math.cube on complex, real part

Percentage Accurate: 83.1% → 96.4%

Time: 8.1s

Alternatives: 9

Speedup: 0.9×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 83.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -6.8 \cdot 10^{+225} \lor \neg \left(x.re \leq 1.3 \cdot 10^{+205}\right):\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right) + 0 \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im, x.re \cdot \left(x.re + x.im\right), x.im \cdot \left(x.re \cdot \left(\left(-x.im\right) - x.im\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.re -6.8e+225) (not (<= x.re 1.3e+205)))
   (+ (* x.re (* x.re x.re)) (* 0.0 x.im))
   (fma
    (- x.re x.im)
    (* x.re (+ x.re x.im))
    (* x.im (* x.re (- (- x.im) x.im))))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -6.8e+225) || !(x_46_re <= 1.3e+205)) {
		tmp = (x_46_re * (x_46_re * x_46_re)) + (0.0 * x_46_im);
	} else {
		tmp = fma((x_46_re - x_46_im), (x_46_re * (x_46_re + x_46_im)), (x_46_im * (x_46_re * (-x_46_im - x_46_im))));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_re <= -6.8e+225) || !(x_46_re <= 1.3e+205))
		tmp = Float64(Float64(x_46_re * Float64(x_46_re * x_46_re)) + Float64(0.0 * x_46_im));
	else
		tmp = fma(Float64(x_46_re - x_46_im), Float64(x_46_re * Float64(x_46_re + x_46_im)), Float64(x_46_im * Float64(x_46_re * Float64(Float64(-x_46_im) - x_46_im))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$re, -6.8e+225], N[Not[LessEqual[x$46$re, 1.3e+205]], $MachinePrecision]], N[(N[(x$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] + N[(0.0 * x$46$im), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re - x$46$im), $MachinePrecision] * N[(x$46$re * N[(x$46$re + x$46$im), $MachinePrecision]), $MachinePrecision] + N[(x$46$im * N[(x$46$re * N[((-x$46$im) - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < -6.80000000000000037e225 or 1.2999999999999999e205 < x.re

   1. Initial program 65.0%

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

   2. Taylor expanded in x.re around inf 65.0%

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

      1. unpow2 65.0%

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

   4. Simplified 65.0%

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

      1. flip-+ 0.0%

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

      2. *-commutative 0.0%

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

      3. *-commutative 0.0%

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

      4. div-sub 0.0%

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

      5. pow2 0.0%

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

      6. *-commutative 0.0%

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

      7. *-commutative 0.0%

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

      8. pow2 0.0%

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

      9. *-commutative 0.0%

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

      10. *-commutative 0.0%

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

   6. Applied egg-rr 0.0%

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

      1. +-inverses 100.0%

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

   8. Simplified 100.0%

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

   if -6.80000000000000037e225 < x.re < 1.2999999999999999e205

   1. Initial program 86.1%

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

      1. sqr-neg 86.1%

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

      2. difference-of-squares 89.8%

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

      3. sub-neg 89.8%

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

      4. associate-*l* 98.4%

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

      5. sub-neg 98.4%

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

      6. remove-double-neg 98.4%

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

      7. +-commutative 98.4%

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

      8. *-commutative 98.4%

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

      9. *-commutative 98.4%

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

      10. distribute-rgt-out 98.4%

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

   3. Simplified 98.4%

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

      1. cancel-sign-sub-inv 98.4%

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

      2. fma-def 98.4%

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

      3. *-commutative 98.4%

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

   5. Applied egg-rr 98.4%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -6.8 \cdot 10^{+225} \lor \neg \left(x.re \leq 1.3 \cdot 10^{+205}\right):\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right) + 0 \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im, x.re \cdot \left(x.re + x.im\right), x.im \cdot \left(x.re \cdot \left(\left(-x.im\right) - x.im\right)\right)\right)\\ \end{array} \]

Alternative 2: 95.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.re \cdot \left(x.re \cdot x.re\right) + 0 \cdot x.im\\ t_1 := \left(x.re - x.im\right) \cdot \left(x.re \cdot \left(x.re + x.im\right)\right) - \left(x.re \cdot \left(x.im \cdot x.im\right)\right) \cdot 2\\ \mathbf{if}\;x.re \leq -2.45 \cdot 10^{+224}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.re \leq -2.4 \cdot 10^{-94}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x.re \leq 3.5 \cdot 10^{-160}:\\ \;\;\;\;x.im \cdot \left(-3 \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{elif}\;x.re \leq 1.3 \cdot 10^{+205}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (+ (* x.re (* x.re x.re)) (* 0.0 x.im)))
        (t_1
         (-
          (* (- x.re x.im) (* x.re (+ x.re x.im)))
          (* (* x.re (* x.im x.im)) 2.0))))
   (if (<= x.re -2.45e+224)
     t_0
     (if (<= x.re -2.4e-94)
       t_1
       (if (<= x.re 3.5e-160)
         (* x.im (* -3.0 (* x.re x.im)))
         (if (<= x.re 1.3e+205) t_1 t_0))))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = (x_46_re * (x_46_re * x_46_re)) + (0.0 * x_46_im);
	double t_1 = ((x_46_re - x_46_im) * (x_46_re * (x_46_re + x_46_im))) - ((x_46_re * (x_46_im * x_46_im)) * 2.0);
	double tmp;
	if (x_46_re <= -2.45e+224) {
		tmp = t_0;
	} else if (x_46_re <= -2.4e-94) {
		tmp = t_1;
	} else if (x_46_re <= 3.5e-160) {
		tmp = x_46_im * (-3.0 * (x_46_re * x_46_im));
	} else if (x_46_re <= 1.3e+205) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x_46re * (x_46re * x_46re)) + (0.0d0 * x_46im)
    t_1 = ((x_46re - x_46im) * (x_46re * (x_46re + x_46im))) - ((x_46re * (x_46im * x_46im)) * 2.0d0)
    if (x_46re <= (-2.45d+224)) then
        tmp = t_0
    else if (x_46re <= (-2.4d-94)) then
        tmp = t_1
    else if (x_46re <= 3.5d-160) then
        tmp = x_46im * ((-3.0d0) * (x_46re * x_46im))
    else if (x_46re <= 1.3d+205) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double t_0 = (x_46_re * (x_46_re * x_46_re)) + (0.0 * x_46_im);
	double t_1 = ((x_46_re - x_46_im) * (x_46_re * (x_46_re + x_46_im))) - ((x_46_re * (x_46_im * x_46_im)) * 2.0);
	double tmp;
	if (x_46_re <= -2.45e+224) {
		tmp = t_0;
	} else if (x_46_re <= -2.4e-94) {
		tmp = t_1;
	} else if (x_46_re <= 3.5e-160) {
		tmp = x_46_im * (-3.0 * (x_46_re * x_46_im));
	} else if (x_46_re <= 1.3e+205) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	t_0 = (x_46_re * (x_46_re * x_46_re)) + (0.0 * x_46_im)
	t_1 = ((x_46_re - x_46_im) * (x_46_re * (x_46_re + x_46_im))) - ((x_46_re * (x_46_im * x_46_im)) * 2.0)
	tmp = 0
	if x_46_re <= -2.45e+224:
		tmp = t_0
	elif x_46_re <= -2.4e-94:
		tmp = t_1
	elif x_46_re <= 3.5e-160:
		tmp = x_46_im * (-3.0 * (x_46_re * x_46_im))
	elif x_46_re <= 1.3e+205:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(Float64(x_46_re * Float64(x_46_re * x_46_re)) + Float64(0.0 * x_46_im))
	t_1 = Float64(Float64(Float64(x_46_re - x_46_im) * Float64(x_46_re * Float64(x_46_re + x_46_im))) - Float64(Float64(x_46_re * Float64(x_46_im * x_46_im)) * 2.0))
	tmp = 0.0
	if (x_46_re <= -2.45e+224)
		tmp = t_0;
	elseif (x_46_re <= -2.4e-94)
		tmp = t_1;
	elseif (x_46_re <= 3.5e-160)
		tmp = Float64(x_46_im * Float64(-3.0 * Float64(x_46_re * x_46_im)));
	elseif (x_46_re <= 1.3e+205)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	t_0 = (x_46_re * (x_46_re * x_46_re)) + (0.0 * x_46_im);
	t_1 = ((x_46_re - x_46_im) * (x_46_re * (x_46_re + x_46_im))) - ((x_46_re * (x_46_im * x_46_im)) * 2.0);
	tmp = 0.0;
	if (x_46_re <= -2.45e+224)
		tmp = t_0;
	elseif (x_46_re <= -2.4e-94)
		tmp = t_1;
	elseif (x_46_re <= 3.5e-160)
		tmp = x_46_im * (-3.0 * (x_46_re * x_46_im));
	elseif (x_46_re <= 1.3e+205)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(N[(x$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] + N[(0.0 * x$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x$46$re - x$46$im), $MachinePrecision] * N[(x$46$re * N[(x$46$re + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(x$46$re * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$re, -2.45e+224], t$95$0, If[LessEqual[x$46$re, -2.4e-94], t$95$1, If[LessEqual[x$46$re, 3.5e-160], N[(x$46$im * N[(-3.0 * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 1.3e+205], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -2.44999999999999992e224 or 1.2999999999999999e205 < x.re

