math.cube on complex, real part

Percentage Accurate: 82.7% → 99.7%

Time: 8.5s

Alternatives: 10

Speedup: 0.9×

• 44.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

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_re) - (((x_46_re * x_46_im) + (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_46re) - (x_46im * x_46im)) * x_46re) - (((x_46re * x_46im) + (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_re) - (x_46_im * x_46_im)) * x_46_re) - (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im);
}

Python

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

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

Initial Program: 82.7% accurate, 1.0× speedup

Math

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

FPCore

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

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_re) - (((x_46_re * x_46_im) + (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_46re) - (x_46im * x_46im)) * x_46re) - (((x_46re * x_46im) + (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_re) - (x_46_im * x_46_im)) * x_46_re) - (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im);
}

Python

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

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

Alternative 1: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ \begin{array}{l} t_0 := x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) - x.im\_m \cdot \left(x.re\_m \cdot x.im\_m + x.re\_m \cdot x.im\_m\right)\\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+78}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m - x.im\_m, x.re\_m \cdot x.im\_m, x.im\_m \cdot \left(\left(x.re\_m \cdot x.im\_m\right) \cdot -2\right)\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;{x.re\_m}^{3} + x.re\_m \cdot \left(x.im\_m \cdot \left(x.im\_m \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(x.re\_m \cdot \left(x.re\_m + x.im\_m \cdot -2\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (let* ((t_0
         (-
          (* x.re_m (- (* x.re_m x.re_m) (* x.im_m x.im_m)))
          (* x.im_m (+ (* x.re_m x.im_m) (* x.re_m x.im_m))))))
   (*
    x.re_s
    (if (<= t_0 -1e+78)
      (fma
       (- x.re_m x.im_m)
       (* x.re_m x.im_m)
       (* x.im_m (* (* x.re_m x.im_m) -2.0)))
      (if (<= t_0 INFINITY)
        (+ (pow x.re_m 3.0) (* x.re_m (* x.im_m (* x.im_m -3.0))))
        (* x.im_m (* x.re_m (+ x.re_m (* x.im_m -2.0)))))))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double t_0 = (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)));
	double tmp;
	if (t_0 <= -1e+78) {
		tmp = fma((x_46_re_m - x_46_im_m), (x_46_re_m * x_46_im_m), (x_46_im_m * ((x_46_re_m * x_46_im_m) * -2.0)));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = pow(x_46_re_m, 3.0) + (x_46_re_m * (x_46_im_m * (x_46_im_m * -3.0)));
	} else {
		tmp = x_46_im_m * (x_46_re_m * (x_46_re_m + (x_46_im_m * -2.0)));
	}
	return x_46_re_s * tmp;
}

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	t_0 = Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m))) - Float64(x_46_im_m * Float64(Float64(x_46_re_m * x_46_im_m) + Float64(x_46_re_m * x_46_im_m))))
	tmp = 0.0
	if (t_0 <= -1e+78)
		tmp = fma(Float64(x_46_re_m - x_46_im_m), Float64(x_46_re_m * x_46_im_m), Float64(x_46_im_m * Float64(Float64(x_46_re_m * x_46_im_m) * -2.0)));
	elseif (t_0 <= Inf)
		tmp = Float64((x_46_re_m ^ 3.0) + Float64(x_46_re_m * Float64(x_46_im_m * Float64(x_46_im_m * -3.0))));
	else
		tmp = Float64(x_46_im_m * Float64(x_46_re_m * Float64(x_46_re_m + Float64(x_46_im_m * -2.0))));
	end
	return Float64(x_46_re_s * tmp)
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im$95$m * N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$re$95$s * If[LessEqual[t$95$0, -1e+78], N[(N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision] * N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[Power[x$46$re$95$m, 3.0], $MachinePrecision] + N[(x$46$re$95$m * N[(x$46$im$95$m * N[(x$46$im$95$m * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im$95$m * N[(x$46$re$95$m * N[(x$46$re$95$m + N[(x$46$im$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -1.00000000000000001e78

   1. Initial program 89.8%

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

      1. difference-of-squares 89.8%

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

      2. *-commutative 89.8%

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

   4. Applied egg-rr 89.8%

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

      1. *-commutative 89.8%

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

      2. count-2 89.8%

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

      3. *-commutative 89.8%

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

   6. Applied egg-rr 89.8%

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

      1. associate-*l* 99.8%

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

      2. fma-neg 99.9%

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

      3. *-commutative 99.9%

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

      4. *-commutative 99.9%

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

      5. distribute-rgt-neg-in 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. metadata-eval 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in x.re around 0 58.2%

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

   if -1.00000000000000001e78 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < +inf.0

