math.cube on complex, imaginary part

Percentage Accurate: 83.4% → 99.7%

Time: 9.3s

Alternatives: 7

Speedup: 1.3×

• 45.1% 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: 83.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.4× speedup

Math

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

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 x.re) (* x.im_m x.re)) x.re)
        (* (- (* x.re x.re) (* x.im_m x.im_m)) x.im_m))
       INFINITY)
    (+
     (* (* (+ x.im_m x.im_m) x.re) x.re)
     (* (* (- x.re x.im_m) x.im_m) (+ x.im_m x.re)))
    (+
     (* (* (fma (/ (- x.im_m) x.re) (/ x.im_m x.re) 1.0) (* 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 * x_46_re) + (x_46_im_m * x_46_re)) * x_46_re) + (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m)) <= ((double) INFINITY)) {
		tmp = (((x_46_im_m + x_46_im_m) * x_46_re) * x_46_re) + (((x_46_re - x_46_im_m) * x_46_im_m) * (x_46_im_m + x_46_re));
	} else {
		tmp = ((fma((-x_46_im_m / x_46_re), (x_46_im_m / x_46_re), 1.0) * (x_46_re * x_46_re)) * 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 (Float64(Float64(Float64(Float64(x_46_im_m * x_46_re) + Float64(x_46_im_m * x_46_re)) * x_46_re) + Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m)) <= Inf)
		tmp = Float64(Float64(Float64(Float64(x_46_im_m + x_46_im_m) * x_46_re) * x_46_re) + Float64(Float64(Float64(x_46_re - x_46_im_m) * x_46_im_m) * Float64(x_46_im_m + x_46_re)));
	else
		tmp = Float64(Float64(Float64(fma(Float64(Float64(-x_46_im_m) / x_46_re), Float64(x_46_im_m / x_46_re), 1.0) * Float64(x_46_re * x_46_re)) * 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[N[(N[(N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision] + 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]), $MachinePrecision], Infinity], N[(N[(N[(N[(x$46$im$95$m + x$46$im$95$m), $MachinePrecision] * x$46$re), $MachinePrecision] * x$46$re), $MachinePrecision] + N[(N[(N[(x$46$re - x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m + x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[((-x$46$im$95$m) / x$46$re), $MachinePrecision] * N[(x$46$im$95$m / x$46$re), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m + x$46$im$95$m), $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)) < +inf.0

   1. Initial program 93.8%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. difference-of-squares N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 99.7

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

   6. Applied rewrites 99.7%

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

   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.im around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 35.9

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

   5. Applied rewrites 35.9%

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

   6. Taylor expanded in x.re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 35.9

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

   8. Applied rewrites 35.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. flip-+ N/A

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

      8. +-inverses N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-x.im}{x.re}, \frac{x.im}{x.re}, 1\right) \cdot \left(x.re \cdot x.re\right)\right) \cdot x.im + x.re \cdot \frac{\color{blue}{0}}{x.re \cdot x.im - x.re \cdot x.im} \]

      9. +-inverses N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-x.im}{x.re}, \frac{x.im}{x.re}, 1\right) \cdot \left(x.re \cdot x.re\right)\right) \cdot x.im + x.re \cdot \frac{0}{\color{blue}{0}} \]

      10. associate-*r/ N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-x.im}{x.re}, \frac{x.im}{x.re}, 1\right) \cdot \left(x.re \cdot x.re\right)\right) \cdot x.im + \color{blue}{\frac{x.re \cdot 0}{0}} \]

      11. +-inverses N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-x.im}{x.re}, \frac{x.im}{x.re}, 1\right) \cdot \left(x.re \cdot x.re\right)\right) \cdot x.im + \frac{x.re \cdot \color{blue}{\left(x.im - x.im\right)}}{0} \]

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

          \[\leadsto \left(\mathsf{fma}\left(\frac{-x.im}{x.re}, \frac{x.im}{x.re}, 1\right) \cdot \left(x.re \cdot x.re\right)\right) \cdot x.im + \frac{\color{blue}{x.re \cdot x.im - x.re \cdot x.im}}{0} \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-x.im}{x.re}, \frac{x.im}{x.re}, 1\right) \cdot \left(x.re \cdot x.re\right)\right) \cdot x.im + \frac{\color{blue}{x.re \cdot x.im} - x.re \cdot x.im}{0} \]

