math.cube on complex, imaginary part

Percentage Accurate: 82.4% → 99.5%

Time: 9.9s

Alternatives: 14

Speedup: 1.1×

• 45.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

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_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
}

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_46im) + (((x_46re * x_46im) + (x_46im * x_46re)) * x_46re)
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_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
}

Python

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

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_im) + Float64(Float64(Float64(x_46_re * x_46_im) + Float64(x_46_im * x_46_re)) * x_46_re))
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_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
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$46$im), $MachinePrecision] + N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]

Initial Program: 82.4% accurate, 1.0× speedup

Math

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

FPCore

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

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_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
}

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_46im) + (((x_46re * x_46im) + (x_46im * x_46re)) * x_46re)
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_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
}

Python

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

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_im) + Float64(Float64(Float64(x_46_re * x_46_im) + Float64(x_46_im * x_46_re)) * x_46_re))
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_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
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$46$im), $MachinePrecision] + N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := x.re\_m \cdot \left(x.re\_m - x.im\_m\right)\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 1.4 \cdot 10^{+89}:\\ \;\;\;\;\mathsf{fma}\left(x.im\_m \cdot x.re\_m, x.re\_m \cdot 3, -{x.im\_m}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(t\_0 + \frac{x.im\_m}{x.re\_m} \cdot t\_0\right) + -3\\ \end{array} \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (let* ((t_0 (* x.re_m (- x.re_m x.im_m))))
   (*
    x.im_s
    (if (<= x.im_m 1.4e+89)
      (fma (* x.im_m x.re_m) (* x.re_m 3.0) (- (pow x.im_m 3.0)))
      (+ (* x.im_m (+ t_0 (* (/ x.im_m x.re_m) t_0))) -3.0)))))

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double t_0 = x_46_re_m * (x_46_re_m - x_46_im_m);
	double tmp;
	if (x_46_im_m <= 1.4e+89) {
		tmp = fma((x_46_im_m * x_46_re_m), (x_46_re_m * 3.0), -pow(x_46_im_m, 3.0));
	} else {
		tmp = (x_46_im_m * (t_0 + ((x_46_im_m / x_46_re_m) * t_0))) + -3.0;
	}
	return x_46_im_s * tmp;
}

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	t_0 = Float64(x_46_re_m * Float64(x_46_re_m - x_46_im_m))
	tmp = 0.0
	if (x_46_im_m <= 1.4e+89)
		tmp = fma(Float64(x_46_im_m * x_46_re_m), Float64(x_46_re_m * 3.0), Float64(-(x_46_im_m ^ 3.0)));
	else
		tmp = Float64(Float64(x_46_im_m * Float64(t_0 + Float64(Float64(x_46_im_m / x_46_re_m) * t_0))) + -3.0);
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re$95$m_, x$46$im$95$m_] := Block[{t$95$0 = N[(x$46$re$95$m * N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 1.4e+89], N[(N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision] * N[(x$46$re$95$m * 3.0), $MachinePrecision] + (-N[Power[x$46$im$95$m, 3.0], $MachinePrecision])), $MachinePrecision], N[(N[(x$46$im$95$m * N[(t$95$0 + N[(N[(x$46$im$95$m / x$46$re$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -3.0), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.3999999999999999e89

   1. Initial program 87.1%

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

   3. Simplified 92.5%

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

      1. associate-*r* 92.6%

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

      2. fma-neg 93.5%

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

   6. Applied egg-rr 93.5%

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

   if 1.3999999999999999e89 < x.im

   1. Initial program 65.8%

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

      1. difference-of-squares 73.7%

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

      2. *-commutative 73.7%

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

   4. Applied egg-rr 73.7%

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

   5. Taylor expanded in x.re around inf 73.7%

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

      1. associate-*r* 73.7%

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

      2. +-commutative 73.7%

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

      3. distribute-rgt-in 63.2%

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

      4. *-un-lft-identity 63.2%

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

   7. Applied egg-rr 63.2%

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

   8. Taylor expanded in x.re around 0 63.2%

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

   10. Simplified 73.7%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 1.4 \cdot 10^{+89}:\\ \;\;\;\;\mathsf{fma}\left(x.im \cdot x.re, x.re \cdot 3, -{x.im}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.re - x.im\right) + \frac{x.im}{x.re} \cdot \left(x.re \cdot \left(x.re - x.im\right)\right)\right) + -3\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := x.re\_m \cdot \left(x.re\_m - x.im\_m\right)\\ t_1 := x.im\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) + x.re\_m \cdot \left(x.im\_m \cdot x.re\_m + x.im\_m \cdot x.re\_m\right)\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 10^{+280}:\\ \;\;\;\;x.im\_m \cdot \left(\left(x.re\_m - x.im\_m\right) \cdot \left(x.im\_m + x.re\_m\right)\right) + x.re\_m \cdot \left(\left(x.im\_m \cdot x.re\_m\right) \cdot 2\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(x.im\_m \cdot x.re\_m, x.re\_m \cdot 3, -1\right)\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(t\_0 + \frac{x.im\_m}{x.re\_m} \cdot t\_0\right) + -3\\ \end{array} \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (let* ((t_0 (* x.re_m (- x.re_m x.im_m)))
        (t_1
         (+
          (* x.im_m (- (* x.re_m x.re_m) (* x.im_m x.im_m)))
          (* x.re_m (+ (* x.im_m x.re_m) (* x.im_m x.re_m))))))
   (*
    x.im_s
    (if (<= t_1 1e+280)
      (+
       (* x.im_m (* (- x.re_m x.im_m) (+ x.im_m x.re_m)))
       (* x.re_m (* (* x.im_m x.re_m) 2.0)))
      (if (<= t_1 INFINITY)
        (fma (* x.im_m x.re_m) (* x.re_m 3.0) -1.0)
        (+ (* x.im_m (+ t_0 (* (/ x.im_m x.re_m) t_0))) -3.0))))))

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double t_0 = x_46_re_m * (x_46_re_m - x_46_im_m);
	double t_1 = (x_46_im_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) + (x_46_re_m * ((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)));
	double tmp;
	if (t_1 <= 1e+280) {
		tmp = (x_46_im_m * ((x_46_re_m - x_46_im_m) * (x_46_im_m + x_46_re_m))) + (x_46_re_m * ((x_46_im_m * x_46_re_m) * 2.0));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma((x_46_im_m * x_46_re_m), (x_46_re_m * 3.0), -1.0);
	} else {
		tmp = (x_46_im_m * (t_0 + ((x_46_im_m / x_46_re_m) * t_0))) + -3.0;
	}
	return x_46_im_s * tmp;
}

