math.cube on complex, real part

Percentage Accurate: 83.1% → 99.7%

Time: 14.1s

Alternatives: 14

Speedup: 1.2×

• 43.7% 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: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ \begin{array}{l} t_0 := x.re\_m \cdot \left(x.re\_m - x.im\_m\right)\\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) - x.im\_m \cdot \left(x.re\_m \cdot x.im\_m + x.re\_m \cdot x.im\_m\right) \leq 10^{+177}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m \cdot \left(x.im\_m + x.im\_m\right), -x.im\_m, \left(x.re\_m + x.im\_m\right) \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.im\_m + x.im\_m, -x.im\_m, t\_0 \cdot \mathsf{fma}\left(\frac{1}{x.im\_m}, x.re\_m \cdot x.im\_m, x.im\_m\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (let* ((t_0 (* x.re_m (- x.re_m x.im_m))))
   (*
    x.re_s
    (if (<=
         (-
          (* x.re_m (- (* x.re_m x.re_m) (* x.im_m x.im_m)))
          (* x.im_m (+ (* x.re_m x.im_m) (* x.re_m x.im_m))))
         1e+177)
      (fma (* x.re_m (+ x.im_m x.im_m)) (- x.im_m) (* (+ x.re_m x.im_m) t_0))
      (fma
       (+ x.im_m x.im_m)
       (- x.im_m)
       (* t_0 (fma (/ 1.0 x.im_m) (* x.re_m x.im_m) x.im_m)))))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double t_0 = x_46_re_m * (x_46_re_m - x_46_im_m);
	double tmp;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)))) <= 1e+177) {
		tmp = fma((x_46_re_m * (x_46_im_m + x_46_im_m)), -x_46_im_m, ((x_46_re_m + x_46_im_m) * t_0));
	} else {
		tmp = fma((x_46_im_m + x_46_im_m), -x_46_im_m, (t_0 * fma((1.0 / x_46_im_m), (x_46_re_m * x_46_im_m), x_46_im_m)));
	}
	return x_46_re_s * tmp;
}

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	t_0 = Float64(x_46_re_m * Float64(x_46_re_m - x_46_im_m))
	tmp = 0.0
	if (Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m))) - Float64(x_46_im_m * Float64(Float64(x_46_re_m * x_46_im_m) + Float64(x_46_re_m * x_46_im_m)))) <= 1e+177)
		tmp = fma(Float64(x_46_re_m * Float64(x_46_im_m + x_46_im_m)), Float64(-x_46_im_m), Float64(Float64(x_46_re_m + x_46_im_m) * t_0));
	else
		tmp = fma(Float64(x_46_im_m + x_46_im_m), Float64(-x_46_im_m), Float64(t_0 * fma(Float64(1.0 / x_46_im_m), Float64(x_46_re_m * x_46_im_m), x_46_im_m)));
	end
	return Float64(x_46_re_s * tmp)
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := Block[{t$95$0 = N[(x$46$re$95$m * N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision]), $MachinePrecision]}, N[(x$46$re$95$s * If[LessEqual[N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im$95$m * N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+177], N[(N[(x$46$re$95$m * N[(x$46$im$95$m + x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * (-x$46$im$95$m) + N[(N[(x$46$re$95$m + x$46$im$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im$95$m + x$46$im$95$m), $MachinePrecision] * (-x$46$im$95$m) + N[(t$95$0 * N[(N[(1.0 / x$46$im$95$m), $MachinePrecision] * N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] + x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < 1e177

   1. Initial program 92.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

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

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

      5. *-commutative N/A

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

      6. distribute-rgt-out N/A

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

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

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

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

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

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

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

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

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

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

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

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

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

      15. --lowering--.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   if 1e177 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im))

