math.cube on complex, real part

Percentage Accurate: 83.0% → 99.1%

Time: 6.0s

Alternatives: 9

Speedup: 0.5×

• 45.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 83.0% 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: 99.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} t_0 := \frac{x.re}{-x.re}\\ \mathbf{if}\;x.re \leq -5 \cdot 10^{+102}:\\ \;\;\;\;x.re \cdot \left(\left(x.re + -27\right) \cdot \left(x.re + x.im\right)\right) + t_0\\ \mathbf{elif}\;x.re \leq 6300:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.im \cdot -3\right) + {x.re}^{3}\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) + t_0\\ \end{array} \end{array} \]

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))))
   (if (<= x.re -5e+102)
     (+ (* x.re (* (+ x.re -27.0) (+ x.re x.im))) t_0)
     (if (<= x.re 6300.0)
       (+ (* (* x.re x.im) (* x.im -3.0)) (pow x.re 3.0))
       (+ (* x.re (* (- x.re x.im) (+ x.re x.im))) t_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;
	double tmp;
	if (x_46_re <= -5e+102) {
		tmp = (x_46_re * ((x_46_re + -27.0) * (x_46_re + x_46_im))) + t_0;
	} else if (x_46_re <= 6300.0) {
		tmp = ((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))) + t_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
    if (x_46re <= (-5d+102)) then
        tmp = (x_46re * ((x_46re + (-27.0d0)) * (x_46re + x_46im))) + t_0
    else if (x_46re <= 6300.0d0) then
        tmp = ((x_46re * x_46im) * (x_46im * (-3.0d0))) + (x_46re ** 3.0d0)
    else
        tmp = (x_46re * ((x_46re - x_46im) * (x_46re + x_46im))) + t_0
    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;
	double tmp;
	if (x_46_re <= -5e+102) {
		tmp = (x_46_re * ((x_46_re + -27.0) * (x_46_re + x_46_im))) + t_0;
	} else if (x_46_re <= 6300.0) {
		tmp = ((x_46_re * x_46_im) * (x_46_im * -3.0)) + Math.pow(x_46_re, 3.0);
	} else {
		tmp = (x_46_re * ((x_46_re - x_46_im) * (x_46_re + x_46_im))) + t_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
	tmp = 0
	if x_46_re <= -5e+102:
		tmp = (x_46_re * ((x_46_re + -27.0) * (x_46_re + x_46_im))) + t_0
	elif x_46_re <= 6300.0:
		tmp = ((x_46_re * x_46_im) * (x_46_im * -3.0)) + math.pow(x_46_re, 3.0)
	else:
		tmp = (x_46_re * ((x_46_re - x_46_im) * (x_46_re + x_46_im))) + t_0
	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))
	tmp = 0.0
	if (x_46_re <= -5e+102)
		tmp = Float64(Float64(x_46_re * Float64(Float64(x_46_re + -27.0) * Float64(x_46_re + x_46_im))) + t_0);
	elseif (x_46_re <= 6300.0)
		tmp = Float64(Float64(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))) + t_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;
	tmp = 0.0;
	if (x_46_re <= -5e+102)
		tmp = (x_46_re * ((x_46_re + -27.0) * (x_46_re + x_46_im))) + t_0;
	elseif (x_46_re <= 6300.0)
		tmp = ((x_46_re * x_46_im) * (x_46_im * -3.0)) + (x_46_re ^ 3.0);
	else
		tmp = (x_46_re * ((x_46_re - x_46_im) * (x_46_re + x_46_im))) + t_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 / (-x$46$re)), $MachinePrecision]}, If[LessEqual[x$46$re, -5e+102], N[(N[(x$46$re * N[(N[(x$46$re + -27.0), $MachinePrecision] * N[(x$46$re + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[x$46$re, 6300.0], N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(x$46$im * -3.0), $MachinePrecision]), $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] + t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -5e102

   1. Initial program 75.0%

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

      1. *-commutative 75.0%

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

      2. flip3-+ 40.9%

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

      3. associate-*r/ 40.9%

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

      4. *-commutative 40.9%

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

      5. count-2 40.9%

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

      6. pow2 40.9%

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

      7. *-commutative 40.9%

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

      8. *-commutative 40.9%

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

      9. *-commutative 40.9%

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

      10. +-inverses 40.9%

          \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \frac{x.im \cdot \left(2 \cdot {\left(x.re \cdot x.im\right)}^{3}\right)}{{\left(x.re \cdot x.im\right)}^{2} + \color{blue}{0}} \]