      14. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-x.im}{x.re}, \frac{x.im}{x.re}, 1\right) \cdot \left(x.re \cdot x.re\right)\right) \cdot x.im + \frac{x.re \cdot x.im - \color{blue}{x.re \cdot x.im}}{0} \]

      15. +-inverses N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-x.im}{x.re}, \frac{x.im}{x.re}, 1\right) \cdot \left(x.re \cdot x.re\right)\right) \cdot x.im + \frac{\color{blue}{0}}{0} \]

      16. +-inverses N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-x.im}{x.re}, \frac{x.im}{x.re}, 1\right) \cdot \left(x.re \cdot x.re\right)\right) \cdot x.im + \frac{\color{blue}{x.im \cdot x.im - x.im \cdot x.im}}{0} \]

      17. +-inverses N/A

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

      18. flip-+ N/A

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

      19. lift-+.f64 100.0

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

   10. Applied rewrites 100.0%

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

4. Final simplification 99.8%

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

Alternative 2: 95.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(\left(-x.im\_m\right) \cdot x.im\_m\right) \cdot x.im\_m\\ t_1 := \left(x.im\_m \cdot x.re + x.im\_m \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-286}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left(3 \cdot \left(x.im\_m \cdot x.re\right)\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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.im_m) x.im_m))
        (t_1
         (+
          (* (+ (* x.im_m x.re) (* x.im_m x.re)) x.re)
          (* (- (* x.re x.re) (* x.im_m x.im_m)) x.im_m))))
   (*
    x.im_s
    (if (<= t_1 -2e-286)
      t_0
      (if (<= t_1 INFINITY) (* (* 3.0 (* x.im_m x.re)) x.re) t_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_im_m) * x_46_im_m;
	double t_1 = (((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)) * x_46_re) + (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m);
	double tmp;
	if (t_1 <= -2e-286) {
		tmp = t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (3.0 * (x_46_im_m * x_46_re)) * x_46_re;
	} else {
		tmp = t_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_im_m) * x_46_im_m;
	double t_1 = (((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)) * x_46_re) + (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m);
	double tmp;
	if (t_1 <= -2e-286) {
		tmp = t_0;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (3.0 * (x_46_im_m * x_46_re)) * x_46_re;
	} else {
		tmp = t_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_im_m) * x_46_im_m
	t_1 = (((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)) * x_46_re) + (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m)
	tmp = 0
	if t_1 <= -2e-286:
		tmp = t_0
	elif t_1 <= math.inf:
		tmp = (3.0 * (x_46_im_m * x_46_re)) * x_46_re
	else:
		tmp = t_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(Float64(-x_46_im_m) * x_46_im_m) * x_46_im_m)
	t_1 = Float64(Float64(Float64(Float64(x_46_im_m * x_46_re) + Float64(x_46_im_m * x_46_re)) * x_46_re) + Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m))
	tmp = 0.0
	if (t_1 <= -2e-286)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(3.0 * Float64(x_46_im_m * x_46_re)) * x_46_re);
	else
		tmp = t_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_im_m) * x_46_im_m;
	t_1 = (((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)) * x_46_re) + (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m);
	tmp = 0.0;
	if (t_1 <= -2e-286)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = (3.0 * (x_46_im_m * x_46_re)) * x_46_re;
	else
		tmp = t_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) * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision] + 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]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$1, -2e-286], t$95$0, If[LessEqual[t$95$1, Infinity], N[(N[(3.0 * N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision], t$95$0]]), $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)) < -2.0000000000000001e-286 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 63.6%

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

   3. Taylor expanded in x.im around inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left({x.im}^{3}\right)} \]

      2. lower-neg.f64 N/A

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

      3. lower-pow.f64 49.6

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

   5. Applied rewrites 49.6%

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

   7. Applied rewrites 49.5%

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

   if -2.0000000000000001e-286 < (+.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 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. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. *-inverses N/A

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

      4. associate-/l* N/A

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

      5. unpow2 N/A

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

      6. cube-mult N/A

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

      7. associate-/l* N/A

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

      8. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {x.re}^{2} + {x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3}} \]

      9. distribute-lft1-in N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{3} \cdot {x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3} \]

