math.cube on complex, imaginary part

Percentage Accurate: 82.8% → 96.3%

Time: 9.6s

Alternatives: 10

Speedup: 0.8×

• 44.0% 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: 82.8% 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: 96.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := x.im\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) + x.re\_m \cdot \left(x.re\_m \cdot x.im\_m + x.re\_m \cdot x.im\_m\right)\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 10^{+75}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot \left(x.im\_m \cdot \left(--3\right)\right)\right) - {x.im\_m}^{3}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;3 \cdot \left(x.re\_m \cdot \left(x.re\_m \cdot x.im\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 - x.im\_m \cdot \left(x.re\_m \cdot x.im\_m\right)\\ \end{array} \end{array} \end{array} \]

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.im_m (- (* x.re_m x.re_m) (* x.im_m x.im_m)))
          (* x.re_m (+ (* x.re_m x.im_m) (* x.re_m x.im_m))))))
   (*
    x.im_s
    (if (<= t_0 1e+75)
      (- (* x.re_m (* x.re_m (* x.im_m (- -3.0)))) (pow x.im_m 3.0))
      (if (<= t_0 INFINITY)
        (* 3.0 (* x.re_m (* x.re_m x.im_m)))
        (- -1.0 (* x.im_m (* x.re_m x.im_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 t_0 = (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_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)));
	double tmp;
	if (t_0 <= 1e+75) {
		tmp = (x_46_re_m * (x_46_re_m * (x_46_im_m * -(-3.0)))) - pow(x_46_im_m, 3.0);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = 3.0 * (x_46_re_m * (x_46_re_m * x_46_im_m));
	} else {
		tmp = -1.0 - (x_46_im_m * (x_46_re_m * x_46_im_m));
	}
	return x_46_im_s * tmp;
}

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_im_m * ((x_46_re_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 * x_46_im_m)));
	double tmp;
	if (t_0 <= 1e+75) {
		tmp = (x_46_re_m * (x_46_re_m * (x_46_im_m * -(-3.0)))) - Math.pow(x_46_im_m, 3.0);
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = 3.0 * (x_46_re_m * (x_46_re_m * x_46_im_m));
	} else {
		tmp = -1.0 - (x_46_im_m * (x_46_re_m * x_46_im_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):
	t_0 = (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_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)))
	tmp = 0
	if t_0 <= 1e+75:
		tmp = (x_46_re_m * (x_46_re_m * (x_46_im_m * -(-3.0)))) - math.pow(x_46_im_m, 3.0)
	elif t_0 <= math.inf:
		tmp = 3.0 * (x_46_re_m * (x_46_re_m * x_46_im_m))
	else:
		tmp = -1.0 - (x_46_im_m * (x_46_re_m * x_46_im_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)
	t_0 = 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_re_m * x_46_im_m) + Float64(x_46_re_m * x_46_im_m))))
	tmp = 0.0
	if (t_0 <= 1e+75)
		tmp = Float64(Float64(x_46_re_m * Float64(x_46_re_m * Float64(x_46_im_m * Float64(-(-3.0))))) - (x_46_im_m ^ 3.0));
	elseif (t_0 <= Inf)
		tmp = Float64(3.0 * Float64(x_46_re_m * Float64(x_46_re_m * x_46_im_m)));
	else
		tmp = Float64(-1.0 - Float64(x_46_im_m * Float64(x_46_re_m * x_46_im_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)
	t_0 = (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_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)));
	tmp = 0.0;
	if (t_0 <= 1e+75)
		tmp = (x_46_re_m * (x_46_re_m * (x_46_im_m * -(-3.0)))) - (x_46_im_m ^ 3.0);
	elseif (t_0 <= Inf)
		tmp = 3.0 * (x_46_re_m * (x_46_re_m * x_46_im_m));
	else
		tmp = -1.0 - (x_46_im_m * (x_46_re_m * x_46_im_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_] := Block[{t$95$0 = 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$46$re$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$0, 1e+75], N[(N[(x$46$re$95$m * N[(x$46$re$95$m * N[(x$46$im$95$m * (--3.0)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[x$46$im$95$m, 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(3.0 * N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 - N[(x$46$im$95$m * N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $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)) < 9.99999999999999927e74

   1. Initial program 91.3%

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

      1. difference-of-squares 91.3%

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

      2. *-commutative 91.3%

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

   4. Applied egg-rr 91.3%

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

   5. Taylor expanded in x.re around 0 99.2%

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3} + x.re \cdot \left(x.im \cdot \left(x.im + -1 \cdot x.im\right) + x.re \cdot \left(x.im + 2 \cdot x.im\right)\right)} \]

   6. Taylor expanded in x.re around -inf 99.2%

      \[\leadsto -1 \cdot {x.im}^{3} + x.re \cdot \color{blue}{\left(-1 \cdot \left(x.re \cdot \left(-1 \cdot \left(x.im + 2 \cdot x.im\right) + -1 \cdot \frac{x.im \cdot \left(x.im + -1 \cdot x.im\right)}{x.re}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 99.2%

         \[\leadsto -1 \cdot {x.im}^{3} + x.re \cdot \color{blue}{\left(-x.re \cdot \left(-1 \cdot \left(x.im + 2 \cdot x.im\right) + -1 \cdot \frac{x.im \cdot \left(x.im + -1 \cdot x.im\right)}{x.re}\right)\right)} \]

      2. distribute-rgt-neg-in 99.2%

         \[\leadsto -1 \cdot {x.im}^{3} + x.re \cdot \color{blue}{\left(x.re \cdot \left(-\left(-1 \cdot \left(x.im + 2 \cdot x.im\right) + -1 \cdot \frac{x.im \cdot \left(x.im + -1 \cdot x.im\right)}{x.re}\right)\right)\right)} \]

