math.cube on complex, imaginary part

Percentage Accurate: 82.2% → 96.5%

Time: 8.5s

Alternatives: 6

Speedup: 0.5×

• 44.6% 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: 82.2% 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: 96.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := x.im\_m \cdot \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) + x.re \cdot \left(x.re \cdot x.im\_m + x.re \cdot x.im\_m\right)\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 10^{+15}:\\ \;\;\;\;\left(x.re \cdot \left(x.re \cdot x.im\_m\right)\right) \cdot 3 - {x.im\_m}^{3}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left(x.re \cdot x.im\_m\right) \cdot \left(x.re \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;-{x.im\_m}^{3}\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0
         (+
          (* x.im_m (- (* x.re x.re) (* x.im_m x.im_m)))
          (* x.re (+ (* x.re x.im_m) (* x.re x.im_m))))))
   (*
    x.im_s
    (if (<= t_0 1e+15)
      (- (* (* x.re (* x.re x.im_m)) 3.0) (pow x.im_m 3.0))
      (if (<= t_0 INFINITY)
        (* (* x.re x.im_m) (* x.re 3.0))
        (- (pow x.im_m 3.0)))))))

C

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, double x_46_im_m) {
	double t_0 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)));
	double tmp;
	if (t_0 <= 1e+15) {
		tmp = ((x_46_re * (x_46_re * x_46_im_m)) * 3.0) - pow(x_46_im_m, 3.0);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (x_46_re * x_46_im_m) * (x_46_re * 3.0);
	} else {
		tmp = -pow(x_46_im_m, 3.0);
	}
	return x_46_im_s * tmp;
}

Java

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, double x_46_im_m) {
	double t_0 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)));
	double tmp;
	if (t_0 <= 1e+15) {
		tmp = ((x_46_re * (x_46_re * x_46_im_m)) * 3.0) - Math.pow(x_46_im_m, 3.0);
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (x_46_re * x_46_im_m) * (x_46_re * 3.0);
	} else {
		tmp = -Math.pow(x_46_im_m, 3.0);
	}
	return x_46_im_s * tmp;
}

Python

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, x_46_im_m):
	t_0 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)))
	tmp = 0
	if t_0 <= 1e+15:
		tmp = ((x_46_re * (x_46_re * x_46_im_m)) * 3.0) - math.pow(x_46_im_m, 3.0)
	elif t_0 <= math.inf:
		tmp = (x_46_re * x_46_im_m) * (x_46_re * 3.0)
	else:
		tmp = -math.pow(x_46_im_m, 3.0)
	return x_46_im_s * tmp

Julia

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, x_46_im_m)
	t_0 = Float64(Float64(x_46_im_m * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m))) + Float64(x_46_re * Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_re * x_46_im_m))))
	tmp = 0.0
	if (t_0 <= 1e+15)
		tmp = Float64(Float64(Float64(x_46_re * Float64(x_46_re * x_46_im_m)) * 3.0) - (x_46_im_m ^ 3.0));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(x_46_re * x_46_im_m) * Float64(x_46_re * 3.0));
	else
		tmp = Float64(-(x_46_im_m ^ 3.0));
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

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, x_46_im_m)
	t_0 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)));
	tmp = 0.0;
	if (t_0 <= 1e+15)
		tmp = ((x_46_re * (x_46_re * x_46_im_m)) * 3.0) - (x_46_im_m ^ 3.0);
	elseif (t_0 <= Inf)
		tmp = (x_46_re * x_46_im_m) * (x_46_re * 3.0);
	else
		tmp = -(x_46_im_m ^ 3.0);
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

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_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(x$46$im$95$m * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$re * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$0, 1e+15], N[(N[(N[(x$46$re * N[(x$46$re * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision] - N[Power[x$46$im$95$m, 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] * N[(x$46$re * 3.0), $MachinePrecision]), $MachinePrecision], (-N[Power[x$46$im$95$m, 3.0], $MachinePrecision])]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < 1e15