      11. associate-*r/ N/A

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

      12. associate-*l* N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

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

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

      16. *-commutative N/A

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

      17. distribute-rgt-neg-in N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

   5. Applied rewrites 63.6%

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

   7. Applied rewrites 69.4%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im \leq -2 \cdot 10^{-286}:\\ \;\;\;\;\left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \mathbf{elif}\;\left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im \leq \infty:\\ \;\;\;\;\left(3 \cdot \left(x.im \cdot x.re\right)\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(\left(-x.im\_m\right) \cdot x.im\_m\right) \cdot x.im\_m\\ t_1 := \left(x.im\_m \cdot x.re + x.im\_m \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-286}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;x.im\_m \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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.im_m) x.im_m))
        (t_1
         (+
          (* (+ (* x.im_m x.re) (* x.im_m x.re)) x.re)
          (* (- (* x.re x.re) (* x.im_m x.im_m)) x.im_m))))
   (*
    x.im_s
    (if (<= t_1 -2e-286)
      t_0
      (if (<= t_1 INFINITY) (* x.im_m (* x.re x.re)) t_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_im_m) * x_46_im_m;
	double t_1 = (((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)) * x_46_re) + (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m);
	double tmp;
	if (t_1 <= -2e-286) {
		tmp = t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = x_46_im_m * (x_46_re * x_46_re);
	} else {
		tmp = t_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_im_m) * x_46_im_m;
	double t_1 = (((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)) * x_46_re) + (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m);
	double tmp;
	if (t_1 <= -2e-286) {
		tmp = t_0;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = x_46_im_m * (x_46_re * x_46_re);
	} else {
		tmp = t_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_im_m) * x_46_im_m
	t_1 = (((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)) * x_46_re) + (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m)
	tmp = 0
	if t_1 <= -2e-286:
		tmp = t_0
	elif t_1 <= math.inf:
		tmp = x_46_im_m * (x_46_re * x_46_re)
	else:
		tmp = t_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(Float64(-x_46_im_m) * x_46_im_m) * x_46_im_m)
	t_1 = Float64(Float64(Float64(Float64(x_46_im_m * x_46_re) + Float64(x_46_im_m * x_46_re)) * x_46_re) + Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m))
	tmp = 0.0
	if (t_1 <= -2e-286)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = Float64(x_46_im_m * Float64(x_46_re * x_46_re));
	else
		tmp = t_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_im_m) * x_46_im_m;
	t_1 = (((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)) * x_46_re) + (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_im_m);
	tmp = 0.0;
	if (t_1 <= -2e-286)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = x_46_im_m * (x_46_re * x_46_re);
	else
		tmp = t_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) * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision] + 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]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$1, -2e-286], t$95$0, If[LessEqual[t$95$1, Infinity], N[(x$46$im$95$m * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision], t$95$0]]), $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)) < -2.0000000000000001e-286 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 63.6%

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

   3. Taylor expanded in x.im around inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left({x.im}^{3}\right)} \]

      2. lower-neg.f64 N/A

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

      3. lower-pow.f64 49.6

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

   5. Applied rewrites 49.6%

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

   7. Applied rewrites 49.5%

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

   if -2.0000000000000001e-286 < (+.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 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. Taylor expanded in x.im around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 63.7

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

   5. Applied rewrites 63.7%

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

   6. Taylor expanded in x.re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 79.9

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

   8. Applied rewrites 79.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. flip-+ N/A

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

      8. +-inverses N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-x.im}{x.re}, \frac{x.im}{x.re}, 1\right) \cdot \left(x.re \cdot x.re\right)\right) \cdot x.im + x.re \cdot \frac{\color{blue}{0}}{x.re \cdot x.im - x.re \cdot x.im} \]

      9. +-inverses N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-x.im}{x.re}, \frac{x.im}{x.re}, 1\right) \cdot \left(x.re \cdot x.re\right)\right) \cdot x.im + x.re \cdot \frac{0}{\color{blue}{0}} \]

      10. associate-*r/ N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-x.im}{x.re}, \frac{x.im}{x.re}, 1\right) \cdot \left(x.re \cdot x.re\right)\right) \cdot x.im + \color{blue}{\frac{x.re \cdot 0}{0}} \]