      3. distribute-lft-out 99.2%

         \[\leadsto -1 \cdot {x.im}^{3} + x.re \cdot \left(x.re \cdot \left(-\color{blue}{-1 \cdot \left(\left(x.im + 2 \cdot x.im\right) + \frac{x.im \cdot \left(x.im + -1 \cdot x.im\right)}{x.re}\right)}\right)\right) \]

      4. associate-+r+ 99.2%

         \[\leadsto -1 \cdot {x.im}^{3} + x.re \cdot \left(x.re \cdot \left(--1 \cdot \color{blue}{\left(x.im + \left(2 \cdot x.im + \frac{x.im \cdot \left(x.im + -1 \cdot x.im\right)}{x.re}\right)\right)}\right)\right) \]

      5. distribute-lft-in 99.2%

         \[\leadsto -1 \cdot {x.im}^{3} + x.re \cdot \left(x.re \cdot \left(-\color{blue}{\left(-1 \cdot x.im + -1 \cdot \left(2 \cdot x.im + \frac{x.im \cdot \left(x.im + -1 \cdot x.im\right)}{x.re}\right)\right)}\right)\right) \]

      6. *-commutative 99.2%

         \[\leadsto -1 \cdot {x.im}^{3} + x.re \cdot \left(x.re \cdot \left(-\left(-1 \cdot x.im + -1 \cdot \left(\color{blue}{x.im \cdot 2} + \frac{x.im \cdot \left(x.im + -1 \cdot x.im\right)}{x.re}\right)\right)\right)\right) \]

      7. distribute-rgt1-in 99.2%

         \[\leadsto -1 \cdot {x.im}^{3} + x.re \cdot \left(x.re \cdot \left(-\left(-1 \cdot x.im + -1 \cdot \left(x.im \cdot 2 + \frac{x.im \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot x.im\right)}}{x.re}\right)\right)\right)\right) \]

      8. metadata-eval 99.2%

         \[\leadsto -1 \cdot {x.im}^{3} + x.re \cdot \left(x.re \cdot \left(-\left(-1 \cdot x.im + -1 \cdot \left(x.im \cdot 2 + \frac{x.im \cdot \left(\color{blue}{0} \cdot x.im\right)}{x.re}\right)\right)\right)\right) \]

      9. associate-/l* 99.2%

         \[\leadsto -1 \cdot {x.im}^{3} + x.re \cdot \left(x.re \cdot \left(-\left(-1 \cdot x.im + -1 \cdot \left(x.im \cdot 2 + \color{blue}{x.im \cdot \frac{0 \cdot x.im}{x.re}}\right)\right)\right)\right) \]

      10. distribute-lft-out 99.2%

          \[\leadsto -1 \cdot {x.im}^{3} + x.re \cdot \left(x.re \cdot \left(-\left(-1 \cdot x.im + -1 \cdot \color{blue}{\left(x.im \cdot \left(2 + \frac{0 \cdot x.im}{x.re}\right)\right)}\right)\right)\right) \]

      11. mul0-lft 99.2%

          \[\leadsto -1 \cdot {x.im}^{3} + x.re \cdot \left(x.re \cdot \left(-\left(-1 \cdot x.im + -1 \cdot \left(x.im \cdot \left(2 + \frac{\color{blue}{0}}{x.re}\right)\right)\right)\right)\right) \]

      12. div0 99.2%

          \[\leadsto -1 \cdot {x.im}^{3} + x.re \cdot \left(x.re \cdot \left(-\left(-1 \cdot x.im + -1 \cdot \left(x.im \cdot \left(2 + \frac{\color{blue}{\frac{0}{x.im}}}{x.re}\right)\right)\right)\right)\right) \]

      13. associate-/r* 82.2%

          \[\leadsto -1 \cdot {x.im}^{3} + x.re \cdot \left(x.re \cdot \left(-\left(-1 \cdot x.im + -1 \cdot \left(x.im \cdot \left(2 + \color{blue}{\frac{0}{x.im \cdot x.re}}\right)\right)\right)\right)\right) \]

      14. div0 99.2%

          \[\leadsto -1 \cdot {x.im}^{3} + x.re \cdot \left(x.re \cdot \left(-\left(-1 \cdot x.im + -1 \cdot \left(x.im \cdot \left(2 + \color{blue}{0}\right)\right)\right)\right)\right) \]

      15. metadata-eval 99.2%

          \[\leadsto -1 \cdot {x.im}^{3} + x.re \cdot \left(x.re \cdot \left(-\left(-1 \cdot x.im + -1 \cdot \left(x.im \cdot \color{blue}{2}\right)\right)\right)\right) \]

   8. Simplified 99.2%

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

   if 9.99999999999999927e74 < (+.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 84.3%

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

   3. Simplified 94.8%

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

   5. Taylor expanded in x.re around inf 44.8%

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

      1. add-sqr-sqrt 44.5%

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

      2. pow2 44.5%

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

      3. *-commutative 44.5%

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

      4. sqrt-prod 44.3%

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

      5. sqrt-pow1 59.6%

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

      6. metadata-eval 59.6%

         \[\leadsto 3 \cdot {\left({x.re}^{\color{blue}{1}} \cdot \sqrt{x.im}\right)}^{2} \]