   3. Applied egg-rr 40.9%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \color{blue}{\frac{x.im \cdot \left(2 \cdot {\left(x.re \cdot x.im\right)}^{3}\right)}{{\left(x.re \cdot x.im\right)}^{2} + 0}} \]

   5. Simplified 88.6%

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

      1. difference-of-squares 100.0%

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

   7. Applied egg-rr 100.0%

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

   9. Simplified 93.2%

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

   if -5e102 < x.re < 6300

   1. Initial program 88.1%

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

   3. Simplified 88.2%

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

      1. associate-*r* 88.2%

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

      2. associate-*l* 88.1%

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

      3. +-commutative 88.1%

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

      4. associate-*l* 88.2%

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

      5. associate-*r* 88.2%

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

      6. associate-*r* 99.8%

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

      7. 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%

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

      1. fma-udef 99.8%

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

   7. Applied egg-rr 99.8%

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

   if 6300 < x.re

   1. Initial program 79.2%

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

      1. *-commutative 79.2%

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

      2. flip3-+ 30.2%

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

      3. associate-*r/ 30.2%

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

      4. *-commutative 30.2%

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

      5. count-2 30.2%

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

      6. pow2 30.2%

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

      7. *-commutative 30.2%

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

      8. *-commutative 30.2%

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

      9. *-commutative 30.2%

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

      10. +-inverses 30.2%

          \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \frac{x.im \cdot \left(2 \cdot {\left(x.re \cdot x.im\right)}^{3}\right)}{{\left(x.re \cdot x.im\right)}^{2} + \color{blue}{0}} \]

   3. Applied egg-rr 30.2%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \color{blue}{\frac{x.im \cdot \left(2 \cdot {\left(x.re \cdot x.im\right)}^{3}\right)}{{\left(x.re \cdot x.im\right)}^{2} + 0}} \]

   5. Simplified 92.5%

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

      1. difference-of-squares 100.0%

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

      2. *-commutative 100.0%

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

   7. Applied egg-rr 100.0%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -5 \cdot 10^{+102}:\\ \;\;\;\;x.re \cdot \left(\left(x.re + -27\right) \cdot \left(x.re + x.im\right)\right) + \frac{x.re}{-x.re}\\ \mathbf{elif}\;x.re \leq 6300:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.im \cdot -3\right) + {x.re}^{3}\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) + \frac{x.re}{-x.re}\\ \end{array} \]

Alternative 2: 94.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} t_0 := x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right)\\ \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq 2 \cdot 10^{+281}:\\ \;\;\;\;t_0 - x.im \cdot \left(\left(x.re \cdot x.im\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{x.re}{-x.re}\\ \end{array} \end{array} \]

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))))
        2e+281)
     (- t_0 (* x.im (* (* x.re x.im) 2.0)))
     (+ t_0 (/ x.re (- x.re))))))

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)))) <= 2e+281) {
		tmp = t_0 - (x_46_im * ((x_46_re * x_46_im) * 2.0));
	} else {
		tmp = t_0 + (x_46_re / -x_46_re);
	}
	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)))) <= 2d+281) then
        tmp = t_0 - (x_46im * ((x_46re * x_46im) * 2.0d0))
    else
        tmp = t_0 + (x_46re / -x_46re)
    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)))) <= 2e+281) {
		tmp = t_0 - (x_46_im * ((x_46_re * x_46_im) * 2.0));
	} else {
		tmp = t_0 + (x_46_re / -x_46_re);
	}
	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)))) <= 2e+281:
		tmp = t_0 - (x_46_im * ((x_46_re * x_46_im) * 2.0))
	else:
		tmp = t_0 + (x_46_re / -x_46_re)
	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)))) <= 2e+281)
		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_re / Float64(-x_46_re)));
	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)))) <= 2e+281)
		tmp = t_0 - (x_46_im * ((x_46_re * x_46_im) * 2.0));
	else
		tmp = t_0 + (x_46_re / -x_46_re);
	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], 2e+281], 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$46$re / (-x$46$re)), $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)) < 2.0000000000000001e281

