math.cube on complex, real part

Percentage Accurate: 82.6% → 99.8%

Time: 12.6s

Alternatives: 11

Speedup: 1.2×

• 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: 82.6% 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: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ \begin{array}{l} \mathbf{if}\;x.im\_m \leq 10^{+127}:\\ \;\;\;\;x.re \cdot \mathsf{fma}\left(x.im\_m \cdot x.im\_m, -3, x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(x.re \cdot \mathsf{fma}\left(x.re, \frac{x.re}{x.im\_m}, x.im\_m \cdot -3\right)\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
(FPCore (x.re x.im_m)
 :precision binary64
 (if (<= x.im_m 1e+127)
   (* x.re (fma (* x.im_m x.im_m) -3.0 (* x.re x.re)))
   (* x.im_m (* x.re (fma x.re (/ x.re x.im_m) (* x.im_m -3.0))))))

C

x.im_m = fabs(x_46_im);
double code(double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 1e+127) {
		tmp = x_46_re * fma((x_46_im_m * x_46_im_m), -3.0, (x_46_re * x_46_re));
	} else {
		tmp = x_46_im_m * (x_46_re * fma(x_46_re, (x_46_re / x_46_im_m), (x_46_im_m * -3.0)));
	}
	return tmp;
}

Julia

x.im_m = abs(x_46_im)
function code(x_46_re, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 1e+127)
		tmp = Float64(x_46_re * fma(Float64(x_46_im_m * x_46_im_m), -3.0, Float64(x_46_re * x_46_re)));
	else
		tmp = Float64(x_46_im_m * Float64(x_46_re * fma(x_46_re, Float64(x_46_re / x_46_im_m), Float64(x_46_im_m * -3.0))));
	end
	return tmp
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
code[x$46$re_, x$46$im$95$m_] := If[LessEqual[x$46$im$95$m, 1e+127], N[(x$46$re * N[(N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision] * -3.0 + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im$95$m * N[(x$46$re * N[(x$46$re * N[(x$46$re / x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 9.99999999999999955e126

   1. Initial program 88.6%

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

   3. Taylor expanded in x.re around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate--l+ N/A

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

      4. unpow2 N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. distribute-rgt-out-- N/A

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

      7. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. metadata-eval 96.3

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

   5. Simplified 96.3%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 95.4

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

   7. Applied egg-rr 95.4%

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

   if 9.99999999999999955e126 < x.im

   1. Initial program 50.7%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-out N/A

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

      7. *-lowering-*.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. neg-lowering-neg.f64 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. *-lowering-*.f64 N/A

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

      13. +-lowering-+.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. --lowering--.f64 85.6

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

   4. Applied egg-rr 85.6%

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

   5. Taylor expanded in x.im around inf

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

   7. Simplified 91.3%

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

   8. Taylor expanded in x.im around inf

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

   10. Simplified 99.9%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 10^{+127}:\\ \;\;\;\;x.re \cdot \mathsf{fma}\left(x.im \cdot x.im, -3, x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \mathsf{fma}\left(x.re, \frac{x.re}{x.im}, x.im \cdot -3\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ \begin{array}{l} t_0 := x.re \cdot \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) - x.im\_m \cdot \left(x.im\_m \cdot x.re + x.im\_m \cdot x.re\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;x.im\_m \cdot \left(x.re \cdot \left(x.im\_m \cdot -3\right)\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+100}:\\ \;\;\;\;x.re \cdot \mathsf{fma}\left(x.im\_m \cdot x.im\_m, -3, x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.im\_m + x.re, x.re \cdot \left(x.re - x.im\_m\right), x.im\_m + x.im\_m\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
(FPCore (x.re x.im_m)
 :precision binary64
 (let* ((t_0
         (-
          (* x.re (- (* x.re x.re) (* x.im_m x.im_m)))
          (* x.im_m (+ (* x.im_m x.re) (* x.im_m x.re))))))
   (if (<= t_0 (- INFINITY))
     (* x.im_m (* x.re (* x.im_m -3.0)))
     (if (<= t_0 5e+100)
       (* x.re (fma (* x.im_m x.im_m) -3.0 (* x.re x.re)))
       (fma (+ x.im_m x.re) (* x.re (- x.re x.im_m)) (+ x.im_m x.im_m))))))

C

x.im_m = fabs(x_46_im);
double code(double x_46_re, double x_46_im_m) {
	double t_0 = (x_46_re * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = x_46_im_m * (x_46_re * (x_46_im_m * -3.0));
	} else if (t_0 <= 5e+100) {
		tmp = x_46_re * fma((x_46_im_m * x_46_im_m), -3.0, (x_46_re * x_46_re));
	} else {
		tmp = fma((x_46_im_m + x_46_re), (x_46_re * (x_46_re - x_46_im_m)), (x_46_im_m + x_46_im_m));
	}
	return tmp;
}

Julia

x.im_m = abs(x_46_im)
function code(x_46_re, x_46_im_m)
	t_0 = Float64(Float64(x_46_re * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m))) - Float64(x_46_im_m * Float64(Float64(x_46_im_m * x_46_re) + Float64(x_46_im_m * x_46_re))))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(x_46_im_m * Float64(x_46_re * Float64(x_46_im_m * -3.0)));
	elseif (t_0 <= 5e+100)
		tmp = Float64(x_46_re * fma(Float64(x_46_im_m * x_46_im_m), -3.0, Float64(x_46_re * x_46_re)));
	else
		tmp = fma(Float64(x_46_im_m + x_46_re), Float64(x_46_re * Float64(x_46_re - x_46_im_m)), Float64(x_46_im_m + x_46_im_m));
	end
	return tmp
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
code[x$46$re_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(x$46$re * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im$95$m * N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(x$46$im$95$m * N[(x$46$re * N[(x$46$im$95$m * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+100], N[(x$46$re * N[(N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision] * -3.0 + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im$95$m + x$46$re), $MachinePrecision] * N[(x$46$re * N[(x$46$re - x$46$im$95$m), $MachinePrecision]), $MachinePrecision] + N[(x$46$im$95$m + x$46$im$95$m), $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)) < -inf.0