      11. +-inverses N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-x.im}{x.re}, \frac{x.im}{x.re}, 1\right) \cdot \left(x.re \cdot x.re\right)\right) \cdot x.im + \frac{x.re \cdot \color{blue}{\left(x.im - x.im\right)}}{0} \]

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

          \[\leadsto \left(\mathsf{fma}\left(\frac{-x.im}{x.re}, \frac{x.im}{x.re}, 1\right) \cdot \left(x.re \cdot x.re\right)\right) \cdot x.im + \frac{\color{blue}{x.re \cdot x.im - x.re \cdot x.im}}{0} \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-x.im}{x.re}, \frac{x.im}{x.re}, 1\right) \cdot \left(x.re \cdot x.re\right)\right) \cdot x.im + \frac{\color{blue}{x.re \cdot x.im} - x.re \cdot x.im}{0} \]

      14. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-x.im}{x.re}, \frac{x.im}{x.re}, 1\right) \cdot \left(x.re \cdot x.re\right)\right) \cdot x.im + \frac{x.re \cdot x.im - \color{blue}{x.re \cdot x.im}}{0} \]

      15. +-inverses N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-x.im}{x.re}, \frac{x.im}{x.re}, 1\right) \cdot \left(x.re \cdot x.re\right)\right) \cdot x.im + \frac{\color{blue}{0}}{0} \]

      16. +-inverses N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-x.im}{x.re}, \frac{x.im}{x.re}, 1\right) \cdot \left(x.re \cdot x.re\right)\right) \cdot x.im + \frac{\color{blue}{x.im \cdot x.im - x.im \cdot x.im}}{0} \]

      17. +-inverses N/A

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

      18. flip-+ N/A

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

      19. lift-+.f64 30.0

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

   10. Applied rewrites 30.0%

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

   11. Taylor expanded in x.re around inf

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

       1. *-commutative N/A

          \[\leadsto \color{blue}{{x.re}^{2} \cdot x.im} \]

       2. lower-*.f64 N/A

          \[\leadsto \color{blue}{{x.re}^{2} \cdot x.im} \]

       3. unpow2 N/A

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

       4. lower-*.f64 48.3

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

   13. Applied rewrites 48.3%

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

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im \leq -2 \cdot 10^{-286}:\\ \;\;\;\;\left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \mathbf{elif}\;\left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im \leq \infty:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 1.85 \cdot 10^{+195}:\\ \;\;\;\;\left(\left(x.im\_m + x.im\_m\right) \cdot x.re\right) \cdot x.re + \left(\left(x.re - x.im\_m\right) \cdot x.im\_m\right) \cdot \left(x.im\_m + x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-x.im\_m\right) \cdot x.im\_m\right) \cdot x.im\_m\\ \end{array} \end{array} \]

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 1.85e+195)
    (+
     (* (* (+ x.im_m x.im_m) x.re) x.re)
     (* (* (- x.re x.im_m) x.im_m) (+ x.im_m 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 <= 1.85e+195) {
		tmp = (((x_46_im_m + x_46_im_m) * x_46_re) * x_46_re) + (((x_46_re - x_46_im_m) * x_46_im_m) * (x_46_im_m + x_46_re));
	} else {
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	}
	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) :: tmp
    if (x_46im_m <= 1.85d+195) then
        tmp = (((x_46im_m + x_46im_m) * x_46re) * x_46re) + (((x_46re - x_46im_m) * x_46im_m) * (x_46im_m + x_46re))
    else
        tmp = (-x_46im_m * x_46im_m) * x_46im_m
    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 tmp;
	if (x_46_im_m <= 1.85e+195) {
		tmp = (((x_46_im_m + x_46_im_m) * x_46_re) * x_46_re) + (((x_46_re - x_46_im_m) * x_46_im_m) * (x_46_im_m + x_46_re));
	} else {
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	}
	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):
	tmp = 0
	if x_46_im_m <= 1.85e+195:
		tmp = (((x_46_im_m + x_46_im_m) * x_46_re) * x_46_re) + (((x_46_re - x_46_im_m) * x_46_im_m) * (x_46_im_m + 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 <= 1.85e+195)
		tmp = Float64(Float64(Float64(Float64(x_46_im_m + x_46_im_m) * x_46_re) * x_46_re) + Float64(Float64(Float64(x_46_re - x_46_im_m) * x_46_im_m) * Float64(x_46_im_m + x_46_re)));
	else
		tmp = Float64(Float64(Float64(-x_46_im_m) * x_46_im_m) * x_46_im_m);
	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)
	tmp = 0.0;
	if (x_46_im_m <= 1.85e+195)
		tmp = (((x_46_im_m + x_46_im_m) * x_46_re) * x_46_re) + (((x_46_re - x_46_im_m) * x_46_im_m) * (x_46_im_m + x_46_re));
	else
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	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_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 1.85e+195], N[(N[(N[(N[(x$46$im$95$m + x$46$im$95$m), $MachinePrecision] * x$46$re), $MachinePrecision] * x$46$re), $MachinePrecision] + N[(N[(N[(x$46$re - x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m + x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-x$46$im$95$m) * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.85e195