      7. pow1 59.6%

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

   7. Applied egg-rr 59.6%

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

      1. unpow2 59.6%

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

      2. swap-sqr 44.4%

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

      3. add-sqr-sqrt 44.8%

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

      4. *-commutative 44.8%

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

      5. associate-*r* 60.2%

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

   9. Applied egg-rr 60.2%

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

   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 22.6%

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

      2. *-commutative 22.6%

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

   4. Applied egg-rr 22.6%

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

   5. Taylor expanded in x.re around 0 77.4%

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

   7. Simplified 54.8%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq 10^{+75}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot \left(--3\right)\right)\right) - {x.im}^{3}\\ \mathbf{elif}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq \infty:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 - x.im \cdot \left(x.re \cdot x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := x.im\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) + x.re\_m \cdot \left(x.re\_m \cdot x.im\_m + x.re\_m \cdot x.im\_m\right)\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 10^{+75}:\\ \;\;\;\;x.re\_m \cdot \left(\left(x.re\_m \cdot x.im\_m\right) \cdot 3\right) - {x.im\_m}^{3}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;3 \cdot \left(x.re\_m \cdot \left(x.re\_m \cdot x.im\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 - x.im\_m \cdot \left(x.re\_m \cdot x.im\_m\right)\\ \end{array} \end{array} \end{array} \]

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.im_m (- (* x.re_m x.re_m) (* x.im_m x.im_m)))
          (* x.re_m (+ (* x.re_m x.im_m) (* x.re_m x.im_m))))))
   (*
    x.im_s
    (if (<= t_0 1e+75)
      (- (* x.re_m (* (* x.re_m x.im_m) 3.0)) (pow x.im_m 3.0))
      (if (<= t_0 INFINITY)
        (* 3.0 (* x.re_m (* x.re_m x.im_m)))
        (- -1.0 (* x.im_m (* x.re_m x.im_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 t_0 = (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_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)));
	double tmp;
	if (t_0 <= 1e+75) {
		tmp = (x_46_re_m * ((x_46_re_m * x_46_im_m) * 3.0)) - pow(x_46_im_m, 3.0);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = 3.0 * (x_46_re_m * (x_46_re_m * x_46_im_m));
	} else {
		tmp = -1.0 - (x_46_im_m * (x_46_re_m * x_46_im_m));
	}
	return x_46_im_s * tmp;
}

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_im_m * ((x_46_re_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 * x_46_im_m)));
	double tmp;
	if (t_0 <= 1e+75) {
		tmp = (x_46_re_m * ((x_46_re_m * x_46_im_m) * 3.0)) - Math.pow(x_46_im_m, 3.0);
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = 3.0 * (x_46_re_m * (x_46_re_m * x_46_im_m));
	} else {
		tmp = -1.0 - (x_46_im_m * (x_46_re_m * x_46_im_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):
	t_0 = (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_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)))
	tmp = 0
	if t_0 <= 1e+75:
		tmp = (x_46_re_m * ((x_46_re_m * x_46_im_m) * 3.0)) - math.pow(x_46_im_m, 3.0)
	elif t_0 <= math.inf:
		tmp = 3.0 * (x_46_re_m * (x_46_re_m * x_46_im_m))
	else:
		tmp = -1.0 - (x_46_im_m * (x_46_re_m * x_46_im_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)
	t_0 = 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_re_m * x_46_im_m) + Float64(x_46_re_m * x_46_im_m))))
	tmp = 0.0
	if (t_0 <= 1e+75)
		tmp = Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_im_m) * 3.0)) - (x_46_im_m ^ 3.0));
	elseif (t_0 <= Inf)
		tmp = Float64(3.0 * Float64(x_46_re_m * Float64(x_46_re_m * x_46_im_m)));
	else
		tmp = Float64(-1.0 - Float64(x_46_im_m * Float64(x_46_re_m * x_46_im_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)
	t_0 = (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_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)));
	tmp = 0.0;
	if (t_0 <= 1e+75)
		tmp = (x_46_re_m * ((x_46_re_m * x_46_im_m) * 3.0)) - (x_46_im_m ^ 3.0);
	elseif (t_0 <= Inf)
		tmp = 3.0 * (x_46_re_m * (x_46_re_m * x_46_im_m));
	else
		tmp = -1.0 - (x_46_im_m * (x_46_re_m * x_46_im_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_] := Block[{t$95$0 = 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$46$re$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$0, 1e+75], N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] - N[Power[x$46$im$95$m, 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(3.0 * N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 - N[(x$46$im$95$m * N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $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)) < 9.99999999999999927e74

   1. Initial program 91.3%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in x.im around 0 99.2%

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

   if 9.99999999999999927e74 < (+.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 84.3%

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

   3. Simplified 94.8%

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

   5. Taylor expanded in x.re around inf 44.8%

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

      1. add-sqr-sqrt 44.5%

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

      2. pow2 44.5%

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

      3. *-commutative 44.5%

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

      4. sqrt-prod 44.3%

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

      5. sqrt-pow1 59.6%

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

      6. metadata-eval 59.6%

         \[\leadsto 3 \cdot {\left({x.re}^{\color{blue}{1}} \cdot \sqrt{x.im}\right)}^{2} \]

      7. pow1 59.6%

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

   7. Applied egg-rr 59.6%

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

      1. unpow2 59.6%

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

      2. swap-sqr 44.4%

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

      3. add-sqr-sqrt 44.8%

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

      4. *-commutative 44.8%

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

      5. associate-*r* 60.2%

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

   9. Applied egg-rr 60.2%

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

   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 22.6%

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

      2. *-commutative 22.6%

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

   4. Applied egg-rr 22.6%

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

   5. Taylor expanded in x.re around 0 77.4%

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

   7. Simplified 54.8%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq 10^{+75}:\\ \;\;\;\;x.re \cdot \left(\left(x.re \cdot x.im\right) \cdot 3\right) - {x.im}^{3}\\ \mathbf{elif}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq \infty:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 - x.im \cdot \left(x.re \cdot x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := x.im\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) + x.re\_m \cdot \left(x.re\_m \cdot x.im\_m + x.re\_m \cdot x.im\_m\right)\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-305}:\\ \;\;\;\;-{x.im\_m}^{3}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;3 \cdot \left(x.re\_m \cdot \left(x.re\_m \cdot x.im\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 - x.im\_m \cdot \left(x.re\_m \cdot x.im\_m\right)\\ \end{array} \end{array} \end{array} \]