   1. Initial program 65.0%

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

   2. Taylor expanded in x.re around inf 65.0%

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

      1. unpow2 65.0%

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

   4. Simplified 65.0%

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

      1. flip-+ 0.0%

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

      2. *-commutative 0.0%

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

      3. *-commutative 0.0%

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

      4. div-sub 0.0%

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

      5. pow2 0.0%

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

      6. *-commutative 0.0%

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

      7. *-commutative 0.0%

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

      8. pow2 0.0%

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

      9. *-commutative 0.0%

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

      10. *-commutative 0.0%

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

   6. Applied egg-rr 0.0%

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

      1. +-inverses 100.0%

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

   8. Simplified 100.0%

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

   if -2.44999999999999992e224 < x.re < -2.4e-94 or 3.5000000000000003e-160 < x.re < 1.2999999999999999e205

   1. Initial program 90.7%

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

      1. sqr-neg 90.7%

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

      2. difference-of-squares 96.8%

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

      3. sub-neg 96.8%

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

      4. associate-*l* 97.5%

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

      5. sub-neg 97.5%

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

      6. remove-double-neg 97.5%

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

      7. +-commutative 97.5%

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

      8. *-commutative 97.5%

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

      9. *-commutative 97.5%

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

      10. distribute-rgt-out 97.5%

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

   3. Simplified 97.5%

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

   4. Taylor expanded in x.im around 0 96.8%

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

      1. unpow2 96.8%

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

   6. Simplified 96.8%

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

   if -2.4e-94 < x.re < 3.5000000000000003e-160

   1. Initial program 78.9%

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

      1. sqr-neg 78.9%

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

      2. difference-of-squares 78.9%

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

      3. sub-neg 78.9%

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

      4. associate-*l* 99.7%

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

      5. sub-neg 99.7%

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

      6. remove-double-neg 99.7%

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

      7. +-commutative 99.7%

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

      8. *-commutative 99.7%

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

      9. *-commutative 99.7%

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

      10. distribute-rgt-out 99.7%

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

   3. Simplified 99.7%

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

      1. cancel-sign-sub-inv 99.7%

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

      2. fma-def 99.8%

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

      3. *-commutative 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in x.re around 0 78.8%

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

      1. distribute-rgt-out 78.8%

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

      2. unpow2 78.8%

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

      3. metadata-eval 78.8%

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

      4. associate-*l* 78.8%

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

      5. associate-*l* 99.7%

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

      6. *-commutative 99.7%

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

      7. associate-*r* 99.7%

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

   8. Simplified 99.7%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -2.45 \cdot 10^{+224}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right) + 0 \cdot x.im\\ \mathbf{elif}\;x.re \leq -2.4 \cdot 10^{-94}:\\ \;\;\;\;\left(x.re - x.im\right) \cdot \left(x.re \cdot \left(x.re + x.im\right)\right) - \left(x.re \cdot \left(x.im \cdot x.im\right)\right) \cdot 2\\ \mathbf{elif}\;x.re \leq 3.5 \cdot 10^{-160}:\\ \;\;\;\;x.im \cdot \left(-3 \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{elif}\;x.re \leq 1.3 \cdot 10^{+205}:\\ \;\;\;\;\left(x.re - x.im\right) \cdot \left(x.re \cdot \left(x.re + x.im\right)\right) - \left(x.re \cdot \left(x.im \cdot x.im\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right) + 0 \cdot x.im\\ \end{array} \]

