math.cube on complex, real part

Percentage Accurate: 83.1% → 99.3%

Time: 11.3s

Alternatives: 11

Speedup: 1.0×

• 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: 83.1% 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: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ \begin{array}{l} \mathbf{if}\;x.im\_m \leq 2.2 \cdot 10^{-115}:\\ \;\;\;\;{x.re}^{3}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{x.re}{x.im\_m}, x.re, -3 \cdot x.im\_m\right) \cdot x.re\right) \cdot x.im\_m\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
(FPCore (x.re x.im_m)
 :precision binary64
 (if (<= x.im_m 2.2e-115)
   (pow x.re 3.0)
   (* (* (fma (/ x.re x.im_m) x.re (* -3.0 x.im_m)) x.re) 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_im_m <= 2.2e-115) {
		tmp = pow(x_46_re, 3.0);
	} else {
		tmp = (fma((x_46_re / x_46_im_m), x_46_re, (-3.0 * x_46_im_m)) * x_46_re) * 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_im_m <= 2.2e-115)
		tmp = x_46_re ^ 3.0;
	else
		tmp = Float64(Float64(fma(Float64(x_46_re / x_46_im_m), x_46_re, Float64(-3.0 * x_46_im_m)) * x_46_re) * 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$46$im$95$m, 2.2e-115], N[Power[x$46$re, 3.0], $MachinePrecision], N[(N[(N[(N[(x$46$re / x$46$im$95$m), $MachinePrecision] * x$46$re + N[(-3.0 * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 2.1999999999999999e-115

   1. Initial program 88.3%

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

   3. Taylor expanded in x.im around 0

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 71.9

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

   5. Applied rewrites 71.9%

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

   7. Applied rewrites 72.1%

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

   if 2.1999999999999999e-115 < x.im

   1. Initial program 66.7%

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

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

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

      3. flip3-- N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 12.0%

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

   5. Taylor expanded in x.im around inf

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

   7. Applied rewrites 92.9%

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

   8. Taylor expanded in x.im around 0

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

   10. Applied rewrites 33.1%

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

   11. Taylor expanded in x.re around 0

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

   13. Applied rewrites 99.7%

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

Alternative 2: 84.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ \begin{array}{l} t_0 := \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.re - \left(x.re \cdot x.im\_m + x.re \cdot x.im\_m\right) \cdot x.im\_m\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\left(-3 \cdot x.im\_m\right) \cdot x.re\right) \cdot x.im\_m\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(\left(x.im\_m \cdot x.im\_m\right) \cdot -3, x.re, \left(x.re \cdot x.re\right) \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im\_m, \left(x.re + x.im\_m\right) \cdot x.re, x.re + x.re\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
(FPCore (x.re x.im_m)
 :precision binary64
 (let* ((t_0
         (-
          (* (- (* x.re x.re) (* x.im_m x.im_m)) x.re)
          (* (+ (* x.re x.im_m) (* x.re x.im_m)) x.im_m))))
   (if (<= t_0 (- INFINITY))
     (* (* (* -3.0 x.im_m) x.re) x.im_m)
     (if (<= t_0 5e+20)
       (fma (* (* x.im_m x.im_m) -3.0) x.re (* (* x.re x.re) x.re))
       (fma (- x.re 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 t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_re) - (((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)) * x_46_im_m);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = ((-3.0 * x_46_im_m) * x_46_re) * x_46_im_m;
	} else if (t_0 <= 5e+20) {
		tmp = fma(((x_46_im_m * x_46_im_m) * -3.0), x_46_re, ((x_46_re * x_46_re) * x_46_re));
	} else {
		tmp = fma((x_46_re - x_46_im_m), ((x_46_re + x_46_im_m) * 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)
	t_0 = Float64(Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_re) - Float64(Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_re * x_46_im_m)) * x_46_im_m))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(-3.0 * x_46_im_m) * x_46_re) * x_46_im_m);
	elseif (t_0 <= 5e+20)
		tmp = fma(Float64(Float64(x_46_im_m * x_46_im_m) * -3.0), x_46_re, Float64(Float64(x_46_re * x_46_re) * x_46_re));
	else
		tmp = fma(Float64(x_46_re - x_46_im_m), Float64(Float64(x_46_re + x_46_im_m) * x_46_re), Float64(x_46_re + x_46_re));
	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[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision] - N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$re * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(-3.0 * x$46$im$95$m), $MachinePrecision] * x$46$re), $MachinePrecision] * x$46$im$95$m), $MachinePrecision], If[LessEqual[t$95$0, 5e+20], N[(N[(N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision] * -3.0), $MachinePrecision] * x$46$re + N[(N[(x$46$re * x$46$re), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re - x$46$im$95$m), $MachinePrecision] * N[(N[(x$46$re + x$46$im$95$m), $MachinePrecision] * x$46$re), $MachinePrecision] + N[(x$46$re + x$46$re), $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 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. lift-*.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. lift--.f64 N/A