   1. Initial program 92.7%

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

      1. difference-of-squares 33.1%

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

      2. *-commutative 33.1%

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

   3. Applied egg-rr 92.7%

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

   4. Taylor expanded in x.re around 0 92.7%

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

   if 2.0000000000000001e281 < (-.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 67.2%

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

      1. *-commutative 67.2%

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

      2. flip3-+ 23.5%

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

      3. associate-*r/ 23.5%

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

      4. *-commutative 23.5%

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

      5. count-2 23.5%

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

      6. pow2 23.5%

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

      7. *-commutative 23.5%

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

      8. *-commutative 23.5%

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

      9. *-commutative 23.5%

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

      10. +-inverses 23.5%

          \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \frac{x.im \cdot \left(2 \cdot {\left(x.re \cdot x.im\right)}^{3}\right)}{{\left(x.re \cdot x.im\right)}^{2} + \color{blue}{0}} \]

   3. Applied egg-rr 23.5%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \color{blue}{\frac{x.im \cdot \left(2 \cdot {\left(x.re \cdot x.im\right)}^{3}\right)}{{\left(x.re \cdot x.im\right)}^{2} + 0}} \]

   5. Simplified 82.2%

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

      1. difference-of-squares 92.5%

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

      2. *-commutative 92.5%

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

   7. Applied egg-rr 92.5%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq 2 \cdot 10^{+281}:\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) - x.im \cdot \left(\left(x.re \cdot x.im\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) + \frac{x.re}{-x.re}\\ \end{array} \]

Alternative 3: 54.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.re \leq -1.75 \cdot 10^{-41} \lor \neg \left(x.re \leq 9 \cdot 10^{-59}\right):\\ \;\;\;\;x.re \cdot \left(\left(x.re + -27\right) \cdot \left(x.re + x.im\right)\right) + \frac{x.re}{-x.re}\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot -27\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.re -1.75e-41) (not (<= x.re 9e-59)))
   (+ (* x.re (* (+ x.re -27.0) (+ x.re x.im))) (/ x.re (- 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_re <= -1.75e-41) || !(x_46_re <= 9e-59)) {
		tmp = (x_46_re * ((x_46_re + -27.0) * (x_46_re + x_46_im))) + (x_46_re / -x_46_re);
	} else {
		tmp = 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_46re <= (-1.75d-41)) .or. (.not. (x_46re <= 9d-59))) then
        tmp = (x_46re * ((x_46re + (-27.0d0)) * (x_46re + x_46im))) + (x_46re / -x_46re)
    else
        tmp = 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_re <= -1.75e-41) || !(x_46_re <= 9e-59)) {
		tmp = (x_46_re * ((x_46_re + -27.0) * (x_46_re + x_46_im))) + (x_46_re / -x_46_re);
	} else {
		tmp = 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_re <= -1.75e-41) or not (x_46_re <= 9e-59):
		tmp = (x_46_re * ((x_46_re + -27.0) * (x_46_re + x_46_im))) + (x_46_re / -x_46_re)
	else:
		tmp = 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_re <= -1.75e-41) || !(x_46_re <= 9e-59))
		tmp = Float64(Float64(x_46_re * Float64(Float64(x_46_re + -27.0) * Float64(x_46_re + x_46_im))) + Float64(x_46_re / Float64(-x_46_re)));
	else
		tmp = Float64(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_re <= -1.75e-41) || ~((x_46_re <= 9e-59)))
		tmp = (x_46_re * ((x_46_re + -27.0) * (x_46_re + x_46_im))) + (x_46_re / -x_46_re);
	else
		tmp = 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[Or[LessEqual[x$46$re, -1.75e-41], N[Not[LessEqual[x$46$re, 9e-59]], $MachinePrecision]], N[(N[(x$46$re * N[(N[(x$46$re + -27.0), $MachinePrecision] * N[(x$46$re + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re / (-x$46$re)), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(x$46$re * -27.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < -1.75e-41 or 9.00000000000000023e-59 < x.re

   1. Initial program 83.4%

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

      1. *-commutative 83.4%

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

      2. flip3-+ 35.6%

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

      3. associate-*r/ 33.6%

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

      4. *-commutative 33.6%

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

      5. count-2 33.6%

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

      6. pow2 33.6%

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

      7. *-commutative 33.6%

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

      8. *-commutative 33.6%

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

      9. *-commutative 33.6%

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

      10. +-inverses 33.6%

          \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \frac{x.im \cdot \left(2 \cdot {\left(x.re \cdot x.im\right)}^{3}\right)}{{\left(x.re \cdot x.im\right)}^{2} + \color{blue}{0}} \]