   1. Initial program 94.8%

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

   3. Simplified 98.5%

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

   if 1e15 < (+.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 86.6%

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

      1. remove-double-neg 86.6%

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

      2. distribute-rgt-neg-in 86.6%

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

      3. distribute-lft-neg-in 86.6%

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

      4. *-commutative 86.6%

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

      5. distribute-neg-out 86.6%

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

      6. distribute-lft-neg-out 86.6%

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

      7. distribute-lft-neg-out 86.6%

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

      8. *-commutative 86.6%

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

      9. fma-define 86.6%

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

      10. *-commutative 86.6%

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

      11. distribute-lft-neg-in 86.6%

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

      12. distribute-rgt-neg-in 86.6%

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

      13. distribute-lft-neg-out 86.6%

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

      14. distribute-rgt-neg-out 86.6%

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

      15. *-commutative 86.6%

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

      16. distribute-neg-out 86.6%

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

   3. Simplified 86.6%

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

   5. Taylor expanded in x.re around inf 29.4%

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

      1. add-sqr-sqrt 29.0%

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

      2. pow2 29.0%

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

      3. sqrt-prod 28.7%

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

      4. sqrt-pow1 41.9%

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

      5. metadata-eval 41.9%

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

      6. pow1 41.9%

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

      7. *-un-lft-identity 41.9%

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

      8. distribute-rgt-out 41.9%

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

      9. metadata-eval 41.9%

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

   7. Applied egg-rr 41.9%

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

      1. unpow2 41.9%

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

      2. *-commutative 41.9%

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

      3. associate-*r* 41.9%

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

      4. associate-*r* 41.8%

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

      5. add-sqr-sqrt 42.6%

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

      6. associate-*r* 42.6%

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

      7. associate-*l* 42.6%

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

   9. Applied egg-rr 42.6%

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

   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%

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

   3. Simplified 0.0%

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

      1. associate-*r* 0.0%

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

      2. fma-neg 17.9%

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

   6. Applied egg-rr 17.9%

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

   7. Taylor expanded in x.re around 0 75.0%

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

      1. neg-mul-1 75.0%

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

   9. Simplified 75.0%

      \[\leadsto \color{blue}{-{x.im}^{3}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq 10^{+15}:\\ \;\;\;\;\left(x.re \cdot \left(x.re \cdot x.im\right)\right) \cdot 3 - {x.im}^{3}\\ \mathbf{elif}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq \infty:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;-{x.im}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := x.im\_m \cdot \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) + x.re \cdot \left(x.re \cdot x.im\_m + x.re \cdot x.im\_m\right)\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-309} \lor \neg \left(t\_0 \leq \infty\right):\\ \;\;\;\;-{x.im\_m}^{3}\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re \cdot x.im\_m\right) \cdot 3\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0
         (+
          (* x.im_m (- (* x.re x.re) (* x.im_m x.im_m)))
          (* x.re (+ (* x.re x.im_m) (* x.re x.im_m))))))
   (*
    x.im_s
    (if (or (<= t_0 -5e-309) (not (<= t_0 INFINITY)))
      (- (pow x.im_m 3.0))
      (* x.re (* (* x.re x.im_m) 3.0))))))

C

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, double x_46_im_m) {
	double t_0 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)));
	double tmp;
	if ((t_0 <= -5e-309) || !(t_0 <= ((double) INFINITY))) {
		tmp = -pow(x_46_im_m, 3.0);
	} else {
		tmp = x_46_re * ((x_46_re * x_46_im_m) * 3.0);
	}
	return x_46_im_s * tmp;
}

Java

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, double x_46_im_m) {
	double t_0 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)));
	double tmp;
	if ((t_0 <= -5e-309) || !(t_0 <= Double.POSITIVE_INFINITY)) {
		tmp = -Math.pow(x_46_im_m, 3.0);
	} else {
		tmp = x_46_re * ((x_46_re * x_46_im_m) * 3.0);
	}
	return x_46_im_s * tmp;
}

Python

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, x_46_im_m):
	t_0 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)))
	tmp = 0
	if (t_0 <= -5e-309) or not (t_0 <= math.inf):
		tmp = -math.pow(x_46_im_m, 3.0)
	else:
		tmp = x_46_re * ((x_46_re * x_46_im_m) * 3.0)
	return x_46_im_s * tmp

Julia

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, x_46_im_m)
	t_0 = Float64(Float64(x_46_im_m * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m))) + Float64(x_46_re * Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_re * x_46_im_m))))
	tmp = 0.0
	if ((t_0 <= -5e-309) || !(t_0 <= Inf))
		tmp = Float64(-(x_46_im_m ^ 3.0));
	else
		tmp = Float64(x_46_re * Float64(Float64(x_46_re * x_46_im_m) * 3.0));
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