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

      3. flip3-- N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in x.im around inf

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 45.9

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

   7. Applied rewrites 45.9%

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

   9. Applied rewrites 45.9%

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

   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)) < 5e20

   1. Initial program 99.6%

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

   3. Taylor expanded in x.im around 0

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

      1. +-commutative N/A

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

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

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

      9. distribute-lft-in N/A

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

      10. associate--l+ N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 99.6%

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

   7. Applied rewrites 99.6%

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

   if 5e20 < (-.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 59.0%

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

      1. lift--.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. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-lft-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-neg.f64 67.8

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift--.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. difference-of-squares N/A

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

      21. associate-*r* N/A

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

      22. lower-*.f64 N/A

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

   4. Applied rewrites 82.3%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. distribute-lft-in N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-+.f64 N/A

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

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

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

      21. *-commutative N/A

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

      22. lift-+.f64 N/A

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

      23. lift-*.f64 N/A

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

      24. *-commutative N/A

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

      25. lift-*.f64 N/A

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

   6. Applied rewrites 87.3%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.re \cdot x.im\right) \cdot x.im \leq -\infty:\\ \;\;\;\;\left(\left(-3 \cdot x.im\right) \cdot x.re\right) \cdot x.im\\ \mathbf{elif}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.re \cdot x.im\right) \cdot x.im \leq 5 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(\left(x.im \cdot x.im\right) \cdot -3, x.re, \left(x.re \cdot x.re\right) \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im, \left(x.re + x.im\right) \cdot x.re, x.re + x.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ \begin{array}{l} t_0 := \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.re - \left(x.re \cdot x.im\_m + x.re \cdot x.im\_m\right) \cdot x.im\_m\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\left(-3 \cdot x.im\_m\right) \cdot x.re\right) \cdot x.im\_m\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(-3, x.im\_m \cdot x.im\_m, x.re \cdot x.re\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im\_m, \left(x.re + x.im\_m\right) \cdot x.re, x.re + x.re\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
(FPCore (x.re x.im_m)
 :precision binary64
 (let* ((t_0
         (-
          (* (- (* x.re x.re) (* x.im_m x.im_m)) x.re)
          (* (+ (* x.re x.im_m) (* x.re x.im_m)) x.im_m))))
   (if (<= t_0 (- INFINITY))
     (* (* (* -3.0 x.im_m) x.re) x.im_m)
     (if (<= t_0 5e+20)
       (* (fma -3.0 (* x.im_m x.im_m) (* x.re x.re)) x.re)
       (fma (- x.re 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 t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_re) - (((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)) * x_46_im_m);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = ((-3.0 * x_46_im_m) * x_46_re) * x_46_im_m;
	} else if (t_0 <= 5e+20) {
		tmp = fma(-3.0, (x_46_im_m * x_46_im_m), (x_46_re * x_46_re)) * x_46_re;
	} else {
		tmp = fma((x_46_re - x_46_im_m), ((x_46_re + x_46_im_m) * 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)
	t_0 = Float64(Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_re) - Float64(Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_re * x_46_im_m)) * x_46_im_m))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(-3.0 * x_46_im_m) * x_46_re) * x_46_im_m);
	elseif (t_0 <= 5e+20)
		tmp = Float64(fma(-3.0, Float64(x_46_im_m * x_46_im_m), Float64(x_46_re * x_46_re)) * x_46_re);
	else
		tmp = fma(Float64(x_46_re - x_46_im_m), Float64(Float64(x_46_re + x_46_im_m) * x_46_re), Float64(x_46_re + x_46_re));
	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[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision] - N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$re * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(-3.0 * x$46$im$95$m), $MachinePrecision] * x$46$re), $MachinePrecision] * x$46$im$95$m), $MachinePrecision], If[LessEqual[t$95$0, 5e+20], N[(N[(-3.0 * N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision], N[(N[(x$46$re - x$46$im$95$m), $MachinePrecision] * N[(N[(x$46$re + x$46$im$95$m), $MachinePrecision] * x$46$re), $MachinePrecision] + N[(x$46$re + x$46$re), $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 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. lift-*.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. lift--.f64 N/A