   1. Initial program 95.8%

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

   3. Simplified 93.4%

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

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

   1. Initial program 0.0%

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

      1. difference-of-squares 24.1%

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

      2. *-commutative 24.1%

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

   4. Applied egg-rr 24.1%

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

      1. *-commutative 24.1%

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

      2. count-2 24.1%

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

      3. *-commutative 24.1%

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

   6. Applied egg-rr 24.1%

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

   7. Taylor expanded in x.re around inf 13.8%

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

   8. Taylor expanded in x.re around 0 34.5%

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

      1. cancel-sign-sub-inv 34.5%

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

      2. *-commutative 34.5%

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

      3. metadata-eval 34.5%

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

      4. distribute-rgt-in 13.8%

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

      5. *-commutative 13.8%

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

      6. associate-*r* 13.8%

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

      7. unpow2 13.8%

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

      8. associate-*r* 13.8%

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

      9. *-commutative 13.8%

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

      10. unpow2 13.8%

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

      11. associate-*r* 13.8%

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

      12. associate-*l* 13.8%

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

      13. *-commutative 13.8%

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

      14. *-commutative 13.8%

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

      15. distribute-lft-in 20.7%

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

      16. unpow2 20.7%

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

      17. associate-*r* 20.7%

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

      18. metadata-eval 20.7%

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

      19. distribute-lft-neg-in 20.7%

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

      20. *-commutative 20.7%

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

      21. distribute-rgt-out 55.2%

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

   10. Simplified 55.2%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -1 \cdot 10^{+78}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im, x.re \cdot x.im, x.im \cdot \left(\left(x.re \cdot x.im\right) \cdot -2\right)\right)\\ \mathbf{elif}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq \infty:\\ \;\;\;\;{x.re}^{3} + x.re \cdot \left(x.im \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.re + x.im \cdot -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) - x.im\_m \cdot \left(x.re\_m \cdot x.im\_m + x.re\_m \cdot x.im\_m\right) \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m - x.im\_m, x.re\_m \cdot \left(x.re\_m + x.im\_m\right), x.im\_m \cdot \left(\left(x.re\_m \cdot x.im\_m\right) \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(x.re\_m \cdot \left(x.re\_m + x.im\_m \cdot -2\right)\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<=
       (-
        (* x.re_m (- (* x.re_m x.re_m) (* x.im_m x.im_m)))
        (* x.im_m (+ (* x.re_m x.im_m) (* x.re_m x.im_m))))
       INFINITY)
    (fma
     (- x.re_m x.im_m)
     (* x.re_m (+ x.re_m x.im_m))
     (* x.im_m (* (* x.re_m x.im_m) -2.0)))
    (* x.im_m (* x.re_m (+ x.re_m (* x.im_m -2.0)))))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)))) <= ((double) INFINITY)) {
		tmp = fma((x_46_re_m - x_46_im_m), (x_46_re_m * (x_46_re_m + x_46_im_m)), (x_46_im_m * ((x_46_re_m * x_46_im_m) * -2.0)));
	} else {
		tmp = x_46_im_m * (x_46_re_m * (x_46_re_m + (x_46_im_m * -2.0)));
	}
	return x_46_re_s * tmp;
}

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m))) - Float64(x_46_im_m * Float64(Float64(x_46_re_m * x_46_im_m) + Float64(x_46_re_m * x_46_im_m)))) <= Inf)
		tmp = fma(Float64(x_46_re_m - x_46_im_m), Float64(x_46_re_m * Float64(x_46_re_m + x_46_im_m)), Float64(x_46_im_m * Float64(Float64(x_46_re_m * x_46_im_m) * -2.0)));
	else
		tmp = Float64(x_46_im_m * Float64(x_46_re_m * Float64(x_46_re_m + Float64(x_46_im_m * -2.0))));
	end
	return Float64(x_46_re_s * tmp)
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * If[LessEqual[N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im$95$m * N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision] * N[(x$46$re$95$m * N[(x$46$re$95$m + x$46$im$95$m), $MachinePrecision]), $MachinePrecision] + N[(x$46$im$95$m * N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im$95$m * N[(x$46$re$95$m * N[(x$46$re$95$m + N[(x$46$im$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $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.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < +inf.0