   3. Applied egg-rr 33.6%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \color{blue}{\frac{x.im \cdot \left(2 \cdot {\left(x.re \cdot x.im\right)}^{3}\right)}{{\left(x.re \cdot x.im\right)}^{2} + 0}} \]

   5. Simplified 79.8%

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

      1. difference-of-squares 86.4%

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

   7. Applied egg-rr 86.4%

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

   9. Simplified 71.1%

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

   if -1.75e-41 < x.re < 9.00000000000000023e-59

   1. Initial program 84.8%

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

      1. *-commutative 84.8%

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

      2. flip3-+ 19.1%

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

      3. associate-*r/ 18.3%

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

      4. *-commutative 18.3%

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

      5. count-2 18.3%

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

      6. pow2 18.3%

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

      7. *-commutative 18.3%

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

      8. *-commutative 18.3%

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

      9. *-commutative 18.3%

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

      10. +-inverses 18.3%

          \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \frac{x.im \cdot \left(2 \cdot {\left(x.re \cdot x.im\right)}^{3}\right)}{{\left(x.re \cdot x.im\right)}^{2} + \color{blue}{0}} \]

   3. Applied egg-rr 18.3%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \color{blue}{\frac{x.im \cdot \left(2 \cdot {\left(x.re \cdot x.im\right)}^{3}\right)}{{\left(x.re \cdot x.im\right)}^{2} + 0}} \]

   5. Simplified 14.7%

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

      1. difference-of-squares 14.7%

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

   7. Applied egg-rr 14.7%

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

   9. Simplified 3.2%

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

   10. Taylor expanded in x.re around 0 3.2%

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

   11. Taylor expanded in x.im around inf 33.1%

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

       1. *-commutative 33.1%

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

       2. associate-*l* 33.1%

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

   13. Simplified 33.1%

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

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -1.75 \cdot 10^{-41} \lor \neg \left(x.re \leq 9 \cdot 10^{-59}\right):\\ \;\;\;\;x.re \cdot \left(\left(x.re + -27\right) \cdot \left(x.re + x.im\right)\right) + \frac{x.re}{-x.re}\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot -27\right)\\ \end{array} \]

Alternative 4: 65.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.re \leq -2.1 \cdot 10^{-114} \lor \neg \left(x.re \leq 2.45 \cdot 10^{-107}\right):\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) + \frac{x.re}{-x.re}\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot -27\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.re -2.1e-114) (not (<= x.re 2.45e-107)))
   (+ (* x.re (* (- x.re x.im) (+ x.re x.im))) (/ x.re (- 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_re <= -2.1e-114) || !(x_46_re <= 2.45e-107)) {
		tmp = (x_46_re * ((x_46_re - x_46_im) * (x_46_re + x_46_im))) + (x_46_re / -x_46_re);
	} else {
		tmp = 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_46re <= (-2.1d-114)) .or. (.not. (x_46re <= 2.45d-107))) then
        tmp = (x_46re * ((x_46re - x_46im) * (x_46re + x_46im))) + (x_46re / -x_46re)
    else
        tmp = 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_re <= -2.1e-114) || !(x_46_re <= 2.45e-107)) {
		tmp = (x_46_re * ((x_46_re - x_46_im) * (x_46_re + x_46_im))) + (x_46_re / -x_46_re);
	} else {
		tmp = 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_re <= -2.1e-114) or not (x_46_re <= 2.45e-107):
		tmp = (x_46_re * ((x_46_re - x_46_im) * (x_46_re + x_46_im))) + (x_46_re / -x_46_re)
	else:
		tmp = 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_re <= -2.1e-114) || !(x_46_re <= 2.45e-107))
		tmp = Float64(Float64(x_46_re * Float64(Float64(x_46_re - x_46_im) * Float64(x_46_re + x_46_im))) + Float64(x_46_re / Float64(-x_46_re)));
	else
		tmp = Float64(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_re <= -2.1e-114) || ~((x_46_re <= 2.45e-107)))
		tmp = (x_46_re * ((x_46_re - x_46_im) * (x_46_re + x_46_im))) + (x_46_re / -x_46_re);
	else
		tmp = 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[Or[LessEqual[x$46$re, -2.1e-114], N[Not[LessEqual[x$46$re, 2.45e-107]], $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$46$re / (-x$46$re)), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(x$46$re * -27.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < -2.09999999999999993e-114 or 2.4499999999999999e-107 < x.re