   1. Initial program 61.6%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

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

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

      5. *-commutative N/A

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

      6. distribute-rgt-out N/A

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

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

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

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

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

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

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

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

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

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

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

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

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

      15. --lowering--.f64 80.4

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

   4. Applied egg-rr 80.4%

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

   5. Taylor expanded in x.im around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

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

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

      5. /-lowering-/.f64 77.3

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

   7. Simplified 77.3%

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

      1. distribute-lft-in N/A

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

      2. flip-+ N/A

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

      3. +-inverses N/A

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

      4. +-inverses N/A

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

      5. +-inverses N/A

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

      6. +-inverses N/A

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

      7. flip-+ N/A

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

      8. +-lowering-+.f64 84.3

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

   9. Applied egg-rr 84.3%

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

       1. associate-*r/ N/A

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

       2. div-inv N/A

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

       3. *-commutative N/A

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

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

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

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

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

       6. *-lowering-*.f64 78.1

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

   11. Applied egg-rr 78.1%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq 10^{+177}:\\ \;\;\;\;\mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), -x.im, \left(x.re + x.im\right) \cdot \left(x.re \cdot \left(x.re - x.im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.im + x.im, -x.im, \left(x.re \cdot \left(x.re - x.im\right)\right) \cdot \mathsf{fma}\left(\frac{1}{x.im}, x.re \cdot x.im, x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ \begin{array}{l} t_0 := x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) - x.im\_m \cdot \left(x.re\_m \cdot x.im\_m + x.re\_m \cdot x.im\_m\right)\\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-161}:\\ \;\;\;\;-3 \cdot \left(x.im\_m \cdot \left(x.re\_m \cdot x.im\_m\right)\right)\\ \mathbf{elif}\;t\_0 \leq 10^{-99}:\\ \;\;\;\;x.re\_m \cdot \mathsf{fma}\left(x.re\_m, x.re\_m, \left(x.im\_m \cdot x.im\_m\right) \cdot -3\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m - x.im\_m, x.re\_m \cdot \left(x.re\_m + x.im\_m\right), x.im\_m + x.im\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (let* ((t_0
         (-
          (* x.re_m (- (* x.re_m x.re_m) (* x.im_m x.im_m)))
          (* x.im_m (+ (* x.re_m x.im_m) (* x.re_m x.im_m))))))
   (*
    x.re_s
    (if (<= t_0 -2e-161)
      (* -3.0 (* x.im_m (* x.re_m x.im_m)))
      (if (<= t_0 1e-99)
        (* x.re_m (fma x.re_m x.re_m (* (* x.im_m x.im_m) -3.0)))
        (fma
         (- x.re_m x.im_m)
         (* x.re_m (+ x.re_m x.im_m))
         (+ x.im_m x.im_m)))))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double t_0 = (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)));
	double tmp;
	if (t_0 <= -2e-161) {
		tmp = -3.0 * (x_46_im_m * (x_46_re_m * x_46_im_m));
	} else if (t_0 <= 1e-99) {
		tmp = x_46_re_m * fma(x_46_re_m, x_46_re_m, ((x_46_im_m * x_46_im_m) * -3.0));
	} else {
		tmp = fma((x_46_re_m - x_46_im_m), (x_46_re_m * (x_46_re_m + x_46_im_m)), (x_46_im_m + x_46_im_m));
	}
	return x_46_re_s * tmp;
}

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	t_0 = Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m))) - Float64(x_46_im_m * Float64(Float64(x_46_re_m * x_46_im_m) + Float64(x_46_re_m * x_46_im_m))))
	tmp = 0.0
	if (t_0 <= -2e-161)
		tmp = Float64(-3.0 * Float64(x_46_im_m * Float64(x_46_re_m * x_46_im_m)));
	elseif (t_0 <= 1e-99)
		tmp = Float64(x_46_re_m * fma(x_46_re_m, x_46_re_m, Float64(Float64(x_46_im_m * x_46_im_m) * -3.0)));
	else
		tmp = fma(Float64(x_46_re_m - x_46_im_m), Float64(x_46_re_m * Float64(x_46_re_m + x_46_im_m)), Float64(x_46_im_m + x_46_im_m));
	end
	return Float64(x_46_re_s * tmp)
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im$95$m * N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$re$95$s * If[LessEqual[t$95$0, -2e-161], N[(-3.0 * N[(x$46$im$95$m * N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-99], N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$re$95$m + N[(N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision] * N[(x$46$re$95$m * N[(x$46$re$95$m + x$46$im$95$m), $MachinePrecision]), $MachinePrecision] + N[(x$46$im$95$m + x$46$im$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -2.00000000000000006e-161

   1. Initial program 86.2%

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

   3. Taylor expanded in x.re around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

      6. metadata-eval 42.5

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

   5. Simplified 42.5%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 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. *-lowering-*.f64 56.1