   1. Initial program 88.4%

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

   3. Taylor expanded in x.re around 0

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

      1. *-lowering-*.f64 N/A

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

      2. distribute-rgt-out-- N/A

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

      3. *-lowering-*.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. metadata-eval 42.9

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

   5. Simplified 42.9%

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

      1. associate-*l* N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 54.6

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

   7. Applied egg-rr 54.6%

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

   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)) < 4.9999999999999999e100

   1. Initial program 99.7%

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

   3. Taylor expanded in x.re around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate--l+ N/A

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

      4. unpow2 N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. distribute-rgt-out-- N/A

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

      7. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. metadata-eval 99.7

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

   5. Simplified 99.7%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 99.7

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

   7. Applied egg-rr 99.7%

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

   if 4.9999999999999999e100 < (-.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 61.6%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-out N/A

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

      7. *-lowering-*.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. neg-lowering-neg.f64 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. *-lowering-*.f64 N/A

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

      13. +-lowering-+.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. --lowering--.f64 83.1

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

   4. Applied egg-rr 83.1%

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

      1. +-commutative N/A

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

      2. distribute-rgt-neg-out N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. flip-+ N/A

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

      7. +-inverses N/A

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

      8. +-inverses N/A

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

      9. associate-*r/ N/A

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

      10. +-inverses N/A

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

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

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

      12. +-inverses N/A

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

      13. distribute-neg-frac2 N/A

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

      14. +-inverses N/A

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

      15. metadata-eval N/A

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

      16. +-inverses N/A

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

      17. flip-+ N/A

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

      18. distribute-lft-in N/A

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

      19. accelerator-lowering-fma.f64 N/A

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

   6. Applied egg-rr 85.7%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right) \leq -\infty:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{elif}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right) \leq 5 \cdot 10^{+100}:\\ \;\;\;\;x.re \cdot \mathsf{fma}\left(x.im \cdot x.im, -3, x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.im + x.re, x.re \cdot \left(x.re - x.im\right), x.im + x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ \begin{array}{l} t_0 := x.re \cdot \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) - x.im\_m \cdot \left(x.im\_m \cdot x.re + x.im\_m \cdot x.re\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;x.im\_m \cdot \left(x.re \cdot \left(x.im\_m \cdot -3\right)\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+100}:\\ \;\;\;\;x.re \cdot \mathsf{fma}\left(x.im\_m \cdot -3, x.im\_m, x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.im\_m + x.re, x.re \cdot \left(x.re - x.im\_m\right), x.im\_m + x.im\_m\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
(FPCore (x.re x.im_m)
 :precision binary64
 (let* ((t_0
         (-
          (* x.re (- (* x.re x.re) (* x.im_m x.im_m)))
          (* x.im_m (+ (* x.im_m x.re) (* x.im_m x.re))))))
   (if (<= t_0 (- INFINITY))
     (* x.im_m (* x.re (* x.im_m -3.0)))
     (if (<= t_0 5e+100)
       (* x.re (fma (* x.im_m -3.0) x.im_m (* x.re x.re)))
       (fma (+ x.im_m x.re) (* x.re (- x.re x.im_m)) (+ x.im_m x.im_m))))))

C

x.im_m = fabs(x_46_im);
double code(double x_46_re, double x_46_im_m) {
	double t_0 = (x_46_re * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = x_46_im_m * (x_46_re * (x_46_im_m * -3.0));
	} else if (t_0 <= 5e+100) {
		tmp = x_46_re * fma((x_46_im_m * -3.0), x_46_im_m, (x_46_re * x_46_re));
	} else {
		tmp = fma((x_46_im_m + x_46_re), (x_46_re * (x_46_re - x_46_im_m)), (x_46_im_m + x_46_im_m));
	}
	return tmp;
}

Julia

x.im_m = abs(x_46_im)
function code(x_46_re, x_46_im_m)
	t_0 = Float64(Float64(x_46_re * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m))) - Float64(x_46_im_m * Float64(Float64(x_46_im_m * x_46_re) + Float64(x_46_im_m * x_46_re))))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(x_46_im_m * Float64(x_46_re * Float64(x_46_im_m * -3.0)));
	elseif (t_0 <= 5e+100)
		tmp = Float64(x_46_re * fma(Float64(x_46_im_m * -3.0), x_46_im_m, Float64(x_46_re * x_46_re)));
	else
		tmp = fma(Float64(x_46_im_m + x_46_re), Float64(x_46_re * Float64(x_46_re - x_46_im_m)), Float64(x_46_im_m + x_46_im_m));
	end
	return tmp
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
code[x$46$re_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(x$46$re * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im$95$m * N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(x$46$im$95$m * N[(x$46$re * N[(x$46$im$95$m * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+100], N[(x$46$re * N[(N[(x$46$im$95$m * -3.0), $MachinePrecision] * x$46$im$95$m + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im$95$m + x$46$re), $MachinePrecision] * N[(x$46$re * N[(x$46$re - x$46$im$95$m), $MachinePrecision]), $MachinePrecision] + N[(x$46$im$95$m + x$46$im$95$m), $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)) < -inf.0

   1. Initial program 88.4%

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

   3. Taylor expanded in x.re around 0

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

      1. *-lowering-*.f64 N/A

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

      2. distribute-rgt-out-- N/A

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

      3. *-lowering-*.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. metadata-eval 42.9