   1. Initial program 84.8%

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

      1. *-commutative 84.8%

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

      2. flip3-+ 37.4%

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

      3. associate-*r/ 35.1%

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

      4. *-commutative 35.1%

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

      5. count-2 35.1%

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

      6. pow2 35.1%

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

      7. *-commutative 35.1%

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

      8. *-commutative 35.1%

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

      9. *-commutative 35.1%

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

      10. +-inverses 35.1%

          \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \frac{x.im \cdot \left(2 \cdot {\left(x.re \cdot x.im\right)}^{3}\right)}{{\left(x.re \cdot x.im\right)}^{2} + \color{blue}{0}} \]

   3. Applied egg-rr 35.1%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \color{blue}{\frac{x.im \cdot \left(2 \cdot {\left(x.re \cdot x.im\right)}^{3}\right)}{{\left(x.re \cdot x.im\right)}^{2} + 0}} \]

   5. Simplified 73.5%

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

      1. difference-of-squares 79.0%

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

      2. *-commutative 79.0%

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

   7. Applied egg-rr 79.0%

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

   if -2.09999999999999993e-114 < x.re < 2.4499999999999999e-107

   1. Initial program 82.6%

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

      1. *-commutative 82.6%

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

      2. flip3-+ 11.3%

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

      3. associate-*r/ 11.3%

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

      4. *-commutative 11.3%

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

      5. count-2 11.3%

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

      6. pow2 11.3%

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

      7. *-commutative 11.3%

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

      8. *-commutative 11.3%

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

      9. *-commutative 11.3%

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

      10. +-inverses 11.3%

          \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \frac{x.im \cdot \left(2 \cdot {\left(x.re \cdot x.im\right)}^{3}\right)}{{\left(x.re \cdot x.im\right)}^{2} + \color{blue}{0}} \]

   3. Applied egg-rr 11.3%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \color{blue}{\frac{x.im \cdot \left(2 \cdot {\left(x.re \cdot x.im\right)}^{3}\right)}{{\left(x.re \cdot x.im\right)}^{2} + 0}} \]

   5. Simplified 7.6%

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

      1. difference-of-squares 7.6%

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

   7. Applied egg-rr 7.6%

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

   9. Simplified 3.4%

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

   10. Taylor expanded in x.re around 0 3.4%

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

   11. Taylor expanded in x.im around inf 41.2%

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

       1. *-commutative 41.2%

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

       2. associate-*l* 41.2%

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

   13. Simplified 41.2%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -2.1 \cdot 10^{-114} \lor \neg \left(x.re \leq 2.45 \cdot 10^{-107}\right):\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) + \frac{x.re}{-x.re}\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot -27\right)\\ \end{array} \]

Alternative 5: 47.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.re \leq -3.7 \cdot 10^{-16} \lor \neg \left(x.re \leq 1.06 \cdot 10^{+69}\right):\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.re - 27\right)\right) + \frac{x.re}{-x.re}\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot -27\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.re -3.7e-16) (not (<= x.re 1.06e+69)))
   (+ (* x.re (* x.re (- x.re 27.0))) (/ x.re (- 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_re <= -3.7e-16) || !(x_46_re <= 1.06e+69)) {
		tmp = (x_46_re * (x_46_re * (x_46_re - 27.0))) + (x_46_re / -x_46_re);
	} else {
		tmp = 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_46re <= (-3.7d-16)) .or. (.not. (x_46re <= 1.06d+69))) then
        tmp = (x_46re * (x_46re * (x_46re - 27.0d0))) + (x_46re / -x_46re)
    else
        tmp = 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_re <= -3.7e-16) || !(x_46_re <= 1.06e+69)) {
		tmp = (x_46_re * (x_46_re * (x_46_re - 27.0))) + (x_46_re / -x_46_re);
	} else {
		tmp = 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_re <= -3.7e-16) or not (x_46_re <= 1.06e+69):
		tmp = (x_46_re * (x_46_re * (x_46_re - 27.0))) + (x_46_re / -x_46_re)
	else:
		tmp = 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_re <= -3.7e-16) || !(x_46_re <= 1.06e+69))
		tmp = Float64(Float64(x_46_re * Float64(x_46_re * Float64(x_46_re - 27.0))) + Float64(x_46_re / Float64(-x_46_re)));
	else
		tmp = Float64(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_re <= -3.7e-16) || ~((x_46_re <= 1.06e+69)))
		tmp = (x_46_re * (x_46_re * (x_46_re - 27.0))) + (x_46_re / -x_46_re);
	else
		tmp = 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[Or[LessEqual[x$46$re, -3.7e-16], N[Not[LessEqual[x$46$re, 1.06e+69]], $MachinePrecision]], N[(N[(x$46$re * N[(x$46$re * N[(x$46$re - 27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re / (-x$46$re)), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(x$46$re * -27.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < -3.7e-16 or 1.06000000000000004e69 < x.re

