - 27\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re + x.im\right) \cdot \left(x.re \cdot -27\right)\\ \end{array} \]

Alternative 12: 28.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.im \leq 5.6 \cdot 10^{+235}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot -27\right)\\ \mathbf{else}:\\ \;\;\;\;-27 \cdot \left(x.re \cdot x.im\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im 5.6e+235) (* x.re (* x.re -27.0)) (* -27.0 (* x.re x.im))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 5.6e+235) {
		tmp = x_46_re * (x_46_re * -27.0);
	} else {
		tmp = -27.0 * (x_46_re * x_46_im);
	}
	return tmp;
}

Fortran

NOTE: x.im should be positive before calling this function
real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46im <= 5.6d+235) then
        tmp = x_46re * (x_46re * (-27.0d0))
    else
        tmp = (-27.0d0) * (x_46re * x_46im)
    end if
    code = tmp
end function

Java

x.im = Math.abs(x.im);
public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 5.6e+235) {
		tmp = x_46_re * (x_46_re * -27.0);
	} else {
		tmp = -27.0 * (x_46_re * x_46_im);
	}
	return tmp;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= 5.6e+235:
		tmp = x_46_re * (x_46_re * -27.0)
	else:
		tmp = -27.0 * (x_46_re * x_46_im)
	return tmp

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= 5.6e+235)
		tmp = Float64(x_46_re * Float64(x_46_re * -27.0));
	else
		tmp = Float64(-27.0 * Float64(x_46_re * x_46_im));
	end
	return tmp
end

MATLAB

x.im = abs(x.im)
function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_im <= 5.6e+235)
		tmp = x_46_re * (x_46_re * -27.0);
	else
		tmp = -27.0 * (x_46_re * x_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, 5.6e+235], N[(x$46$re * N[(x$46$re * -27.0), $MachinePrecision]), $MachinePrecision], N[(-27.0 * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 5.60000000000000052e235

   1. Initial program 80.1%

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

      1. difference-of-squares 83.0%

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

   3. Applied egg-rr 83.0%

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

   5. Simplified 48.6%

      \[\leadsto \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   7. Applied egg-rr 50.4%

      \[\leadsto \color{blue}{\left(\left(x.re + x.im\right) \cdot \left(x.re + -27\right)\right) \cdot x.re} \]

   8. Taylor expanded in x.im around 0 48.4%

      \[\leadsto \color{blue}{\left(x.re \cdot \left(x.re - 27\right)\right)} \cdot x.re \]

   9. Taylor expanded in x.re around 0 28.5%

      \[\leadsto \color{blue}{\left(-27 \cdot x.re\right)} \cdot x.re \]
   10. Step-by-step derivation

       1. *-commutative 28.5%

          \[\leadsto \color{blue}{\left(x.re \cdot -27\right)} \cdot x.re \]

   11. Simplified 28.5%

       \[\leadsto \color{blue}{\left(x.re \cdot -27\right)} \cdot x.re \]

   if 5.60000000000000052e235 < x.im

   1. Initial program 58.1%

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

      1. difference-of-squares 72.4%

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

   3. Applied egg-rr 72.4%

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

   5. Simplified 38.6%

      \[\leadsto \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

   7. Applied egg-rr 30.2%

      \[\leadsto \color{blue}{\left(\left(x.re + x.im\right) \cdot \left(x.re + -27\right)\right) \cdot x.re} \]

   8. Taylor expanded in x.re around 0 51.6%

      \[\leadsto \color{blue}{-27 \cdot \left(x.im \cdot x.re\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 5.6 \cdot 10^{+235}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot -27\right)\\ \mathbf{else}:\\ \;\;\;\;-27 \cdot \left(x.re \cdot x.im\right)\\ \end{array} \]

Alternative 13: 19.8% accurate, 3.8× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ -27 \cdot \left(x.re \cdot x.im\right) \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im) :precision binary64 (* -27.0 (* x.re x.im)))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	return -27.0 * (x_46_re * x_46_im);
}

Fortran

NOTE: x.im should be positive before calling this function
real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = (-27.0d0) * (x_46re * x_46im)
end function

Java

x.im = Math.abs(x.im);
public static double code(double x_46_re, double x_46_im) {
	return -27.0 * (x_46_re * x_46_im);
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	return -27.0 * (x_46_re * x_46_im)

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	return Float64(-27.0 * Float64(x_46_re * x_46_im))
end

MATLAB

x.im = abs(x.im)
function tmp = code(x_46_re, x_46_im)
	tmp = -27.0 * (x_46_re * x_46_im);
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := N[(-27.0 * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.9%

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

   1. difference-of-squares 82.4%

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

3. Applied egg-rr 82.4%

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

5. Simplified 48.1%

   \[\leadsto \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]

7. Applied egg-rr 49.3%

   \[\leadsto \color{blue}{\left(\left(x.re + x.im\right) \cdot \left(x.re + -27\right)\right) \cdot x.re} \]

8. Taylor expanded in x.re around 0 22.2%

   \[\leadsto \color{blue}{-27 \cdot \left(x.im \cdot x.re\right)} \]

9. Final simplification 22.2%

   \[\leadsto -27 \cdot \left(x.re \cdot x.im\right) \]

Alternative 14: 3.1% accurate, 9.5× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: x.im should be positive before calling this function
real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = -x_46re
end function

Java

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

Python

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

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	return Float64(-x_46_re)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

3. Simplified 73.8%

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

4. Taylor expanded in x.re around 0 52.5%

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

6. Simplified 3.3%

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

7. Final simplification 3.3%

   \[\leadsto -x.re \]

Developer target: 87.3% 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 2023305 
(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)))