   1. Initial program 94.1%

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

      1. difference-of-squares 94.1%

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

      2. *-commutative 94.1%

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

   4. Applied egg-rr 94.1%

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

      1. *-commutative 94.1%

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

      2. count-2 94.1%

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

      3. *-commutative 94.1%

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

   6. Applied egg-rr 94.1%

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

      1. associate-*l* 99.8%

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

      2. fma-neg 99.8%

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

      3. *-commutative 99.8%

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

      4. *-commutative 99.8%

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

      5. distribute-rgt-neg-in 99.8%

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

      6. distribute-rgt-neg-in 99.8%

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

      7. metadata-eval 99.8%

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

   8. Applied egg-rr 99.8%

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

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

   1. Initial program 0.0%

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

      1. difference-of-squares 24.1%

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

      2. *-commutative 24.1%

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

   4. Applied egg-rr 24.1%

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

      1. *-commutative 24.1%

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

      2. count-2 24.1%

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

      3. *-commutative 24.1%

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

   6. Applied egg-rr 24.1%

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

   7. Taylor expanded in x.re around inf 13.8%

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

   8. Taylor expanded in x.re around 0 34.5%

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

      1. cancel-sign-sub-inv 34.5%

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

      2. *-commutative 34.5%

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

      3. metadata-eval 34.5%

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

      4. distribute-rgt-in 13.8%

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

      5. *-commutative 13.8%

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

      6. associate-*r* 13.8%

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

      7. unpow2 13.8%

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

      8. associate-*r* 13.8%

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

      9. *-commutative 13.8%

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

      10. unpow2 13.8%

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

      11. associate-*r* 13.8%

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

      12. associate-*l* 13.8%

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

      13. *-commutative 13.8%

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

      14. *-commutative 13.8%

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

      15. distribute-lft-in 20.7%

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

      16. unpow2 20.7%

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

      17. associate-*r* 20.7%

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

      18. metadata-eval 20.7%

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

      19. distribute-lft-neg-in 20.7%

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

      20. *-commutative 20.7%

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

      21. distribute-rgt-out 55.2%

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

   10. Simplified 55.2%

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

4. Final simplification 94.8%

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

Alternative 3: 93.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)))) <= ((double) INFINITY)) {
		tmp = (x_46_re_m * ((x_46_re_m - x_46_im_m) * (x_46_re_m + x_46_im_m))) - (x_46_im_m * ((x_46_re_m * x_46_im_m) * 2.0));
	} else {
		tmp = x_46_im_m * (x_46_re_m * (x_46_re_m + (x_46_im_m * -2.0)));
	}
	return x_46_re_s * tmp;
}

Java

x.im_m = Math.abs(x_46_im);
x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)))) <= Double.POSITIVE_INFINITY) {
		tmp = (x_46_re_m * ((x_46_re_m - x_46_im_m) * (x_46_re_m + x_46_im_m))) - (x_46_im_m * ((x_46_re_m * x_46_im_m) * 2.0));
	} else {
		tmp = x_46_im_m * (x_46_re_m * (x_46_re_m + (x_46_im_m * -2.0)));
	}
	return x_46_re_s * tmp;
}

Python

x.im_m = math.fabs(x_46_im)
x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if ((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)))) <= math.inf:
		tmp = (x_46_re_m * ((x_46_re_m - x_46_im_m) * (x_46_re_m + x_46_im_m))) - (x_46_im_m * ((x_46_re_m * x_46_im_m) * 2.0))
	else:
		tmp = x_46_im_m * (x_46_re_m * (x_46_re_m + (x_46_im_m * -2.0)))
	return x_46_re_s * tmp

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m))) - Float64(x_46_im_m * Float64(Float64(x_46_re_m * x_46_im_m) + Float64(x_46_re_m * x_46_im_m)))) <= Inf)
		tmp = Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m - x_46_im_m) * Float64(x_46_re_m + x_46_im_m))) - Float64(x_46_im_m * Float64(Float64(x_46_re_m * x_46_im_m) * 2.0)));
	else
		tmp = Float64(x_46_im_m * Float64(x_46_re_m * Float64(x_46_re_m + Float64(x_46_im_m * -2.0))));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.im_m = abs(x_46_im);
x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)))) <= Inf)
		tmp = (x_46_re_m * ((x_46_re_m - x_46_im_m) * (x_46_re_m + x_46_im_m))) - (x_46_im_m * ((x_46_re_m * x_46_im_m) * 2.0));
	else
		tmp = x_46_im_m * (x_46_re_m * (x_46_re_m + (x_46_im_m * -2.0)));
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * If[LessEqual[N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im$95$m * N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision] * N[(x$46$re$95$m + x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im$95$m * N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im$95$m * N[(x$46$re$95$m * N[(x$46$re$95$m + N[(x$46$im$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $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.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < +inf.0