   1. Initial program 80.6%

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

      1. *-commutative 80.6%

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

      2. flip3-+ 35.7%

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

      3. associate-*r/ 35.7%

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

      4. *-commutative 35.7%

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

      5. count-2 35.7%

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

      6. pow2 35.7%

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

      7. *-commutative 35.7%

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

      8. *-commutative 35.7%

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

      9. *-commutative 35.7%

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

      10. +-inverses 35.7%

          \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \frac{x.im \cdot \left(2 \cdot {\left(x.re \cdot x.im\right)}^{3}\right)}{{\left(x.re \cdot x.im\right)}^{2} + \color{blue}{0}} \]

   3. Applied egg-rr 35.7%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \color{blue}{\frac{x.im \cdot \left(2 \cdot {\left(x.re \cdot x.im\right)}^{3}\right)}{{\left(x.re \cdot x.im\right)}^{2} + 0}} \]

   5. Simplified 87.2%

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

      1. difference-of-squares 94.8%

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

   7. Applied egg-rr 94.8%

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

   9. Simplified 81.9%

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

   10. Taylor expanded in x.im around 0 71.6%

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

   if -3.7e-16 < x.re < 1.06000000000000004e69

   1. Initial program 87.0%

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

      1. *-commutative 87.0%

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

      2. flip3-+ 21.4%

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

      3. associate-*r/ 18.7%

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

      4. *-commutative 18.7%

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

      5. count-2 18.7%

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

      6. pow2 18.7%

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

      7. *-commutative 18.7%

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

      8. *-commutative 18.7%

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

      9. *-commutative 18.7%

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

      10. +-inverses 18.7%

          \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \frac{x.im \cdot \left(2 \cdot {\left(x.re \cdot x.im\right)}^{3}\right)}{{\left(x.re \cdot x.im\right)}^{2} + \color{blue}{0}} \]

   3. Applied egg-rr 18.7%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \color{blue}{\frac{x.im \cdot \left(2 \cdot {\left(x.re \cdot x.im\right)}^{3}\right)}{{\left(x.re \cdot x.im\right)}^{2} + 0}} \]

   5. Simplified 17.8%

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

      1. difference-of-squares 17.8%

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

   7. Applied egg-rr 17.8%

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

   9. Simplified 3.9%

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

   10. Taylor expanded in x.re around 0 3.3%

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

   11. Taylor expanded in x.im around inf 28.9%

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

       1. *-commutative 28.9%

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

       2. associate-*l* 28.9%

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

   13. Simplified 28.9%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -3.7 \cdot 10^{-16} \lor \neg \left(x.re \leq 1.06 \cdot 10^{+69}\right):\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.re - 27\right)\right) + \frac{x.re}{-x.re}\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot -27\right)\\ \end{array} \]