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

   5. Simplified 42.9%

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

      1. associate-*l* N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 54.6

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

   7. Applied egg-rr 54.6%

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

   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)) < 4.9999999999999999e100

   1. Initial program 99.7%

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

   3. Taylor expanded in x.re around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate--l+ N/A

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

      4. unpow2 N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. distribute-rgt-out-- N/A

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

      7. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. metadata-eval 99.7

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

   5. Simplified 99.7%

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

      1. +-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 99.7

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

   7. Applied egg-rr 99.7%

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

   if 4.9999999999999999e100 < (-.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 61.6%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-out N/A

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

      7. *-lowering-*.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. neg-lowering-neg.f64 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. *-lowering-*.f64 N/A

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

      13. +-lowering-+.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. --lowering--.f64 83.1

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

   4. Applied egg-rr 83.1%

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

      1. +-commutative N/A

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

      2. distribute-rgt-neg-out N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. flip-+ N/A

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

      7. +-inverses N/A

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

      8. +-inverses N/A

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

      9. associate-*r/ N/A

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

      10. +-inverses N/A

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

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

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

      12. +-inverses N/A

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

      13. distribute-neg-frac2 N/A

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

      14. +-inverses N/A

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

      15. metadata-eval N/A

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

      16. +-inverses N/A

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

      17. flip-+ N/A

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

      18. distribute-lft-in N/A

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

      19. accelerator-lowering-fma.f64 N/A

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

   6. Applied egg-rr 85.7%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right) \leq -\infty:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{elif}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right) \leq 5 \cdot 10^{+100}:\\ \;\;\;\;x.re \cdot \mathsf{fma}\left(x.im \cdot -3, x.im, x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.im + x.re, x.re \cdot \left(x.re - x.im\right), x.im + x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ \begin{array}{l} t_0 := x.re \cdot \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) - x.im\_m \cdot \left(x.im\_m \cdot x.re + x.im\_m \cdot x.re\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;x.im\_m \cdot \left(x.re \cdot \left(x.im\_m \cdot -3\right)\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+100}:\\ \;\;\;\;x.re \cdot \mathsf{fma}\left(x.re, x.re, \left(x.im\_m \cdot x.im\_m\right) \cdot -3\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.im\_m + x.re, x.re \cdot \left(x.re - x.im\_m\right), x.im\_m + x.im\_m\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
(FPCore (x.re x.im_m)
 :precision binary64
 (let* ((t_0
         (-
          (* x.re (- (* x.re x.re) (* x.im_m x.im_m)))
          (* x.im_m (+ (* x.im_m x.re) (* x.im_m x.re))))))
   (if (<= t_0 (- INFINITY))
     (* x.im_m (* x.re (* x.im_m -3.0)))
     (if (<= t_0 5e+100)
       (* x.re (fma x.re x.re (* (* x.im_m x.im_m) -3.0)))
       (fma (+ x.im_m x.re) (* x.re (- x.re x.im_m)) (+ x.im_m x.im_m))))))

C

x.im_m = fabs(x_46_im);
double code(double x_46_re, double x_46_im_m) {
	double t_0 = (x_46_re * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = x_46_im_m * (x_46_re * (x_46_im_m * -3.0));
	} else if (t_0 <= 5e+100) {
		tmp = x_46_re * fma(x_46_re, x_46_re, ((x_46_im_m * x_46_im_m) * -3.0));
	} else {
		tmp = fma((x_46_im_m + x_46_re), (x_46_re * (x_46_re - x_46_im_m)), (x_46_im_m + x_46_im_m));
	}
	return tmp;
}

Julia

x.im_m = abs(x_46_im)
function code(x_46_re, x_46_im_m)
	t_0 = Float64(Float64(x_46_re * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m))) - Float64(x_46_im_m * Float64(Float64(x_46_im_m * x_46_re) + Float64(x_46_im_m * x_46_re))))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(x_46_im_m * Float64(x_46_re * Float64(x_46_im_m * -3.0)));
	elseif (t_0 <= 5e+100)
		tmp = Float64(x_46_re * fma(x_46_re, x_46_re, Float64(Float64(x_46_im_m * x_46_im_m) * -3.0)));
	else
		tmp = fma(Float64(x_46_im_m + x_46_re), Float64(x_46_re * Float64(x_46_re - x_46_im_m)), Float64(x_46_im_m + x_46_im_m));
	end
	return tmp
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
code[x$46$re_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(x$46$re * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im$95$m * N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(x$46$im$95$m * N[(x$46$re * N[(x$46$im$95$m * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+100], N[(x$46$re * N[(x$46$re * x$46$re + N[(N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im$95$m + x$46$re), $MachinePrecision] * N[(x$46$re * N[(x$46$re - x$46$im$95$m), $MachinePrecision]), $MachinePrecision] + N[(x$46$im$95$m + x$46$im$95$m), $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)) < -inf.0

   1. Initial program 88.4%

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

   3. Taylor expanded in x.re around 0

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

      1. *-lowering-*.f64 N/A

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

      2. distribute-rgt-out-- N/A

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

      3. *-lowering-*.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. metadata-eval 42.9

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

   5. Simplified 42.9%

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

      1. associate-*l* N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 54.6

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

   7. Applied egg-rr 54.6%

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

   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)) < 4.9999999999999999e100

   1. Initial program 99.7%

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

   3. Taylor expanded in x.re around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate--l+ N/A

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

      4. unpow2 N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. distribute-rgt-out-- N/A

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

      7. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. metadata-eval 99.7

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

   5. Simplified 99.7%

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

   if 4.9999999999999999e100 < (-.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 61.6%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-out N/A

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

      7. *-lowering-*.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. neg-lowering-neg.f64 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. *-lowering-*.f64 N/A

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

      13. +-lowering-+.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. --lowering--.f64 83.1

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

   4. Applied egg-rr 83.1%

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

      1. +-commutative N/A

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

      2. distribute-rgt-neg-out N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. flip-+ N/A