   1. Initial program 83.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. difference-of-squares N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 93.3

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

   4. Applied rewrites 93.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 93.3

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

   6. Applied rewrites 93.3%

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

   if 1.85e195 < x.im

   1. Initial program 43.5%

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

   3. Taylor expanded in x.im around inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left({x.im}^{3}\right)} \]

      2. lower-neg.f64 N/A

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

      3. lower-pow.f64 87.0

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

   5. Applied rewrites 87.0%

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

   7. Applied rewrites 87.0%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 1.85 \cdot 10^{+195}:\\ \;\;\;\;\left(\left(x.im + x.im\right) \cdot x.re\right) \cdot x.re + \left(\left(x.re - x.im\right) \cdot x.im\right) \cdot \left(x.im + x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.re \leq 1.22 \cdot 10^{+160}:\\ \;\;\;\;\mathsf{fma}\left(3 \cdot x.re, x.re, \left(-x.im\_m\right) \cdot x.im\_m\right) \cdot x.im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot \left(x.im\_m \cdot x.re\right)\right) \cdot x.re\\ \end{array} \end{array} \]

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.re 1.22e+160)
    (* (fma (* 3.0 x.re) x.re (* (- 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 tmp;
	if (x_46_re <= 1.22e+160) {
		tmp = fma((3.0 * x_46_re), x_46_re, (-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)
	tmp = 0.0
	if (x_46_re <= 1.22e+160)
		tmp = Float64(fma(Float64(3.0 * x_46_re), x_46_re, Float64(Float64(-x_46_im_m) * x_46_im_m)) * x_46_im_m);
	else
		tmp = Float64(Float64(3.0 * Float64(x_46_im_m * 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_] := N[(x$46$im$95$s * If[LessEqual[x$46$re, 1.22e+160], N[(N[(N[(3.0 * x$46$re), $MachinePrecision] * x$46$re + N[((-x$46$im$95$m) * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision], N[(N[(3.0 * N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.22e160

   1. Initial program 83.6%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip3-- N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. pow-pow N/A

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

      11. lower-pow.f64 N/A

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

      12. metadata-eval N/A

          \[\leadsto \frac{\left({x.re}^{\color{blue}{6}} - {\left(x.im \cdot x.im\right)}^{3}\right) \cdot x.im}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\left({x.re}^{6} - {\color{blue}{\left(x.im \cdot x.im\right)}}^{3}\right) \cdot x.im}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

      14. pow2 N/A

          \[\leadsto \frac{\left({x.re}^{6} - {\color{blue}{\left({x.im}^{2}\right)}}^{3}\right) \cdot x.im}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

      15. pow-pow N/A

          \[\leadsto \frac{\left({x.re}^{6} - \color{blue}{{x.im}^{\left(2 \cdot 3\right)}}\right) \cdot x.im}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

      16. lower-pow.f64 N/A

          \[\leadsto \frac{\left({x.re}^{6} - \color{blue}{{x.im}^{\left(2 \cdot 3\right)}}\right) \cdot x.im}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

      17. metadata-eval N/A

          \[\leadsto \frac{\left({x.re}^{6} - {x.im}^{\color{blue}{6}}\right) \cdot x.im}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

   4. Applied rewrites 14.6%

      \[\leadsto \color{blue}{\frac{\left({x.re}^{6} - {x.im}^{6}\right) \cdot x.im}{\mathsf{fma}\left(x.im \cdot x.im, \mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right), {x.re}^{4}\right)}} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