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	t_0 = Float64(x_46_re_m * Float64(x_46_re_m - x_46_im_m))
	t_1 = Float64(Float64(x_46_im_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m))) + Float64(x_46_re_m * Float64(Float64(x_46_im_m * x_46_re_m) + Float64(x_46_im_m * x_46_re_m))))
	tmp = 0.0
	if (t_1 <= 1e+280)
		tmp = Float64(Float64(x_46_im_m * Float64(Float64(x_46_re_m - x_46_im_m) * Float64(x_46_im_m + x_46_re_m))) + Float64(x_46_re_m * Float64(Float64(x_46_im_m * x_46_re_m) * 2.0)));
	elseif (t_1 <= Inf)
		tmp = fma(Float64(x_46_im_m * x_46_re_m), Float64(x_46_re_m * 3.0), -1.0);
	else
		tmp = Float64(Float64(x_46_im_m * Float64(t_0 + Float64(Float64(x_46_im_m / x_46_re_m) * t_0))) + -3.0);
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re$95$m_, x$46$im$95$m_] := Block[{t$95$0 = N[(x$46$re$95$m * N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re$95$m * N[(N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$1, 1e+280], N[(N[(x$46$im$95$m * N[(N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m + x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re$95$m * N[(N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision] * N[(x$46$re$95$m * 3.0), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(x$46$im$95$m * N[(t$95$0 + N[(N[(x$46$im$95$m / x$46$re$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -3.0), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < 1e280

   1. Initial program 95.2%

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

      1. difference-of-squares 95.2%

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

      2. *-commutative 95.2%

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

   4. Applied egg-rr 95.2%

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

      1. *-commutative 95.2%

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

      2. count-2 95.2%

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

      3. *-commutative 95.2%

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

   6. Applied egg-rr 95.2%

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

   if 1e280 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   1. Initial program 85.1%

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

   3. Simplified 92.0%

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

      1. associate-*r* 92.0%

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

      2. fma-neg 92.0%

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

   6. Applied egg-rr 92.0%

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

   7. Taylor expanded in x.im around 0 92.0%

      \[\leadsto \mathsf{fma}\left(x.re \cdot x.im, x.re \cdot 3, \color{blue}{-1 \cdot {x.im}^{3}}\right) \]

   9. Simplified 50.3%

      \[\leadsto \mathsf{fma}\left(x.re \cdot x.im, x.re \cdot 3, \color{blue}{-1}\right) \]

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

   1. Initial program 0.0%

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

      1. difference-of-squares 20.0%

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

      2. *-commutative 20.0%

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

   4. Applied egg-rr 20.0%

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

   5. Taylor expanded in x.re around inf 20.0%

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

      1. associate-*r* 20.0%

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

      2. +-commutative 20.0%

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

      3. distribute-rgt-in 12.0%

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

      4. *-un-lft-identity 12.0%

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

   7. Applied egg-rr 12.0%

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

   8. Taylor expanded in x.re around 0 12.0%

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

   10. Simplified 56.0%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right) \leq 10^{+280}:\\ \;\;\;\;x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right) + x.re \cdot \left(\left(x.im \cdot x.re\right) \cdot 2\right)\\ \mathbf{elif}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right) \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(x.im \cdot x.re, x.re \cdot 3, -1\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.re - x.im\right) + \frac{x.im}{x.re} \cdot \left(x.re \cdot \left(x.re - x.im\right)\right)\right) + -3\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := x.re\_m \cdot \left(x.re\_m - x.im\_m\right)\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 1.4 \cdot 10^{+89}:\\ \;\;\;\;x.re\_m \cdot \left(\left(x.im\_m \cdot x.re\_m\right) \cdot 3\right) - {x.im\_m}^{3}\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(t\_0 + \frac{x.im\_m}{x.re\_m} \cdot t\_0\right) + -3\\ \end{array} \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (let* ((t_0 (* x.re_m (- x.re_m x.im_m))))
   (*
    x.im_s
    (if (<= x.im_m 1.4e+89)
      (- (* x.re_m (* (* x.im_m x.re_m) 3.0)) (pow x.im_m 3.0))
      (+ (* x.im_m (+ t_0 (* (/ x.im_m x.re_m) t_0))) -3.0)))))

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double t_0 = x_46_re_m * (x_46_re_m - x_46_im_m);
	double tmp;
	if (x_46_im_m <= 1.4e+89) {
		tmp = (x_46_re_m * ((x_46_im_m * x_46_re_m) * 3.0)) - pow(x_46_im_m, 3.0);
	} else {
		tmp = (x_46_im_m * (t_0 + ((x_46_im_m / x_46_re_m) * t_0))) + -3.0;
	}
	return x_46_im_s * tmp;
}

Fortran

x.re_m = abs(x_46re)
x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x_46re_m * (x_46re_m - x_46im_m)
    if (x_46im_m <= 1.4d+89) then
        tmp = (x_46re_m * ((x_46im_m * x_46re_m) * 3.0d0)) - (x_46im_m ** 3.0d0)
    else
        tmp = (x_46im_m * (t_0 + ((x_46im_m / x_46re_m) * t_0))) + (-3.0d0)
    end if
    code = x_46im_s * tmp
end function

Java

x.re_m = Math.abs(x_46_re);
x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double t_0 = x_46_re_m * (x_46_re_m - x_46_im_m);
	double tmp;
	if (x_46_im_m <= 1.4e+89) {
		tmp = (x_46_re_m * ((x_46_im_m * x_46_re_m) * 3.0)) - Math.pow(x_46_im_m, 3.0);
	} else {
		tmp = (x_46_im_m * (t_0 + ((x_46_im_m / x_46_re_m) * t_0))) + -3.0;
	}
	return x_46_im_s * tmp;
}

Python

x.re_m = math.fabs(x_46_re)
x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re_m, x_46_im_m):
	t_0 = x_46_re_m * (x_46_re_m - x_46_im_m)
	tmp = 0
	if x_46_im_m <= 1.4e+89:
		tmp = (x_46_re_m * ((x_46_im_m * x_46_re_m) * 3.0)) - math.pow(x_46_im_m, 3.0)
	else:
		tmp = (x_46_im_m * (t_0 + ((x_46_im_m / x_46_re_m) * t_0))) + -3.0
	return x_46_im_s * tmp