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.im_m (- (* x.re_m x.re_m) (* x.im_m x.im_m)))
          (* x.re_m (+ (* x.re_m x.im_m) (* x.re_m x.im_m))))))
   (*
    x.im_s
    (if (<= t_0 -5e-305)
      (- (pow x.im_m 3.0))
      (if (<= t_0 INFINITY)
        (* 3.0 (* x.re_m (* x.re_m x.im_m)))
        (- -1.0 (* x.im_m (* x.re_m x.im_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 t_0 = (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_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)));
	double tmp;
	if (t_0 <= -5e-305) {
		tmp = -pow(x_46_im_m, 3.0);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = 3.0 * (x_46_re_m * (x_46_re_m * x_46_im_m));
	} else {
		tmp = -1.0 - (x_46_im_m * (x_46_re_m * x_46_im_m));
	}
	return x_46_im_s * tmp;
}

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_im_m * ((x_46_re_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 * x_46_im_m)));
	double tmp;
	if (t_0 <= -5e-305) {
		tmp = -Math.pow(x_46_im_m, 3.0);
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = 3.0 * (x_46_re_m * (x_46_re_m * x_46_im_m));
	} else {
		tmp = -1.0 - (x_46_im_m * (x_46_re_m * x_46_im_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):
	t_0 = (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_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)))
	tmp = 0
	if t_0 <= -5e-305:
		tmp = -math.pow(x_46_im_m, 3.0)
	elif t_0 <= math.inf:
		tmp = 3.0 * (x_46_re_m * (x_46_re_m * x_46_im_m))
	else:
		tmp = -1.0 - (x_46_im_m * (x_46_re_m * x_46_im_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)
	t_0 = 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_re_m * x_46_im_m) + Float64(x_46_re_m * x_46_im_m))))
	tmp = 0.0
	if (t_0 <= -5e-305)
		tmp = Float64(-(x_46_im_m ^ 3.0));
	elseif (t_0 <= Inf)
		tmp = Float64(3.0 * Float64(x_46_re_m * Float64(x_46_re_m * x_46_im_m)));
	else
		tmp = Float64(-1.0 - Float64(x_46_im_m * Float64(x_46_re_m * x_46_im_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)
	t_0 = (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_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)));
	tmp = 0.0;
	if (t_0 <= -5e-305)
		tmp = -(x_46_im_m ^ 3.0);
	elseif (t_0 <= Inf)
		tmp = 3.0 * (x_46_re_m * (x_46_re_m * x_46_im_m));
	else
		tmp = -1.0 - (x_46_im_m * (x_46_re_m * x_46_im_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_] := Block[{t$95$0 = 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$46$re$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$0, -5e-305], (-N[Power[x$46$im$95$m, 3.0], $MachinePrecision]), If[LessEqual[t$95$0, Infinity], N[(3.0 * N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 - N[(x$46$im$95$m * N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $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)) < -4.99999999999999985e-305

   1. Initial program 85.6%

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

      1. difference-of-squares 85.6%

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

      2. *-commutative 85.6%

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

   4. Applied egg-rr 85.6%

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

   5. Taylor expanded in x.re around 0 98.8%

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3} + x.re \cdot \left(x.im \cdot \left(x.im + -1 \cdot x.im\right) + x.re \cdot \left(x.im + 2 \cdot x.im\right)\right)} \]

   6. Taylor expanded in x.im around inf 44.6%

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   7. Step-by-step derivation

      1. neg-mul-1 44.6%

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

   8. Simplified 44.6%

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

   if -4.99999999999999985e-305 < (+.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 92.5%

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

   3. Simplified 97.4%

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

   5. Taylor expanded in x.re around inf 63.1%

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

      1. add-sqr-sqrt 62.8%

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

      2. pow2 62.8%

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

      3. *-commutative 62.8%

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

      4. sqrt-prod 45.6%

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

      5. sqrt-pow1 52.9%

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

      6. metadata-eval 52.9%

         \[\leadsto 3 \cdot {\left({x.re}^{\color{blue}{1}} \cdot \sqrt{x.im}\right)}^{2} \]

      7. pow1 52.9%

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

   7. Applied egg-rr 52.9%

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

      1. unpow2 52.9%

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

      2. swap-sqr 45.6%

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

      3. add-sqr-sqrt 63.1%

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

      4. *-commutative 63.1%

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

      5. associate-*r* 70.5%

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

   9. Applied egg-rr 70.5%

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

   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 22.6%

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

      2. *-commutative 22.6%

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

   4. Applied egg-rr 22.6%

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

   5. Taylor expanded in x.re around 0 77.4%

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

   7. Simplified 54.8%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -5 \cdot 10^{-305}:\\ \;\;\;\;-{x.im}^{3}\\ \mathbf{elif}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq \infty:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 - x.im \cdot \left(x.re \cdot x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := x.im\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right)\\ t_1 := t\_0 + x.re\_m \cdot \left(x.re\_m \cdot x.im\_m + x.re\_m \cdot x.im\_m\right)\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-134}:\\ \;\;\;\;t\_0 + x.re\_m \cdot \left(\left(x.re\_m \cdot x.im\_m\right) \cdot 2\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;3 \cdot \left(x.re\_m \cdot \left(x.re\_m \cdot x.im\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 - x.im\_m \cdot \left(x.re\_m \cdot x.im\_m\right)\\ \end{array} \end{array} \end{array} \]