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, x_46_im_m)
	t_0 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)));
	tmp = 0.0;
	if ((t_0 <= -5e-309) || ~((t_0 <= Inf)))
		tmp = -(x_46_im_m ^ 3.0);
	else
		tmp = x_46_re * ((x_46_re * x_46_im_m) * 3.0);
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

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_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(x$46$im$95$m * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$re * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[Or[LessEqual[t$95$0, -5e-309], N[Not[LessEqual[t$95$0, Infinity]], $MachinePrecision]], (-N[Power[x$46$im$95$m, 3.0], $MachinePrecision]), N[(x$46$re * N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < -4.9999999999999995e-309 or +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 70.7%

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

   3. Simplified 75.3%

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

      1. associate-*r* 75.3%

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

      2. fma-neg 79.4%

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

   6. Applied egg-rr 79.4%

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

   7. Taylor expanded in x.re around 0 57.5%

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

      1. neg-mul-1 57.5%

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

   9. Simplified 57.5%

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

   if -4.9999999999999995e-309 < (+.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 92.3%

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

      1. remove-double-neg 92.3%

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

      2. distribute-rgt-neg-in 92.3%

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

      3. distribute-lft-neg-in 92.3%

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

      4. *-commutative 92.3%

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

      5. distribute-neg-out 92.3%

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

      6. distribute-lft-neg-out 92.3%

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

      7. distribute-lft-neg-out 92.3%

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

      8. *-commutative 92.3%

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

      9. fma-define 92.3%

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

      10. *-commutative 92.3%

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

      11. distribute-lft-neg-in 92.3%

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

      12. distribute-rgt-neg-in 92.3%

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

      13. distribute-lft-neg-out 92.3%

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

      14. distribute-rgt-neg-out 92.3%

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

      15. *-commutative 92.3%

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

      16. distribute-neg-out 92.3%

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

   3. Simplified 92.3%

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

   5. Taylor expanded in x.re around inf 51.5%

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

      1. add-sqr-sqrt 51.1%

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

      2. pow2 51.1%

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

      3. sqrt-prod 38.6%

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

      4. sqrt-pow1 46.1%

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

      5. metadata-eval 46.1%

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

      6. pow1 46.1%

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

      7. *-un-lft-identity 46.1%

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

      8. distribute-rgt-out 46.1%

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

      9. metadata-eval 46.1%

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

   7. Applied egg-rr 46.1%

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

      1. unpow2 46.1%

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

      2. swap-sqr 38.5%

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

      3. add-sqr-sqrt 51.5%

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

      4. associate-*l* 58.9%

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

      5. *-commutative 58.9%

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

   9. Applied egg-rr 58.9%

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

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

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -5 \cdot 10^{-309} \lor \neg \left(x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq \infty\right):\\ \;\;\;\;-{x.im}^{3}\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re \cdot x.im\right) \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := x.im\_m \cdot \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) + x.re \cdot \left(x.re \cdot x.im\_m + x.re \cdot x.im\_m\right)\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-224}:\\ \;\;\;\;x.re \cdot \left(x.im\_m \cdot \left(x.re \cdot 3\right)\right) - {x.im\_m}^{3}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left(x.re \cdot \left(x.re \cdot x.im\_m\right)\right) \cdot 3\\ \mathbf{else}:\\ \;\;\;\;-{x.im\_m}^{3}\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0
         (+
          (* x.im_m (- (* x.re x.re) (* x.im_m x.im_m)))
          (* x.re (+ (* x.re x.im_m) (* x.re x.im_m))))))
   (*
    x.im_s
    (if (<= t_0 2e-224)
      (- (* x.re (* x.im_m (* x.re 3.0))) (pow x.im_m 3.0))
      (if (<= t_0 INFINITY)
        (* (* x.re (* x.re x.im_m)) 3.0)
        (- (pow x.im_m 3.0)))))))

C

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, double x_46_im_m) {
	double t_0 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)));
	double tmp;
	if (t_0 <= 2e-224) {
		tmp = (x_46_re * (x_46_im_m * (x_46_re * 3.0))) - pow(x_46_im_m, 3.0);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (x_46_re * (x_46_re * x_46_im_m)) * 3.0;
	} else {
		tmp = -pow(x_46_im_m, 3.0);
	}
	return x_46_im_s * tmp;
}