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	t_0 = Float64(x_46_re_m * Float64(x_46_re_m - x_46_im_m))
	tmp = 0.0
	if (x_46_im_m <= 1.4e+89)
		tmp = Float64(Float64(x_46_re_m * Float64(Float64(x_46_im_m * x_46_re_m) * 3.0)) - (x_46_im_m ^ 3.0));
	else
		tmp = Float64(Float64(x_46_im_m * Float64(t_0 + Float64(Float64(x_46_im_m / x_46_re_m) * t_0))) + -3.0);
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.re_m = abs(x_46_re);
x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re_m, x_46_im_m)
	t_0 = x_46_re_m * (x_46_re_m - x_46_im_m);
	tmp = 0.0;
	if (x_46_im_m <= 1.4e+89)
		tmp = (x_46_re_m * ((x_46_im_m * x_46_re_m) * 3.0)) - (x_46_im_m ^ 3.0);
	else
		tmp = (x_46_im_m * (t_0 + ((x_46_im_m / x_46_re_m) * t_0))) + -3.0;
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re$95$m_, x$46$im$95$m_] := Block[{t$95$0 = N[(x$46$re$95$m * N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 1.4e+89], N[(N[(x$46$re$95$m * N[(N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] - N[Power[x$46$im$95$m, 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im$95$m * N[(t$95$0 + N[(N[(x$46$im$95$m / x$46$re$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -3.0), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.3999999999999999e89

   1. Initial program 87.1%

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

   3. Simplified 92.5%

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

   5. Taylor expanded in x.im around 0 92.5%

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

   if 1.3999999999999999e89 < x.im

   1. Initial program 65.8%

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

      1. difference-of-squares 73.7%

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

      2. *-commutative 73.7%

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

   4. Applied egg-rr 73.7%

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

   5. Taylor expanded in x.re around inf 73.7%

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

      1. associate-*r* 73.7%

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

      2. +-commutative 73.7%

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

      3. distribute-rgt-in 63.2%

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

      4. *-un-lft-identity 63.2%

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

   7. Applied egg-rr 63.2%

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

   8. Taylor expanded in x.re around 0 63.2%

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

   10. Simplified 73.7%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 1.4 \cdot 10^{+89}:\\ \;\;\;\;x.re \cdot \left(\left(x.im \cdot x.re\right) \cdot 3\right) - {x.im}^{3}\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.re - x.im\right) + \frac{x.im}{x.re} \cdot \left(x.re \cdot \left(x.re - x.im\right)\right)\right) + -3\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := x.re\_m \cdot \left(x.re\_m - x.im\_m\right)\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 1.4 \cdot 10^{+89}:\\ \;\;\;\;x.im\_m \cdot \left(\left(x.re\_m - x.im\_m\right) \cdot \left(x.im\_m + x.re\_m\right)\right) + x.re\_m \cdot \left(\left(x.im\_m \cdot x.re\_m\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(t\_0 + \frac{x.im\_m}{x.re\_m} \cdot t\_0\right) + -3\\ \end{array} \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (let* ((t_0 (* x.re_m (- x.re_m x.im_m))))
   (*
    x.im_s
    (if (<= x.im_m 1.4e+89)
      (+
       (* x.im_m (* (- x.re_m x.im_m) (+ x.im_m x.re_m)))
       (* x.re_m (* (* x.im_m x.re_m) 2.0)))
      (+ (* x.im_m (+ t_0 (* (/ x.im_m x.re_m) t_0))) -3.0)))))

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double t_0 = x_46_re_m * (x_46_re_m - x_46_im_m);
	double tmp;
	if (x_46_im_m <= 1.4e+89) {
		tmp = (x_46_im_m * ((x_46_re_m - x_46_im_m) * (x_46_im_m + x_46_re_m))) + (x_46_re_m * ((x_46_im_m * x_46_re_m) * 2.0));
	} else {
		tmp = (x_46_im_m * (t_0 + ((x_46_im_m / x_46_re_m) * t_0))) + -3.0;
	}
	return x_46_im_s * tmp;
}

Fortran

x.re_m = abs(x_46re)
x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x_46re_m * (x_46re_m - x_46im_m)
    if (x_46im_m <= 1.4d+89) then
        tmp = (x_46im_m * ((x_46re_m - x_46im_m) * (x_46im_m + x_46re_m))) + (x_46re_m * ((x_46im_m * x_46re_m) * 2.0d0))
    else
        tmp = (x_46im_m * (t_0 + ((x_46im_m / x_46re_m) * t_0))) + (-3.0d0)
    end if
    code = x_46im_s * tmp
end function

Java

x.re_m = Math.abs(x_46_re);
x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double t_0 = x_46_re_m * (x_46_re_m - x_46_im_m);
	double tmp;
	if (x_46_im_m <= 1.4e+89) {
		tmp = (x_46_im_m * ((x_46_re_m - x_46_im_m) * (x_46_im_m + x_46_re_m))) + (x_46_re_m * ((x_46_im_m * x_46_re_m) * 2.0));
	} else {
		tmp = (x_46_im_m * (t_0 + ((x_46_im_m / x_46_re_m) * t_0))) + -3.0;
	}
	return x_46_im_s * tmp;
}

Python

x.re_m = math.fabs(x_46_re)
x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re_m, x_46_im_m):
	t_0 = x_46_re_m * (x_46_re_m - x_46_im_m)
	tmp = 0
	if x_46_im_m <= 1.4e+89:
		tmp = (x_46_im_m * ((x_46_re_m - x_46_im_m) * (x_46_im_m + x_46_re_m))) + (x_46_re_m * ((x_46_im_m * x_46_re_m) * 2.0))
	else:
		tmp = (x_46_im_m * (t_0 + ((x_46_im_m / x_46_re_m) * t_0))) + -3.0
	return x_46_im_s * tmp

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	t_0 = Float64(x_46_re_m * Float64(x_46_re_m - x_46_im_m))
	tmp = 0.0
	if (x_46_im_m <= 1.4e+89)
		tmp = Float64(Float64(x_46_im_m * Float64(Float64(x_46_re_m - x_46_im_m) * Float64(x_46_im_m + x_46_re_m))) + Float64(x_46_re_m * Float64(Float64(x_46_im_m * x_46_re_m) * 2.0)));
	else
		tmp = Float64(Float64(x_46_im_m * Float64(t_0 + Float64(Float64(x_46_im_m / x_46_re_m) * t_0))) + -3.0);
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.re_m = abs(x_46_re);
x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re_m, x_46_im_m)
	t_0 = x_46_re_m * (x_46_re_m - x_46_im_m);
	tmp = 0.0;
	if (x_46_im_m <= 1.4e+89)
		tmp = (x_46_im_m * ((x_46_re_m - x_46_im_m) * (x_46_im_m + x_46_re_m))) + (x_46_re_m * ((x_46_im_m * x_46_re_m) * 2.0));
	else
		tmp = (x_46_im_m * (t_0 + ((x_46_im_m / x_46_re_m) * t_0))) + -3.0;
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re$95$m_, x$46$im$95$m_] := Block[{t$95$0 = N[(x$46$re$95$m * N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 1.4e+89], N[(N[(x$46$im$95$m * N[(N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m + x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re$95$m * N[(N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im$95$m * N[(t$95$0 + N[(N[(x$46$im$95$m / x$46$re$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -3.0), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.3999999999999999e89