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

   7. Applied egg-rr 56.1%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. *-lowering-*.f64 56.1

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

   9. Applied egg-rr 56.1%

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

   if -2.00000000000000006e-161 < (-.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)) < 1e-99

   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.re around 0

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

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

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

      2. +-commutative N/A

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

      3. associate--l+ N/A

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

      4. unpow2 N/A

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

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

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

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

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

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

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

      8. unpow2 N/A

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

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

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

      10. metadata-eval 99.8

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

   5. Simplified 99.8%

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

   if 1e-99 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im))

   1. Initial program 69.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

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

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

      5. *-commutative N/A

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

      6. distribute-rgt-out N/A

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

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

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

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

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

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

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

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

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

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

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

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

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

      15. --lowering--.f64 84.1

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

   4. Applied egg-rr 84.1%

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

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

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. flip-+ N/A

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

      10. +-inverses N/A

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

      11. +-inverses N/A

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

      12. associate-*r/ N/A

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

      13. +-inverses N/A

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

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

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

      15. +-inverses N/A

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

      16. distribute-neg-frac2 N/A

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

      17. +-inverses N/A

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

      18. metadata-eval N/A

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

      19. +-inverses N/A

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

      20. flip-+ N/A

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

      21. distribute-lft-in N/A

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

   6. Applied egg-rr 82.4%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -2 \cdot 10^{-161}:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{elif}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq 10^{-99}:\\ \;\;\;\;x.re \cdot \mathsf{fma}\left(x.re, x.re, \left(x.im \cdot x.im\right) \cdot -3\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im, x.re \cdot \left(x.re + x.im\right), x.im + x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ \begin{array}{l} t_0 := x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) - x.im\_m \cdot \left(x.re\_m \cdot x.im\_m + x.re\_m \cdot x.im\_m\right)\\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-316}:\\ \;\;\;\;-3 \cdot \left(x.im\_m \cdot \left(x.re\_m \cdot x.im\_m\right)\right)\\ \mathbf{elif}\;t\_0 \leq 10^{-99}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m - x.im\_m, x.re\_m \cdot \left(x.re\_m + x.im\_m\right), x.im\_m + x.im\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (let* ((t_0
         (-
          (* x.re_m (- (* x.re_m x.re_m) (* x.im_m x.im_m)))
          (* x.im_m (+ (* x.re_m x.im_m) (* x.re_m x.im_m))))))
   (*
    x.re_s
    (if (<= t_0 -2e-316)
      (* -3.0 (* x.im_m (* x.re_m x.im_m)))
      (if (<= t_0 1e-99)
        (* x.re_m (* x.re_m x.re_m))
        (fma
         (- x.re_m x.im_m)
         (* x.re_m (+ x.re_m x.im_m))
         (+ x.im_m x.im_m)))))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double t_0 = (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)));
	double tmp;
	if (t_0 <= -2e-316) {
		tmp = -3.0 * (x_46_im_m * (x_46_re_m * x_46_im_m));
	} else if (t_0 <= 1e-99) {
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	} else {
		tmp = fma((x_46_re_m - x_46_im_m), (x_46_re_m * (x_46_re_m + x_46_im_m)), (x_46_im_m + x_46_im_m));
	}
	return x_46_re_s * tmp;
}

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	t_0 = Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m))) - Float64(x_46_im_m * Float64(Float64(x_46_re_m * x_46_im_m) + Float64(x_46_re_m * x_46_im_m))))
	tmp = 0.0
	if (t_0 <= -2e-316)
		tmp = Float64(-3.0 * Float64(x_46_im_m * Float64(x_46_re_m * x_46_im_m)));
	elseif (t_0 <= 1e-99)
		tmp = Float64(x_46_re_m * Float64(x_46_re_m * x_46_re_m));
	else
		tmp = fma(Float64(x_46_re_m - x_46_im_m), Float64(x_46_re_m * Float64(x_46_re_m + x_46_im_m)), Float64(x_46_im_m + x_46_im_m));
	end
	return Float64(x_46_re_s * tmp)
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im$95$m * N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$re$95$s * If[LessEqual[t$95$0, -2e-316], N[(-3.0 * N[(x$46$im$95$m * N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-99], N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision] * N[(x$46$re$95$m * N[(x$46$re$95$m + x$46$im$95$m), $MachinePrecision]), $MachinePrecision] + N[(x$46$im$95$m + x$46$im$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -2.000000017e-316