Java

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, double x_46_im_m) {
	double t_0 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)));
	double tmp;
	if (t_0 <= 2e-224) {
		tmp = (x_46_re * (x_46_im_m * (x_46_re * 3.0))) - Math.pow(x_46_im_m, 3.0);
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (x_46_re * (x_46_re * x_46_im_m)) * 3.0;
	} else {
		tmp = -Math.pow(x_46_im_m, 3.0);
	}
	return x_46_im_s * tmp;
}

Python

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, x_46_im_m):
	t_0 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)))
	tmp = 0
	if t_0 <= 2e-224:
		tmp = (x_46_re * (x_46_im_m * (x_46_re * 3.0))) - math.pow(x_46_im_m, 3.0)
	elif t_0 <= math.inf:
		tmp = (x_46_re * (x_46_re * x_46_im_m)) * 3.0
	else:
		tmp = -math.pow(x_46_im_m, 3.0)
	return x_46_im_s * tmp

Julia

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, x_46_im_m)
	t_0 = Float64(Float64(x_46_im_m * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m))) + Float64(x_46_re * Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_re * x_46_im_m))))
	tmp = 0.0
	if (t_0 <= 2e-224)
		tmp = Float64(Float64(x_46_re * Float64(x_46_im_m * Float64(x_46_re * 3.0))) - (x_46_im_m ^ 3.0));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(x_46_re * Float64(x_46_re * x_46_im_m)) * 3.0);
	else
		tmp = Float64(-(x_46_im_m ^ 3.0));
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

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, x_46_im_m)
	t_0 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)));
	tmp = 0.0;
	if (t_0 <= 2e-224)
		tmp = (x_46_re * (x_46_im_m * (x_46_re * 3.0))) - (x_46_im_m ^ 3.0);
	elseif (t_0 <= Inf)
		tmp = (x_46_re * (x_46_re * x_46_im_m)) * 3.0;
	else
		tmp = -(x_46_im_m ^ 3.0);
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

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_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(x$46$im$95$m * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$re * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$0, 2e-224], N[(N[(x$46$re * N[(x$46$im$95$m * N[(x$46$re * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[x$46$im$95$m, 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(x$46$re * N[(x$46$re * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision], (-N[Power[x$46$im$95$m, 3.0], $MachinePrecision])]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < 2e-224

   1. Initial program 93.8%

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

   3. Simplified 98.2%

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

   if 2e-224 < (+.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 90.0%

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

   3. Simplified 97.8%

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

   5. Taylor expanded in x.re around inf 40.0%

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

      1. add-sqr-sqrt 39.5%

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

      2. pow2 39.5%

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

      3. *-commutative 39.5%

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

      4. sqrt-prod 39.0%

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

      5. sqrt-pow1 48.6%

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

      6. metadata-eval 48.6%

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

      7. pow1 48.6%

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

   7. Applied egg-rr 48.6%

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

      1. unpow2 48.6%

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

      2. *-commutative 48.6%

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

      3. associate-*r* 48.7%

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

      4. associate-*r* 48.6%

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

      5. add-sqr-sqrt 49.6%

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

   9. Applied egg-rr 49.6%

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

   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%

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

   3. Simplified 0.0%

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

      1. associate-*r* 0.0%

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

      2. fma-neg 17.9%

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

   6. Applied egg-rr 17.9%

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

   7. Taylor expanded in x.re around 0 75.0%

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

      1. neg-mul-1 75.0%

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

   9. Simplified 75.0%

      \[\leadsto \color{blue}{-{x.im}^{3}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq 2 \cdot 10^{-224}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right) - {x.im}^{3}\\ \mathbf{elif}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq \infty:\\ \;\;\;\;\left(x.re \cdot \left(x.re \cdot x.im\right)\right) \cdot 3\\ \mathbf{else}:\\ \;\;\;\;-{x.im}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := x.im\_m \cdot \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) + x.re \cdot \left(x.re \cdot x.im\_m + x.re \cdot x.im\_m\right)\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 5 \cdot 10^{+286}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re \cdot x.im\_m\right) \cdot 3\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0
         (+
          (* x.im_m (- (* x.re x.re) (* x.im_m x.im_m)))
          (* x.re (+ (* x.re x.im_m) (* x.re x.im_m))))))
   (* x.im_s (if (<= t_0 5e+286) t_0 (* x.re (* (* x.re x.im_m) 3.0))))))