   1. Initial program 87.1%

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

      1. difference-of-squares 88.0%

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

      2. *-commutative 88.0%

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

   4. Applied egg-rr 88.0%

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

      1. *-commutative 88.0%

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

      2. count-2 88.0%

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

      3. *-commutative 88.0%

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

   6. Applied egg-rr 88.0%

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

   if 1.3999999999999999e89 < x.im

   1. Initial program 65.8%

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

      1. difference-of-squares 73.7%

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

      2. *-commutative 73.7%

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

   4. Applied egg-rr 73.7%

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

   5. Taylor expanded in x.re around inf 73.7%

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

      1. associate-*r* 73.7%

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

      2. +-commutative 73.7%

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

      3. distribute-rgt-in 63.2%

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

      4. *-un-lft-identity 63.2%

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

   7. Applied egg-rr 63.2%

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

   8. Taylor expanded in x.re around 0 63.2%

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

   10. Simplified 73.7%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 1.4 \cdot 10^{+89}:\\ \;\;\;\;x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right) + x.re \cdot \left(\left(x.im \cdot x.re\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.re - x.im\right) + \frac{x.im}{x.re} \cdot \left(x.re \cdot \left(x.re - x.im\right)\right)\right) + -3\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 1.5 \cdot 10^{-89}:\\ \;\;\;\;3 \cdot \left(x.im\_m \cdot \left(x.re\_m \cdot x.re\_m\right)\right)\\ \mathbf{elif}\;x.im\_m \leq 7.5 \cdot 10^{+63}:\\ \;\;\;\;x.re\_m \cdot \left(\left(x.im\_m \cdot x.re\_m\right) \cdot 2\right) - x.im\_m \cdot \left(x.im\_m \cdot x.im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(x.im\_m \cdot \left(x.re\_m - x.im\_m\right)\right) + x.re\_m \cdot -3\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.im_m 1.5e-89)
    (* 3.0 (* x.im_m (* x.re_m x.re_m)))
    (if (<= x.im_m 7.5e+63)
      (- (* x.re_m (* (* x.im_m x.re_m) 2.0)) (* x.im_m (* x.im_m x.im_m)))
      (+ (* x.im_m (* x.im_m (- x.re_m x.im_m))) (* x.re_m -3.0))))))

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 1.5e-89) {
		tmp = 3.0 * (x_46_im_m * (x_46_re_m * x_46_re_m));
	} else if (x_46_im_m <= 7.5e+63) {
		tmp = (x_46_re_m * ((x_46_im_m * x_46_re_m) * 2.0)) - (x_46_im_m * (x_46_im_m * x_46_im_m));
	} else {
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + (x_46_re_m * -3.0);
	}
	return x_46_im_s * tmp;
}

Fortran

x.re_m = abs(x_46re)
x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46im_m <= 1.5d-89) then
        tmp = 3.0d0 * (x_46im_m * (x_46re_m * x_46re_m))
    else if (x_46im_m <= 7.5d+63) then
        tmp = (x_46re_m * ((x_46im_m * x_46re_m) * 2.0d0)) - (x_46im_m * (x_46im_m * x_46im_m))
    else
        tmp = (x_46im_m * (x_46im_m * (x_46re_m - x_46im_m))) + (x_46re_m * (-3.0d0))
    end if
    code = x_46im_s * tmp
end function

Java

x.re_m = Math.abs(x_46_re);
x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 1.5e-89) {
		tmp = 3.0 * (x_46_im_m * (x_46_re_m * x_46_re_m));
	} else if (x_46_im_m <= 7.5e+63) {
		tmp = (x_46_re_m * ((x_46_im_m * x_46_re_m) * 2.0)) - (x_46_im_m * (x_46_im_m * x_46_im_m));
	} else {
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + (x_46_re_m * -3.0);
	}
	return x_46_im_s * tmp;
}

Python

x.re_m = math.fabs(x_46_re)
x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if x_46_im_m <= 1.5e-89:
		tmp = 3.0 * (x_46_im_m * (x_46_re_m * x_46_re_m))
	elif x_46_im_m <= 7.5e+63:
		tmp = (x_46_re_m * ((x_46_im_m * x_46_re_m) * 2.0)) - (x_46_im_m * (x_46_im_m * x_46_im_m))
	else:
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + (x_46_re_m * -3.0)
	return x_46_im_s * tmp

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 1.5e-89)
		tmp = Float64(3.0 * Float64(x_46_im_m * Float64(x_46_re_m * x_46_re_m)));
	elseif (x_46_im_m <= 7.5e+63)
		tmp = Float64(Float64(x_46_re_m * Float64(Float64(x_46_im_m * x_46_re_m) * 2.0)) - Float64(x_46_im_m * Float64(x_46_im_m * x_46_im_m)));
	else
		tmp = Float64(Float64(x_46_im_m * Float64(x_46_im_m * Float64(x_46_re_m - x_46_im_m))) + Float64(x_46_re_m * -3.0));
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.re_m = abs(x_46_re);
x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (x_46_im_m <= 1.5e-89)
		tmp = 3.0 * (x_46_im_m * (x_46_re_m * x_46_re_m));
	elseif (x_46_im_m <= 7.5e+63)
		tmp = (x_46_re_m * ((x_46_im_m * x_46_re_m) * 2.0)) - (x_46_im_m * (x_46_im_m * x_46_im_m));
	else
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + (x_46_re_m * -3.0);
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 1.5e-89], N[(3.0 * N[(x$46$im$95$m * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im$95$m, 7.5e+63], N[(N[(x$46$re$95$m * N[(N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] - N[(x$46$im$95$m * N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im$95$m * N[(x$46$im$95$m * N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re$95$m * -3.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x.im < 1.5e-89