Alternative 3: 87.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.re \cdot \left(x.re \cdot x.re\right)\\ t_1 := t_0 + 0 \cdot x.im\\ t_2 := t_0 - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\\ \mathbf{if}\;x.re \leq -2.1 \cdot 10^{+118}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x.re \leq -1.4 \cdot 10^{-44}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x.re \leq 3.5 \cdot 10^{-72}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(\left(-x.im\right) - x.im\right)\right) - x.im \cdot \left(x.re \cdot x.im\right)\\ \mathbf{elif}\;x.re \leq 1.2 \cdot 10^{+102}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (* x.re (* x.re x.re)))
        (t_1 (+ t_0 (* 0.0 x.im)))
        (t_2 (- t_0 (* x.im (+ (* x.re x.im) (* x.re x.im))))))
   (if (<= x.re -2.1e+118)
     t_1
     (if (<= x.re -1.4e-44)
       t_2
       (if (<= x.re 3.5e-72)
         (- (* x.im (* x.re (- (- x.im) x.im))) (* x.im (* x.re x.im)))
         (if (<= x.re 1.2e+102) t_2 t_1))))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_re * (x_46_re * x_46_re);
	double t_1 = t_0 + (0.0 * x_46_im);
	double t_2 = t_0 - (x_46_im * ((x_46_re * x_46_im) + (x_46_re * x_46_im)));
	double tmp;
	if (x_46_re <= -2.1e+118) {
		tmp = t_1;
	} else if (x_46_re <= -1.4e-44) {
		tmp = t_2;
	} else if (x_46_re <= 3.5e-72) {
		tmp = (x_46_im * (x_46_re * (-x_46_im - x_46_im))) - (x_46_im * (x_46_re * x_46_im));
	} else if (x_46_re <= 1.2e+102) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = x_46re * (x_46re * x_46re)
    t_1 = t_0 + (0.0d0 * x_46im)
    t_2 = t_0 - (x_46im * ((x_46re * x_46im) + (x_46re * x_46im)))
    if (x_46re <= (-2.1d+118)) then
        tmp = t_1
    else if (x_46re <= (-1.4d-44)) then
        tmp = t_2
    else if (x_46re <= 3.5d-72) then
        tmp = (x_46im * (x_46re * (-x_46im - x_46im))) - (x_46im * (x_46re * x_46im))
    else if (x_46re <= 1.2d+102) then
        tmp = t_2
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_re * (x_46_re * x_46_re);
	double t_1 = t_0 + (0.0 * x_46_im);
	double t_2 = t_0 - (x_46_im * ((x_46_re * x_46_im) + (x_46_re * x_46_im)));
	double tmp;
	if (x_46_re <= -2.1e+118) {
		tmp = t_1;
	} else if (x_46_re <= -1.4e-44) {
		tmp = t_2;
	} else if (x_46_re <= 3.5e-72) {
		tmp = (x_46_im * (x_46_re * (-x_46_im - x_46_im))) - (x_46_im * (x_46_re * x_46_im));
	} else if (x_46_re <= 1.2e+102) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	t_0 = x_46_re * (x_46_re * x_46_re)
	t_1 = t_0 + (0.0 * x_46_im)
	t_2 = t_0 - (x_46_im * ((x_46_re * x_46_im) + (x_46_re * x_46_im)))
	tmp = 0
	if x_46_re <= -2.1e+118:
		tmp = t_1
	elif x_46_re <= -1.4e-44:
		tmp = t_2
	elif x_46_re <= 3.5e-72:
		tmp = (x_46_im * (x_46_re * (-x_46_im - x_46_im))) - (x_46_im * (x_46_re * x_46_im))
	elif x_46_re <= 1.2e+102:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(x_46_re * Float64(x_46_re * x_46_re))
	t_1 = Float64(t_0 + Float64(0.0 * x_46_im))
	t_2 = Float64(t_0 - Float64(x_46_im * Float64(Float64(x_46_re * x_46_im) + Float64(x_46_re * x_46_im))))
	tmp = 0.0
	if (x_46_re <= -2.1e+118)
		tmp = t_1;
	elseif (x_46_re <= -1.4e-44)
		tmp = t_2;
	elseif (x_46_re <= 3.5e-72)
		tmp = Float64(Float64(x_46_im * Float64(x_46_re * Float64(Float64(-x_46_im) - x_46_im))) - Float64(x_46_im * Float64(x_46_re * x_46_im)));
	elseif (x_46_re <= 1.2e+102)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	t_0 = x_46_re * (x_46_re * x_46_re);
	t_1 = t_0 + (0.0 * x_46_im);
	t_2 = t_0 - (x_46_im * ((x_46_re * x_46_im) + (x_46_re * x_46_im)));
	tmp = 0.0;
	if (x_46_re <= -2.1e+118)
		tmp = t_1;
	elseif (x_46_re <= -1.4e-44)
		tmp = t_2;
	elseif (x_46_re <= 3.5e-72)
		tmp = (x_46_im * (x_46_re * (-x_46_im - x_46_im))) - (x_46_im * (x_46_re * x_46_im));
	elseif (x_46_re <= 1.2e+102)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(x$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + N[(0.0 * x$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 - N[(x$46$im * N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$re, -2.1e+118], t$95$1, If[LessEqual[x$46$re, -1.4e-44], t$95$2, If[LessEqual[x$46$re, 3.5e-72], N[(N[(x$46$im * N[(x$46$re * N[((-x$46$im) - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 1.2e+102], t$95$2, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -2.1e118 or 1.19999999999999997e102 < x.re

   1. Initial program 65.8%

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

   2. Taylor expanded in x.re around inf 60.3%

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

      1. unpow2 60.3%

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

   4. Simplified 60.3%

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

      1. flip-+ 0.0%

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

      2. *-commutative 0.0%

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

      3. *-commutative 0.0%

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

      4. div-sub 0.0%

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

      5. pow2 0.0%

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

      6. *-commutative 0.0%

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

      7. *-commutative 0.0%

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

      8. pow2 0.0%

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

      9. *-commutative 0.0%

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

      10. *-commutative 0.0%

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

   6. Applied egg-rr 0.0%

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

      1. +-inverses 83.6%

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

   8. Simplified 83.6%

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

   if -2.1e118 < x.re < -1.4e-44 or 3.5e-72 < x.re < 1.19999999999999997e102

   1. Initial program 99.8%

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

   2. Taylor expanded in x.re around inf 92.4%

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

      1. unpow2 92.4%

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

   4. Simplified 92.4%

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

   if -1.4e-44 < x.re < 3.5e-72

   1. Initial program 83.9%

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

      1. sqr-neg 83.9%

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

      2. difference-of-squares 83.9%

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

      3. sub-neg 83.9%

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

      4. associate-*l* 99.7%

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

      5. sub-neg 99.7%

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

      6. remove-double-neg 99.7%

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

      7. +-commutative 99.7%

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

      8. *-commutative 99.7%

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

      9. *-commutative 99.7%

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

      10. distribute-rgt-out 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x.re around 0 79.6%

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

      1. mul-1-neg 79.6%

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

      2. distribute-rgt-neg-in 79.6%

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

      3. unpow2 79.6%

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

      4. mul-1-neg 79.6%

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

      5. associate-*l* 95.4%

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

      6. mul-1-neg 95.4%

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

   6. Simplified 95.4%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -2.1 \cdot 10^{+118}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right) + 0 \cdot x.im\\ \mathbf{elif}\;x.re \leq -1.4 \cdot 10^{-44}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\\ \mathbf{elif}\;x.re \leq 3.5 \cdot 10^{-72}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(\left(-x.im\right) - x.im\right)\right) - x.im \cdot \left(x.re \cdot x.im\right)\\ \mathbf{elif}\;x.re \leq 1.2 \cdot 10^{+102}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right) + 0 \cdot x.im\\ \end{array} \]