C

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, double x_46_im_m) {
	double t_0 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)));
	double tmp;
	if (t_0 <= 5e+286) {
		tmp = t_0;
	} else {
		tmp = x_46_re * ((x_46_re * x_46_im_m) * 3.0);
	}
	return x_46_im_s * tmp;
}

Fortran

x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_46im_m * ((x_46re * x_46re) - (x_46im_m * x_46im_m))) + (x_46re * ((x_46re * x_46im_m) + (x_46re * x_46im_m)))
    if (t_0 <= 5d+286) then
        tmp = t_0
    else
        tmp = x_46re * ((x_46re * x_46im_m) * 3.0d0)
    end if
    code = x_46im_s * tmp
end function

Java

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, double x_46_im_m) {
	double t_0 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)));
	double tmp;
	if (t_0 <= 5e+286) {
		tmp = t_0;
	} else {
		tmp = x_46_re * ((x_46_re * x_46_im_m) * 3.0);
	}
	return x_46_im_s * tmp;
}

Python

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, x_46_im_m):
	t_0 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)))
	tmp = 0
	if t_0 <= 5e+286:
		tmp = t_0
	else:
		tmp = x_46_re * ((x_46_re * x_46_im_m) * 3.0)
	return x_46_im_s * tmp

Julia

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, x_46_im_m)
	t_0 = Float64(Float64(x_46_im_m * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m))) + Float64(x_46_re * Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_re * x_46_im_m))))
	tmp = 0.0
	if (t_0 <= 5e+286)
		tmp = t_0;
	else
		tmp = Float64(x_46_re * Float64(Float64(x_46_re * x_46_im_m) * 3.0));
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

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, x_46_im_m)
	t_0 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)));
	tmp = 0.0;
	if (t_0 <= 5e+286)
		tmp = t_0;
	else
		tmp = x_46_re * ((x_46_re * x_46_im_m) * 3.0);
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

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_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(x$46$im$95$m * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$re * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$0, 5e+286], t$95$0, N[(x$46$re * N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < 5.0000000000000004e286

   1. Initial program 95.5%

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

   if 5.0000000000000004e286 < (+.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 52.4%

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

      1. remove-double-neg 52.4%

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

      2. distribute-rgt-neg-in 52.4%

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

      3. distribute-lft-neg-in 52.4%

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

      4. *-commutative 52.4%

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

      5. distribute-neg-out 52.4%

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

      6. distribute-lft-neg-out 52.4%

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

      7. distribute-lft-neg-out 52.4%

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

      8. *-commutative 52.4%

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

      9. fma-define 52.4%

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

      10. *-commutative 52.4%

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

      11. distribute-lft-neg-in 52.4%

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

      12. distribute-rgt-neg-in 52.4%

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

      13. distribute-lft-neg-out 52.4%

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

      14. distribute-rgt-neg-out 52.4%

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

      15. *-commutative 52.4%

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

      16. distribute-neg-out 52.4%

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

   3. Simplified 52.4%

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

   5. Taylor expanded in x.re around inf 26.4%

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

      1. add-sqr-sqrt 26.3%

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

      2. pow2 26.3%

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

      3. sqrt-prod 26.1%

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

      4. sqrt-pow1 38.6%

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

      5. metadata-eval 38.6%

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

      6. pow1 38.6%

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

      7. *-un-lft-identity 38.6%

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

      8. distribute-rgt-out 38.6%

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

      9. metadata-eval 38.6%

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

   7. Applied egg-rr 38.6%

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

      1. unpow2 38.6%

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

      2. swap-sqr 26.1%

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

      3. add-sqr-sqrt 26.4%

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

      4. associate-*l* 38.9%

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

      5. *-commutative 38.9%

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

   9. Applied egg-rr 38.9%

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

   10. Taylor expanded in x.re around 0 39.0%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq 5 \cdot 10^{+286}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re \cdot x.im\right) \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 55.6% accurate, 2.7× speedup

Math

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

FPCore

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

C

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, double x_46_im_m) {
	return x_46_im_s * (x_46_re * ((x_46_re * x_46_im_m) * 3.0));
}