   1. Initial program 84.4%

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

   3. Simplified 90.6%

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

   5. Taylor expanded in x.re around inf 58.5%

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

      1. pow2 58.5%

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

   7. Applied egg-rr 58.5%

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

   if 1.5e-89 < x.im < 7.5000000000000005e63

   1. Initial program 97.3%

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

      1. difference-of-squares 97.3%

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

      2. *-commutative 97.3%

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

   4. Applied egg-rr 97.3%

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

      1. *-commutative 97.3%

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

      2. count-2 97.3%

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

      3. *-commutative 97.3%

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

   6. Applied egg-rr 97.3%

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

   7. Taylor expanded in x.re around 0 71.3%

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

   8. Taylor expanded in x.re around 0 76.9%

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

   10. Simplified 76.9%

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

   if 7.5000000000000005e63 < x.im

   1. Initial program 69.0%

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

      1. difference-of-squares 76.2%

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

      2. *-commutative 76.2%

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

   4. Applied egg-rr 76.2%

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

      1. *-commutative 76.2%

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

      2. count-2 76.2%

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

      3. *-commutative 76.2%

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

   6. Applied egg-rr 76.2%

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

   7. Taylor expanded in x.re around 0 66.6%

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

   8. Taylor expanded in x.re around 0 66.6%

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

   10. Simplified 90.4%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 1.5 \cdot 10^{-89}:\\ \;\;\;\;3 \cdot \left(x.im \cdot \left(x.re \cdot x.re\right)\right)\\ \mathbf{elif}\;x.im \leq 7.5 \cdot 10^{+63}:\\ \;\;\;\;x.re \cdot \left(\left(x.im \cdot x.re\right) \cdot 2\right) - x.im \cdot \left(x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re - x.im\right)\right) + x.re \cdot -3\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 7.5 \cdot 10^{+63}:\\ \;\;\;\;x.im\_m \cdot \left(\left(x.re\_m - x.im\_m\right) \cdot \left(x.im\_m + x.re\_m\right)\right) + x.re\_m \cdot \left(\left(x.im\_m \cdot x.re\_m\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(x.im\_m \cdot \left(x.re\_m - x.im\_m\right)\right) + x.re\_m \cdot -3\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.im_m 7.5e+63)
    (+
     (* x.im_m (* (- x.re_m x.im_m) (+ x.im_m x.re_m)))
     (* x.re_m (* (* x.im_m x.re_m) 2.0)))
    (+ (* x.im_m (* x.im_m (- x.re_m x.im_m))) (* x.re_m -3.0)))))

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 7.5e+63) {
		tmp = (x_46_im_m * ((x_46_re_m - x_46_im_m) * (x_46_im_m + x_46_re_m))) + (x_46_re_m * ((x_46_im_m * x_46_re_m) * 2.0));
	} else {
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + (x_46_re_m * -3.0);
	}
	return x_46_im_s * tmp;
}

Fortran

x.re_m = abs(x_46re)
x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46im_m <= 7.5d+63) then
        tmp = (x_46im_m * ((x_46re_m - x_46im_m) * (x_46im_m + x_46re_m))) + (x_46re_m * ((x_46im_m * x_46re_m) * 2.0d0))
    else
        tmp = (x_46im_m * (x_46im_m * (x_46re_m - x_46im_m))) + (x_46re_m * (-3.0d0))
    end if
    code = x_46im_s * tmp
end function

Java

x.re_m = Math.abs(x_46_re);
x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 7.5e+63) {
		tmp = (x_46_im_m * ((x_46_re_m - x_46_im_m) * (x_46_im_m + x_46_re_m))) + (x_46_re_m * ((x_46_im_m * x_46_re_m) * 2.0));
	} else {
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + (x_46_re_m * -3.0);
	}
	return x_46_im_s * tmp;
}

Python

x.re_m = math.fabs(x_46_re)
x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if x_46_im_m <= 7.5e+63:
		tmp = (x_46_im_m * ((x_46_re_m - x_46_im_m) * (x_46_im_m + x_46_re_m))) + (x_46_re_m * ((x_46_im_m * x_46_re_m) * 2.0))
	else:
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + (x_46_re_m * -3.0)
	return x_46_im_s * tmp

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 7.5e+63)
		tmp = Float64(Float64(x_46_im_m * Float64(Float64(x_46_re_m - x_46_im_m) * Float64(x_46_im_m + x_46_re_m))) + Float64(x_46_re_m * Float64(Float64(x_46_im_m * x_46_re_m) * 2.0)));
	else
		tmp = Float64(Float64(x_46_im_m * Float64(x_46_im_m * Float64(x_46_re_m - x_46_im_m))) + Float64(x_46_re_m * -3.0));
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.re_m = abs(x_46_re);
x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (x_46_im_m <= 7.5e+63)
		tmp = (x_46_im_m * ((x_46_re_m - x_46_im_m) * (x_46_im_m + x_46_re_m))) + (x_46_re_m * ((x_46_im_m * x_46_re_m) * 2.0));
	else
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + (x_46_re_m * -3.0);
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 7.5e+63], N[(N[(x$46$im$95$m * N[(N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m + x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re$95$m * N[(N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im$95$m * N[(x$46$im$95$m * N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re$95$m * -3.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 7.5000000000000005e63

   1. Initial program 86.8%

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

      1. difference-of-squares 87.8%

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

      2. *-commutative 87.8%

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

   4. Applied egg-rr 87.8%

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

      1. *-commutative 87.8%

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

      2. count-2 87.8%

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

      3. *-commutative 87.8%

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

   6. Applied egg-rr 87.8%

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

   if 7.5000000000000005e63 < x.im

   1. Initial program 69.0%

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

      1. difference-of-squares 76.2%

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

      2. *-commutative 76.2%

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

   4. Applied egg-rr 76.2%

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

      1. *-commutative 76.2%

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

      2. count-2 76.2%

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

      3. *-commutative 76.2%

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

   6. Applied egg-rr 76.2%

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

   7. Taylor expanded in x.re around 0 66.6%

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

   8. Taylor expanded in x.re around 0 66.6%

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

   10. Simplified 90.4%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 7.5 \cdot 10^{+63}:\\ \;\;\;\;x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right) + x.re \cdot \left(\left(x.im \cdot x.re\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re - x.im\right)\right) + x.re \cdot -3\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 135000:\\ \;\;\;\;3 \cdot \left(x.im\_m \cdot \left(x.re\_m \cdot x.re\_m\right)\right)\\ \mathbf{elif}\;x.im\_m \leq 7.5 \cdot 10^{+63}:\\ \;\;\;\;-3 + x.im\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(x.im\_m \cdot \left(x.re\_m - x.im\_m\right)\right) + x.re\_m \cdot -3\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.im_m 135000.0)
    (* 3.0 (* x.im_m (* x.re_m x.re_m)))
    (if (<= x.im_m 7.5e+63)
      (+ -3.0 (* x.im_m (- (* x.re_m x.re_m) (* x.im_m x.im_m))))
      (+ (* x.im_m (* x.im_m (- x.re_m x.im_m))) (* x.re_m -3.0))))))

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 135000.0) {
		tmp = 3.0 * (x_46_im_m * (x_46_re_m * x_46_re_m));
	} else if (x_46_im_m <= 7.5e+63) {
		tmp = -3.0 + (x_46_im_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)));
	} else {
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + (x_46_re_m * -3.0);
	}
	return x_46_im_s * tmp;
}