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

      3. flip3-- N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in x.im around inf

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 45.9

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

   7. Applied rewrites 45.9%

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

   9. Applied rewrites 45.9%

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

   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)) < 5e20

   1. Initial program 99.6%

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

   3. Taylor expanded in x.im around 0

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

      1. +-commutative N/A

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

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

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

      9. distribute-lft-in N/A

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

      10. associate--l+ N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 99.6%

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

   if 5e20 < (-.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 59.0%

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

      1. lift--.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. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-lft-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-neg.f64 67.8

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift--.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. difference-of-squares N/A

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

      21. associate-*r* N/A

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

      22. lower-*.f64 N/A

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

   4. Applied rewrites 82.3%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. distribute-lft-in N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-+.f64 N/A

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

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

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

      21. *-commutative N/A

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

      22. lift-+.f64 N/A

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

      23. lift-*.f64 N/A

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

      24. *-commutative N/A

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

      25. lift-*.f64 N/A

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

   6. Applied rewrites 87.3%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.re \cdot x.im\right) \cdot x.im \leq -\infty:\\ \;\;\;\;\left(\left(-3 \cdot x.im\right) \cdot x.re\right) \cdot x.im\\ \mathbf{elif}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.re \cdot x.im\right) \cdot x.im \leq 5 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(-3, x.im \cdot x.im, x.re \cdot x.re\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im, \left(x.re + x.im\right) \cdot x.re, x.re + x.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ \begin{array}{l} t_0 := \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.re - \left(x.re \cdot x.im\_m + x.re \cdot x.im\_m\right) \cdot x.im\_m\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-298}:\\ \;\;\;\;\left(\left(x.re \cdot x.im\_m\right) \cdot -3\right) \cdot x.im\_m\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+20}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im\_m, \left(x.re + x.im\_m\right) \cdot x.re, x.re + x.re\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
(FPCore (x.re x.im_m)
 :precision binary64
 (let* ((t_0
         (-
          (* (- (* x.re x.re) (* x.im_m x.im_m)) x.re)
          (* (+ (* x.re x.im_m) (* x.re x.im_m)) x.im_m))))
   (if (<= t_0 -4e-298)
     (* (* (* x.re x.im_m) -3.0) x.im_m)
     (if (<= t_0 5e+20)
       (* (* x.re x.re) x.re)
       (fma (- x.re 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 t_0 = (((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m)) * x_46_re) - (((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)) * x_46_im_m);
	double tmp;
	if (t_0 <= -4e-298) {
		tmp = ((x_46_re * x_46_im_m) * -3.0) * x_46_im_m;
	} else if (t_0 <= 5e+20) {
		tmp = (x_46_re * x_46_re) * x_46_re;
	} else {
		tmp = fma((x_46_re - x_46_im_m), ((x_46_re + x_46_im_m) * 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)
	t_0 = Float64(Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_re) - Float64(Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_re * x_46_im_m)) * x_46_im_m))
	tmp = 0.0
	if (t_0 <= -4e-298)
		tmp = Float64(Float64(Float64(x_46_re * x_46_im_m) * -3.0) * x_46_im_m);
	elseif (t_0 <= 5e+20)
		tmp = Float64(Float64(x_46_re * x_46_re) * x_46_re);
	else
		tmp = fma(Float64(x_46_re - x_46_im_m), Float64(Float64(x_46_re + x_46_im_m) * x_46_re), Float64(x_46_re + x_46_re));
	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[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision] - N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$re * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-298], N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] * -3.0), $MachinePrecision] * x$46$im$95$m), $MachinePrecision], If[LessEqual[t$95$0, 5e+20], N[(N[(x$46$re * x$46$re), $MachinePrecision] * x$46$re), $MachinePrecision], N[(N[(x$46$re - x$46$im$95$m), $MachinePrecision] * N[(N[(x$46$re + x$46$im$95$m), $MachinePrecision] * x$46$re), $MachinePrecision] + N[(x$46$re + x$46$re), $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)) < -3.99999999999999965e-298