Alternative 4: 96.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -3.7 \cdot 10^{+224} \lor \neg \left(x.re \leq 1.4 \cdot 10^{+205}\right):\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right) + 0 \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\left(x.re - x.im\right) \cdot \left(x.re \cdot \left(x.re + x.im\right)\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.re -3.7e+224) (not (<= x.re 1.4e+205)))
   (+ (* x.re (* x.re x.re)) (* 0.0 x.im))
   (-
    (* (- x.re x.im) (* x.re (+ x.re x.im)))
    (* x.im (* x.re (+ x.im x.im))))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -3.7e+224) || !(x_46_re <= 1.4e+205)) {
		tmp = (x_46_re * (x_46_re * x_46_re)) + (0.0 * x_46_im);
	} else {
		tmp = ((x_46_re - x_46_im) * (x_46_re * (x_46_re + x_46_im))) - (x_46_im * (x_46_re * (x_46_im + x_46_im)));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46re <= (-3.7d+224)) .or. (.not. (x_46re <= 1.4d+205))) then
        tmp = (x_46re * (x_46re * x_46re)) + (0.0d0 * x_46im)
    else
        tmp = ((x_46re - x_46im) * (x_46re * (x_46re + x_46im))) - (x_46im * (x_46re * (x_46im + x_46im)))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -3.7e+224) || !(x_46_re <= 1.4e+205)) {
		tmp = (x_46_re * (x_46_re * x_46_re)) + (0.0 * x_46_im);
	} else {
		tmp = ((x_46_re - x_46_im) * (x_46_re * (x_46_re + x_46_im))) - (x_46_im * (x_46_re * (x_46_im + x_46_im)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_re <= -3.7e+224) or not (x_46_re <= 1.4e+205):
		tmp = (x_46_re * (x_46_re * x_46_re)) + (0.0 * x_46_im)
	else:
		tmp = ((x_46_re - x_46_im) * (x_46_re * (x_46_re + x_46_im))) - (x_46_im * (x_46_re * (x_46_im + x_46_im)))
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_re <= -3.7e+224) || !(x_46_re <= 1.4e+205))
		tmp = Float64(Float64(x_46_re * Float64(x_46_re * x_46_re)) + Float64(0.0 * x_46_im));
	else
		tmp = Float64(Float64(Float64(x_46_re - x_46_im) * Float64(x_46_re * Float64(x_46_re + x_46_im))) - Float64(x_46_im * Float64(x_46_re * Float64(x_46_im + x_46_im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_re <= -3.7e+224) || ~((x_46_re <= 1.4e+205)))
		tmp = (x_46_re * (x_46_re * x_46_re)) + (0.0 * x_46_im);
	else
		tmp = ((x_46_re - x_46_im) * (x_46_re * (x_46_re + x_46_im))) - (x_46_im * (x_46_re * (x_46_im + x_46_im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$re, -3.7e+224], N[Not[LessEqual[x$46$re, 1.4e+205]], $MachinePrecision]], N[(N[(x$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] + N[(0.0 * x$46$im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$46$re - x$46$im), $MachinePrecision] * N[(x$46$re * N[(x$46$re + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(x$46$re * N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < -3.70000000000000003e224 or 1.39999999999999996e205 < x.re

   1. Initial program 65.0%

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

   2. Taylor expanded in x.re around inf 65.0%

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

      1. unpow2 65.0%

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

   4. Simplified 65.0%

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

      1. flip-+ 0.0%

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

      2. *-commutative 0.0%

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

      3. *-commutative 0.0%

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

      4. div-sub 0.0%

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

      5. pow2 0.0%

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

      6. *-commutative 0.0%

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

      7. *-commutative 0.0%

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

      8. pow2 0.0%

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

      9. *-commutative 0.0%

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

      10. *-commutative 0.0%

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

   6. Applied egg-rr 0.0%

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

      1. +-inverses 100.0%

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

   8. Simplified 100.0%

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

   if -3.70000000000000003e224 < x.re < 1.39999999999999996e205

   1. Initial program 86.1%

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

      1. sqr-neg 86.1%

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

      2. difference-of-squares 89.8%

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

      3. sub-neg 89.8%

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

      4. associate-*l* 98.4%

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

      5. sub-neg 98.4%

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

      6. remove-double-neg 98.4%

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

      7. +-commutative 98.4%

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

      8. *-commutative 98.4%

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

      9. *-commutative 98.4%

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

      10. distribute-rgt-out 98.4%

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

   3. Simplified 98.4%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -3.7 \cdot 10^{+224} \lor \neg \left(x.re \leq 1.4 \cdot 10^{+205}\right):\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right) + 0 \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\left(x.re - x.im\right) \cdot \left(x.re \cdot \left(x.re + x.im\right)\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\\ \end{array} \]

Alternative 5: 70.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.re \cdot \left(x.re \cdot x.re\right) + 0 \cdot x.im\\ \mathbf{if}\;x.im \leq 1.36 \cdot 10^{-24}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.im \leq 480000000:\\ \;\;\;\;\left(x.re \cdot \left(x.im \cdot x.im\right)\right) \cdot -3\\ \mathbf{elif}\;x.im \leq 2.45 \cdot 10^{+68}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(-3 \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (+ (* x.re (* x.re x.re)) (* 0.0 x.im))))
   (if (<= x.im 1.36e-24)
     t_0
     (if (<= x.im 480000000.0)
       (* (* x.re (* x.im x.im)) -3.0)
       (if (<= x.im 2.45e+68) t_0 (* x.im (* -3.0 (* x.re x.im))))))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = (x_46_re * (x_46_re * x_46_re)) + (0.0 * x_46_im);
	double tmp;
	if (x_46_im <= 1.36e-24) {
		tmp = t_0;
	} else if (x_46_im <= 480000000.0) {
		tmp = (x_46_re * (x_46_im * x_46_im)) * -3.0;
	} else if (x_46_im <= 2.45e+68) {
		tmp = t_0;
	} else {
		tmp = x_46_im * (-3.0 * (x_46_re * x_46_im));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_46re * (x_46re * x_46re)) + (0.0d0 * x_46im)
    if (x_46im <= 1.36d-24) then
        tmp = t_0
    else if (x_46im <= 480000000.0d0) then
        tmp = (x_46re * (x_46im * x_46im)) * (-3.0d0)
    else if (x_46im <= 2.45d+68) then
        tmp = t_0
    else
        tmp = x_46im * ((-3.0d0) * (x_46re * x_46im))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double t_0 = (x_46_re * (x_46_re * x_46_re)) + (0.0 * x_46_im);
	double tmp;
	if (x_46_im <= 1.36e-24) {
		tmp = t_0;
	} else if (x_46_im <= 480000000.0) {
		tmp = (x_46_re * (x_46_im * x_46_im)) * -3.0;
	} else if (x_46_im <= 2.45e+68) {
		tmp = t_0;
	} else {
		tmp = x_46_im * (-3.0 * (x_46_re * x_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	t_0 = (x_46_re * (x_46_re * x_46_re)) + (0.0 * x_46_im)
	tmp = 0
	if x_46_im <= 1.36e-24:
		tmp = t_0
	elif x_46_im <= 480000000.0:
		tmp = (x_46_re * (x_46_im * x_46_im)) * -3.0
	elif x_46_im <= 2.45e+68:
		tmp = t_0
	else:
		tmp = x_46_im * (-3.0 * (x_46_re * x_46_im))
	return tmp