   1. Initial program 94.1%

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

      1. difference-of-squares 94.1%

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

      2. *-commutative 94.1%

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

   4. Applied egg-rr 94.1%

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

      1. *-commutative 94.1%

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

      2. count-2 94.1%

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

      3. *-commutative 94.1%

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

   6. Applied egg-rr 94.1%

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

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

   1. Initial program 0.0%

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

      1. difference-of-squares 24.1%

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

      2. *-commutative 24.1%

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

   4. Applied egg-rr 24.1%

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

      1. *-commutative 24.1%

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

      2. count-2 24.1%

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

      3. *-commutative 24.1%

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

   6. Applied egg-rr 24.1%

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

   7. Taylor expanded in x.re around inf 13.8%

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

   8. Taylor expanded in x.re around 0 34.5%

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

      1. cancel-sign-sub-inv 34.5%

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

      2. *-commutative 34.5%

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

      3. metadata-eval 34.5%

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

      4. distribute-rgt-in 13.8%

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

      5. *-commutative 13.8%

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

      6. associate-*r* 13.8%

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

      7. unpow2 13.8%

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

      8. associate-*r* 13.8%

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

      9. *-commutative 13.8%

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

      10. unpow2 13.8%

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

      11. associate-*r* 13.8%

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

      12. associate-*l* 13.8%

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

      13. *-commutative 13.8%

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

      14. *-commutative 13.8%

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

      15. distribute-lft-in 20.7%

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

      16. unpow2 20.7%

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

      17. associate-*r* 20.7%

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

      18. metadata-eval 20.7%

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

      19. distribute-lft-neg-in 20.7%

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

      20. *-commutative 20.7%

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

      21. distribute-rgt-out 55.2%

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

   10. Simplified 55.2%

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

4. Final simplification 89.7%

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

Alternative 4: 84.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 42:\\ \;\;\;\;x.re\_m \cdot \left(\left(x.im\_m \cdot x.im\_m\right) \cdot -3\right)\\ \mathbf{elif}\;x.re\_m \leq 4.5 \cdot 10^{+149}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) - 0.5\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(x.re\_m \cdot \left(x.re\_m + x.im\_m \cdot -2\right)\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 42.0)
    (* x.re_m (* (* x.im_m x.im_m) -3.0))
    (if (<= x.re_m 4.5e+149)
      (- (* x.re_m (- (* x.re_m x.re_m) (* x.im_m x.im_m))) 0.5)
      (* x.im_m (* x.re_m (+ x.re_m (* x.im_m -2.0))))))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_re_m <= 42.0) {
		tmp = x_46_re_m * ((x_46_im_m * x_46_im_m) * -3.0);
	} else if (x_46_re_m <= 4.5e+149) {
		tmp = (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) - 0.5;
	} else {
		tmp = x_46_im_m * (x_46_re_m * (x_46_re_m + (x_46_im_m * -2.0)));
	}
	return x_46_re_s * tmp;
}

Fortran

x.im_m = abs(x_46im)
x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46re_m <= 42.0d0) then
        tmp = x_46re_m * ((x_46im_m * x_46im_m) * (-3.0d0))
    else if (x_46re_m <= 4.5d+149) then
        tmp = (x_46re_m * ((x_46re_m * x_46re_m) - (x_46im_m * x_46im_m))) - 0.5d0
    else
        tmp = x_46im_m * (x_46re_m * (x_46re_m + (x_46im_m * (-2.0d0))))
    end if
    code = x_46re_s * tmp
end function

Java

x.im_m = Math.abs(x_46_im);
x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_re_m <= 42.0) {
		tmp = x_46_re_m * ((x_46_im_m * x_46_im_m) * -3.0);
	} else if (x_46_re_m <= 4.5e+149) {
		tmp = (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) - 0.5;
	} else {
		tmp = x_46_im_m * (x_46_re_m * (x_46_re_m + (x_46_im_m * -2.0)));
	}
	return x_46_re_s * tmp;
}

Python

x.im_m = math.fabs(x_46_im)
x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if x_46_re_m <= 42.0:
		tmp = x_46_re_m * ((x_46_im_m * x_46_im_m) * -3.0)
	elif x_46_re_m <= 4.5e+149:
		tmp = (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) - 0.5
	else:
		tmp = x_46_im_m * (x_46_re_m * (x_46_re_m + (x_46_im_m * -2.0)))
	return x_46_re_s * tmp