   1. Initial program 89.8%

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

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

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

      3. flip3-- N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 13.9%

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

   5. Taylor expanded in x.im around inf

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 50.6

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

   7. Applied rewrites 50.6%

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

   if -3.99999999999999965e-298 < (-.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)) < 5e20

   1. Initial program 99.8%

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

   3. Taylor expanded in x.im around 0

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 74.2

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

   5. Applied rewrites 74.2%

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

   if 5e20 < (-.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 59.0%

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

      1. lift--.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. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-lft-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-neg.f64 67.8

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift--.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. difference-of-squares N/A

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

      21. associate-*r* N/A

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

      22. lower-*.f64 N/A

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

   4. Applied rewrites 82.3%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. distribute-lft-in N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-+.f64 N/A

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

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

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

      21. *-commutative N/A

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

      22. lift-+.f64 N/A

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

      23. lift-*.f64 N/A

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

      24. *-commutative N/A

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

      25. lift-*.f64 N/A

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

   6. Applied rewrites 87.3%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.re \cdot x.im\right) \cdot x.im \leq -4 \cdot 10^{-298}:\\ \;\;\;\;\left(\left(x.re \cdot x.im\right) \cdot -3\right) \cdot x.im\\ \mathbf{elif}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.re \cdot x.im\right) \cdot x.im \leq 5 \cdot 10^{+20}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im, \left(x.re + x.im\right) \cdot x.re, x.re + x.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.re - \left(x.re \cdot x.im\_m + x.re \cdot x.im\_m\right) \cdot x.im\_m \leq 5 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(\left(x.re + x.re\right) \cdot x.im\_m, -x.im\_m, \left(x.re - x.im\_m\right) \cdot \left(\mathsf{fma}\left(x.re, \frac{x.re}{x.im\_m}, x.re\right) \cdot x.im\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im\_m, \left(x.re + x.im\_m\right) \cdot x.re, x.re + x.re\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
(FPCore (x.re x.im_m)
 :precision binary64
 (if (<=
      (-
       (* (- (* x.re x.re) (* x.im_m x.im_m)) x.re)
       (* (+ (* x.re x.im_m) (* x.re x.im_m)) x.im_m))
      5e+20)
   (fma
    (* (+ x.re x.re) x.im_m)
    (- x.im_m)
    (* (- x.re x.im_m) (* (fma x.re (/ x.re x.im_m) x.re) x.im_m)))
   (fma (- x.re 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_im_m * x_46_im_m)) * x_46_re) - (((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)) * x_46_im_m)) <= 5e+20) {
		tmp = fma(((x_46_re + x_46_re) * x_46_im_m), -x_46_im_m, ((x_46_re - x_46_im_m) * (fma(x_46_re, (x_46_re / x_46_im_m), x_46_re) * x_46_im_m)));
	} else {
		tmp = fma((x_46_re - x_46_im_m), ((x_46_re + x_46_im_m) * 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(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_re) - Float64(Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_re * x_46_im_m)) * x_46_im_m)) <= 5e+20)
		tmp = fma(Float64(Float64(x_46_re + x_46_re) * x_46_im_m), Float64(-x_46_im_m), Float64(Float64(x_46_re - x_46_im_m) * Float64(fma(x_46_re, Float64(x_46_re / x_46_im_m), x_46_re) * x_46_im_m)));
	else
		tmp = fma(Float64(x_46_re - x_46_im_m), Float64(Float64(x_46_re + x_46_im_m) * x_46_re), Float64(x_46_re + x_46_re));
	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[N[(N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision] - N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$re * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], 5e+20], N[(N[(N[(x$46$re + x$46$re), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] * (-x$46$im$95$m) + N[(N[(x$46$re - x$46$im$95$m), $MachinePrecision] * N[(N[(x$46$re * N[(x$46$re / x$46$im$95$m), $MachinePrecision] + x$46$re), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re - x$46$im$95$m), $MachinePrecision] * N[(N[(x$46$re + x$46$im$95$m), $MachinePrecision] * x$46$re), $MachinePrecision] + 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)) < 5e20