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.im_m (- (* x.re_m x.re_m) (* x.im_m x.im_m))))
        (t_1 (+ t_0 (* x.re_m (+ (* x.re_m x.im_m) (* x.re_m x.im_m))))))
   (*
    x.im_s
    (if (<= t_1 2e-134)
      (+ t_0 (* x.re_m (* (* x.re_m x.im_m) 2.0)))
      (if (<= t_1 INFINITY)
        (* 3.0 (* x.re_m (* x.re_m x.im_m)))
        (- -1.0 (* x.im_m (* x.re_m x.im_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 t_0 = x_46_im_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m));
	double t_1 = t_0 + (x_46_re_m * ((x_46_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)));
	double tmp;
	if (t_1 <= 2e-134) {
		tmp = t_0 + (x_46_re_m * ((x_46_re_m * x_46_im_m) * 2.0));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = 3.0 * (x_46_re_m * (x_46_re_m * x_46_im_m));
	} else {
		tmp = -1.0 - (x_46_im_m * (x_46_re_m * x_46_im_m));
	}
	return x_46_im_s * tmp;
}

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_im_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m));
	double t_1 = t_0 + (x_46_re_m * ((x_46_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)));
	double tmp;
	if (t_1 <= 2e-134) {
		tmp = t_0 + (x_46_re_m * ((x_46_re_m * x_46_im_m) * 2.0));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = 3.0 * (x_46_re_m * (x_46_re_m * x_46_im_m));
	} else {
		tmp = -1.0 - (x_46_im_m * (x_46_re_m * x_46_im_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):
	t_0 = x_46_im_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))
	t_1 = t_0 + (x_46_re_m * ((x_46_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)))
	tmp = 0
	if t_1 <= 2e-134:
		tmp = t_0 + (x_46_re_m * ((x_46_re_m * x_46_im_m) * 2.0))
	elif t_1 <= math.inf:
		tmp = 3.0 * (x_46_re_m * (x_46_re_m * x_46_im_m))
	else:
		tmp = -1.0 - (x_46_im_m * (x_46_re_m * x_46_im_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)
	t_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)))
	t_1 = Float64(t_0 + Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_im_m) + Float64(x_46_re_m * x_46_im_m))))
	tmp = 0.0
	if (t_1 <= 2e-134)
		tmp = Float64(t_0 + Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_im_m) * 2.0)));
	elseif (t_1 <= Inf)
		tmp = Float64(3.0 * Float64(x_46_re_m * Float64(x_46_re_m * x_46_im_m)));
	else
		tmp = Float64(-1.0 - Float64(x_46_im_m * Float64(x_46_re_m * x_46_im_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)
	t_0 = x_46_im_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m));
	t_1 = t_0 + (x_46_re_m * ((x_46_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)));
	tmp = 0.0;
	if (t_1 <= 2e-134)
		tmp = t_0 + (x_46_re_m * ((x_46_re_m * x_46_im_m) * 2.0));
	elseif (t_1 <= Inf)
		tmp = 3.0 * (x_46_re_m * (x_46_re_m * x_46_im_m));
	else
		tmp = -1.0 - (x_46_im_m * (x_46_re_m * x_46_im_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_] := Block[{t$95$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]}, Block[{t$95$1 = N[(t$95$0 + N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$1, 2e-134], N[(t$95$0 + N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(3.0 * N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 - N[(x$46$im$95$m * N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $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)) < 2.00000000000000008e-134

   1. Initial program 90.2%

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

      1. *-commutative 90.2%

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

      2. count-2 90.2%

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

      3. *-commutative 90.2%

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

   4. Applied egg-rr 90.2%

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

   if 2.00000000000000008e-134 < (+.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 88.0%

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

   3. Simplified 95.9%

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

   5. Taylor expanded in x.re around inf 48.2%

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

      1. add-sqr-sqrt 47.8%

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

      2. pow2 47.8%

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

      3. *-commutative 47.8%

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

      4. sqrt-prod 47.5%

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

      5. sqrt-pow1 59.1%

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

      6. metadata-eval 59.1%

         \[\leadsto 3 \cdot {\left({x.re}^{\color{blue}{1}} \cdot \sqrt{x.im}\right)}^{2} \]

      7. pow1 59.1%

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

   7. Applied egg-rr 59.1%

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

      1. unpow2 59.1%

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

      2. swap-sqr 47.6%

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

      3. add-sqr-sqrt 48.2%

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

      4. *-commutative 48.2%

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

      5. associate-*r* 59.9%

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

   9. Applied egg-rr 59.9%

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

   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 22.6%

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

      2. *-commutative 22.6%

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

   4. Applied egg-rr 22.6%

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

   5. Taylor expanded in x.re around 0 77.4%

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

   7. Simplified 54.8%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq 2 \cdot 10^{-134}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(\left(x.re \cdot x.im\right) \cdot 2\right)\\ \mathbf{elif}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq \infty:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 - x.im \cdot \left(x.re \cdot x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 3 \cdot 10^{-29}:\\ \;\;\;\;3 \cdot \left(x.re\_m \cdot \left(x.re\_m \cdot x.im\_m\right)\right)\\ \mathbf{elif}\;x.im\_m \leq 3 \cdot 10^{+210}:\\ \;\;\;\;x.re\_m \cdot \left(\left(x.re\_m \cdot x.im\_m\right) \cdot 2\right) + x.im\_m \cdot \left(x.im\_m \cdot \left(x.re\_m - x.im\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 - x.im\_m \cdot \left(x.re\_m \cdot x.im\_m\right)\\ \end{array} \end{array} \]