Fortran

x.re_m = abs(x_46re)
x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46im_m <= 135000.0d0) then
        tmp = 3.0d0 * (x_46im_m * (x_46re_m * x_46re_m))
    else if (x_46im_m <= 7.5d+63) then
        tmp = (-3.0d0) + (x_46im_m * ((x_46re_m * x_46re_m) - (x_46im_m * x_46im_m)))
    else
        tmp = (x_46im_m * (x_46im_m * (x_46re_m - x_46im_m))) + (x_46re_m * (-3.0d0))
    end if
    code = x_46im_s * tmp
end function

Java

x.re_m = Math.abs(x_46_re);
x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 135000.0) {
		tmp = 3.0 * (x_46_im_m * (x_46_re_m * x_46_re_m));
	} else if (x_46_im_m <= 7.5e+63) {
		tmp = -3.0 + (x_46_im_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)));
	} else {
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + (x_46_re_m * -3.0);
	}
	return x_46_im_s * tmp;
}

Python

x.re_m = math.fabs(x_46_re)
x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if x_46_im_m <= 135000.0:
		tmp = 3.0 * (x_46_im_m * (x_46_re_m * x_46_re_m))
	elif x_46_im_m <= 7.5e+63:
		tmp = -3.0 + (x_46_im_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)))
	else:
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + (x_46_re_m * -3.0)
	return x_46_im_s * tmp

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 135000.0)
		tmp = Float64(3.0 * Float64(x_46_im_m * Float64(x_46_re_m * x_46_re_m)));
	elseif (x_46_im_m <= 7.5e+63)
		tmp = Float64(-3.0 + Float64(x_46_im_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m))));
	else
		tmp = Float64(Float64(x_46_im_m * Float64(x_46_im_m * Float64(x_46_re_m - x_46_im_m))) + Float64(x_46_re_m * -3.0));
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.re_m = abs(x_46_re);
x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (x_46_im_m <= 135000.0)
		tmp = 3.0 * (x_46_im_m * (x_46_re_m * x_46_re_m));
	elseif (x_46_im_m <= 7.5e+63)
		tmp = -3.0 + (x_46_im_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)));
	else
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + (x_46_re_m * -3.0);
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 135000.0], N[(3.0 * N[(x$46$im$95$m * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im$95$m, 7.5e+63], N[(-3.0 + N[(x$46$im$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im$95$m * N[(x$46$im$95$m * N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re$95$m * -3.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x.im < 135000

   1. Initial program 85.9%

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

   3. Simplified 91.8%

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

   5. Taylor expanded in x.re around inf 57.0%

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

      1. pow2 57.0%

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

   7. Applied egg-rr 57.0%

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

   if 135000 < x.im < 7.5000000000000005e63

   1. Initial program 99.8%

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

   3. Taylor expanded in x.re around 0 99.8%

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

   5. Simplified 93.9%

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

   if 7.5000000000000005e63 < x.im

   1. Initial program 69.0%

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

      1. difference-of-squares 76.2%

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

      2. *-commutative 76.2%

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

   4. Applied egg-rr 76.2%

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

      1. *-commutative 76.2%

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

      2. count-2 76.2%

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

      3. *-commutative 76.2%

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

   6. Applied egg-rr 76.2%

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

   7. Taylor expanded in x.re around 0 66.6%

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

   8. Taylor expanded in x.re around 0 66.6%

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

   10. Simplified 90.4%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 135000:\\ \;\;\;\;3 \cdot \left(x.im \cdot \left(x.re \cdot x.re\right)\right)\\ \mathbf{elif}\;x.im \leq 7.5 \cdot 10^{+63}:\\ \;\;\;\;-3 + x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re - x.im\right)\right) + x.re \cdot -3\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 86.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 9 \cdot 10^{-61}:\\ \;\;\;\;3 \cdot \left(x.im\_m \cdot \left(x.re\_m \cdot x.re\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(\left(x.re\_m - x.im\_m\right) \cdot \left(x.im\_m + x.re\_m\right)\right) + x.re\_m \cdot -3\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.im_m 9e-61)
    (* 3.0 (* x.im_m (* x.re_m x.re_m)))
    (+ (* x.im_m (* (- x.re_m x.im_m) (+ x.im_m x.re_m))) (* x.re_m -3.0)))))

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 9e-61) {
		tmp = 3.0 * (x_46_im_m * (x_46_re_m * x_46_re_m));
	} else {
		tmp = (x_46_im_m * ((x_46_re_m - x_46_im_m) * (x_46_im_m + x_46_re_m))) + (x_46_re_m * -3.0);
	}
	return x_46_im_s * tmp;
}

Fortran

x.re_m = abs(x_46re)
x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46im_m <= 9d-61) then
        tmp = 3.0d0 * (x_46im_m * (x_46re_m * x_46re_m))
    else
        tmp = (x_46im_m * ((x_46re_m - x_46im_m) * (x_46im_m + x_46re_m))) + (x_46re_m * (-3.0d0))
    end if
    code = x_46im_s * tmp
end function

Java

x.re_m = Math.abs(x_46_re);
x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 9e-61) {
		tmp = 3.0 * (x_46_im_m * (x_46_re_m * x_46_re_m));
	} else {
		tmp = (x_46_im_m * ((x_46_re_m - x_46_im_m) * (x_46_im_m + x_46_re_m))) + (x_46_re_m * -3.0);
	}
	return x_46_im_s * tmp;
}

Python

x.re_m = math.fabs(x_46_re)
x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if x_46_im_m <= 9e-61:
		tmp = 3.0 * (x_46_im_m * (x_46_re_m * x_46_re_m))
	else:
		tmp = (x_46_im_m * ((x_46_re_m - x_46_im_m) * (x_46_im_m + x_46_re_m))) + (x_46_re_m * -3.0)
	return x_46_im_s * tmp

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 9e-61)
		tmp = Float64(3.0 * Float64(x_46_im_m * Float64(x_46_re_m * x_46_re_m)));
	else
		tmp = Float64(Float64(x_46_im_m * Float64(Float64(x_46_re_m - x_46_im_m) * Float64(x_46_im_m + x_46_re_m))) + Float64(x_46_re_m * -3.0));
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.re_m = abs(x_46_re);
x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (x_46_im_m <= 9e-61)
		tmp = 3.0 * (x_46_im_m * (x_46_re_m * x_46_re_m));
	else
		tmp = (x_46_im_m * ((x_46_re_m - x_46_im_m) * (x_46_im_m + x_46_re_m))) + (x_46_re_m * -3.0);
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 9e-61], N[(3.0 * N[(x$46$im$95$m * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im$95$m * N[(N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m + x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re$95$m * -3.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 9e-61