   1. Initial program 92.9%

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

      1. lift--.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. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-lft-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-neg.f64 93.0

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift--.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. difference-of-squares N/A

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

      21. associate-*r* N/A

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

      22. lower-*.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x.im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 97.5

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

   7. Applied rewrites 97.5%

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

   if 5e20 < (-.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 59.0%

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

      1. lift--.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. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-lft-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-neg.f64 67.8

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift--.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. difference-of-squares N/A

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

      21. associate-*r* N/A

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

      22. lower-*.f64 N/A

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

   4. Applied rewrites 82.3%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. distribute-lft-in N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-+.f64 N/A

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

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

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

      21. *-commutative N/A

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

      22. lift-+.f64 N/A

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

      23. lift-*.f64 N/A

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

      24. *-commutative N/A

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

      25. lift-*.f64 N/A

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

   6. Applied rewrites 87.3%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.re \cdot x.im\right) \cdot x.im \leq 5 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(\left(x.re + x.re\right) \cdot x.im, -x.im, \left(x.re - x.im\right) \cdot \left(\mathsf{fma}\left(x.re, \frac{x.re}{x.im}, x.re\right) \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im, \left(x.re + x.im\right) \cdot x.re, x.re + x.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.re - \left(x.re \cdot x.im\_m + x.re \cdot x.im\_m\right) \cdot x.im\_m \leq -4 \cdot 10^{-298}:\\ \;\;\;\;\left(\left(x.re \cdot x.im\_m\right) \cdot -3\right) \cdot x.im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.re\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
(FPCore (x.re x.im_m)
 :precision binary64
 (if (<=
      (-
       (* (- (* x.re x.re) (* x.im_m x.im_m)) x.re)
       (* (+ (* x.re x.im_m) (* x.re x.im_m)) x.im_m))
      -4e-298)
   (* (* (* x.re x.im_m) -3.0) 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_im_m * x_46_im_m)) * x_46_re) - (((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)) * x_46_im_m)) <= -4e-298) {
		tmp = ((x_46_re * x_46_im_m) * -3.0) * 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_46im_m * x_46im_m)) * x_46re) - (((x_46re * x_46im_m) + (x_46re * x_46im_m)) * x_46im_m)) <= (-4d-298)) then
        tmp = ((x_46re * x_46im_m) * (-3.0d0)) * 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_im_m * x_46_im_m)) * x_46_re) - (((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)) * x_46_im_m)) <= -4e-298) {
		tmp = ((x_46_re * x_46_im_m) * -3.0) * 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_im_m * x_46_im_m)) * x_46_re) - (((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)) * x_46_im_m)) <= -4e-298:
		tmp = ((x_46_re * x_46_im_m) * -3.0) * 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(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_re) - Float64(Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_re * x_46_im_m)) * x_46_im_m)) <= -4e-298)
		tmp = Float64(Float64(Float64(x_46_re * x_46_im_m) * -3.0) * x_46_im_m);
	else
		tmp = Float64(Float64(x_46_re * 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_im_m * x_46_im_m)) * x_46_re) - (((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)) * x_46_im_m)) <= -4e-298)
		tmp = ((x_46_re * x_46_im_m) * -3.0) * 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[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision] - N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$re * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], -4e-298], N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] * -3.0), $MachinePrecision] * x$46$im$95$m), $MachinePrecision], N[(N[(x$46$re * x$46$re), $MachinePrecision] * x$46$re), $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)) < -3.99999999999999965e-298