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(Float64(x_46_re * Float64(x_46_re * x_46_re)) + Float64(0.0 * x_46_im))
	tmp = 0.0
	if (x_46_im <= 1.36e-24)
		tmp = t_0;
	elseif (x_46_im <= 480000000.0)
		tmp = Float64(Float64(x_46_re * Float64(x_46_im * x_46_im)) * -3.0);
	elseif (x_46_im <= 2.45e+68)
		tmp = t_0;
	else
		tmp = Float64(x_46_im * Float64(-3.0 * Float64(x_46_re * x_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	t_0 = (x_46_re * (x_46_re * x_46_re)) + (0.0 * x_46_im);
	tmp = 0.0;
	if (x_46_im <= 1.36e-24)
		tmp = t_0;
	elseif (x_46_im <= 480000000.0)
		tmp = (x_46_re * (x_46_im * x_46_im)) * -3.0;
	elseif (x_46_im <= 2.45e+68)
		tmp = t_0;
	else
		tmp = x_46_im * (-3.0 * (x_46_re * x_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(N[(x$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] + N[(0.0 * x$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$im, 1.36e-24], t$95$0, If[LessEqual[x$46$im, 480000000.0], N[(N[(x$46$re * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] * -3.0), $MachinePrecision], If[LessEqual[x$46$im, 2.45e+68], t$95$0, N[(x$46$im * N[(-3.0 * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x.im < 1.36000000000000001e-24 or 4.8e8 < x.im < 2.44999999999999989e68

   1. Initial program 85.7%

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

   2. Taylor expanded in x.re around inf 73.5%

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

      1. unpow2 73.5%

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

   4. Simplified 73.5%

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

      1. flip-+ 0.0%

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

      2. *-commutative 0.0%

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

      3. *-commutative 0.0%

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

      4. div-sub 0.0%

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

      5. pow2 0.0%

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

      6. *-commutative 0.0%

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

      7. *-commutative 0.0%

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

      8. pow2 0.0%

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

      9. *-commutative 0.0%

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

      10. *-commutative 0.0%

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

   6. Applied egg-rr 0.0%

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

      1. +-inverses 67.7%

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

   8. Simplified 67.7%

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

   if 1.36000000000000001e-24 < x.im < 4.8e8

   1. Initial program 99.6%

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

      1. sqr-neg 99.6%

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

      2. difference-of-squares 99.6%

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

      3. sub-neg 99.6%

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

      4. associate-*l* 99.6%

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

      5. sub-neg 99.6%

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

      6. remove-double-neg 99.6%

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

      7. +-commutative 99.6%

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

      8. *-commutative 99.6%

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

      9. *-commutative 99.6%

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

      10. distribute-rgt-out 99.6%

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

   3. Simplified 99.6%

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

      1. cancel-sign-sub-inv 99.6%

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

      2. fma-def 99.7%

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

      3. *-commutative 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in x.re around 0 78.2%

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

      1. distribute-rgt-out 78.2%

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

      2. unpow2 78.2%

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

      3. metadata-eval 78.2%

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

      4. associate-*l* 78.5%

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

      5. associate-*l* 78.5%

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

      6. *-commutative 78.5%

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

      7. associate-*r* 78.2%

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

   8. Simplified 78.2%

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

   9. Taylor expanded in x.im around 0 78.3%

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

       1. *-commutative 78.3%

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

       2. *-commutative 78.3%

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

       3. unpow2 78.3%

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

   11. Simplified 78.3%

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

   if 2.44999999999999989e68 < x.im

   1. Initial program 63.2%

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

      1. sqr-neg 63.2%

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

      2. difference-of-squares 70.3%

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

      3. sub-neg 70.3%

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

      4. associate-*l* 90.2%

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

      5. sub-neg 90.2%

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

      6. remove-double-neg 90.2%

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

      7. +-commutative 90.2%

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

      8. *-commutative 90.2%

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

      9. *-commutative 90.2%

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

      10. distribute-rgt-out 90.2%

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

   3. Simplified 90.2%

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

      1. cancel-sign-sub-inv 90.2%

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

      2. fma-def 90.4%

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

      3. *-commutative 90.4%

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

   5. Applied egg-rr 90.4%

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

   6. Taylor expanded in x.re around 0 70.1%

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

      1. distribute-rgt-out 70.1%

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

      2. unpow2 70.1%

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

      3. metadata-eval 70.1%

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

      4. associate-*l* 70.1%

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

      5. associate-*l* 90.3%

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

      6. *-commutative 90.3%

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

      7. associate-*r* 90.3%

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

   8. Simplified 90.3%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 1.36 \cdot 10^{-24}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right) + 0 \cdot x.im\\ \mathbf{elif}\;x.im \leq 480000000:\\ \;\;\;\;\left(x.re \cdot \left(x.im \cdot x.im\right)\right) \cdot -3\\ \mathbf{elif}\;x.im \leq 2.45 \cdot 10^{+68}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right) + 0 \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(-3 \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \]

Alternative 6: 59.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -1.2 \cdot 10^{+201} \lor \neg \left(x.re \leq 1.7 \cdot 10^{+205}\right):\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(-3 \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.re -1.2e+201) (not (<= x.re 1.7e+205)))
   (* x.im (* x.re x.im))
   (* x.im (* -3.0 (* x.re x.im)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -1.2e+201) || !(x_46_re <= 1.7e+205)) {
		tmp = x_46_im * (x_46_re * x_46_im);
	} else {
		tmp = x_46_im * (-3.0 * (x_46_re * x_46_im));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46re <= (-1.2d+201)) .or. (.not. (x_46re <= 1.7d+205))) then
        tmp = x_46im * (x_46re * x_46im)
    else
        tmp = x_46im * ((-3.0d0) * (x_46re * x_46im))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -1.2e+201) || !(x_46_re <= 1.7e+205)) {
		tmp = x_46_im * (x_46_re * x_46_im);
	} else {
		tmp = x_46_im * (-3.0 * (x_46_re * x_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_re <= -1.2e+201) or not (x_46_re <= 1.7e+205):
		tmp = x_46_im * (x_46_re * x_46_im)
	else:
		tmp = x_46_im * (-3.0 * (x_46_re * x_46_im))
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_re <= -1.2e+201) || !(x_46_re <= 1.7e+205))
		tmp = Float64(x_46_im * Float64(x_46_re * x_46_im));
	else
		tmp = Float64(x_46_im * Float64(-3.0 * Float64(x_46_re * x_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_re <= -1.2e+201) || ~((x_46_re <= 1.7e+205)))
		tmp = x_46_im * (x_46_re * x_46_im);
	else
		tmp = x_46_im * (-3.0 * (x_46_re * x_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$re, -1.2e+201], N[Not[LessEqual[x$46$re, 1.7e+205]], $MachinePrecision]], N[(x$46$im * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(-3.0 * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < -1.19999999999999996e201 or 1.7e205 < x.re