Fortran

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

Java

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, double x_46_im_m) {
	return x_46_im_s * (x_46_re * ((x_46_re * x_46_im_m) * 3.0));
}

Python

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, x_46_im_m):
	return x_46_im_s * (x_46_re * ((x_46_re * x_46_im_m) * 3.0))

Julia

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, x_46_im_m)
	return Float64(x_46_im_s * Float64(x_46_re * Float64(Float64(x_46_re * x_46_im_m) * 3.0)))
end

MATLAB

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, x_46_im_m)
	tmp = x_46_im_s * (x_46_re * ((x_46_re * x_46_im_m) * 3.0));
end

Wolfram

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_, x$46$im$95$m_] := N[(x$46$im$95$s * N[(x$46$re * N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.0%

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

   1. remove-double-neg 82.0%

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

   2. distribute-rgt-neg-in 82.0%

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

   3. distribute-lft-neg-in 82.0%

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

   4. *-commutative 82.0%

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

   5. distribute-neg-out 82.0%

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

   6. distribute-lft-neg-out 82.0%

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

   7. distribute-lft-neg-out 82.0%

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

   8. *-commutative 82.0%

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

   9. fma-define 82.0%

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

   10. *-commutative 82.0%

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

   11. distribute-lft-neg-in 82.0%

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

   12. distribute-rgt-neg-in 82.0%

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

   13. distribute-lft-neg-out 82.0%

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

   14. distribute-rgt-neg-out 82.0%

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

   15. *-commutative 82.0%

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

   16. distribute-neg-out 82.0%

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

3. Simplified 82.0%

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

5. Taylor expanded in x.re around inf 45.0%

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

   1. add-sqr-sqrt 29.8%

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

   2. pow2 29.8%

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

   3. sqrt-prod 23.3%

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

   4. sqrt-pow1 27.2%

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

   5. metadata-eval 27.2%

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

   6. pow1 27.2%

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

   7. *-un-lft-identity 27.2%

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

   8. distribute-rgt-out 27.2%

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

   9. metadata-eval 27.2%

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

7. Applied egg-rr 27.2%

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

   1. unpow2 27.2%

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

   2. swap-sqr 23.2%

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

   3. add-sqr-sqrt 45.0%

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

   4. associate-*l* 51.8%

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

   5. *-commutative 51.8%

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

9. Applied egg-rr 51.8%

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

10. Taylor expanded in x.re around 0 51.8%

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

11. Final simplification 51.8%

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

Alternative 6: 55.6% accurate, 2.7× speedup

Math

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

FPCore

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

C

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, double x_46_im_m) {
	return x_46_im_s * ((x_46_re * (x_46_re * x_46_im_m)) * 3.0);
}

Fortran

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

Java

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, double x_46_im_m) {
	return x_46_im_s * ((x_46_re * (x_46_re * x_46_im_m)) * 3.0);
}

Python

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, x_46_im_m):
	return x_46_im_s * ((x_46_re * (x_46_re * x_46_im_m)) * 3.0)

Julia

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, x_46_im_m)
	return Float64(x_46_im_s * Float64(Float64(x_46_re * Float64(x_46_re * x_46_im_m)) * 3.0))
end

MATLAB

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, x_46_im_m)
	tmp = x_46_im_s * ((x_46_re * (x_46_re * x_46_im_m)) * 3.0);
end

Wolfram

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_, x$46$im$95$m_] := N[(x$46$im$95$s * N[(N[(x$46$re * N[(x$46$re * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.0%

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

3. Simplified 87.3%

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

5. Taylor expanded in x.re around inf 45.0%

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

   1. add-sqr-sqrt 29.8%

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

   2. pow2 29.8%

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

   3. *-commutative 29.8%

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

   4. sqrt-prod 23.2%

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

   5. sqrt-pow1 27.1%

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

   6. metadata-eval 27.1%

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

   7. pow1 27.1%

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

7. Applied egg-rr 27.1%

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

   1. unpow2 27.1%

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

   2. *-commutative 27.1%

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

   3. associate-*r* 27.1%

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

   4. associate-*r* 27.1%

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

   5. add-sqr-sqrt 51.8%

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

9. Applied egg-rr 51.8%

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

10. Final simplification 51.8%

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

Developer target: 91.2% 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 2024111 
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"
  :precision binary64

  :alt
  (+ (* (* x.re x.im) (* 2.0 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)))