   1. Initial program 89.8%

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

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

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

      3. flip3-- N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 13.9%

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

   5. Taylor expanded in x.im around inf

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 50.6

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

   7. Applied rewrites 50.6%

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

   if -3.99999999999999965e-298 < (-.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 73.8%

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

   3. Taylor expanded in x.im around 0

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 69.1

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

   5. Applied rewrites 69.1%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.re \cdot x.im\right) \cdot x.im \leq -4 \cdot 10^{-298}:\\ \;\;\;\;\left(\left(x.re \cdot x.im\right) \cdot -3\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 60.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.re - \left(x.re \cdot x.im\_m + x.re \cdot x.im\_m\right) \cdot x.im\_m \leq -4 \cdot 10^{-298}:\\ \;\;\;\;\left(\left(x.re \cdot x.im\_m\right) \cdot x.im\_m\right) \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.re\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
(FPCore (x.re x.im_m)
 :precision binary64
 (if (<=
      (-
       (* (- (* x.re x.re) (* x.im_m x.im_m)) x.re)
       (* (+ (* x.re x.im_m) (* x.re x.im_m)) x.im_m))
      -4e-298)
   (* (* (* x.re x.im_m) 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_im_m * x_46_im_m)) * x_46_re) - (((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)) * x_46_im_m)) <= -4e-298) {
		tmp = ((x_46_re * x_46_im_m) * 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_46im_m * x_46im_m)) * x_46re) - (((x_46re * x_46im_m) + (x_46re * x_46im_m)) * x_46im_m)) <= (-4d-298)) then
        tmp = ((x_46re * x_46im_m) * 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_im_m * x_46_im_m)) * x_46_re) - (((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)) * x_46_im_m)) <= -4e-298) {
		tmp = ((x_46_re * x_46_im_m) * 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_im_m * x_46_im_m)) * x_46_re) - (((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)) * x_46_im_m)) <= -4e-298:
		tmp = ((x_46_re * x_46_im_m) * 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(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_re) - Float64(Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_re * x_46_im_m)) * x_46_im_m)) <= -4e-298)
		tmp = Float64(Float64(Float64(x_46_re * x_46_im_m) * x_46_im_m) * -3.0);
	else
		tmp = Float64(Float64(x_46_re * 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_im_m * x_46_im_m)) * x_46_re) - (((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)) * x_46_im_m)) <= -4e-298)
		tmp = ((x_46_re * x_46_im_m) * 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[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision] - N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$re * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], -4e-298], N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] * -3.0), $MachinePrecision], N[(N[(x$46$re * x$46$re), $MachinePrecision] * x$46$re), $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)) < -3.99999999999999965e-298

   1. Initial program 89.8%

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

   3. Taylor expanded in x.im around inf

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

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

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 50.6

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

   5. Applied rewrites 50.6%

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

   if -3.99999999999999965e-298 < (-.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 73.8%

