math.cube on complex, imaginary part

Percentage Accurate: 83.3% → 99.7%

Time: 7.7s

Alternatives: 6

Speedup: 1.3×

• 44.5% 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: 83.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m\\ t_1 := t\_0 + \left(x.re \cdot x.im\_m + x.im\_m \cdot x.re\right) \cdot x.re\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+83}:\\ \;\;\;\;t\_0 + \left(x.re \cdot \left(x.im\_m + x.im\_m\right)\right) \cdot x.re\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left(\left(x.im\_m \cdot x.re\right) \cdot 3\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{x.im\_m}{-x.re}, \frac{x.im\_m}{x.re}, 3\right) \cdot x.im\_m\right) \cdot \left(x.re \cdot x.re\right)\\ \end{array} \end{array} \end{array} \]

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.re x.re) (* x.im_m x.im_m)) x.im_m))
        (t_1 (+ t_0 (* (+ (* x.re x.im_m) (* x.im_m x.re)) x.re))))
   (*
    x.im_s
    (if (<= t_1 5e+83)
      (+ t_0 (* (* x.re (+ x.im_m x.im_m)) x.re))
      (if (<= t_1 INFINITY)
        (* (* (* x.im_m x.re) 3.0) x.re)
        (*
         (* (fma (/ x.im_m (- x.re)) (/ x.im_m x.re) 3.0) x.im_m)
         (* x.re x.re)))))))

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_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m;
	double t_1 = t_0 + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	double tmp;
	if (t_1 <= 5e+83) {
		tmp = t_0 + ((x_46_re * (x_46_im_m + x_46_im_m)) * x_46_re);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = ((x_46_im_m * x_46_re) * 3.0) * x_46_re;
	} else {
		tmp = (fma((x_46_im_m / -x_46_re), (x_46_im_m / x_46_re), 3.0) * x_46_im_m) * (x_46_re * x_46_re);
	}
	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(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m)
	t_1 = Float64(t_0 + Float64(Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_im_m * x_46_re)) * x_46_re))
	tmp = 0.0
	if (t_1 <= 5e+83)
		tmp = Float64(t_0 + Float64(Float64(x_46_re * Float64(x_46_im_m + x_46_im_m)) * x_46_re));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(x_46_im_m * x_46_re) * 3.0) * x_46_re);
	else
		tmp = Float64(Float64(fma(Float64(x_46_im_m / Float64(-x_46_re)), Float64(x_46_im_m / x_46_re), 3.0) * x_46_im_m) * Float64(x_46_re * x_46_re));
	end
	return Float64(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[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$1, 5e+83], N[(t$95$0 + N[(N[(x$46$re * N[(x$46$im$95$m + x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] * 3.0), $MachinePrecision] * x$46$re), $MachinePrecision], N[(N[(N[(N[(x$46$im$95$m / (-x$46$re)), $MachinePrecision] * N[(x$46$im$95$m / x$46$re), $MachinePrecision] + 3.0), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] * N[(x$46$re * x$46$re), $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)) < 5.00000000000000029e83

   1. Initial program 94.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 94.1

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

   4. Applied rewrites 94.1%

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

   if 5.00000000000000029e83 < (+.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 91.6%

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

   3. Taylor expanded in x.re around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt1-in N/A

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

      7. metadata-eval N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 52.9

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

   5. Applied rewrites 52.9%

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

   7. Applied rewrites 53.0%

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

   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. Taylor expanded in x.re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower--.f64 N/A

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

      9. distribute-rgt1-in N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-/r* N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 85.3

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

   5. Applied rewrites 85.3%

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

   6. Taylor expanded in x.re around 0

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

   8. Applied rewrites 100.0%

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

Alternative 2: 96.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m + \left(x.re \cdot x.im\_m + x.im\_m \cdot x.re\right) \cdot x.re\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-304} \lor \neg \left(t\_0 \leq \infty\right):\\ \;\;\;\;\left(-x.im\_m\right) \cdot \left(x.im\_m \cdot x.im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x.im\_m \cdot x.re\right) \cdot 3\right) \cdot x.re\\ \end{array} \end{array} \end{array} \]