   1. Initial program 66.7%

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

      1. sqr-neg 66.7%

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

      2. difference-of-squares 66.7%

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

      3. sub-neg 66.7%

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

      4. associate-*l* 66.7%

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

      5. sub-neg 66.7%

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

      6. remove-double-neg 66.7%

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

      7. +-commutative 66.7%

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

      8. *-commutative 66.7%

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

      9. *-commutative 66.7%

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

      10. distribute-rgt-out 66.7%

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

   3. Simplified 66.7%

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

   4. Taylor expanded in x.re around 0 0.5%

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

      1. mul-1-neg 0.5%

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

      2. distribute-rgt-neg-in 0.5%

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

      3. unpow2 0.5%

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

      4. mul-1-neg 0.5%

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

      5. associate-*l* 0.5%

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

      6. mul-1-neg 0.5%

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

   6. Simplified 0.5%

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

      1. distribute-lft-out-- 0.5%

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

      2. add-sqr-sqrt 0.2%

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

      3. sqrt-unprod 35.7%

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

      4. sqr-neg 35.7%

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

      5. sqrt-unprod 0.3%

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

      6. add-sqr-sqrt 0.5%

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

      7. *-commutative 0.5%

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

   8. Applied egg-rr 0.5%

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

      1. distribute-lft-out-- 0.5%

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

      2. associate--r+ 0.5%

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

      3. unsub-neg 0.5%

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

      4. mul-1-neg 0.5%

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

      5. distribute-rgt1-in 0.5%

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

      6. metadata-eval 0.5%

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

      7. mul0-lft 0.5%

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

      8. neg-sub0 0.5%

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

      9. distribute-rgt-neg-in 0.5%

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

      10. *-commutative 0.5%

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

      11. distribute-rgt-neg-in 0.5%

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

   10. Simplified 0.5%

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

       1. expm1-log1p-u 0.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x.im \cdot \left(x.im \cdot \left(-x.re\right)\right)\right)\right)} \]

       2. expm1-udef 0.6%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x.im \cdot \left(x.im \cdot \left(-x.re\right)\right)\right)} - 1} \]

       3. add-sqr-sqrt 0.3%

          \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \left(x.im \cdot \color{blue}{\left(\sqrt{-x.re} \cdot \sqrt{-x.re}\right)}\right)\right)} - 1 \]

       4. sqrt-unprod 45.2%

          \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \left(x.im \cdot \color{blue}{\sqrt{\left(-x.re\right) \cdot \left(-x.re\right)}}\right)\right)} - 1 \]

       5. sqr-neg 45.2%

          \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \left(x.im \cdot \sqrt{\color{blue}{x.re \cdot x.re}}\right)\right)} - 1 \]

       6. sqrt-unprod 17.5%

          \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \left(x.im \cdot \color{blue}{\left(\sqrt{x.re} \cdot \sqrt{x.re}\right)}\right)\right)} - 1 \]

       7. add-sqr-sqrt 17.8%

          \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \left(x.im \cdot \color{blue}{x.re}\right)\right)} - 1 \]

       8. *-commutative 17.8%

          \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \color{blue}{\left(x.re \cdot x.im\right)}\right)} - 1 \]

   12. Applied egg-rr 17.8%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x.im \cdot \left(x.re \cdot x.im\right)\right)} - 1} \]
   13. Step-by-step derivation

       1. expm1-def 18.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x.im \cdot \left(x.re \cdot x.im\right)\right)\right)} \]

       2. expm1-log1p 36.1%

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

   14. Simplified 36.1%

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

   if -1.19999999999999996e201 < x.re < 1.7e205

   1. Initial program 85.9%

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

      1. sqr-neg 85.9%

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

      2. difference-of-squares 89.7%

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

      3. sub-neg 89.7%

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

      4. associate-*l* 98.4%

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

      5. sub-neg 98.4%

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

      6. remove-double-neg 98.4%

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

      7. +-commutative 98.4%

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

      8. *-commutative 98.4%

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

      9. *-commutative 98.4%

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

      10. distribute-rgt-out 98.4%

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

   3. Simplified 98.4%

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

      1. cancel-sign-sub-inv 98.4%

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

      2. fma-def 98.4%

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

      3. *-commutative 98.4%

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

   5. Applied egg-rr 98.4%

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

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

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

      1. distribute-rgt-out 63.4%

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

      2. unpow2 63.4%

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

      3. metadata-eval 63.4%

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

      4. associate-*l* 63.4%

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

      5. associate-*l* 72.2%

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

      6. *-commutative 72.2%

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

      7. associate-*r* 72.2%

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

   8. Simplified 72.2%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -1.2 \cdot 10^{+201} \lor \neg \left(x.re \leq 1.7 \cdot 10^{+205}\right):\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(-3 \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \]

Alternative 7: 59.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -1.35 \cdot 10^{+209} \lor \neg \left(x.re \leq 2.4 \cdot 10^{+205}\right):\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.im \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(-3 \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.re -1.35e+209) (not (<= x.re 2.4e+205)))
   (* (* x.re x.im) (* x.im 3.0))
   (* x.im (* -3.0 (* x.re x.im)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -1.35e+209) || !(x_46_re <= 2.4e+205)) {
		tmp = (x_46_re * x_46_im) * (x_46_im * 3.0);
	} else {
		tmp = x_46_im * (-3.0 * (x_46_re * x_46_im));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46re <= (-1.35d+209)) .or. (.not. (x_46re <= 2.4d+205))) then
        tmp = (x_46re * x_46im) * (x_46im * 3.0d0)
    else
        tmp = x_46im * ((-3.0d0) * (x_46re * x_46im))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -1.35e+209) || !(x_46_re <= 2.4e+205)) {
		tmp = (x_46_re * x_46_im) * (x_46_im * 3.0);
	} else {
		tmp = x_46_im * (-3.0 * (x_46_re * x_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_re <= -1.35e+209) or not (x_46_re <= 2.4e+205):
		tmp = (x_46_re * x_46_im) * (x_46_im * 3.0)
	else:
		tmp = x_46_im * (-3.0 * (x_46_re * x_46_im))
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_re <= -1.35e+209) || !(x_46_re <= 2.4e+205))
		tmp = Float64(Float64(x_46_re * x_46_im) * Float64(x_46_im * 3.0));
	else
		tmp = Float64(x_46_im * Float64(-3.0 * Float64(x_46_re * x_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_re <= -1.35e+209) || ~((x_46_re <= 2.4e+205)))
		tmp = (x_46_re * x_46_im) * (x_46_im * 3.0);
	else
		tmp = x_46_im * (-3.0 * (x_46_re * x_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$re, -1.35e+209], N[Not[LessEqual[x$46$re, 2.4e+205]], $MachinePrecision]], N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(x$46$im * 3.0), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(-3.0 * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < -1.35e209 or 2.39999999999999986e205 < x.re