   1. Initial program 87.4%

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

   3. Taylor expanded in x.re around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

      6. metadata-eval 44.2

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

   5. Simplified 44.2%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 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. *-lowering-*.f64 56.6

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

   7. Applied egg-rr 56.6%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. *-lowering-*.f64 56.6

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

   9. Applied egg-rr 56.6%

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

   if -2.000000017e-316 < (-.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)) < 1e-99

   1. Initial program 99.9%

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

   3. Taylor expanded in x.re around inf

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

      1. cube-mult N/A

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

      2. unpow2 N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 79.0

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

   5. Simplified 79.0%

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

   if 1e-99 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im))

   1. Initial program 69.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

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

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

      5. *-commutative N/A

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

      6. distribute-rgt-out N/A

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

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

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

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

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

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

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

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

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

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

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

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

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

      15. --lowering--.f64 84.1

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

   4. Applied egg-rr 84.1%

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

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

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. flip-+ N/A

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

      10. +-inverses N/A

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

      11. +-inverses N/A

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

      12. associate-*r/ N/A

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

      13. +-inverses N/A

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

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

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

      15. +-inverses N/A

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

      16. distribute-neg-frac2 N/A

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

      17. +-inverses N/A

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

      18. metadata-eval N/A

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

      19. +-inverses N/A

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

      20. flip-+ N/A

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

      21. distribute-lft-in N/A

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

   6. Applied egg-rr 82.4%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -2 \cdot 10^{-316}:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{elif}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq 10^{-99}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im, x.re \cdot \left(x.re + x.im\right), x.im + x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) - x.im\_m \cdot \left(x.re\_m \cdot x.im\_m + x.re\_m \cdot x.im\_m\right) \leq 10^{-99}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m \cdot \left(x.im\_m + x.im\_m\right), -x.im\_m, \left(x.re\_m + x.im\_m\right) \cdot \left(x.re\_m \cdot \left(x.re\_m - x.im\_m\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m - x.im\_m, x.re\_m \cdot \left(x.re\_m + x.im\_m\right), x.im\_m + x.im\_m\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<=
       (-
        (* x.re_m (- (* x.re_m x.re_m) (* x.im_m x.im_m)))
        (* x.im_m (+ (* x.re_m x.im_m) (* x.re_m x.im_m))))
       1e-99)
    (fma
     (* x.re_m (+ x.im_m x.im_m))
     (- x.im_m)
     (* (+ x.re_m x.im_m) (* x.re_m (- x.re_m x.im_m))))
    (fma (- x.re_m x.im_m) (* x.re_m (+ x.re_m x.im_m)) (+ x.im_m x.im_m)))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)))) <= 1e-99) {
		tmp = fma((x_46_re_m * (x_46_im_m + x_46_im_m)), -x_46_im_m, ((x_46_re_m + x_46_im_m) * (x_46_re_m * (x_46_re_m - x_46_im_m))));
	} else {
		tmp = fma((x_46_re_m - x_46_im_m), (x_46_re_m * (x_46_re_m + x_46_im_m)), (x_46_im_m + x_46_im_m));
	}
	return x_46_re_s * tmp;
}

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m))) - Float64(x_46_im_m * Float64(Float64(x_46_re_m * x_46_im_m) + Float64(x_46_re_m * x_46_im_m)))) <= 1e-99)
		tmp = fma(Float64(x_46_re_m * Float64(x_46_im_m + x_46_im_m)), Float64(-x_46_im_m), Float64(Float64(x_46_re_m + x_46_im_m) * Float64(x_46_re_m * Float64(x_46_re_m - x_46_im_m))));
	else
		tmp = fma(Float64(x_46_re_m - x_46_im_m), Float64(x_46_re_m * Float64(x_46_re_m + x_46_im_m)), Float64(x_46_im_m + x_46_im_m));
	end
	return Float64(x_46_re_s * tmp)
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * If[LessEqual[N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im$95$m * N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-99], N[(N[(x$46$re$95$m * N[(x$46$im$95$m + x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * (-x$46$im$95$m) + N[(N[(x$46$re$95$m + x$46$im$95$m), $MachinePrecision] * N[(x$46$re$95$m * N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision] * N[(x$46$re$95$m * N[(x$46$re$95$m + x$46$im$95$m), $MachinePrecision]), $MachinePrecision] + N[(x$46$im$95$m + x$46$im$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < 1e-99