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.re x.re) (* x.im_m x.im_m)) x.im_m)
          (* (+ (* x.re x.im_m) (* x.im_m x.re)) x.re))))
   (*
    x.im_s
    (if (or (<= t_0 -4e-304) (not (<= t_0 INFINITY)))
      (* (- x.im_m) (* x.im_m x.im_m))
      (* (* (* x.im_m x.re) 3.0) x.re)))))

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_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	double tmp;
	if ((t_0 <= -4e-304) || !(t_0 <= ((double) INFINITY))) {
		tmp = -x_46_im_m * (x_46_im_m * x_46_im_m);
	} else {
		tmp = ((x_46_im_m * x_46_re) * 3.0) * x_46_re;
	}
	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_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	double tmp;
	if ((t_0 <= -4e-304) || !(t_0 <= Double.POSITIVE_INFINITY)) {
		tmp = -x_46_im_m * (x_46_im_m * x_46_im_m);
	} else {
		tmp = ((x_46_im_m * x_46_re) * 3.0) * x_46_re;
	}
	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_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re)
	tmp = 0
	if (t_0 <= -4e-304) or not (t_0 <= math.inf):
		tmp = -x_46_im_m * (x_46_im_m * x_46_im_m)
	else:
		tmp = ((x_46_im_m * x_46_re) * 3.0) * x_46_re
	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(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m) + Float64(Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_im_m * x_46_re)) * x_46_re))
	tmp = 0.0
	if ((t_0 <= -4e-304) || !(t_0 <= Inf))
		tmp = Float64(Float64(-x_46_im_m) * Float64(x_46_im_m * x_46_im_m));
	else
		tmp = Float64(Float64(Float64(x_46_im_m * x_46_re) * 3.0) * x_46_re);
	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_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	tmp = 0.0;
	if ((t_0 <= -4e-304) || ~((t_0 <= Inf)))
		tmp = -x_46_im_m * (x_46_im_m * x_46_im_m);
	else
		tmp = ((x_46_im_m * x_46_re) * 3.0) * x_46_re;
	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[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] + N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[Or[LessEqual[t$95$0, -4e-304], N[Not[LessEqual[t$95$0, Infinity]], $MachinePrecision]], N[((-x$46$im$95$m) * N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] * 3.0), $MachinePrecision] * x$46$re), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < -3.99999999999999988e-304 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 67.2%

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

   3. Taylor expanded in x.re around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*r* N/A

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

      4. count-2-rev N/A

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

      5. distribute-lft-in N/A

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

      6. count-2-rev N/A

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

      7. distribute-lft-in N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. mul-1-neg N/A

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

      10. cube-mult N/A

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

      11. unpow2 N/A

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

      12. distribute-lft-neg-in N/A

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

      13. distribute-lft-out-- N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. distribute-lft1-in N/A

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

   5. Applied rewrites 84.4%

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

   6. Taylor expanded in x.re around 0

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

   8. Applied rewrites 58.5%

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

   if -3.99999999999999988e-304 < (+.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 96.0%

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

   3. Taylor expanded in x.re around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt1-in N/A

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

      7. metadata-eval N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 66.4

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

   5. Applied rewrites 66.4%

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

   7. Applied rewrites 66.4%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq -4 \cdot 10^{-304} \lor \neg \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq \infty\right):\\ \;\;\;\;\left(-x.im\right) \cdot \left(x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x.im \cdot x.re\right) \cdot 3\right) \cdot x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m + \left(x.re \cdot x.im\_m + x.im\_m \cdot x.re\right) \cdot x.re\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-304} \lor \neg \left(t\_0 \leq \infty\right):\\ \;\;\;\;\left(-x.im\_m\right) \cdot \left(x.im\_m \cdot x.im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\left(x.im\_m \cdot x.re\right) \cdot x.re\right)\\ \end{array} \end{array} \end{array} \]