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

   3. Taylor expanded in x.im around 0

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 69.1

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

   5. Applied rewrites 69.1%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.re \cdot x.im\right) \cdot x.im \leq -4 \cdot 10^{-298}:\\ \;\;\;\;\left(\left(x.re \cdot x.im\right) \cdot x.im\right) \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) \cdot x.re - \left(x.re \cdot x.im\_m + x.re \cdot x.im\_m\right) \cdot x.im\_m \leq -4 \cdot 10^{-298}:\\ \;\;\;\;\left(\left(-x.im\_m\right) \cdot x.im\_m\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.re\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
(FPCore (x.re x.im_m)
 :precision binary64
 (if (<=
      (-
       (* (- (* x.re x.re) (* x.im_m x.im_m)) x.re)
       (* (+ (* x.re x.im_m) (* x.re x.im_m)) x.im_m))
      -4e-298)
   (* (* (- 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_im_m * x_46_im_m)) * x_46_re) - (((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)) * x_46_im_m)) <= -4e-298) {
		tmp = (-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_46im_m * x_46im_m)) * x_46re) - (((x_46re * x_46im_m) + (x_46re * x_46im_m)) * x_46im_m)) <= (-4d-298)) then
        tmp = (-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_im_m * x_46_im_m)) * x_46_re) - (((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)) * x_46_im_m)) <= -4e-298) {
		tmp = (-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_im_m * x_46_im_m)) * x_46_re) - (((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)) * x_46_im_m)) <= -4e-298:
		tmp = (-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(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m)) * x_46_re) - Float64(Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_re * x_46_im_m)) * x_46_im_m)) <= -4e-298)
		tmp = Float64(Float64(Float64(-x_46_im_m) * x_46_im_m) * x_46_re);
	else
		tmp = Float64(Float64(x_46_re * 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_im_m * x_46_im_m)) * x_46_re) - (((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)) * x_46_im_m)) <= -4e-298)
		tmp = (-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[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision] - N[(N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$re * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], -4e-298], N[(N[((-x$46$im$95$m) * x$46$im$95$m), $MachinePrecision] * x$46$re), $MachinePrecision], N[(N[(x$46$re * x$46$re), $MachinePrecision] * x$46$re), $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)) < -3.99999999999999965e-298

   1. Initial program 89.8%

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

      1. lift--.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. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-lft-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-neg.f64 89.8

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift--.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. difference-of-squares N/A

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

      21. associate-*r* N/A

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

      22. lower-*.f64 N/A

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

   4. Applied rewrites 99.7%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. distribute-lft-in N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-+.f64 N/A

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

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

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

      21. *-commutative N/A

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

      22. lift-+.f64 N/A

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

      23. lift-*.f64 N/A

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

      24. *-commutative N/A

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

      25. lift-*.f64 N/A

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

   6. Applied rewrites 66.0%

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

   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. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 22.2

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

   9. Applied rewrites 22.2%

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

   if -3.99999999999999965e-298 < (-.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 73.8%

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

   3. Taylor expanded in x.im around 0

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 69.1

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

   5. Applied rewrites 69.1%

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

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.re \cdot x.im\right) \cdot x.im \leq -4 \cdot 10^{-298}:\\ \;\;\;\;\left(\left(-x.im\right) \cdot x.im\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ \begin{array}{l} \mathbf{if}\;x.im\_m \leq 1.18 \cdot 10^{+85}:\\ \;\;\;\;\mathsf{fma}\left(\left(x.re + x.re\right) \cdot x.im\_m, -x.im\_m, \left(\left(x.re + x.im\_m\right) \cdot x.re\right) \cdot \left(x.re - x.im\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{x.re}{x.im\_m}, x.re, -3 \cdot x.im\_m\right) \cdot x.re\right) \cdot x.im\_m\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
(FPCore (x.re x.im_m)
 :precision binary64
 (if (<= x.im_m 1.18e+85)
   (fma
    (* (+ x.re x.re) x.im_m)
    (- x.im_m)
    (* (* (+ x.re x.im_m) x.re) (- x.re x.im_m)))
   (* (* (fma (/ x.re x.im_m) x.re (* -3.0 x.im_m)) x.re) 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_im_m <= 1.18e+85) {
		tmp = fma(((x_46_re + x_46_re) * x_46_im_m), -x_46_im_m, (((x_46_re + x_46_im_m) * x_46_re) * (x_46_re - x_46_im_m)));
	} else {
		tmp = (fma((x_46_re / x_46_im_m), x_46_re, (-3.0 * x_46_im_m)) * x_46_re) * 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_im_m <= 1.18e+85)
		tmp = fma(Float64(Float64(x_46_re + x_46_re) * x_46_im_m), Float64(-x_46_im_m), Float64(Float64(Float64(x_46_re + x_46_im_m) * x_46_re) * Float64(x_46_re - x_46_im_m)));
	else
		tmp = Float64(Float64(fma(Float64(x_46_re / x_46_im_m), x_46_re, Float64(-3.0 * x_46_im_m)) * x_46_re) * 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$46$im$95$m, 1.18e+85], N[(N[(N[(x$46$re + x$46$re), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] * (-x$46$im$95$m) + N[(N[(N[(x$46$re + x$46$im$95$m), $MachinePrecision] * x$46$re), $MachinePrecision] * N[(x$46$re - x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x$46$re / x$46$im$95$m), $MachinePrecision] * x$46$re + N[(-3.0 * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.17999999999999997e85