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 3e-29)
    (* 3.0 (* x.re_m (* x.re_m x.im_m)))
    (if (<= x.im_m 3e+210)
      (+
       (* x.re_m (* (* x.re_m x.im_m) 2.0))
       (* x.im_m (* x.im_m (- x.re_m x.im_m))))
      (- -1.0 (* x.im_m (* x.re_m x.im_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 <= 3e-29) {
		tmp = 3.0 * (x_46_re_m * (x_46_re_m * x_46_im_m));
	} else if (x_46_im_m <= 3e+210) {
		tmp = (x_46_re_m * ((x_46_re_m * x_46_im_m) * 2.0)) + (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m)));
	} else {
		tmp = -1.0 - (x_46_im_m * (x_46_re_m * x_46_im_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 <= 3d-29) then
        tmp = 3.0d0 * (x_46re_m * (x_46re_m * x_46im_m))
    else if (x_46im_m <= 3d+210) then
        tmp = (x_46re_m * ((x_46re_m * x_46im_m) * 2.0d0)) + (x_46im_m * (x_46im_m * (x_46re_m - x_46im_m)))
    else
        tmp = (-1.0d0) - (x_46im_m * (x_46re_m * x_46im_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 <= 3e-29) {
		tmp = 3.0 * (x_46_re_m * (x_46_re_m * x_46_im_m));
	} else if (x_46_im_m <= 3e+210) {
		tmp = (x_46_re_m * ((x_46_re_m * x_46_im_m) * 2.0)) + (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m)));
	} else {
		tmp = -1.0 - (x_46_im_m * (x_46_re_m * x_46_im_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 <= 3e-29:
		tmp = 3.0 * (x_46_re_m * (x_46_re_m * x_46_im_m))
	elif x_46_im_m <= 3e+210:
		tmp = (x_46_re_m * ((x_46_re_m * x_46_im_m) * 2.0)) + (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m)))
	else:
		tmp = -1.0 - (x_46_im_m * (x_46_re_m * x_46_im_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 <= 3e-29)
		tmp = Float64(3.0 * Float64(x_46_re_m * Float64(x_46_re_m * x_46_im_m)));
	elseif (x_46_im_m <= 3e+210)
		tmp = Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_im_m) * 2.0)) + Float64(x_46_im_m * Float64(x_46_im_m * Float64(x_46_re_m - x_46_im_m))));
	else
		tmp = Float64(-1.0 - Float64(x_46_im_m * Float64(x_46_re_m * x_46_im_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 <= 3e-29)
		tmp = 3.0 * (x_46_re_m * (x_46_re_m * x_46_im_m));
	elseif (x_46_im_m <= 3e+210)
		tmp = (x_46_re_m * ((x_46_re_m * x_46_im_m) * 2.0)) + (x_46_im_m * (x_46_im_m * (x_46_re_m - x_46_im_m)));
	else
		tmp = -1.0 - (x_46_im_m * (x_46_re_m * x_46_im_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, 3e-29], N[(3.0 * N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im$95$m, 3e+210], N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] + 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]), $MachinePrecision], N[(-1.0 - N[(x$46$im$95$m * N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x.im < 3.0000000000000003e-29

   1. Initial program 77.6%

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

   3. Simplified 89.6%

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

   5. Taylor expanded in x.re around inf 60.4%

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

      1. add-sqr-sqrt 35.0%

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

      2. pow2 35.0%

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

      3. *-commutative 35.0%

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

      4. sqrt-prod 23.4%

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

      5. sqrt-pow1 28.3%

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

      6. metadata-eval 28.3%

         \[\leadsto 3 \cdot {\left({x.re}^{\color{blue}{1}} \cdot \sqrt{x.im}\right)}^{2} \]

      7. pow1 28.3%

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

   7. Applied egg-rr 28.3%

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

      1. unpow2 28.3%

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

      2. swap-sqr 23.4%

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

      3. add-sqr-sqrt 60.4%

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

      4. *-commutative 60.4%

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

      5. associate-*r* 72.9%

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

   9. Applied egg-rr 72.9%

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

   if 3.0000000000000003e-29 < x.im < 3.00000000000000022e210

   1. Initial program 91.9%

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

      1. difference-of-squares 95.9%

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

      2. *-commutative 95.9%

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

   4. Applied egg-rr 95.9%

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

   5. Taylor expanded in x.re around 0 80.5%

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

      1. *-commutative 91.9%

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

      2. count-2 91.9%

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

      3. *-commutative 91.9%

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

   7. Applied egg-rr 80.5%

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

   if 3.00000000000000022e210 < x.im

   1. Initial program 52.6%

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

      1. difference-of-squares 52.6%

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

      2. *-commutative 52.6%

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

   4. Applied egg-rr 52.6%

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

   5. Taylor expanded in x.re around 0 100.0%

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

   7. Simplified 42.1%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 3 \cdot 10^{-29}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{elif}\;x.im \leq 3 \cdot 10^{+210}:\\ \;\;\;\;x.re \cdot \left(\left(x.re \cdot x.im\right) \cdot 2\right) + x.im \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 - x.im \cdot \left(x.re \cdot x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 7.2 \cdot 10^{-30}:\\ \;\;\;\;3 \cdot \left(x.re\_m \cdot \left(x.re\_m \cdot x.im\_m\right)\right)\\ \mathbf{elif}\;x.im\_m \leq 1.4 \cdot 10^{+210}:\\ \;\;\;\;x.re\_m \cdot \left(\left(x.re\_m \cdot x.im\_m\right) \cdot 2\right) - x.im\_m \cdot \left(x.im\_m \cdot x.im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;-1 - x.im\_m \cdot \left(x.re\_m \cdot x.im\_m\right)\\ \end{array} \end{array} \]