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.re x.re) (* x.im_m x.im_m)) x.im_m)
          (* (+ (* x.re x.im_m) (* x.im_m x.re)) x.re))))
   (*
    x.im_s
    (if (or (<= t_0 -4e-304) (not (<= t_0 INFINITY)))
      (* (- x.im_m) (* x.im_m x.im_m))
      (* 3.0 (* (* x.im_m x.re) x.re))))))

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_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	double tmp;
	if ((t_0 <= -4e-304) || !(t_0 <= ((double) INFINITY))) {
		tmp = -x_46_im_m * (x_46_im_m * x_46_im_m);
	} else {
		tmp = 3.0 * ((x_46_im_m * x_46_re) * x_46_re);
	}
	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_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	double tmp;
	if ((t_0 <= -4e-304) || !(t_0 <= Double.POSITIVE_INFINITY)) {
		tmp = -x_46_im_m * (x_46_im_m * x_46_im_m);
	} else {
		tmp = 3.0 * ((x_46_im_m * x_46_re) * x_46_re);
	}
	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_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re)
	tmp = 0
	if (t_0 <= -4e-304) or not (t_0 <= math.inf):
		tmp = -x_46_im_m * (x_46_im_m * x_46_im_m)
	else:
		tmp = 3.0 * ((x_46_im_m * x_46_re) * x_46_re)
	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(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m) + Float64(Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_im_m * x_46_re)) * x_46_re))
	tmp = 0.0
	if ((t_0 <= -4e-304) || !(t_0 <= Inf))
		tmp = Float64(Float64(-x_46_im_m) * Float64(x_46_im_m * x_46_im_m));
	else
		tmp = Float64(3.0 * Float64(Float64(x_46_im_m * x_46_re) * x_46_re));
	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_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re * x_46_im_m) + (x_46_im_m * x_46_re)) * x_46_re);
	tmp = 0.0;
	if ((t_0 <= -4e-304) || ~((t_0 <= Inf)))
		tmp = -x_46_im_m * (x_46_im_m * x_46_im_m);
	else
		tmp = 3.0 * ((x_46_im_m * x_46_re) * x_46_re);
	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[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] + N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[Or[LessEqual[t$95$0, -4e-304], N[Not[LessEqual[t$95$0, Infinity]], $MachinePrecision]], N[((-x$46$im$95$m) * N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], N[(3.0 * N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] * x$46$re), $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)) < -3.99999999999999988e-304 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 67.2%

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

   3. Taylor expanded in x.re around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*r* N/A

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

      4. count-2-rev N/A

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

      5. distribute-lft-in N/A

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

      6. count-2-rev N/A

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

      7. distribute-lft-in N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. mul-1-neg N/A

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

      10. cube-mult N/A

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

      11. unpow2 N/A

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

      12. distribute-lft-neg-in N/A

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

      13. distribute-lft-out-- N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. distribute-lft1-in N/A

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

   5. Applied rewrites 84.4%

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

   6. Taylor expanded in x.re around 0

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

   8. Applied rewrites 58.5%

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

   if -3.99999999999999988e-304 < (+.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 96.0%

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

   3. Taylor expanded in x.re around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt1-in N/A

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

      7. metadata-eval N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 66.4

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

   5. Applied rewrites 66.4%

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

   7. Applied rewrites 66.3%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq -4 \cdot 10^{-304} \lor \neg \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq \infty\right):\\ \;\;\;\;\left(-x.im\right) \cdot \left(x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\left(x.im \cdot x.re\right) \cdot x.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 3.6 \cdot 10^{-110}:\\ \;\;\;\;\left(\left(x.im\_m \cdot x.re\right) \cdot 3\right) \cdot x.re\\ \mathbf{elif}\;x.im\_m \leq 4 \cdot 10^{+202}:\\ \;\;\;\;\left(-x.im\_m\right) \cdot \mathsf{fma}\left(x.im\_m, x.im\_m, -3 \cdot \left(x.re \cdot x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-x.im\_m\right) \cdot \left(x.im\_m \cdot x.im\_m\right)\\ \end{array} \end{array} \]