   1. Initial program 84.8%

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

   3. Simplified 91.2%

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

   5. Taylor expanded in x.re around inf 57.9%

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

      1. pow2 57.9%

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

   7. Applied egg-rr 57.9%

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

   if 9e-61 < x.im

   1. Initial program 81.5%

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

      1. difference-of-squares 85.7%

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

      2. *-commutative 85.7%

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

   4. Applied egg-rr 85.7%

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

      1. *-commutative 85.7%

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

      2. count-2 85.7%

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

      3. *-commutative 85.7%

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

   6. Applied egg-rr 85.7%

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

   7. Taylor expanded in x.re around 0 85.7%

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

   9. Simplified 92.9%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 9 \cdot 10^{-61}:\\ \;\;\;\;3 \cdot \left(x.im \cdot \left(x.re \cdot x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right) + x.re \cdot -3\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 2.2 \cdot 10^{-57}:\\ \;\;\;\;3 \cdot \left(x.im\_m \cdot \left(x.re\_m \cdot x.re\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(x.im\_m \cdot \left(x.re\_m - x.im\_m\right)\right) + x.re\_m \cdot -3\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.im_m 2.2e-57)
    (* 3.0 (* x.im_m (* x.re_m x.re_m)))
    (+ (* x.im_m (* x.im_m (- x.re_m x.im_m))) (* x.re_m -3.0)))))

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 2.2e-57) {
		tmp = 3.0 * (x_46_im_m * (x_46_re_m * x_46_re_m));
	} else {
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + (x_46_re_m * -3.0);
	}
	return x_46_im_s * tmp;
}

Fortran

x.re_m = abs(x_46re)
x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46im_m <= 2.2d-57) then
        tmp = 3.0d0 * (x_46im_m * (x_46re_m * x_46re_m))
    else
        tmp = (x_46im_m * (x_46im_m * (x_46re_m - x_46im_m))) + (x_46re_m * (-3.0d0))
    end if
    code = x_46im_s * tmp
end function

Java

x.re_m = Math.abs(x_46_re);
x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 2.2e-57) {
		tmp = 3.0 * (x_46_im_m * (x_46_re_m * x_46_re_m));
	} else {
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + (x_46_re_m * -3.0);
	}
	return x_46_im_s * tmp;
}

Python

x.re_m = math.fabs(x_46_re)
x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if x_46_im_m <= 2.2e-57:
		tmp = 3.0 * (x_46_im_m * (x_46_re_m * x_46_re_m))
	else:
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + (x_46_re_m * -3.0)
	return x_46_im_s * tmp

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 2.2e-57)
		tmp = Float64(3.0 * Float64(x_46_im_m * Float64(x_46_re_m * x_46_re_m)));
	else
		tmp = Float64(Float64(x_46_im_m * Float64(x_46_im_m * Float64(x_46_re_m - x_46_im_m))) + Float64(x_46_re_m * -3.0));
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.re_m = abs(x_46_re);
x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (x_46_im_m <= 2.2e-57)
		tmp = 3.0 * (x_46_im_m * (x_46_re_m * x_46_re_m));
	else
		tmp = (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m))) + (x_46_re_m * -3.0);
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 2.2e-57], N[(3.0 * N[(x$46$im$95$m * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im$95$m * N[(x$46$im$95$m * N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re$95$m * -3.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 2.19999999999999999e-57

   1. Initial program 84.9%

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

   3. Simplified 91.2%

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

   5. Taylor expanded in x.re around inf 57.6%

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

      1. pow2 57.6%

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

   7. Applied egg-rr 57.6%

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

   if 2.19999999999999999e-57 < x.im

   1. Initial program 81.3%

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

      1. difference-of-squares 85.5%

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

      2. *-commutative 85.5%

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

   4. Applied egg-rr 85.5%

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

      1. *-commutative 85.5%

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

      2. count-2 85.5%

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

      3. *-commutative 85.5%

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

   6. Applied egg-rr 85.5%

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

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

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

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

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

   10. Simplified 76.1%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 2.2 \cdot 10^{-57}:\\ \;\;\;\;3 \cdot \left(x.im \cdot \left(x.re \cdot x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re - x.im\right)\right) + x.re \cdot -3\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 1.15 \cdot 10^{+156}:\\ \;\;\;\;3 \cdot \left(x.im\_m \cdot \left(x.re\_m \cdot x.re\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 - x.im\_m \cdot \left(x.im\_m \cdot x.re\_m\right)\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.im_m 1.15e+156)
    (* 3.0 (* x.im_m (* x.re_m x.re_m)))
    (- -1.0 (* x.im_m (* x.im_m x.re_m))))))

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 1.15e+156) {
		tmp = 3.0 * (x_46_im_m * (x_46_re_m * x_46_re_m));
	} else {
		tmp = -1.0 - (x_46_im_m * (x_46_im_m * x_46_re_m));
	}
	return x_46_im_s * tmp;
}

Fortran

x.re_m = abs(x_46re)
x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46im_m <= 1.15d+156) then
        tmp = 3.0d0 * (x_46im_m * (x_46re_m * x_46re_m))
    else
        tmp = (-1.0d0) - (x_46im_m * (x_46im_m * x_46re_m))
    end if
    code = x_46im_s * tmp
end function

Java

x.re_m = Math.abs(x_46_re);
x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 1.15e+156) {
		tmp = 3.0 * (x_46_im_m * (x_46_re_m * x_46_re_m));
	} else {
		tmp = -1.0 - (x_46_im_m * (x_46_im_m * x_46_re_m));
	}
	return x_46_im_s * tmp;
}

Python

x.re_m = math.fabs(x_46_re)
x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if x_46_im_m <= 1.15e+156:
		tmp = 3.0 * (x_46_im_m * (x_46_re_m * x_46_re_m))
	else:
		tmp = -1.0 - (x_46_im_m * (x_46_im_m * x_46_re_m))
	return x_46_im_s * tmp

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 1.15e+156)
		tmp = Float64(3.0 * Float64(x_46_im_m * Float64(x_46_re_m * x_46_re_m)));
	else
		tmp = Float64(-1.0 - Float64(x_46_im_m * Float64(x_46_im_m * x_46_re_m)));
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.re_m = abs(x_46_re);
x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (x_46_im_m <= 1.15e+156)
		tmp = 3.0 * (x_46_im_m * (x_46_re_m * x_46_re_m));
	else
		tmp = -1.0 - (x_46_im_m * (x_46_im_m * x_46_re_m));
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 1.15e+156], N[(3.0 * N[(x$46$im$95$m * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 - N[(x$46$im$95$m * N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.1499999999999999e156