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

      7. +-inverses N/A

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

      8. +-inverses N/A

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

      9. associate-*r/ N/A

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

      10. +-inverses N/A

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

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

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

      12. +-inverses N/A

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

      13. distribute-neg-frac2 N/A

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

      14. +-inverses N/A

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

      15. metadata-eval N/A

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

      16. +-inverses N/A

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

      17. flip-+ N/A

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

      18. distribute-lft-in N/A

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

      19. accelerator-lowering-fma.f64 N/A

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

   6. Applied egg-rr 85.7%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right) \leq -\infty:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{elif}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right) \leq 5 \cdot 10^{+100}:\\ \;\;\;\;x.re \cdot \mathsf{fma}\left(x.re, x.re, \left(x.im \cdot x.im\right) \cdot -3\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.im + x.re, x.re \cdot \left(x.re - x.im\right), x.im + x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ \begin{array}{l} t_0 := x.re \cdot \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) - x.im\_m \cdot \left(x.im\_m \cdot x.re + x.im\_m \cdot x.re\right)\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-304}:\\ \;\;\;\;x.im\_m \cdot \left(x.re \cdot \left(x.im\_m \cdot -3\right)\right)\\ \mathbf{elif}\;t\_0 \leq 10^{-96}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.im\_m + x.re, x.re \cdot \left(x.re - x.im\_m\right), x.im\_m + x.im\_m\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
(FPCore (x.re x.im_m)
 :precision binary64
 (let* ((t_0
         (-
          (* x.re (- (* x.re x.re) (* x.im_m x.im_m)))
          (* x.im_m (+ (* x.im_m x.re) (* x.im_m x.re))))))
   (if (<= t_0 -4e-304)
     (* x.im_m (* x.re (* x.im_m -3.0)))
     (if (<= t_0 1e-96)
       (* x.re (* x.re x.re))
       (fma (+ x.im_m x.re) (* x.re (- x.re x.im_m)) (+ x.im_m x.im_m))))))

C

x.im_m = fabs(x_46_im);
double code(double x_46_re, double x_46_im_m) {
	double t_0 = (x_46_re * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)));
	double tmp;
	if (t_0 <= -4e-304) {
		tmp = x_46_im_m * (x_46_re * (x_46_im_m * -3.0));
	} else if (t_0 <= 1e-96) {
		tmp = x_46_re * (x_46_re * x_46_re);
	} else {
		tmp = fma((x_46_im_m + x_46_re), (x_46_re * (x_46_re - x_46_im_m)), (x_46_im_m + x_46_im_m));
	}
	return tmp;
}

Julia

x.im_m = abs(x_46_im)
function code(x_46_re, x_46_im_m)
	t_0 = Float64(Float64(x_46_re * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m))) - Float64(x_46_im_m * Float64(Float64(x_46_im_m * x_46_re) + Float64(x_46_im_m * x_46_re))))
	tmp = 0.0
	if (t_0 <= -4e-304)
		tmp = Float64(x_46_im_m * Float64(x_46_re * Float64(x_46_im_m * -3.0)));
	elseif (t_0 <= 1e-96)
		tmp = Float64(x_46_re * Float64(x_46_re * x_46_re));
	else
		tmp = fma(Float64(x_46_im_m + x_46_re), Float64(x_46_re * Float64(x_46_re - x_46_im_m)), Float64(x_46_im_m + x_46_im_m));
	end
	return tmp
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
code[x$46$re_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(x$46$re * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im$95$m * N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-304], N[(x$46$im$95$m * N[(x$46$re * N[(x$46$im$95$m * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-96], N[(x$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im$95$m + x$46$re), $MachinePrecision] * N[(x$46$re * N[(x$46$re - x$46$im$95$m), $MachinePrecision]), $MachinePrecision] + N[(x$46$im$95$m + x$46$im$95$m), $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)) < -3.99999999999999988e-304

   1. Initial program 93.1%

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

   3. Taylor expanded in x.re around 0

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

      1. *-lowering-*.f64 N/A

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

      2. distribute-rgt-out-- N/A

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

      3. *-lowering-*.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. metadata-eval 44.4

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

   5. Simplified 44.4%

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

      1. associate-*l* N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 51.0

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

   7. Applied egg-rr 51.0%

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

   if -3.99999999999999988e-304 < (-.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)) < 9.9999999999999991e-97

   1. Initial program 100.0%

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

   3. Taylor expanded in x.re around inf

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

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 89.4

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

   5. Simplified 89.4%

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

   if 9.9999999999999991e-97 < (-.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 69.3%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-out N/A

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

      7. *-lowering-*.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. neg-lowering-neg.f64 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. *-lowering-*.f64 N/A

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

      13. +-lowering-+.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. --lowering--.f64 86.4

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

   4. Applied egg-rr 86.4%

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

      1. +-commutative N/A

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

      2. distribute-rgt-neg-out N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. flip-+ N/A

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

      7. +-inverses N/A

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

      8. +-inverses N/A

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

      9. associate-*r/ N/A

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

      10. +-inverses N/A

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

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

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

      12. +-inverses N/A

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

      13. distribute-neg-frac2 N/A

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

      14. +-inverses N/A

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

      15. metadata-eval N/A

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

      16. +-inverses N/A

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

      17. flip-+ N/A

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

      18. distribute-lft-in N/A

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

      19. accelerator-lowering-fma.f64 N/A

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

   6. Applied egg-rr 79.0%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right) \leq -4 \cdot 10^{-304}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{elif}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right) \leq 10^{-96}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.im + x.re, x.re \cdot \left(x.re - x.im\right), x.im + x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) - x.im\_m \cdot \left(x.im\_m \cdot x.re + x.im\_m \cdot x.re\right) \leq -4 \cdot 10^{-304}:\\ \;\;\;\;x.im\_m \cdot \left(x.re \cdot \left(x.im\_m \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
(FPCore (x.re x.im_m)
 :precision binary64
 (if (<=
      (-
       (* x.re (- (* x.re x.re) (* x.im_m x.im_m)))
       (* x.im_m (+ (* x.im_m x.re) (* x.im_m x.re))))
      -4e-304)
   (* x.im_m (* x.re (* x.im_m -3.0)))
   (* x.re (* x.re x.re))))