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
 (*
  x.im_s
  (if (<= x.im_m 3.6e-110)
    (* (* (* x.im_m x.re) 3.0) x.re)
    (if (<= x.im_m 4e+202)
      (* (- x.im_m) (fma x.im_m x.im_m (* -3.0 (* x.re x.re))))
      (* (- x.im_m) (* x.im_m x.im_m))))))

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 tmp;
	if (x_46_im_m <= 3.6e-110) {
		tmp = ((x_46_im_m * x_46_re) * 3.0) * x_46_re;
	} else if (x_46_im_m <= 4e+202) {
		tmp = -x_46_im_m * fma(x_46_im_m, x_46_im_m, (-3.0 * (x_46_re * x_46_re)));
	} else {
		tmp = -x_46_im_m * (x_46_im_m * x_46_im_m);
	}
	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)
	tmp = 0.0
	if (x_46_im_m <= 3.6e-110)
		tmp = Float64(Float64(Float64(x_46_im_m * x_46_re) * 3.0) * x_46_re);
	elseif (x_46_im_m <= 4e+202)
		tmp = Float64(Float64(-x_46_im_m) * fma(x_46_im_m, x_46_im_m, Float64(-3.0 * Float64(x_46_re * x_46_re))));
	else
		tmp = Float64(Float64(-x_46_im_m) * Float64(x_46_im_m * x_46_im_m));
	end
	return Float64(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_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 3.6e-110], N[(N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] * 3.0), $MachinePrecision] * x$46$re), $MachinePrecision], If[LessEqual[x$46$im$95$m, 4e+202], N[((-x$46$im$95$m) * N[(x$46$im$95$m * x$46$im$95$m + N[(-3.0 * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-x$46$im$95$m) * N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x.im < 3.59999999999999995e-110

   1. Initial program 81.9%

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

   3. Taylor expanded in x.re around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt1-in N/A

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

      7. metadata-eval N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 66.6

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

   5. Applied rewrites 66.6%

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

   7. Applied rewrites 66.7%

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

   if 3.59999999999999995e-110 < x.im < 3.9999999999999996e202

   1. Initial program 93.7%

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

   3. Taylor expanded in x.re around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*r* N/A

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

      4. count-2-rev N/A

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

      5. distribute-lft-in N/A

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

      6. count-2-rev N/A

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

      7. distribute-lft-in N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. mul-1-neg N/A

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

      10. cube-mult N/A

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

      11. unpow2 N/A

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

      12. distribute-lft-neg-in N/A

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

      13. distribute-lft-out-- N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. distribute-lft1-in N/A

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

   5. Applied rewrites 96.9%

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

   if 3.9999999999999996e202 < x.im

   1. Initial program 46.4%

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

   3. Taylor expanded in x.re around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*r* N/A

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

      4. count-2-rev N/A

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

      5. distribute-lft-in N/A

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

      6. count-2-rev N/A

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

      7. distribute-lft-in N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. mul-1-neg N/A

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

      10. cube-mult N/A

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

      11. unpow2 N/A

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

      12. distribute-lft-neg-in N/A

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

      13. distribute-lft-out-- N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. distribute-lft1-in N/A

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

   5. Applied rewrites 75.0%

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

   6. Taylor expanded in x.re around 0

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

   8. Applied rewrites 96.4%

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

Alternative 5: 96.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 4.8 \cdot 10^{-109}:\\ \;\;\;\;\left(\left(x.im\_m \cdot x.re\right) \cdot 3\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\left(-x.im\_m\right) \cdot \mathsf{fma}\left(-3 \cdot x.re, x.re, x.im\_m \cdot x.im\_m\right)\\ \end{array} \end{array} \]