Alternative 6: 29.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.re \leq 7.2 \cdot 10^{+160}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot -27\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right) + \frac{x.re}{-x.re}\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.re 7.2e+160)
   (* x.im (* x.re -27.0))
   (+ (* x.im (* x.re (- x.re 27.0))) (/ x.re (- x.re)))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= 7.2e+160) {
		tmp = x_46_im * (x_46_re * -27.0);
	} else {
		tmp = (x_46_im * (x_46_re * (x_46_re - 27.0))) + (x_46_re / -x_46_re);
	}
	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 <= 7.2d+160) then
        tmp = x_46im * (x_46re * (-27.0d0))
    else
        tmp = (x_46im * (x_46re * (x_46re - 27.0d0))) + (x_46re / -x_46re)
    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 <= 7.2e+160) {
		tmp = x_46_im * (x_46_re * -27.0);
	} else {
		tmp = (x_46_im * (x_46_re * (x_46_re - 27.0))) + (x_46_re / -x_46_re);
	}
	return tmp;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_re <= 7.2e+160:
		tmp = x_46_im * (x_46_re * -27.0)
	else:
		tmp = (x_46_im * (x_46_re * (x_46_re - 27.0))) + (x_46_re / -x_46_re)
	return tmp

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_re <= 7.2e+160)
		tmp = Float64(x_46_im * Float64(x_46_re * -27.0));
	else
		tmp = Float64(Float64(x_46_im * Float64(x_46_re * Float64(x_46_re - 27.0))) + Float64(x_46_re / Float64(-x_46_re)));
	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 <= 7.2e+160)
		tmp = x_46_im * (x_46_re * -27.0);
	else
		tmp = (x_46_im * (x_46_re * (x_46_re - 27.0))) + (x_46_re / -x_46_re);
	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$46$re, 7.2e+160], N[(x$46$im * N[(x$46$re * -27.0), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im * N[(x$46$re * N[(x$46$re - 27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re / (-x$46$re)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 7.20000000000000042e160

   1. Initial program 86.4%

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

      1. *-commutative 86.4%

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

      2. flip3-+ 27.6%

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

      3. associate-*r/ 26.0%

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

      4. *-commutative 26.0%

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

      5. count-2 26.0%

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

      6. pow2 26.0%

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

      7. *-commutative 26.0%

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

      8. *-commutative 26.0%

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

      9. *-commutative 26.0%

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

      10. +-inverses 26.0%

          \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \frac{x.im \cdot \left(2 \cdot {\left(x.re \cdot x.im\right)}^{3}\right)}{{\left(x.re \cdot x.im\right)}^{2} + \color{blue}{0}} \]

   3. Applied egg-rr 26.0%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \color{blue}{\frac{x.im \cdot \left(2 \cdot {\left(x.re \cdot x.im\right)}^{3}\right)}{{\left(x.re \cdot x.im\right)}^{2} + 0}} \]

   5. Simplified 44.7%

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

      1. difference-of-squares 47.3%

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

   7. Applied egg-rr 47.3%

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

   9. Simplified 32.1%

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

   10. Taylor expanded in x.re around 0 9.3%

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

   11. Taylor expanded in x.im around inf 24.7%

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

       1. *-commutative 24.7%

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

       2. associate-*l* 24.7%

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

   13. Simplified 24.7%

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

   if 7.20000000000000042e160 < x.re

   1. Initial program 65.5%

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

      1. *-commutative 65.5%

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

      2. flip3-+ 31.0%

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

      3. associate-*r/ 31.0%

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

      4. *-commutative 31.0%

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

      5. count-2 31.0%

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

      6. pow2 31.0%

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

      7. *-commutative 31.0%

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

      8. *-commutative 31.0%

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

      9. *-commutative 31.0%

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

      10. +-inverses 31.0%

          \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \frac{x.im \cdot \left(2 \cdot {\left(x.re \cdot x.im\right)}^{3}\right)}{{\left(x.re \cdot x.im\right)}^{2} + \color{blue}{0}} \]

   3. Applied egg-rr 31.0%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \color{blue}{\frac{x.im \cdot \left(2 \cdot {\left(x.re \cdot x.im\right)}^{3}\right)}{{\left(x.re \cdot x.im\right)}^{2} + 0}} \]

   5. Simplified 89.7%

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

      1. difference-of-squares 100.0%

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

   7. Applied egg-rr 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in x.im around inf 55.2%

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

4. Final simplification 28.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 7.2 \cdot 10^{+160}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot -27\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right) + \frac{x.re}{-x.re}\\ \end{array} \]