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (x_46_re_m <= 42.0)
		tmp = Float64(x_46_re_m * Float64(Float64(x_46_im_m * x_46_im_m) * -3.0));
	elseif (x_46_re_m <= 4.5e+149)
		tmp = Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m))) - 0.5);
	else
		tmp = Float64(x_46_im_m * Float64(x_46_re_m * Float64(x_46_re_m + Float64(x_46_im_m * -2.0))));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.im_m = abs(x_46_im);
x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (x_46_re_m <= 42.0)
		tmp = x_46_re_m * ((x_46_im_m * x_46_im_m) * -3.0);
	elseif (x_46_re_m <= 4.5e+149)
		tmp = (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) - 0.5;
	else
		tmp = x_46_im_m * (x_46_re_m * (x_46_re_m + (x_46_im_m * -2.0)));
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 42.0], N[(x$46$re$95$m * N[(N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re$95$m, 4.5e+149], N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision], N[(x$46$im$95$m * N[(x$46$re$95$m * N[(x$46$re$95$m + N[(x$46$im$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x.re < 42

   1. Initial program 85.1%

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

      1. difference-of-squares 87.7%

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

      2. *-commutative 87.7%

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

   4. Applied egg-rr 87.7%

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

      1. *-commutative 87.7%

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

      2. count-2 87.7%

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

      3. *-commutative 87.7%

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

   6. Applied egg-rr 87.7%

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

   7. Taylor expanded in x.re around 0 61.7%

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

      1. distribute-rgt-out-- 61.7%

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

      2. metadata-eval 61.7%

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

   9. Simplified 61.7%

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

       1. unpow2 61.7%

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

   11. Applied egg-rr 61.7%

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

   if 42 < x.re < 4.49999999999999982e149

   1. Initial program 92.8%

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

      1. *-commutative 92.8%

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

      2. *-commutative 92.8%

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

      3. flip-+ 0.0%

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

      4. +-inverses 0.0%

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

      5. +-inverses 0.0%

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

      6. associate-*r/ 0.0%

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

   4. Applied egg-rr 0.0%

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

   6. Applied egg-rr 97.0%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \color{blue}{0.5} \]

   if 4.49999999999999982e149 < x.re

   1. Initial program 63.3%

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

      1. difference-of-squares 70.0%

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

      2. *-commutative 70.0%

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

   4. Applied egg-rr 70.0%

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

      1. *-commutative 70.0%

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

      2. count-2 70.0%

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

      3. *-commutative 70.0%

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

   6. Applied egg-rr 70.0%

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

   7. Taylor expanded in x.re around inf 70.0%

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

   8. Taylor expanded in x.re around 0 31.1%

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

      1. cancel-sign-sub-inv 31.1%

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

      2. *-commutative 31.1%

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

      3. metadata-eval 31.1%

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

      4. distribute-rgt-in 21.1%

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

      5. *-commutative 21.1%

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

      6. associate-*r* 36.7%

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

      7. unpow2 36.7%

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

      8. associate-*r* 36.7%

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

      9. *-commutative 36.7%

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

      10. unpow2 36.7%

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

      11. associate-*r* 36.7%

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

      12. associate-*l* 36.7%

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

      13. *-commutative 36.7%

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

      14. *-commutative 36.7%

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

      15. distribute-lft-in 36.7%

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

      16. unpow2 36.7%

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

      17. associate-*r* 36.7%

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

      18. metadata-eval 36.7%

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

      19. distribute-lft-neg-in 36.7%

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

      20. *-commutative 36.7%

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

      21. distribute-rgt-out 50.0%

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

   10. Simplified 50.0%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 42:\\ \;\;\;\;x.re \cdot \left(\left(x.im \cdot x.im\right) \cdot -3\right)\\ \mathbf{elif}\;x.re \leq 4.5 \cdot 10^{+149}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - 0.5\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.re + x.im \cdot -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 1.8 \cdot 10^{+43}:\\ \;\;\;\;x.re\_m \cdot \left(\left(x.im\_m \cdot x.im\_m\right) \cdot -3\right)\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(x.re\_m \cdot \left(x.re\_m + x.im\_m \cdot -2\right)\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 1.8e+43)
    (* x.re_m (* (* x.im_m x.im_m) -3.0))
    (* x.im_m (* x.re_m (+ x.re_m (* x.im_m -2.0)))))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_re_m <= 1.8e+43) {
		tmp = x_46_re_m * ((x_46_im_m * x_46_im_m) * -3.0);
	} else {
		tmp = x_46_im_m * (x_46_re_m * (x_46_re_m + (x_46_im_m * -2.0)));
	}
	return x_46_re_s * tmp;
}

Fortran

x.im_m = abs(x_46im)
x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46re_m <= 1.8d+43) then
        tmp = x_46re_m * ((x_46im_m * x_46im_m) * (-3.0d0))
    else
        tmp = x_46im_m * (x_46re_m * (x_46re_m + (x_46im_m * (-2.0d0))))
    end if
    code = x_46re_s * tmp
end function