C

x.im_m = fabs(x_46_im);
double code(double x_46_re, double x_46_im_m) {
	double tmp;
	if (((x_46_re * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)))) <= -4e-304) {
		tmp = x_46_im_m * (x_46_re * (x_46_im_m * -3.0));
	} else {
		tmp = x_46_re * (x_46_re * x_46_re);
	}
	return tmp;
}

Fortran

x.im_m = abs(x_46im)
real(8) function code(x_46re, x_46im_m)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (((x_46re * ((x_46re * x_46re) - (x_46im_m * x_46im_m))) - (x_46im_m * ((x_46im_m * x_46re) + (x_46im_m * x_46re)))) <= (-4d-304)) then
        tmp = x_46im_m * (x_46re * (x_46im_m * (-3.0d0)))
    else
        tmp = x_46re * (x_46re * x_46re)
    end if
    code = tmp
end function

Java

x.im_m = Math.abs(x_46_im);
public static double code(double x_46_re, double x_46_im_m) {
	double tmp;
	if (((x_46_re * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)))) <= -4e-304) {
		tmp = x_46_im_m * (x_46_re * (x_46_im_m * -3.0));
	} else {
		tmp = x_46_re * (x_46_re * x_46_re);
	}
	return tmp;
}

Python

x.im_m = math.fabs(x_46_im)
def code(x_46_re, x_46_im_m):
	tmp = 0
	if ((x_46_re * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)))) <= -4e-304:
		tmp = x_46_im_m * (x_46_re * (x_46_im_m * -3.0))
	else:
		tmp = x_46_re * (x_46_re * x_46_re)
	return tmp

Julia

x.im_m = abs(x_46_im)
function code(x_46_re, x_46_im_m)
	tmp = 0.0
	if (Float64(Float64(x_46_re * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m))) - Float64(x_46_im_m * Float64(Float64(x_46_im_m * x_46_re) + Float64(x_46_im_m * x_46_re)))) <= -4e-304)
		tmp = Float64(x_46_im_m * Float64(x_46_re * Float64(x_46_im_m * -3.0)));
	else
		tmp = Float64(x_46_re * Float64(x_46_re * x_46_re));
	end
	return tmp
end

MATLAB

x.im_m = abs(x_46_im);
function tmp_2 = code(x_46_re, x_46_im_m)
	tmp = 0.0;
	if (((x_46_re * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)))) <= -4e-304)
		tmp = x_46_im_m * (x_46_re * (x_46_im_m * -3.0));
	else
		tmp = x_46_re * (x_46_re * x_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
code[x$46$re_, x$46$im$95$m_] := If[LessEqual[N[(N[(x$46$re * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im$95$m * N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -4e-304], N[(x$46$im$95$m * N[(x$46$re * N[(x$46$im$95$m * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 93.1%

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

   3. Taylor expanded in x.re around 0

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

      1. *-lowering-*.f64 N/A

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

      2. distribute-rgt-out-- N/A

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

      3. *-lowering-*.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. metadata-eval 44.4

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

   5. Simplified 44.4%

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

      1. associate-*l* N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 51.0

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

   7. Applied egg-rr 51.0%

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

   if -3.99999999999999988e-304 < (-.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 78.7%

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

   3. Taylor expanded in x.re around inf

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

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 65.6

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

   5. Simplified 65.6%

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

4. Final simplification 60.8%

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

Alternative 7: 60.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) - x.im\_m \cdot \left(x.im\_m \cdot x.re + x.im\_m \cdot x.re\right) \leq -4 \cdot 10^{-304}:\\ \;\;\;\;-3 \cdot \left(x.im\_m \cdot \left(x.im\_m \cdot x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
(FPCore (x.re x.im_m)
 :precision binary64
 (if (<=
      (-
       (* x.re (- (* x.re x.re) (* x.im_m x.im_m)))
       (* x.im_m (+ (* x.im_m x.re) (* x.im_m x.re))))
      -4e-304)
   (* -3.0 (* x.im_m (* x.im_m x.re)))
   (* x.re (* x.re x.re))))

C

x.im_m = fabs(x_46_im);
double code(double x_46_re, double x_46_im_m) {
	double tmp;
	if (((x_46_re * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)))) <= -4e-304) {
		tmp = -3.0 * (x_46_im_m * (x_46_im_m * x_46_re));
	} else {
		tmp = x_46_re * (x_46_re * x_46_re);
	}
	return tmp;
}

Fortran

x.im_m = abs(x_46im)
real(8) function code(x_46re, x_46im_m)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (((x_46re * ((x_46re * x_46re) - (x_46im_m * x_46im_m))) - (x_46im_m * ((x_46im_m * x_46re) + (x_46im_m * x_46re)))) <= (-4d-304)) then
        tmp = (-3.0d0) * (x_46im_m * (x_46im_m * x_46re))
    else
        tmp = x_46re * (x_46re * x_46re)
    end if
    code = tmp
end function

Java

x.im_m = Math.abs(x_46_im);
public static double code(double x_46_re, double x_46_im_m) {
	double tmp;
	if (((x_46_re * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)))) <= -4e-304) {
		tmp = -3.0 * (x_46_im_m * (x_46_im_m * x_46_re));
	} else {
		tmp = x_46_re * (x_46_re * x_46_re);
	}
	return tmp;
}