Alternative 7: 20.0% 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 84.0%

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

   1. *-commutative 84.0%

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

   2. flip3-+ 28.0%

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

   3. associate-*r/ 26.6%

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

   4. *-commutative 26.6%

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

   5. count-2 26.6%

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

   6. pow2 26.6%

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

   7. *-commutative 26.6%

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

   8. *-commutative 26.6%

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

   9. *-commutative 26.6%

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

   10. +-inverses 26.6%

       \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \frac{x.im \cdot \left(2 \cdot {\left(x.re \cdot x.im\right)}^{3}\right)}{{\left(x.re \cdot x.im\right)}^{2} + \color{blue}{0}} \]

3. Applied egg-rr 26.6%

   \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \color{blue}{\frac{x.im \cdot \left(2 \cdot {\left(x.re \cdot x.im\right)}^{3}\right)}{{\left(x.re \cdot x.im\right)}^{2} + 0}} \]

5. Simplified 49.8%

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

   1. difference-of-squares 53.3%

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

7. Applied egg-rr 53.3%

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

9. Simplified 39.8%

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

10. Taylor expanded in x.re around 0 10.0%

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

11. Taylor expanded in x.im around inf 23.6%

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

12. Final simplification 23.6%

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

Alternative 8: 20.0% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

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_46im * (x_46re * (-27.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.0%

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

   1. *-commutative 84.0%

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

   2. flip3-+ 28.0%

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

   3. associate-*r/ 26.6%

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

   4. *-commutative 26.6%

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

   5. count-2 26.6%

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

   6. pow2 26.6%

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

   7. *-commutative 26.6%

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

   8. *-commutative 26.6%

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

   9. *-commutative 26.6%

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

   10. +-inverses 26.6%

       \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \frac{x.im \cdot \left(2 \cdot {\left(x.re \cdot x.im\right)}^{3}\right)}{{\left(x.re \cdot x.im\right)}^{2} + \color{blue}{0}} \]

3. Applied egg-rr 26.6%

   \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \color{blue}{\frac{x.im \cdot \left(2 \cdot {\left(x.re \cdot x.im\right)}^{3}\right)}{{\left(x.re \cdot x.im\right)}^{2} + 0}} \]

5. Simplified 49.8%

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

   1. difference-of-squares 53.3%

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

7. Applied egg-rr 53.3%

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

9. Simplified 39.8%

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

10. Taylor expanded in x.re around 0 10.0%

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

11. Taylor expanded in x.im around inf 23.6%

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

    1. *-commutative 23.6%

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

    2. associate-*l* 23.6%

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

13. Simplified 23.6%

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

14. Final simplification 23.6%

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

Alternative 9: 2.7% accurate, 19.0× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ -1 \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im) :precision binary64 -1.0)

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	return -1.0;
}

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 = -1.0d0
end function

Java

x.im = Math.abs(x.im);
public static double code(double x_46_re, double x_46_im) {
	return -1.0;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	return -1.0

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	return -1.0
end

MATLAB

x.im = abs(x.im)
function tmp = code(x_46_re, x_46_im)
	tmp = -1.0;
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := -1.0

Derivation

1. Initial program 84.0%

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

   1. *-commutative 84.0%

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

   2. flip3-+ 28.0%

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

   3. associate-*r/ 26.6%

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

   4. *-commutative 26.6%

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

   5. count-2 26.6%

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

   6. pow2 26.6%

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

   7. *-commutative 26.6%

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

   8. *-commutative 26.6%

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

   9. *-commutative 26.6%

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

   10. +-inverses 26.6%

       \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \frac{x.im \cdot \left(2 \cdot {\left(x.re \cdot x.im\right)}^{3}\right)}{{\left(x.re \cdot x.im\right)}^{2} + \color{blue}{0}} \]

3. Applied egg-rr 26.6%

   \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \color{blue}{\frac{x.im \cdot \left(2 \cdot {\left(x.re \cdot x.im\right)}^{3}\right)}{{\left(x.re \cdot x.im\right)}^{2} + 0}} \]

5. Simplified 49.8%

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

   1. difference-of-squares 53.3%

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

7. Applied egg-rr 53.3%

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

9. Simplified 39.8%

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

10. Taylor expanded in x.re around 0 10.0%

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

11. Taylor expanded in x.im around 0 2.7%

    \[\leadsto \color{blue}{-1} \]

12. Final simplification 2.7%

    \[\leadsto -1 \]

Developer target: 87.6% 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 2023292 
(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)))