Java

x.im_m = Math.abs(x_46_im);
x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_re_m <= 1.8e+43) {
		tmp = x_46_re_m * ((x_46_im_m * x_46_im_m) * -3.0);
	} else {
		tmp = x_46_im_m * (x_46_re_m * (x_46_re_m + (x_46_im_m * -2.0)));
	}
	return x_46_re_s * tmp;
}

Python

x.im_m = math.fabs(x_46_im)
x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if x_46_re_m <= 1.8e+43:
		tmp = x_46_re_m * ((x_46_im_m * x_46_im_m) * -3.0)
	else:
		tmp = x_46_im_m * (x_46_re_m * (x_46_re_m + (x_46_im_m * -2.0)))
	return x_46_re_s * tmp

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (x_46_re_m <= 1.8e+43)
		tmp = Float64(x_46_re_m * Float64(Float64(x_46_im_m * x_46_im_m) * -3.0));
	else
		tmp = Float64(x_46_im_m * Float64(x_46_re_m * Float64(x_46_re_m + Float64(x_46_im_m * -2.0))));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.im_m = abs(x_46_im);
x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (x_46_re_m <= 1.8e+43)
		tmp = x_46_re_m * ((x_46_im_m * x_46_im_m) * -3.0);
	else
		tmp = x_46_im_m * (x_46_re_m * (x_46_re_m + (x_46_im_m * -2.0)));
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 1.8e+43], N[(x$46$re$95$m * N[(N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision], N[(x$46$im$95$m * N[(x$46$re$95$m * N[(x$46$re$95$m + N[(x$46$im$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.80000000000000005e43

   1. Initial program 85.4%

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

      1. difference-of-squares 87.9%

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

      2. *-commutative 87.9%

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

   4. Applied egg-rr 87.9%

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

      1. *-commutative 87.9%

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

      2. count-2 87.9%

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

      3. *-commutative 87.9%

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

   6. Applied egg-rr 87.9%

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

   7. Taylor expanded in x.re around 0 62.3%

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

      1. distribute-rgt-out-- 62.3%

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

      2. metadata-eval 62.3%

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

   9. Simplified 62.3%

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

       1. unpow2 62.3%

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

   11. Applied egg-rr 62.3%

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

   if 1.80000000000000005e43 < x.re

   1. Initial program 76.3%

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

      1. difference-of-squares 80.0%

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

      2. *-commutative 80.0%

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

   4. Applied egg-rr 80.0%

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

      1. *-commutative 80.0%

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

      2. count-2 80.0%

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

      3. *-commutative 80.0%

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

   6. Applied egg-rr 80.0%

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

   7. Taylor expanded in x.re around inf 72.7%

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

   8. Taylor expanded in x.re around 0 26.8%

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

      1. cancel-sign-sub-inv 26.8%

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

      2. *-commutative 26.8%

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

      3. metadata-eval 26.8%

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

      4. distribute-rgt-in 19.6%

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

      5. *-commutative 19.6%

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

      6. associate-*r* 28.1%

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

      7. unpow2 28.1%

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

      8. associate-*r* 28.1%

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

      9. *-commutative 28.1%

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

      10. unpow2 28.1%

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

      11. associate-*r* 28.1%

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

      12. associate-*l* 28.1%

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

      13. *-commutative 28.1%

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

      14. *-commutative 28.1%

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

      15. distribute-lft-in 35.3%

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

      16. unpow2 35.3%

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

      17. associate-*r* 35.3%

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

      18. metadata-eval 35.3%

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

      19. distribute-lft-neg-in 35.3%

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

      20. *-commutative 35.3%

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

      21. distribute-rgt-out 42.6%

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

   10. Simplified 42.6%

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

Alternative 6: 68.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 1.3 \cdot 10^{+154}:\\ \;\;\;\;x.re\_m \cdot \left(\left(x.im\_m \cdot x.im\_m\right) \cdot -3\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot x.re\_m\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 1.3e+154)
    (* x.re_m (* (* x.im_m x.im_m) -3.0))
    (* x.re_m x.re_m))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_re_m <= 1.3e+154) {
		tmp = x_46_re_m * ((x_46_im_m * x_46_im_m) * -3.0);
	} else {
		tmp = x_46_re_m * x_46_re_m;
	}
	return x_46_re_s * tmp;
}

Fortran

x.im_m = abs(x_46im)
x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46re_m <= 1.3d+154) then
        tmp = x_46re_m * ((x_46im_m * x_46im_m) * (-3.0d0))
    else
        tmp = x_46re_m * x_46re_m
    end if
    code = x_46re_s * tmp
end function