   1. Initial program 66.7%

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

      1. sqr-neg 66.7%

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

      2. difference-of-squares 66.7%

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

      3. sub-neg 66.7%

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

      4. associate-*l* 66.7%

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

      5. sub-neg 66.7%

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

      6. remove-double-neg 66.7%

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

      7. +-commutative 66.7%

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

      8. *-commutative 66.7%

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

      9. *-commutative 66.7%

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

      10. distribute-rgt-out 66.7%

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

   3. Simplified 66.7%

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

   4. Taylor expanded in x.re around 0 0.5%

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

      1. mul-1-neg 0.5%

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

      2. unpow2 0.5%

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

   6. Simplified 0.5%

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

      1. cancel-sign-sub-inv 0.5%

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

      2. add-sqr-sqrt 0.4%

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

      3. sqrt-unprod 7.6%

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

      4. sqr-neg 7.6%

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

      5. sqrt-unprod 0.4%

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

      6. add-sqr-sqrt 0.5%

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

      7. *-commutative 0.5%

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

      8. add-sqr-sqrt 0.2%

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

      9. sqrt-unprod 18.7%

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

      10. sqr-neg 18.7%

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

      11. sqrt-prod 18.5%

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

      12. add-sqr-sqrt 36.1%

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

      13. associate-*r* 36.1%

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

      14. *-commutative 36.1%

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

   8. Applied egg-rr 36.1%

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

      1. +-commutative 36.1%

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

      2. associate-*l* 36.0%

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

      3. unpow2 36.0%

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

      4. distribute-lft-out 36.0%

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

      5. *-commutative 36.0%

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

      6. count-2 36.0%

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

      7. associate-*r* 36.0%

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

      8. unpow2 36.0%

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

      9. distribute-lft1-in 36.0%

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

      10. metadata-eval 36.0%

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

      11. *-commutative 36.0%

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

      12. unpow2 36.0%

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

   10. Simplified 36.0%

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

   11. Taylor expanded in x.re around 0 36.0%

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

       1. associate-*r* 36.0%

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

       2. *-commutative 36.0%

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

       3. unpow2 36.0%

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

       4. associate-*r* 36.0%

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

       5. *-commutative 36.0%

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

       6. associate-*r* 36.1%

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

   13. Simplified 36.1%

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

   if -1.35e209 < x.re < 2.39999999999999986e205

   1. Initial program 85.9%

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

      1. sqr-neg 85.9%

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

      2. difference-of-squares 89.7%

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

      3. sub-neg 89.7%

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

      4. associate-*l* 98.4%

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

      5. sub-neg 98.4%

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

      6. remove-double-neg 98.4%

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

      7. +-commutative 98.4%

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

      8. *-commutative 98.4%

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

      9. *-commutative 98.4%

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

      10. distribute-rgt-out 98.4%

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

   3. Simplified 98.4%

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

      1. cancel-sign-sub-inv 98.4%

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

      2. fma-def 98.4%

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

      3. *-commutative 98.4%

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

   5. Applied egg-rr 98.4%

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

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

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

      1. distribute-rgt-out 63.4%

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

      2. unpow2 63.4%

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

      3. metadata-eval 63.4%

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

      4. associate-*l* 63.4%

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

      5. associate-*l* 72.2%

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

      6. *-commutative 72.2%

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

      7. associate-*r* 72.2%

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

   8. Simplified 72.2%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -1.35 \cdot 10^{+209} \lor \neg \left(x.re \leq 2.4 \cdot 10^{+205}\right):\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.im \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(-3 \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \]

Alternative 8: 39.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -4.25 \cdot 10^{+204} \lor \neg \left(x.re \leq 7.2 \cdot 10^{+205}\right):\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(-x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.re -4.25e+204) (not (<= x.re 7.2e+205)))
   (* x.im (* x.re x.im))
   (* x.im (* x.re (- x.im)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -4.25e+204) || !(x_46_re <= 7.2e+205)) {
		tmp = x_46_im * (x_46_re * x_46_im);
	} else {
		tmp = x_46_im * (x_46_re * -x_46_im);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46re <= (-4.25d+204)) .or. (.not. (x_46re <= 7.2d+205))) then
        tmp = x_46im * (x_46re * x_46im)
    else
        tmp = x_46im * (x_46re * -x_46im)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -4.25e+204) || !(x_46_re <= 7.2e+205)) {
		tmp = x_46_im * (x_46_re * x_46_im);
	} else {
		tmp = x_46_im * (x_46_re * -x_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_re <= -4.25e+204) or not (x_46_re <= 7.2e+205):
		tmp = x_46_im * (x_46_re * x_46_im)
	else:
		tmp = x_46_im * (x_46_re * -x_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_re <= -4.25e+204) || !(x_46_re <= 7.2e+205))
		tmp = Float64(x_46_im * Float64(x_46_re * x_46_im));
	else
		tmp = Float64(x_46_im * Float64(x_46_re * Float64(-x_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_re <= -4.25e+204) || ~((x_46_re <= 7.2e+205)))
		tmp = x_46_im * (x_46_re * x_46_im);
	else
		tmp = x_46_im * (x_46_re * -x_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$re, -4.25e+204], N[Not[LessEqual[x$46$re, 7.2e+205]], $MachinePrecision]], N[(x$46$im * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(x$46$re * (-x$46$im)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < -4.25e204 or 7.20000000000000003e205 < x.re

   1. Initial program 66.7%

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

      1. sqr-neg 66.7%

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

      2. difference-of-squares 66.7%

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

      3. sub-neg 66.7%

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

      4. associate-*l* 66.7%

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

      5. sub-neg 66.7%

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

      6. remove-double-neg 66.7%

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

      7. +-commutative 66.7%

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

      8. *-commutative 66.7%

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

      9. *-commutative 66.7%

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

      10. distribute-rgt-out 66.7%

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

   3. Simplified 66.7%

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

   4. Taylor expanded in x.re around 0 0.5%

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

      1. mul-1-neg 0.5%

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

      2. distribute-rgt-neg-in 0.5%

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

      3. unpow2 0.5%

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

      4. mul-1-neg 0.5%

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

      5. associate-*l* 0.5%

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

      6. mul-1-neg 0.5%

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

   6. Simplified 0.5%

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

      1. distribute-lft-out-- 0.5%

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

      2. add-sqr-sqrt 0.2%

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

      3. sqrt-unprod 35.7%

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

      4. sqr-neg 35.7%

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

      5. sqrt-unprod 0.3%

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

      6. add-sqr-sqrt 0.5%

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

      7. *-commutative 0.5%

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

   8. Applied egg-rr 0.5%

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

      1. distribute-lft-out-- 0.5%

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

      2. associate--r+ 0.5%

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

      3. unsub-neg 0.5%

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

      4. mul-1-neg 0.5%

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

      5. distribute-rgt1-in 0.5%

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

      6. metadata-eval 0.5%

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

      7. mul0-lft 0.5%

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

      8. neg-sub0 0.5%

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

      9. distribute-rgt-neg-in 0.5%

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

      10. *-commutative 0.5%

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

      11. distribute-rgt-neg-in 0.5%

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

   10. Simplified 0.5%

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

       1. expm1-log1p-u 0.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x.im \cdot \left(x.im \cdot \left(-x.re\right)\right)\right)\right)} \]

       2. expm1-udef 0.6%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x.im \cdot \left(x.im \cdot \left(-x.re\right)\right)\right)} - 1} \]

       3. add-sqr-sqrt 0.3%

          \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \left(x.im \cdot \color{blue}{\left(\sqrt{-x.re} \cdot \sqrt{-x.re}\right)}\right)\right)} - 1 \]