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
 (*
  x.im_s
  (if (<= x.im_m 4.8e-109)
    (* (* (* x.im_m x.re) 3.0) x.re)
    (* (- x.im_m) (fma (* -3.0 x.re) x.re (* x.im_m x.im_m))))))

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 tmp;
	if (x_46_im_m <= 4.8e-109) {
		tmp = ((x_46_im_m * x_46_re) * 3.0) * x_46_re;
	} else {
		tmp = -x_46_im_m * fma((-3.0 * x_46_re), x_46_re, (x_46_im_m * x_46_im_m));
	}
	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)
	tmp = 0.0
	if (x_46_im_m <= 4.8e-109)
		tmp = Float64(Float64(Float64(x_46_im_m * x_46_re) * 3.0) * x_46_re);
	else
		tmp = Float64(Float64(-x_46_im_m) * fma(Float64(-3.0 * x_46_re), x_46_re, Float64(x_46_im_m * x_46_im_m)));
	end
	return Float64(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_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 4.8e-109], N[(N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] * 3.0), $MachinePrecision] * x$46$re), $MachinePrecision], N[((-x$46$im$95$m) * N[(N[(-3.0 * x$46$re), $MachinePrecision] * x$46$re + N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 4.79999999999999977e-109

   1. Initial program 81.9%

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

   3. Taylor expanded in x.re around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt1-in N/A

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

      7. metadata-eval N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 66.6

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

   5. Applied rewrites 66.6%

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

   7. Applied rewrites 66.7%

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

   if 4.79999999999999977e-109 < x.im

   1. Initial program 79.5%

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

   3. Taylor expanded in x.re around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*r* N/A

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

      4. count-2-rev N/A

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

      5. distribute-lft-in N/A

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

      6. count-2-rev N/A

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

      7. distribute-lft-in N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. mul-1-neg N/A

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

      10. cube-mult N/A

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

      11. unpow2 N/A

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

      12. distribute-lft-neg-in N/A

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

      13. distribute-lft-out-- N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. distribute-lft1-in N/A

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

   5. Applied rewrites 90.3%

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

   7. Applied rewrites 96.7%

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

Alternative 6: 58.3% accurate, 3.1× speedup

Math

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

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.im_m) (* x.im_m x.im_m))))

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_im_m * (x_46_im_m * x_46_im_m));
}

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_46im_m * (x_46im_m * x_46im_m))
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_im_m * (x_46_im_m * x_46_im_m));
}

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_im_m * (x_46_im_m * x_46_im_m))

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_im_m) * Float64(x_46_im_m * x_46_im_m)))
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_im_m * (x_46_im_m * x_46_im_m));
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$46$im$95$m) * N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.0%

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

3. Taylor expanded in x.re around 0

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. associate-*r* N/A

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

   4. count-2-rev N/A

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

   5. distribute-lft-in N/A

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

   6. count-2-rev N/A

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

   7. distribute-lft-in N/A

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

   8. fp-cancel-sign-sub-inv N/A

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

   9. mul-1-neg N/A

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

   10. cube-mult N/A

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

   11. unpow2 N/A

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

   12. distribute-lft-neg-in N/A

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

   13. distribute-lft-out-- N/A

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

   14. lower-*.f64 N/A

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

   15. lower-neg.f64 N/A

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

   16. distribute-lft1-in N/A

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

5. Applied rewrites 90.0%

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

6. Taylor expanded in x.re around 0

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

8. Applied rewrites 59.0%

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

Developer Target 1: 91.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double x_46_re, double x_46_im) {
	return ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = ((x_46re * x_46im) * (2.0d0 * x_46re)) + ((x_46im * (x_46re - x_46im)) * (x_46re + x_46im))
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024337 
(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)))