Python

x.im_m = math.fabs(x_46_im)
def code(x_46_re, x_46_im_m):
	tmp = 0
	if ((x_46_re * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)))) <= -4e-304:
		tmp = -3.0 * (x_46_im_m * (x_46_im_m * x_46_re))
	else:
		tmp = x_46_re * (x_46_re * x_46_re)
	return tmp

Julia

x.im_m = abs(x_46_im)
function code(x_46_re, x_46_im_m)
	tmp = 0.0
	if (Float64(Float64(x_46_re * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m))) - Float64(x_46_im_m * Float64(Float64(x_46_im_m * x_46_re) + Float64(x_46_im_m * x_46_re)))) <= -4e-304)
		tmp = Float64(-3.0 * Float64(x_46_im_m * Float64(x_46_im_m * x_46_re)));
	else
		tmp = Float64(x_46_re * Float64(x_46_re * x_46_re));
	end
	return tmp
end

MATLAB

x.im_m = abs(x_46_im);
function tmp_2 = code(x_46_re, x_46_im_m)
	tmp = 0.0;
	if (((x_46_re * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)))) <= -4e-304)
		tmp = -3.0 * (x_46_im_m * (x_46_im_m * x_46_re));
	else
		tmp = x_46_re * (x_46_re * x_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
code[x$46$re_, x$46$im$95$m_] := If[LessEqual[N[(N[(x$46$re * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im$95$m * N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -4e-304], N[(-3.0 * N[(x$46$im$95$m * N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 93.1%

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

   3. Taylor expanded in x.re around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate--l+ N/A

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

      4. unpow2 N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. distribute-rgt-out-- N/A

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

      7. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. metadata-eval 93.1

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

   5. Simplified 93.1%

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

      1. +-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 93.1

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

   7. Applied egg-rr 93.1%

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

   8. Taylor expanded in x.re around 0

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

      1. *-lowering-*.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 51.0

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

   10. Simplified 51.0%

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

   if -3.99999999999999988e-304 < (-.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 78.7%

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

   3. Taylor expanded in x.re around inf

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

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 65.6

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

   5. Simplified 65.6%

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

4. Final simplification 60.8%

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

Alternative 8: 50.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) - x.im\_m \cdot \left(x.im\_m \cdot x.re + x.im\_m \cdot x.re\right) \leq -4 \cdot 10^{-304}:\\ \;\;\;\;\left(x.im\_m \cdot x.re\right) \cdot \left(-x.im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
(FPCore (x.re x.im_m)
 :precision binary64
 (if (<=
      (-
       (* x.re (- (* x.re x.re) (* x.im_m x.im_m)))
       (* x.im_m (+ (* x.im_m x.re) (* x.im_m x.re))))
      -4e-304)
   (* (* x.im_m x.re) (- x.im_m))
   (* x.re (* x.re x.re))))

C

x.im_m = fabs(x_46_im);
double code(double x_46_re, double x_46_im_m) {
	double tmp;
	if (((x_46_re * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)))) <= -4e-304) {
		tmp = (x_46_im_m * x_46_re) * -x_46_im_m;
	} else {
		tmp = x_46_re * (x_46_re * x_46_re);
	}
	return tmp;
}

Fortran

x.im_m = abs(x_46im)
real(8) function code(x_46re, x_46im_m)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (((x_46re * ((x_46re * x_46re) - (x_46im_m * x_46im_m))) - (x_46im_m * ((x_46im_m * x_46re) + (x_46im_m * x_46re)))) <= (-4d-304)) then
        tmp = (x_46im_m * x_46re) * -x_46im_m
    else
        tmp = x_46re * (x_46re * x_46re)
    end if
    code = tmp
end function

Java

x.im_m = Math.abs(x_46_im);
public static double code(double x_46_re, double x_46_im_m) {
	double tmp;
	if (((x_46_re * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)))) <= -4e-304) {
		tmp = (x_46_im_m * x_46_re) * -x_46_im_m;
	} else {
		tmp = x_46_re * (x_46_re * x_46_re);
	}
	return tmp;
}

Python

x.im_m = math.fabs(x_46_im)
def code(x_46_re, x_46_im_m):
	tmp = 0
	if ((x_46_re * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)))) <= -4e-304:
		tmp = (x_46_im_m * x_46_re) * -x_46_im_m
	else:
		tmp = x_46_re * (x_46_re * x_46_re)
	return tmp

Julia

x.im_m = abs(x_46_im)
function code(x_46_re, x_46_im_m)
	tmp = 0.0
	if (Float64(Float64(x_46_re * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m))) - Float64(x_46_im_m * Float64(Float64(x_46_im_m * x_46_re) + Float64(x_46_im_m * x_46_re)))) <= -4e-304)
		tmp = Float64(Float64(x_46_im_m * x_46_re) * Float64(-x_46_im_m));
	else
		tmp = Float64(x_46_re * Float64(x_46_re * x_46_re));
	end
	return tmp
end

MATLAB

x.im_m = abs(x_46_im);
function tmp_2 = code(x_46_re, x_46_im_m)
	tmp = 0.0;
	if (((x_46_re * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)))) <= -4e-304)
		tmp = (x_46_im_m * x_46_re) * -x_46_im_m;
	else
		tmp = x_46_re * (x_46_re * x_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
code[x$46$re_, x$46$im$95$m_] := If[LessEqual[N[(N[(x$46$re * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im$95$m * N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -4e-304], N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] * (-x$46$im$95$m)), $MachinePrecision], N[(x$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 93.1%

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

      1. --lowering--.f64 N/A

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

      2. difference-of-squares N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-lowering-+.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-rgt-out N/A