Java

x.im_m = Math.abs(x_46_im);
x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_re_m <= 1.3e+154) {
		tmp = x_46_re_m * ((x_46_im_m * x_46_im_m) * -3.0);
	} else {
		tmp = x_46_re_m * x_46_re_m;
	}
	return x_46_re_s * tmp;
}

Python

x.im_m = math.fabs(x_46_im)
x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if x_46_re_m <= 1.3e+154:
		tmp = x_46_re_m * ((x_46_im_m * x_46_im_m) * -3.0)
	else:
		tmp = x_46_re_m * x_46_re_m
	return x_46_re_s * tmp

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (x_46_re_m <= 1.3e+154)
		tmp = Float64(x_46_re_m * Float64(Float64(x_46_im_m * x_46_im_m) * -3.0));
	else
		tmp = Float64(x_46_re_m * x_46_re_m);
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.im_m = abs(x_46_im);
x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (x_46_re_m <= 1.3e+154)
		tmp = x_46_re_m * ((x_46_im_m * x_46_im_m) * -3.0);
	else
		tmp = x_46_re_m * x_46_re_m;
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 1.3e+154], N[(x$46$re$95$m * N[(N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision], N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.29999999999999994e154

   1. Initial program 86.1%

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

      1. difference-of-squares 88.3%

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

      2. *-commutative 88.3%

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

   4. Applied egg-rr 88.3%

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

      1. *-commutative 88.3%

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

      2. count-2 88.3%

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

      3. *-commutative 88.3%

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

   6. Applied egg-rr 88.3%

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

   7. Taylor expanded in x.re around 0 58.6%

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

      1. distribute-rgt-out-- 58.6%

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

      2. metadata-eval 58.6%

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

   9. Simplified 58.6%

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

       1. unpow2 58.6%

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

   11. Applied egg-rr 58.6%

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

   if 1.29999999999999994e154 < x.re

   1. Initial program 63.3%

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

   3. Simplified 63.3%

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

      1. +-commutative 63.3%

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

      2. associate-*r* 63.3%

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

      3. fma-define 73.3%

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

   6. Applied egg-rr 73.3%

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

   8. Applied egg-rr 93.3%

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

Alternative 7: 34.3% accurate, 6.3× speedup

Math

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

FPCore

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

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	return x_46_re_s * (x_46_re_m * x_46_re_m);
}

Fortran

x.im_m = abs(x_46im)
x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    code = x_46re_s * (x_46re_m * x_46re_m)
end function

Java

x.im_m = Math.abs(x_46_im);
x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	return x_46_re_s * (x_46_re_m * x_46_re_m);
}

Python

x.im_m = math.fabs(x_46_im)
x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im_m):
	return x_46_re_s * (x_46_re_m * x_46_re_m)

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	return Float64(x_46_re_s * Float64(x_46_re_m * x_46_re_m))
end

MATLAB

x.im_m = abs(x_46_im);
x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp = code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = x_46_re_s * (x_46_re_m * x_46_re_m);
end

Wolfram

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

Derivation

1. Initial program 83.4%

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

3. Simplified 80.7%

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

   1. +-commutative 80.7%

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

   2. associate-*r* 85.8%

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

   3. fma-define 88.9%

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

6. Applied egg-rr 88.9%

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

8. Applied egg-rr 31.2%

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

Alternative 8: 4.4% accurate, 19.0× speedup

Math

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

FPCore

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

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	return x_46_re_s * x_46_re_m;
}

Fortran

x.im_m = abs(x_46im)
x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    code = x_46re_s * x_46re_m
end function

Java

x.im_m = Math.abs(x_46_im);
x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	return x_46_re_s * x_46_re_m;
}

Python

x.im_m = math.fabs(x_46_im)
x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im_m):
	return x_46_re_s * x_46_re_m

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	return Float64(x_46_re_s * x_46_re_m)
end

MATLAB

x.im_m = abs(x_46_im);
x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp = code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = x_46_re_s * x_46_re_m;
end

Wolfram

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

Derivation

1. Initial program 83.4%

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

3. Simplified 80.7%

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

   1. +-commutative 80.7%

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

   2. associate-*r* 85.8%

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

   3. fma-define 88.9%

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

6. Applied egg-rr 88.9%

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

8. Applied egg-rr 2.1%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\log x.re\right)} \]
9. Step-by-step derivation

   1. expm1-undefine 2.1%

      \[\leadsto \color{blue}{e^{\log x.re} - 1} \]

   2. rem-exp-log 4.0%

      \[\leadsto \color{blue}{x.re} - 1 \]

10. Simplified 4.0%

    \[\leadsto \color{blue}{x.re - 1} \]