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 7.2e-30)
    (* 3.0 (* x.re_m (* x.re_m x.im_m)))
    (if (<= x.im_m 1.4e+210)
      (- (* x.re_m (* (* x.re_m x.im_m) 2.0)) (* x.im_m (* x.im_m x.im_m)))
      (- -1.0 (* x.im_m (* x.re_m x.im_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 <= 7.2e-30) {
		tmp = 3.0 * (x_46_re_m * (x_46_re_m * x_46_im_m));
	} else if (x_46_im_m <= 1.4e+210) {
		tmp = (x_46_re_m * ((x_46_re_m * x_46_im_m) * 2.0)) - (x_46_im_m * (x_46_im_m * x_46_im_m));
	} else {
		tmp = -1.0 - (x_46_im_m * (x_46_re_m * x_46_im_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 <= 7.2d-30) then
        tmp = 3.0d0 * (x_46re_m * (x_46re_m * x_46im_m))
    else if (x_46im_m <= 1.4d+210) then
        tmp = (x_46re_m * ((x_46re_m * x_46im_m) * 2.0d0)) - (x_46im_m * (x_46im_m * x_46im_m))
    else
        tmp = (-1.0d0) - (x_46im_m * (x_46re_m * x_46im_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 <= 7.2e-30) {
		tmp = 3.0 * (x_46_re_m * (x_46_re_m * x_46_im_m));
	} else if (x_46_im_m <= 1.4e+210) {
		tmp = (x_46_re_m * ((x_46_re_m * x_46_im_m) * 2.0)) - (x_46_im_m * (x_46_im_m * x_46_im_m));
	} else {
		tmp = -1.0 - (x_46_im_m * (x_46_re_m * x_46_im_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 <= 7.2e-30:
		tmp = 3.0 * (x_46_re_m * (x_46_re_m * x_46_im_m))
	elif x_46_im_m <= 1.4e+210:
		tmp = (x_46_re_m * ((x_46_re_m * x_46_im_m) * 2.0)) - (x_46_im_m * (x_46_im_m * x_46_im_m))
	else:
		tmp = -1.0 - (x_46_im_m * (x_46_re_m * x_46_im_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 <= 7.2e-30)
		tmp = Float64(3.0 * Float64(x_46_re_m * Float64(x_46_re_m * x_46_im_m)));
	elseif (x_46_im_m <= 1.4e+210)
		tmp = Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_im_m) * 2.0)) - Float64(x_46_im_m * Float64(x_46_im_m * x_46_im_m)));
	else
		tmp = Float64(-1.0 - Float64(x_46_im_m * Float64(x_46_re_m * x_46_im_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 <= 7.2e-30)
		tmp = 3.0 * (x_46_re_m * (x_46_re_m * x_46_im_m));
	elseif (x_46_im_m <= 1.4e+210)
		tmp = (x_46_re_m * ((x_46_re_m * x_46_im_m) * 2.0)) - (x_46_im_m * (x_46_im_m * x_46_im_m));
	else
		tmp = -1.0 - (x_46_im_m * (x_46_re_m * x_46_im_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, 7.2e-30], N[(3.0 * N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im$95$m, 1.4e+210], N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$im$95$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[(-1.0 - N[(x$46$im$95$m * N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x.im < 7.2000000000000006e-30

   1. Initial program 77.6%

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

   3. Simplified 89.6%

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

   5. Taylor expanded in x.re around inf 60.4%

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

      1. add-sqr-sqrt 35.0%

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

      2. pow2 35.0%

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

      3. *-commutative 35.0%

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

      4. sqrt-prod 23.4%

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

      5. sqrt-pow1 28.3%

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

      6. metadata-eval 28.3%

         \[\leadsto 3 \cdot {\left({x.re}^{\color{blue}{1}} \cdot \sqrt{x.im}\right)}^{2} \]

      7. pow1 28.3%

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

   7. Applied egg-rr 28.3%

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

      1. unpow2 28.3%

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

      2. swap-sqr 23.4%

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

      3. add-sqr-sqrt 60.4%

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

      4. *-commutative 60.4%

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

      5. associate-*r* 72.9%

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

   9. Applied egg-rr 72.9%

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

   if 7.2000000000000006e-30 < x.im < 1.4000000000000001e210

   1. Initial program 91.9%

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

      1. difference-of-squares 95.9%

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

      2. *-commutative 95.9%

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

   4. Applied egg-rr 95.9%

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

   5. Taylor expanded in x.re around 0 80.5%

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

      1. *-commutative 91.9%

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

      2. count-2 91.9%

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

      3. *-commutative 91.9%

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

   7. Applied egg-rr 80.5%

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

   8. Taylor expanded in x.re around 0 81.2%

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

   10. Simplified 81.2%

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

   if 1.4000000000000001e210 < x.im

   1. Initial program 52.6%

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

      1. difference-of-squares 52.6%

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

      2. *-commutative 52.6%

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

   4. Applied egg-rr 52.6%

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

   5. Taylor expanded in x.re around 0 100.0%

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

   7. Simplified 42.1%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 7.2 \cdot 10^{-30}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{elif}\;x.im \leq 1.4 \cdot 10^{+210}:\\ \;\;\;\;x.re \cdot \left(\left(x.re \cdot x.im\right) \cdot 2\right) - x.im \cdot \left(x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;-1 - x.im \cdot \left(x.re \cdot x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 4.1 \cdot 10^{+136}:\\ \;\;\;\;3 \cdot \left(x.re\_m \cdot \left(x.re\_m \cdot x.im\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 - x.im\_m \cdot \left(x.re\_m \cdot x.im\_m\right)\\ \end{array} \end{array} \]