   5. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 89.4

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

   7. Applied rewrites 89.4%

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

   9. Applied rewrites 92.5%

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

   if 1.22e160 < x.re

   1. Initial program 48.3%

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

   3. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. *-inverses N/A

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

      4. associate-/l* N/A

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

      5. unpow2 N/A

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

      6. cube-mult N/A

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

      7. associate-/l* N/A

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

      8. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {x.re}^{2} + {x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3}} \]

      9. distribute-lft1-in N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{3} \cdot {x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3} \]

      11. associate-*r/ N/A

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

      12. associate-*l* N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

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

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

      16. *-commutative N/A

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

      17. distribute-rgt-neg-in N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

   5. Applied rewrites 61.7%

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

   7. Applied rewrites 83.3%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 1.22 \cdot 10^{+160}:\\ \;\;\;\;\mathsf{fma}\left(3 \cdot x.re, x.re, \left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot \left(x.im \cdot x.re\right)\right) \cdot x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.re \leq 9 \cdot 10^{+136}:\\ \;\;\;\;\mathsf{fma}\left(-3, x.re \cdot x.re, x.im\_m \cdot x.im\_m\right) \cdot \left(-x.im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 \cdot x.im\_m\right) \cdot x.re\right) \cdot x.re\\ \end{array} \end{array} \]

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.re 9e+136)
    (* (fma -3.0 (* x.re x.re) (* 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 tmp;
	if (x_46_re <= 9e+136) {
		tmp = fma(-3.0, (x_46_re * x_46_re), (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)
	tmp = 0.0
	if (x_46_re <= 9e+136)
		tmp = Float64(fma(-3.0, Float64(x_46_re * x_46_re), Float64(x_46_im_m * x_46_im_m)) * Float64(-x_46_im_m));
	else
		tmp = Float64(Float64(Float64(3.0 * x_46_im_m) * 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_] := N[(x$46$im$95$s * If[LessEqual[x$46$re, 9e+136], N[(N[(-3.0 * 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[(3.0 * x$46$im$95$m), $MachinePrecision] * x$46$re), $MachinePrecision] * x$46$re), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 8.9999999999999999e136

   1. Initial program 85.0%

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

   3. Taylor expanded in x.im around 0

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

   5. Applied rewrites 90.9%

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

   if 8.9999999999999999e136 < x.re

   1. Initial program 44.5%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip3-- N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. pow-pow N/A

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

      11. lower-pow.f64 N/A

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

      12. metadata-eval N/A

          \[\leadsto \frac{\left({x.re}^{\color{blue}{6}} - {\left(x.im \cdot x.im\right)}^{3}\right) \cdot x.im}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\left({x.re}^{6} - {\color{blue}{\left(x.im \cdot x.im\right)}}^{3}\right) \cdot x.im}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

      14. pow2 N/A

          \[\leadsto \frac{\left({x.re}^{6} - {\color{blue}{\left({x.im}^{2}\right)}}^{3}\right) \cdot x.im}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

      15. pow-pow N/A

          \[\leadsto \frac{\left({x.re}^{6} - \color{blue}{{x.im}^{\left(2 \cdot 3\right)}}\right) \cdot x.im}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

      16. lower-pow.f64 N/A

          \[\leadsto \frac{\left({x.re}^{6} - \color{blue}{{x.im}^{\left(2 \cdot 3\right)}}\right) \cdot x.im}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

      17. metadata-eval N/A

          \[\leadsto \frac{\left({x.re}^{6} - {x.im}^{\color{blue}{6}}\right) \cdot x.im}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

   4. Applied rewrites 0.0%

      \[\leadsto \color{blue}{\frac{\left({x.re}^{6} - {x.im}^{6}\right) \cdot x.im}{\mathsf{fma}\left(x.im \cdot x.im, \mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right), {x.re}^{4}\right)}} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

   5. Taylor expanded in x.im around 0

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

      1. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. unpow2 N/A

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

      6. +-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. distribute-rgt1-in N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 77.1

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

   7. Applied rewrites 77.1%

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

Alternative 7: 34.6% accurate, 3.6× speedup

Math

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

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.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) {
	return x_46_im_s * (x_46_im_m * (x_46_re * x_46_re));
}

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_46re * x_46re))
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_re * x_46_re));
}