11. Taylor expanded in x.re around inf 4.5%

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

Alternative 9: 2.7% accurate, 19.0× speedup

Math

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

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m) :precision binary64 (* x.re_s 1.0))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	return x_46_re_s * 1.0;
}

Fortran

x.im_m = abs(x_46im)
x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    code = x_46re_s * 1.0d0
end function

Java

x.im_m = Math.abs(x_46_im);
x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	return x_46_re_s * 1.0;
}

Python

x.im_m = math.fabs(x_46_im)
x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im_m):
	return x_46_re_s * 1.0

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	return Float64(x_46_re_s * 1.0)
end

MATLAB

x.im_m = abs(x_46_im);
x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp = code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = x_46_re_s * 1.0;
end

Wolfram

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

Derivation

1. Initial program 83.4%

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

3. Simplified 80.7%

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

   1. +-commutative 80.7%

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

   2. associate-*r* 85.8%

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

   3. fma-define 88.9%

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

6. Applied egg-rr 88.9%

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

8. Applied egg-rr 2.5%

   \[\leadsto \color{blue}{\frac{x.re}{x.re}} \]
9. Step-by-step derivation

   1. *-inverses 2.5%

      \[\leadsto \color{blue}{1} \]

10. Simplified 2.5%

    \[\leadsto \color{blue}{1} \]
11. Add Preprocessing

Alternative 10: 2.8% accurate, 19.0× speedup

Math

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

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m) :precision binary64 (* x.re_s -1.0))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	return x_46_re_s * -1.0;
}

Fortran

x.im_m = abs(x_46im)
x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    code = x_46re_s * (-1.0d0)
end function

Java

x.im_m = Math.abs(x_46_im);
x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	return x_46_re_s * -1.0;
}

Python

x.im_m = math.fabs(x_46_im)
x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im_m):
	return x_46_re_s * -1.0

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	return Float64(x_46_re_s * -1.0)
end

MATLAB

x.im_m = abs(x_46_im);
x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp = code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = x_46_re_s * -1.0;
end

Wolfram

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

Derivation

1. Initial program 83.4%

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

3. Simplified 80.7%

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

   1. +-commutative 80.7%

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

   2. associate-*r* 85.8%

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

   3. fma-define 88.9%

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

6. Applied egg-rr 88.9%

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

8. Applied egg-rr 2.1%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\log x.re\right)} \]
9. Step-by-step derivation

   1. expm1-undefine 2.1%

      \[\leadsto \color{blue}{e^{\log x.re} - 1} \]

   2. rem-exp-log 4.0%

      \[\leadsto \color{blue}{x.re} - 1 \]

10. Simplified 4.0%

    \[\leadsto \color{blue}{x.re - 1} \]

11. Taylor expanded in x.re around 0 2.7%

    \[\leadsto \color{blue}{-1} \]
12. Add Preprocessing

Developer Target 1: 99.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(x.re \cdot x.re\right) \cdot \left(x.re - x.im\right) + \left(x.re \cdot x.im\right) \cdot \left(x.re - 3 \cdot x.im\right) \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (+ (* (* x.re x.re) (- x.re x.im)) (* (* x.re x.im) (- x.re (* 3.0 x.im)))))

C

double code(double x_46_re, double x_46_im) {
	return ((x_46_re * x_46_re) * (x_46_re - x_46_im)) + ((x_46_re * x_46_im) * (x_46_re - (3.0 * 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_46re) * (x_46re - x_46im)) + ((x_46re * x_46im) * (x_46re - (3.0d0 * x_46im)))
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return ((x_46_re * x_46_re) * (x_46_re - x_46_im)) + ((x_46_re * x_46_im) * (x_46_re - (3.0 * x_46_im)));
}

Python

def code(x_46_re, x_46_im):
	return ((x_46_re * x_46_re) * (x_46_re - x_46_im)) + ((x_46_re * x_46_im) * (x_46_re - (3.0 * x_46_im)))

Julia

function code(x_46_re, x_46_im)
	return Float64(Float64(Float64(x_46_re * x_46_re) * Float64(x_46_re - x_46_im)) + Float64(Float64(x_46_re * x_46_im) * Float64(x_46_re - Float64(3.0 * x_46_im))))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = ((x_46_re * x_46_re) * (x_46_re - x_46_im)) + ((x_46_re * x_46_im) * (x_46_re - (3.0 * x_46_im)));
end

Wolfram

code[x$46$re_, x$46$im_] := N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision] + N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(x$46$re - N[(3.0 * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

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

  :alt
  (! :herbie-platform default (+ (* (* x.re x.re) (- x.re x.im)) (* (* x.re x.im) (- x.re (* 3 x.im)))))

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