       4. sqrt-unprod 45.2%

          \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \left(x.im \cdot \color{blue}{\sqrt{\left(-x.re\right) \cdot \left(-x.re\right)}}\right)\right)} - 1 \]

       5. sqr-neg 45.2%

          \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \left(x.im \cdot \sqrt{\color{blue}{x.re \cdot x.re}}\right)\right)} - 1 \]

       6. sqrt-unprod 17.5%

          \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \left(x.im \cdot \color{blue}{\left(\sqrt{x.re} \cdot \sqrt{x.re}\right)}\right)\right)} - 1 \]

       7. add-sqr-sqrt 17.8%

          \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \left(x.im \cdot \color{blue}{x.re}\right)\right)} - 1 \]

       8. *-commutative 17.8%

          \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \color{blue}{\left(x.re \cdot x.im\right)}\right)} - 1 \]

   12. Applied egg-rr 17.8%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x.im \cdot \left(x.re \cdot x.im\right)\right)} - 1} \]
   13. Step-by-step derivation

       1. expm1-def 18.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x.im \cdot \left(x.re \cdot x.im\right)\right)\right)} \]

       2. expm1-log1p 36.1%

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

   14. Simplified 36.1%

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

   if -4.25e204 < x.re < 7.20000000000000003e205

   1. Initial program 85.9%

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

      1. sqr-neg 85.9%

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

      2. difference-of-squares 89.7%

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

      3. sub-neg 89.7%

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

      4. associate-*l* 98.4%

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

      5. sub-neg 98.4%

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

      6. remove-double-neg 98.4%

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

      7. +-commutative 98.4%

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

      8. *-commutative 98.4%

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

      9. *-commutative 98.4%

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

      10. distribute-rgt-out 98.4%

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

   3. Simplified 98.4%

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

   4. Taylor expanded in x.re around 0 63.5%

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

      1. mul-1-neg 63.5%

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

      2. distribute-rgt-neg-in 63.5%

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

      3. unpow2 63.5%

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

      4. mul-1-neg 63.5%

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

      5. associate-*l* 72.2%

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

      6. mul-1-neg 72.2%

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

   6. Simplified 72.2%

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

      1. distribute-lft-out-- 72.2%

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

      2. add-sqr-sqrt 32.0%

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

      3. sqrt-unprod 50.2%

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

      4. sqr-neg 50.2%

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

      5. sqrt-unprod 23.0%

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

      6. add-sqr-sqrt 38.1%

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

      7. *-commutative 38.1%

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

   8. Applied egg-rr 38.1%

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

      1. distribute-lft-out-- 44.6%

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

      2. associate--r+ 44.6%

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

      3. unsub-neg 44.6%

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

      4. mul-1-neg 44.6%

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

      5. distribute-rgt1-in 44.6%

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

      6. metadata-eval 44.6%

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

      7. mul0-lft 44.6%

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

      8. neg-sub0 44.6%

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

      9. distribute-rgt-neg-in 44.6%

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

      10. *-commutative 44.6%

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

      11. distribute-rgt-neg-in 44.6%

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

   10. Simplified 44.6%

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

4. Final simplification 43.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -4.25 \cdot 10^{+204} \lor \neg \left(x.re \leq 7.2 \cdot 10^{+205}\right):\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(-x.im\right)\right)\\ \end{array} \]

Alternative 9: 23.3% accurate, 3.8× speedup

Math

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

FPCore

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

C

double code(double x_46_re, double x_46_im) {
	return 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_46im * (x_46re * x_46im)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.8%

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

   1. sqr-neg 82.8%

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

   2. difference-of-squares 85.9%

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

   3. sub-neg 85.9%

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

   4. associate-*l* 93.2%

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

   5. sub-neg 93.2%

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

   6. remove-double-neg 93.2%

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

   7. +-commutative 93.2%

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

   8. *-commutative 93.2%

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

   9. *-commutative 93.2%

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

   10. distribute-rgt-out 93.2%

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

3. Simplified 93.2%

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

4. Taylor expanded in x.re around 0 53.1%

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

   1. mul-1-neg 53.1%

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

   2. distribute-rgt-neg-in 53.1%

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

   3. unpow2 53.1%

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

   4. mul-1-neg 53.1%

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

   5. associate-*l* 60.4%

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

   6. mul-1-neg 60.4%

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

6. Simplified 60.4%

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

   1. distribute-lft-out-- 60.4%

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

   2. add-sqr-sqrt 26.8%

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

   3. sqrt-unprod 47.8%

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

   4. sqr-neg 47.8%

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

   5. sqrt-unprod 19.3%

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

   6. add-sqr-sqrt 31.9%

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

   7. *-commutative 31.9%

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

8. Applied egg-rr 31.9%

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

   1. distribute-lft-out-- 37.4%

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

   2. associate--r+ 37.4%

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

   3. unsub-neg 37.4%

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

   4. mul-1-neg 37.4%

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

   5. distribute-rgt1-in 37.4%

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

   6. metadata-eval 37.4%

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

   7. mul0-lft 37.4%

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

   8. neg-sub0 37.4%

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

   9. distribute-rgt-neg-in 37.4%

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

   10. *-commutative 37.4%

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

   11. distribute-rgt-neg-in 37.4%

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

10. Simplified 37.4%

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

    1. expm1-log1p-u 25.8%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x.im \cdot \left(x.im \cdot \left(-x.re\right)\right)\right)\right)} \]

    2. expm1-udef 24.5%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x.im \cdot \left(x.im \cdot \left(-x.re\right)\right)\right)} - 1} \]

    3. add-sqr-sqrt 14.3%

       \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \left(x.im \cdot \color{blue}{\left(\sqrt{-x.re} \cdot \sqrt{-x.re}\right)}\right)\right)} - 1 \]

    4. sqrt-unprod 33.1%

       \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \left(x.im \cdot \color{blue}{\sqrt{\left(-x.re\right) \cdot \left(-x.re\right)}}\right)\right)} - 1 \]

    5. sqr-neg 33.1%

       \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \left(x.im \cdot \sqrt{\color{blue}{x.re \cdot x.re}}\right)\right)} - 1 \]

    6. sqrt-unprod 14.3%

       \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \left(x.im \cdot \color{blue}{\left(\sqrt{x.re} \cdot \sqrt{x.re}\right)}\right)\right)} - 1 \]

    7. add-sqr-sqrt 20.6%

       \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \left(x.im \cdot \color{blue}{x.re}\right)\right)} - 1 \]

    8. *-commutative 20.6%

       \[\leadsto e^{\mathsf{log1p}\left(x.im \cdot \color{blue}{\left(x.re \cdot x.im\right)}\right)} - 1 \]

12. Applied egg-rr 20.6%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x.im \cdot \left(x.re \cdot x.im\right)\right)} - 1} \]
13. Step-by-step derivation

    1. expm1-def 20.5%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x.im \cdot \left(x.re \cdot x.im\right)\right)\right)} \]

    2. expm1-log1p 23.7%

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

14. Simplified 23.7%

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

15. Final simplification 23.7%

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

Developer target: 87.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023293 
(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)))