   1. Initial program 87.4%

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

   3. Simplified 90.7%

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

   5. Taylor expanded in x.re around inf 54.6%

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

      1. pow2 54.6%

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

   7. Applied egg-rr 54.6%

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

   if 1.1499999999999999e156 < x.im

   1. Initial program 52.0%

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

      1. difference-of-squares 60.0%

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

      2. *-commutative 60.0%

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

   4. Applied egg-rr 60.0%

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

   5. Taylor expanded in x.re around 0 92.0%

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3} + x.im \cdot \left(x.re \cdot \left(x.im + -1 \cdot x.im\right)\right)} \]

   7. Simplified 33.0%

      \[\leadsto \color{blue}{-1 + x.im \cdot \left(x.re \cdot \left(-x.im\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 1.15 \cdot 10^{+156}:\\ \;\;\;\;3 \cdot \left(x.im \cdot \left(x.re \cdot x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 - x.im \cdot \left(x.im \cdot x.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 49.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \left(3 \cdot \left(x.im\_m \cdot \left(x.re\_m \cdot x.re\_m\right)\right)\right) \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (* x.im_s (* 3.0 (* x.im_m (* x.re_m x.re_m)))))

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	return x_46_im_s * (3.0 * (x_46_im_m * (x_46_re_m * x_46_re_m)));
}

Fortran

x.re_m = abs(x_46re)
x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    code = x_46im_s * (3.0d0 * (x_46im_m * (x_46re_m * x_46re_m)))
end function

Java

x.re_m = Math.abs(x_46_re);
x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	return x_46_im_s * (3.0 * (x_46_im_m * (x_46_re_m * x_46_re_m)));
}

Python

x.re_m = math.fabs(x_46_re)
x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re_m, x_46_im_m):
	return x_46_im_s * (3.0 * (x_46_im_m * (x_46_re_m * x_46_re_m)))

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	return Float64(x_46_im_s * Float64(3.0 * Float64(x_46_im_m * Float64(x_46_re_m * x_46_re_m))))
end

MATLAB

x.re_m = abs(x_46_re);
x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp = code(x_46_im_s, x_46_re_m, x_46_im_m)
	tmp = x_46_im_s * (3.0 * (x_46_im_m * (x_46_re_m * x_46_re_m)));
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$im$95$s * N[(3.0 * N[(x$46$im$95$m * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.9%

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

3. Simplified 87.0%

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

5. Taylor expanded in x.re around inf 50.1%

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

   1. pow2 50.1%

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

7. Applied egg-rr 50.1%

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

Alternative 12: 2.9% accurate, 9.5× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \left(-x.re\_m\right) \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m) :precision binary64 (* x.im_s (- x.re_m)))

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	return x_46_im_s * -x_46_re_m;
}

Fortran

x.re_m = abs(x_46re)
x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    code = x_46im_s * -x_46re_m
end function

Java

x.re_m = Math.abs(x_46_re);
x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	return x_46_im_s * -x_46_re_m;
}

Python

x.re_m = math.fabs(x_46_re)
x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re_m, x_46_im_m):
	return x_46_im_s * -x_46_re_m

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	return Float64(x_46_im_s * Float64(-x_46_re_m))
end

MATLAB

x.re_m = abs(x_46_re);
x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp = code(x_46_im_s, x_46_re_m, x_46_im_m)
	tmp = x_46_im_s * -x_46_re_m;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$im$95$s * (-x$46$re$95$m)), $MachinePrecision]

Derivation

1. Initial program 83.9%

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

3. Simplified 87.0%

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

5. Taylor expanded in x.im around 0 87.0%

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

6. Taylor expanded in x.re around inf 50.1%

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

8. Simplified 3.5%

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

Alternative 13: 2.7% accurate, 19.0× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot -2 \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m) :precision binary64 (* x.im_s -2.0))

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	return x_46_im_s * -2.0;
}

Fortran

x.re_m = abs(x_46re)
x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    code = x_46im_s * (-2.0d0)
end function

Java

x.re_m = Math.abs(x_46_re);
x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	return x_46_im_s * -2.0;
}

Python

x.re_m = math.fabs(x_46_re)
x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re_m, x_46_im_m):
	return x_46_im_s * -2.0

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	return Float64(x_46_im_s * -2.0)
end

MATLAB

x.re_m = abs(x_46_re);
x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp = code(x_46_im_s, x_46_re_m, x_46_im_m)
	tmp = x_46_im_s * -2.0;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$im$95$s * -2.0), $MachinePrecision]

Derivation

1. Initial program 83.9%

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

3. Taylor expanded in x.re around 0 83.9%

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

5. Simplified 50.0%

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

6. Taylor expanded in x.re around 0 35.0%

   \[\leadsto \color{blue}{-1 \cdot {x.im}^{3} - 3} \]

8. Simplified 2.7%

   \[\leadsto \color{blue}{-2} \]
9. Add Preprocessing

Alternative 14: 2.7% accurate, 19.0× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot -3 \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m) :precision binary64 (* x.im_s -3.0))

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	return x_46_im_s * -3.0;
}

Fortran

x.re_m = abs(x_46re)
x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    code = x_46im_s * (-3.0d0)
end function

Java

x.re_m = Math.abs(x_46_re);
x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	return x_46_im_s * -3.0;
}

Python

x.re_m = math.fabs(x_46_re)
x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re_m, x_46_im_m):
	return x_46_im_s * -3.0

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	return Float64(x_46_im_s * -3.0)
end

MATLAB

x.re_m = abs(x_46_re);
x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp = code(x_46_im_s, x_46_re_m, x_46_im_m)
	tmp = x_46_im_s * -3.0;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$im$95$s * -3.0), $MachinePrecision]

Derivation

1. Initial program 83.9%

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

3. Taylor expanded in x.re around 0 83.9%

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

5. Simplified 50.0%

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

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

   \[\leadsto \color{blue}{-3} \]
7. Add Preprocessing

Developer Target 1: 90.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(x.re \cdot x.im\right) \cdot \left(2 \cdot x.re\right) + \left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.re + x.im\right) \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (+ (* (* x.re x.im) (* 2.0 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im))))

C

double code(double x_46_re, double x_46_im) {
	return ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - 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_46im) * (2.0d0 * x_46re)) + ((x_46im * (x_46re - 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_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
}

Python

def code(x_46_re, x_46_im):
	return ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im))

Julia

function code(x_46_re, x_46_im)
	return Float64(Float64(Float64(x_46_re * x_46_im) * Float64(2.0 * x_46_re)) + Float64(Float64(x_46_im * Float64(x_46_re - x_46_im)) * Float64(x_46_re + x_46_im)))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
end

Wolfram

code[x$46$re_, x$46$im_] := N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(2.0 * x$46$re), $MachinePrecision]), $MachinePrecision] + N[(N[(x$46$im * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision] * N[(x$46$re + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024146 
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (* (* x.re x.im) (* 2 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im))))

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