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

      12. *-lowering-*.f64 N/A

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

      13. +-lowering-+.f64 99.7

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

   4. Applied egg-rr 99.7%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. flip-+ N/A

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

      4. +-inverses N/A

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

      5. +-inverses N/A

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

      6. associate-*r/ N/A

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

      7. +-inverses N/A

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

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

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

      9. +-inverses N/A

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

      10. +-inverses N/A

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

      11. +-inverses N/A

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

      12. flip-+ N/A

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

      13. +-lowering-+.f64 69.2

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

   6. Applied egg-rr 69.2%

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

   7. Taylor expanded in x.im around inf

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

      1. mul-1-neg N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. distribute-rgt-neg-out N/A

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

      5. mul-1-neg N/A

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

      6. *-lowering-*.f64 N/A

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

      7. mul-1-neg N/A

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

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

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

      9. mul-1-neg N/A

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

      10. *-lowering-*.f64 N/A

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

      11. mul-1-neg N/A

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

      12. neg-lowering-neg.f64 29.4

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

   9. Simplified 29.4%

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

   if -3.99999999999999988e-304 < (-.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 78.7%

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

   3. Taylor expanded in x.re around inf

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

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 65.6

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

   5. Simplified 65.6%

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

4. Final simplification 53.7%

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

Alternative 9: 45.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) - x.im\_m \cdot \left(x.im\_m \cdot x.re + x.im\_m \cdot x.re\right) \leq -5 \cdot 10^{-139}:\\ \;\;\;\;x.im\_m \cdot -2\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
(FPCore (x.re x.im_m)
 :precision binary64
 (if (<=
      (-
       (* x.re (- (* x.re x.re) (* x.im_m x.im_m)))
       (* x.im_m (+ (* x.im_m x.re) (* x.im_m x.re))))
      -5e-139)
   (* x.im_m -2.0)
   (* x.re (* x.re x.re))))

C

x.im_m = fabs(x_46_im);
double code(double x_46_re, double x_46_im_m) {
	double tmp;
	if (((x_46_re * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)))) <= -5e-139) {
		tmp = x_46_im_m * -2.0;
	} else {
		tmp = x_46_re * (x_46_re * x_46_re);
	}
	return tmp;
}

Fortran

x.im_m = abs(x_46im)
real(8) function code(x_46re, x_46im_m)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (((x_46re * ((x_46re * x_46re) - (x_46im_m * x_46im_m))) - (x_46im_m * ((x_46im_m * x_46re) + (x_46im_m * x_46re)))) <= (-5d-139)) then
        tmp = x_46im_m * (-2.0d0)
    else
        tmp = x_46re * (x_46re * x_46re)
    end if
    code = tmp
end function

Java

x.im_m = Math.abs(x_46_im);
public static double code(double x_46_re, double x_46_im_m) {
	double tmp;
	if (((x_46_re * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)))) <= -5e-139) {
		tmp = x_46_im_m * -2.0;
	} else {
		tmp = x_46_re * (x_46_re * x_46_re);
	}
	return tmp;
}

Python

x.im_m = math.fabs(x_46_im)
def code(x_46_re, x_46_im_m):
	tmp = 0
	if ((x_46_re * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)))) <= -5e-139:
		tmp = x_46_im_m * -2.0
	else:
		tmp = x_46_re * (x_46_re * x_46_re)
	return tmp

Julia

x.im_m = abs(x_46_im)
function code(x_46_re, x_46_im_m)
	tmp = 0.0
	if (Float64(Float64(x_46_re * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m))) - Float64(x_46_im_m * Float64(Float64(x_46_im_m * x_46_re) + Float64(x_46_im_m * x_46_re)))) <= -5e-139)
		tmp = Float64(x_46_im_m * -2.0);
	else
		tmp = Float64(x_46_re * Float64(x_46_re * x_46_re));
	end
	return tmp
end

MATLAB

x.im_m = abs(x_46_im);
function tmp_2 = code(x_46_re, x_46_im_m)
	tmp = 0.0;
	if (((x_46_re * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)))) <= -5e-139)
		tmp = x_46_im_m * -2.0;
	else
		tmp = x_46_re * (x_46_re * x_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
code[x$46$re_, x$46$im$95$m_] := If[LessEqual[N[(N[(x$46$re * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im$95$m * N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-139], N[(x$46$im$95$m * -2.0), $MachinePrecision], N[(x$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -5.00000000000000034e-139

   1. Initial program 92.7%

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

      1. --lowering--.f64 N/A

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

      2. difference-of-squares N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-lowering-+.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-rgt-out N/A

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

      12. *-lowering-*.f64 N/A

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

      13. +-lowering-+.f64 99.7

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

   4. Applied egg-rr 99.7%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. flip-+ N/A

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

      4. +-inverses N/A

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

      5. +-inverses N/A

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

      6. associate-*r/ N/A

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

      7. +-inverses N/A

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

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

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

      9. +-inverses N/A

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

      10. +-inverses N/A

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

      11. +-inverses N/A

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

      12. flip-+ N/A

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

      13. +-lowering-+.f64 73.3

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

   6. Applied egg-rr 73.3%

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

   7. Taylor expanded in x.re around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 3.0

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

   9. Simplified 3.0%

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

   if -5.00000000000000034e-139 < (-.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 79.3%

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

   3. Taylor expanded in x.re around inf

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

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 65.5

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

   5. Simplified 65.5%

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

4. Final simplification 46.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right) \leq -5 \cdot 10^{-139}:\\ \;\;\;\;x.im \cdot -2\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 93.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ \begin{array}{l} \mathbf{if}\;x.re \leq 10^{+94}:\\ \;\;\;\;\mathsf{fma}\left(x.im\_m, x.re \cdot \left(x.im\_m \cdot -3\right), x.re \cdot \left(x.re \cdot x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.im\_m + x.re, x.re \cdot \left(x.re - x.im\_m\right), x.im\_m + x.im\_m\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
(FPCore (x.re x.im_m)
 :precision binary64
 (if (<= x.re 1e+94)
   (fma x.im_m (* x.re (* x.im_m -3.0)) (* x.re (* x.re x.re)))
   (fma (+ x.im_m x.re) (* x.re (- x.re x.im_m)) (+ x.im_m x.im_m))))