   1. Initial program 89.4%

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

      1. lift--.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. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-lft-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-neg.f64 92.4

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift--.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. difference-of-squares N/A

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

      21. associate-*r* N/A

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

      22. lower-*.f64 N/A

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

   4. Applied rewrites 96.9%

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

   if 1.17999999999999997e85 < x.im

   1. Initial program 45.6%

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

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

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

      3. flip3-- N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in x.im around inf

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

   7. Applied rewrites 87.9%

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

   8. Taylor expanded in x.im around 0

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

   10. Applied rewrites 13.5%

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

   11. Taylor expanded in x.re around 0

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

   13. Applied rewrites 99.8%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 1.18 \cdot 10^{+85}:\\ \;\;\;\;\mathsf{fma}\left(\left(x.re + x.re\right) \cdot x.im, -x.im, \left(\left(x.re + x.im\right) \cdot x.re\right) \cdot \left(x.re - x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{x.re}{x.im}, x.re, -3 \cdot x.im\right) \cdot x.re\right) \cdot x.im\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ \begin{array}{l} \mathbf{if}\;x.im\_m \leq 5 \cdot 10^{-143}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{x.re}{x.im\_m}, x.re, -3 \cdot x.im\_m\right) \cdot x.re\right) \cdot x.im\_m\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
(FPCore (x.re x.im_m)
 :precision binary64
 (if (<= x.im_m 5e-143)
   (* (* x.re x.re) x.re)
   (* (* (fma (/ x.re x.im_m) x.re (* -3.0 x.im_m)) x.re) 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_im_m <= 5e-143) {
		tmp = (x_46_re * x_46_re) * x_46_re;
	} else {
		tmp = (fma((x_46_re / x_46_im_m), x_46_re, (-3.0 * x_46_im_m)) * x_46_re) * 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_im_m <= 5e-143)
		tmp = Float64(Float64(x_46_re * x_46_re) * x_46_re);
	else
		tmp = Float64(Float64(fma(Float64(x_46_re / x_46_im_m), x_46_re, Float64(-3.0 * x_46_im_m)) * x_46_re) * 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$46$im$95$m, 5e-143], N[(N[(x$46$re * x$46$re), $MachinePrecision] * x$46$re), $MachinePrecision], N[(N[(N[(N[(x$46$re / x$46$im$95$m), $MachinePrecision] * x$46$re + N[(-3.0 * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 5.0000000000000002e-143

   1. Initial program 87.7%

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

   3. Taylor expanded in x.im around 0

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 70.6

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

   5. Applied rewrites 70.6%

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

   if 5.0000000000000002e-143 < x.im

   1. Initial program 69.5%

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

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

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

      3. flip3-- N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 14.1%

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

   5. Taylor expanded in x.im around inf

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

   7. Applied rewrites 91.5%

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

   8. Taylor expanded in x.im around 0

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

   10. Applied rewrites 36.7%

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

   11. Taylor expanded in x.re around 0

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

   13. Applied rewrites 99.7%

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

Alternative 11: 58.0% accurate, 3.6× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ \left(x.re \cdot x.re\right) \cdot x.re \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
(FPCore (x.re x.im_m) :precision binary64 (* (* 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) {
	return (x_46_re * x_46_re) * x_46_re;
}

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.9%

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

3. Taylor expanded in x.im around 0

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

   1. unpow3 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 60.9

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

5. Applied rewrites 60.9%

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

Developer Target 1: 87.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = ((x_46re * x_46re) * (x_46re - x_46im)) + ((x_46re * x_46im) * (x_46re - (3.0d0 * x_46im)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024235 
(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)))