   1. Initial program 90.9%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

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

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

      5. *-commutative N/A

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

      6. distribute-rgt-out N/A

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

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

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

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

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

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

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

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

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

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

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

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

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

      15. --lowering--.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   if 1e-99 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im))

   1. Initial program 69.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

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

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

      5. *-commutative N/A

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

      6. distribute-rgt-out N/A

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

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

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

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

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

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

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

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

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

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

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

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

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

      15. --lowering--.f64 84.1

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

   4. Applied egg-rr 84.1%

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

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

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. flip-+ N/A

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

      10. +-inverses N/A

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

      11. +-inverses N/A

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

      12. associate-*r/ N/A

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

      13. +-inverses N/A

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

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

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

      15. +-inverses N/A

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

      16. distribute-neg-frac2 N/A

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

      17. +-inverses N/A

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

      18. metadata-eval N/A

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

      19. +-inverses N/A

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

      20. flip-+ N/A

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

      21. distribute-lft-in N/A

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

   6. Applied egg-rr 82.4%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq 10^{-99}:\\ \;\;\;\;\mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), -x.im, \left(x.re + x.im\right) \cdot \left(x.re \cdot \left(x.re - x.im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im, x.re \cdot \left(x.re + x.im\right), x.im + x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) - x.im\_m \cdot \left(x.re\_m \cdot x.im\_m + x.re\_m \cdot x.im\_m\right) \leq 10^{-99}:\\ \;\;\;\;\left(x.re\_m + x.im\_m\right) \cdot \left(x.re\_m \cdot \left(x.re\_m - x.im\_m\right)\right) - x.im\_m \cdot \left(x.re\_m \cdot \left(x.im\_m + x.im\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m - x.im\_m, x.re\_m \cdot \left(x.re\_m + x.im\_m\right), x.im\_m + x.im\_m\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<=
       (-
        (* x.re_m (- (* x.re_m x.re_m) (* x.im_m x.im_m)))
        (* x.im_m (+ (* x.re_m x.im_m) (* x.re_m x.im_m))))
       1e-99)
    (-
     (* (+ x.re_m x.im_m) (* x.re_m (- x.re_m x.im_m)))
     (* x.im_m (* x.re_m (+ x.im_m x.im_m))))
    (fma (- x.re_m x.im_m) (* x.re_m (+ x.re_m x.im_m)) (+ x.im_m x.im_m)))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)))) <= 1e-99) {
		tmp = ((x_46_re_m + x_46_im_m) * (x_46_re_m * (x_46_re_m - x_46_im_m))) - (x_46_im_m * (x_46_re_m * (x_46_im_m + x_46_im_m)));
	} else {
		tmp = fma((x_46_re_m - x_46_im_m), (x_46_re_m * (x_46_re_m + x_46_im_m)), (x_46_im_m + x_46_im_m));
	}
	return x_46_re_s * tmp;
}

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m))) - Float64(x_46_im_m * Float64(Float64(x_46_re_m * x_46_im_m) + Float64(x_46_re_m * x_46_im_m)))) <= 1e-99)
		tmp = Float64(Float64(Float64(x_46_re_m + x_46_im_m) * Float64(x_46_re_m * Float64(x_46_re_m - x_46_im_m))) - Float64(x_46_im_m * Float64(x_46_re_m * Float64(x_46_im_m + x_46_im_m))));
	else
		tmp = fma(Float64(x_46_re_m - x_46_im_m), Float64(x_46_re_m * Float64(x_46_re_m + x_46_im_m)), Float64(x_46_im_m + x_46_im_m));
	end
	return Float64(x_46_re_s * tmp)
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * If[LessEqual[N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im$95$m * N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-99], N[(N[(N[(x$46$re$95$m + x$46$im$95$m), $MachinePrecision] * N[(x$46$re$95$m * N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im$95$m * N[(x$46$re$95$m * N[(x$46$im$95$m + x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision] * N[(x$46$re$95$m * N[(x$46$re$95$m + x$46$im$95$m), $MachinePrecision]), $MachinePrecision] + N[(x$46$im$95$m + x$46$im$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < 1e-99