C

x.im_m = fabs(x_46_im);
double code(double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_re <= 1e+94) {
		tmp = fma(x_46_im_m, (x_46_re * (x_46_im_m * -3.0)), (x_46_re * (x_46_re * x_46_re)));
	} else {
		tmp = fma((x_46_im_m + x_46_re), (x_46_re * (x_46_re - x_46_im_m)), (x_46_im_m + x_46_im_m));
	}
	return tmp;
}

Julia

x.im_m = abs(x_46_im)
function code(x_46_re, x_46_im_m)
	tmp = 0.0
	if (x_46_re <= 1e+94)
		tmp = fma(x_46_im_m, Float64(x_46_re * Float64(x_46_im_m * -3.0)), Float64(x_46_re * Float64(x_46_re * x_46_re)));
	else
		tmp = fma(Float64(x_46_im_m + x_46_re), Float64(x_46_re * Float64(x_46_re - x_46_im_m)), Float64(x_46_im_m + x_46_im_m));
	end
	return tmp
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
code[x$46$re_, x$46$im$95$m_] := If[LessEqual[x$46$re, 1e+94], N[(x$46$im$95$m * N[(x$46$re * N[(x$46$im$95$m * -3.0), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im$95$m + x$46$re), $MachinePrecision] * N[(x$46$re * N[(x$46$re - x$46$im$95$m), $MachinePrecision]), $MachinePrecision] + N[(x$46$im$95$m + x$46$im$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 1e94

   1. Initial program 84.8%

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

   3. Taylor expanded in x.re around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate--l+ N/A

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

      4. unpow2 N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. distribute-rgt-out-- N/A

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

      7. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. metadata-eval 91.6

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

   5. Simplified 91.6%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 91.1

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

   7. Applied egg-rr 91.1%

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

   if 1e94 < x.re

   1. Initial program 75.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-out N/A

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

      7. *-lowering-*.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. neg-lowering-neg.f64 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. *-lowering-*.f64 N/A

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

      13. +-lowering-+.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. --lowering--.f64 86.1

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

   4. Applied egg-rr 86.1%

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

      1. +-commutative N/A

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

      2. distribute-rgt-neg-out N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. flip-+ N/A

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

      7. +-inverses N/A

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

      8. +-inverses N/A

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

      9. associate-*r/ N/A

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

      10. +-inverses N/A

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

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

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

      12. +-inverses N/A

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

      13. distribute-neg-frac2 N/A

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

      14. +-inverses N/A

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

      15. metadata-eval N/A

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

      16. +-inverses N/A

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

      17. flip-+ N/A

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

      18. distribute-lft-in N/A

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

      19. accelerator-lowering-fma.f64 N/A

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

   6. Applied egg-rr 100.0%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 10^{+94}:\\ \;\;\;\;\mathsf{fma}\left(x.im, x.re \cdot \left(x.im \cdot -3\right), x.re \cdot \left(x.re \cdot x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.im + x.re, x.re \cdot \left(x.re - x.im\right), x.im + x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 3.6% accurate, 6.7× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.im\_m \cdot -2 \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
(FPCore (x.re x.im_m) :precision binary64 (* x.im_m -2.0))

C

x.im_m = fabs(x_46_im);
double code(double x_46_re, double x_46_im_m) {
	return x_46_im_m * -2.0;
}

Fortran

x.im_m = abs(x_46im)
real(8) function code(x_46re, x_46im_m)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    code = x_46im_m * (-2.0d0)
end function

Java

x.im_m = Math.abs(x_46_im);
public static double code(double x_46_re, double x_46_im_m) {
	return x_46_im_m * -2.0;
}

Python

x.im_m = math.fabs(x_46_im)
def code(x_46_re, x_46_im_m):
	return x_46_im_m * -2.0

Julia

x.im_m = abs(x_46_im)
function code(x_46_re, x_46_im_m)
	return Float64(x_46_im_m * -2.0)
end

MATLAB

x.im_m = abs(x_46_im);
function tmp = code(x_46_re, x_46_im_m)
	tmp = x_46_im_m * -2.0;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
code[x$46$re_, x$46$im$95$m_] := N[(x$46$im$95$m * -2.0), $MachinePrecision]

Derivation

1. Initial program 83.4%

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

   1. --lowering--.f64 N/A

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

   2. difference-of-squares N/A

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

   3. associate-*l* N/A

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

   4. *-lowering-*.f64 N/A

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

   5. +-lowering-+.f64 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. --lowering--.f64 N/A

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

   8. *-commutative N/A

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

   9. *-lowering-*.f64 N/A

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

   10. *-commutative N/A

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

   11. distribute-rgt-out N/A

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

   12. *-lowering-*.f64 N/A

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

   13. +-lowering-+.f64 90.4

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

4. Applied egg-rr 90.4%

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. flip-+ N/A

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

   4. +-inverses N/A

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

   5. +-inverses N/A

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

   6. associate-*r/ N/A

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

   7. +-inverses N/A

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

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

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

   9. +-inverses N/A

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

   10. +-inverses N/A

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

   11. +-inverses N/A

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

   12. flip-+ N/A

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

   13. +-lowering-+.f64 60.5

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

6. Applied egg-rr 60.5%

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

7. Taylor expanded in x.re around 0

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 3.5

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

9. Simplified 3.5%

   \[\leadsto \color{blue}{x.im \cdot -2} \]
10. Add Preprocessing

Developer Target 1: 87.7% 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 2024199 
(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)))