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_re * x_46_re))

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_im_m * Float64(x_46_re * x_46_re)))
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_re * x_46_re));
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$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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.im around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 55.7

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

5. Applied rewrites 55.7%

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

6. Taylor expanded in x.re around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. mul-1-neg N/A

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

   5. unpow2 N/A

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

   6. unpow2 N/A

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

   7. times-frac N/A

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

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

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

   9. mul-1-neg N/A

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

   10. lower-fma.f64 N/A

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

   11. associate-*r/ N/A

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

   12. lower-/.f64 N/A

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

   13. mul-1-neg N/A

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

   14. lower-neg.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 71.4

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

8. Applied rewrites 71.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. flip-+ N/A

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

   8. +-inverses N/A

      \[\leadsto \left(\mathsf{fma}\left(\frac{-x.im}{x.re}, \frac{x.im}{x.re}, 1\right) \cdot \left(x.re \cdot x.re\right)\right) \cdot x.im + x.re \cdot \frac{\color{blue}{0}}{x.re \cdot x.im - x.re \cdot x.im} \]

   9. +-inverses N/A

      \[\leadsto \left(\mathsf{fma}\left(\frac{-x.im}{x.re}, \frac{x.im}{x.re}, 1\right) \cdot \left(x.re \cdot x.re\right)\right) \cdot x.im + x.re \cdot \frac{0}{\color{blue}{0}} \]

   10. associate-*r/ N/A

       \[\leadsto \left(\mathsf{fma}\left(\frac{-x.im}{x.re}, \frac{x.im}{x.re}, 1\right) \cdot \left(x.re \cdot x.re\right)\right) \cdot x.im + \color{blue}{\frac{x.re \cdot 0}{0}} \]

   11. +-inverses N/A

       \[\leadsto \left(\mathsf{fma}\left(\frac{-x.im}{x.re}, \frac{x.im}{x.re}, 1\right) \cdot \left(x.re \cdot x.re\right)\right) \cdot x.im + \frac{x.re \cdot \color{blue}{\left(x.im - x.im\right)}}{0} \]

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

       \[\leadsto \left(\mathsf{fma}\left(\frac{-x.im}{x.re}, \frac{x.im}{x.re}, 1\right) \cdot \left(x.re \cdot x.re\right)\right) \cdot x.im + \frac{\color{blue}{x.re \cdot x.im - x.re \cdot x.im}}{0} \]

   13. lift-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\frac{-x.im}{x.re}, \frac{x.im}{x.re}, 1\right) \cdot \left(x.re \cdot x.re\right)\right) \cdot x.im + \frac{\color{blue}{x.re \cdot x.im} - x.re \cdot x.im}{0} \]

   14. lift-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\frac{-x.im}{x.re}, \frac{x.im}{x.re}, 1\right) \cdot \left(x.re \cdot x.re\right)\right) \cdot x.im + \frac{x.re \cdot x.im - \color{blue}{x.re \cdot x.im}}{0} \]

   15. +-inverses N/A

       \[\leadsto \left(\mathsf{fma}\left(\frac{-x.im}{x.re}, \frac{x.im}{x.re}, 1\right) \cdot \left(x.re \cdot x.re\right)\right) \cdot x.im + \frac{\color{blue}{0}}{0} \]

   16. +-inverses N/A

       \[\leadsto \left(\mathsf{fma}\left(\frac{-x.im}{x.re}, \frac{x.im}{x.re}, 1\right) \cdot \left(x.re \cdot x.re\right)\right) \cdot x.im + \frac{\color{blue}{x.im \cdot x.im - x.im \cdot x.im}}{0} \]

   17. +-inverses N/A

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

   18. flip-+ N/A

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

   19. lift-+.f64 48.4

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

10. Applied rewrites 48.4%

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

11. Taylor expanded in x.re around inf

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

    1. *-commutative N/A

       \[\leadsto \color{blue}{{x.re}^{2} \cdot x.im} \]

    2. lower-*.f64 N/A

       \[\leadsto \color{blue}{{x.re}^{2} \cdot x.im} \]

    3. unpow2 N/A

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

    4. lower-*.f64 41.0

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

13. Applied rewrites 41.0%

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

14. Final simplification 41.0%

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

Developer Target 1: 92.1% 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 2024270 
(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)))