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 4.1e+136)
    (* 3.0 (* x.re_m (* x.re_m x.im_m)))
    (- -1.0 (* x.im_m (* x.re_m x.im_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 <= 4.1e+136) {
		tmp = 3.0 * (x_46_re_m * (x_46_re_m * x_46_im_m));
	} else {
		tmp = -1.0 - (x_46_im_m * (x_46_re_m * x_46_im_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 <= 4.1d+136) then
        tmp = 3.0d0 * (x_46re_m * (x_46re_m * x_46im_m))
    else
        tmp = (-1.0d0) - (x_46im_m * (x_46re_m * x_46im_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 <= 4.1e+136) {
		tmp = 3.0 * (x_46_re_m * (x_46_re_m * x_46_im_m));
	} else {
		tmp = -1.0 - (x_46_im_m * (x_46_re_m * x_46_im_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 <= 4.1e+136:
		tmp = 3.0 * (x_46_re_m * (x_46_re_m * x_46_im_m))
	else:
		tmp = -1.0 - (x_46_im_m * (x_46_re_m * x_46_im_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 <= 4.1e+136)
		tmp = Float64(3.0 * Float64(x_46_re_m * Float64(x_46_re_m * x_46_im_m)));
	else
		tmp = Float64(-1.0 - Float64(x_46_im_m * Float64(x_46_re_m * x_46_im_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 <= 4.1e+136)
		tmp = 3.0 * (x_46_re_m * (x_46_re_m * x_46_im_m));
	else
		tmp = -1.0 - (x_46_im_m * (x_46_re_m * x_46_im_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, 4.1e+136], N[(3.0 * N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 - N[(x$46$im$95$m * N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 4.0999999999999998e136

   1. Initial program 80.8%

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

   3. Simplified 89.7%

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

   5. Taylor expanded in x.re around inf 58.1%

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

      1. add-sqr-sqrt 36.4%

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

      2. pow2 36.4%

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

      3. *-commutative 36.4%

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

      4. sqrt-prod 26.5%

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

      5. sqrt-pow1 30.7%

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

      6. metadata-eval 30.7%

         \[\leadsto 3 \cdot {\left({x.re}^{\color{blue}{1}} \cdot \sqrt{x.im}\right)}^{2} \]

      7. pow1 30.7%

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

   7. Applied egg-rr 30.7%

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

      1. unpow2 30.7%

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

      2. swap-sqr 26.5%

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

      3. add-sqr-sqrt 58.1%

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

      4. *-commutative 58.1%

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

      5. associate-*r* 68.7%

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

   9. Applied egg-rr 68.7%

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

   if 4.0999999999999998e136 < 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 71.1%

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

      2. *-commutative 71.1%

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

   4. Applied egg-rr 71.1%

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

   5. Taylor expanded in x.re around 0 94.7%

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

   7. Simplified 40.0%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 4.1 \cdot 10^{+136}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 - x.im \cdot \left(x.re \cdot x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.8% accurate, 2.7× speedup

Math

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

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.re_m (* x.re_m x.im_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_re_m * (x_46_re_m * x_46_im_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_46re_m * (x_46re_m * x_46im_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_re_m * (x_46_re_m * x_46_im_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_re_m * (x_46_re_m * x_46_im_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_re_m * Float64(x_46_re_m * x_46_im_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_re_m * (x_46_re_m * x_46_im_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$46$re$95$m * N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.6%

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

3. Simplified 86.1%

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

5. Taylor expanded in x.re around inf 50.3%

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

   1. add-sqr-sqrt 31.9%

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

   2. pow2 31.9%

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

   3. *-commutative 31.9%

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

   4. sqrt-prod 23.4%

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

   5. sqrt-pow1 27.0%

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

   6. metadata-eval 27.0%

      \[\leadsto 3 \cdot {\left({x.re}^{\color{blue}{1}} \cdot \sqrt{x.im}\right)}^{2} \]

   7. pow1 27.0%

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

7. Applied egg-rr 27.0%

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

   1. unpow2 27.0%

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

   2. swap-sqr 23.4%

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

   3. add-sqr-sqrt 50.3%

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

   4. *-commutative 50.3%

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

   5. associate-*r* 59.4%

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

9. Applied egg-rr 59.4%

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

10. Final simplification 59.4%

    \[\leadsto 3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right) \]
11. Add Preprocessing

Alternative 9: 51.2% accurate, 2.7× speedup

Math

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

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.re_m x.re_m) x.im_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_re_m * x_46_re_m) * x_46_im_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_46re_m * x_46re_m) * x_46im_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_re_m * x_46_re_m) * x_46_im_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_re_m * x_46_re_m) * x_46_im_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(Float64(x_46_re_m * x_46_re_m) * x_46_im_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_re_m * x_46_re_m) * x_46_im_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[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.6%

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

3. Simplified 86.1%

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

5. Taylor expanded in x.re around inf 50.3%

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

   1. unpow2 50.3%

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

7. Applied egg-rr 50.3%

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

8. Final simplification 50.3%

   \[\leadsto 3 \cdot \left(\left(x.re \cdot x.re\right) \cdot x.im\right) \]
9. Add Preprocessing

Alternative 10: 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 78.6%

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

3. Taylor expanded in x.re around 0 55.2%

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

5. Simplified 2.7%

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

Developer Target 1: 91.7% 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 2024137 
(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)))