   1. Initial program 90.9%

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

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

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

      2. difference-of-squares N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

      11. distribute-rgt-out N/A

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

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

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

      13. +-lowering-+.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   if 1e-99 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im))

   1. Initial program 69.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

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

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

      5. *-commutative N/A

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

      6. distribute-rgt-out N/A

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

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

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

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

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

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

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

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

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

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

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

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

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

      15. --lowering--.f64 84.1

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

   4. Applied egg-rr 84.1%

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

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

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. flip-+ N/A

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

      10. +-inverses N/A

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

      11. +-inverses N/A

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

      12. associate-*r/ N/A

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

      13. +-inverses N/A

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

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

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

      15. +-inverses N/A

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

      16. distribute-neg-frac2 N/A

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

      17. +-inverses N/A

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

      18. metadata-eval N/A

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

      19. +-inverses N/A

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

      20. flip-+ N/A

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

      21. distribute-lft-in N/A

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

   6. Applied egg-rr 82.4%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq 10^{-99}:\\ \;\;\;\;\left(x.re + x.im\right) \cdot \left(x.re \cdot \left(x.re - x.im\right)\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im, x.re \cdot \left(x.re + x.im\right), x.im + x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) - x.im\_m \cdot \left(x.re\_m \cdot x.im\_m + x.re\_m \cdot x.im\_m\right) \leq -2 \cdot 10^{-316}:\\ \;\;\;\;-3 \cdot \left(x.im\_m \cdot \left(x.re\_m \cdot x.im\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<=
       (-
        (* x.re_m (- (* x.re_m x.re_m) (* x.im_m x.im_m)))
        (* x.im_m (+ (* x.re_m x.im_m) (* x.re_m x.im_m))))
       -2e-316)
    (* -3.0 (* x.im_m (* x.re_m x.im_m)))
    (* x.re_m (* x.re_m x.re_m)))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)))) <= -2e-316) {
		tmp = -3.0 * (x_46_im_m * (x_46_re_m * x_46_im_m));
	} else {
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	}
	return x_46_re_s * tmp;
}

Fortran

x.im_m = abs(x_46im)
x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (((x_46re_m * ((x_46re_m * x_46re_m) - (x_46im_m * x_46im_m))) - (x_46im_m * ((x_46re_m * x_46im_m) + (x_46re_m * x_46im_m)))) <= (-2d-316)) then
        tmp = (-3.0d0) * (x_46im_m * (x_46re_m * x_46im_m))
    else
        tmp = x_46re_m * (x_46re_m * x_46re_m)
    end if
    code = x_46re_s * tmp
end function

Java

x.im_m = Math.abs(x_46_im);
x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)))) <= -2e-316) {
		tmp = -3.0 * (x_46_im_m * (x_46_re_m * x_46_im_m));
	} else {
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	}
	return x_46_re_s * tmp;
}

Python

x.im_m = math.fabs(x_46_im)
x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if ((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)))) <= -2e-316:
		tmp = -3.0 * (x_46_im_m * (x_46_re_m * x_46_im_m))
	else:
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m)
	return x_46_re_s * tmp

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m))) - Float64(x_46_im_m * Float64(Float64(x_46_re_m * x_46_im_m) + Float64(x_46_re_m * x_46_im_m)))) <= -2e-316)
		tmp = Float64(-3.0 * Float64(x_46_im_m * Float64(x_46_re_m * x_46_im_m)));
	else
		tmp = Float64(x_46_re_m * Float64(x_46_re_m * x_46_re_m));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.im_m = abs(x_46_im);
x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)))) <= -2e-316)
		tmp = -3.0 * (x_46_im_m * (x_46_re_m * x_46_im_m));
	else
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * If[LessEqual[N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im$95$m * N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-316], N[(-3.0 * N[(x$46$im$95$m * N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -2.000000017e-316

   1. Initial program 87.4%

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

   3. Taylor expanded in x.re around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

      6. metadata-eval 44.2

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

   5. Simplified 44.2%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 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. *-lowering-*.f64 56.6

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

   7. Applied egg-rr 56.6%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. *-lowering-*.f64 56.6

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

   9. Applied egg-rr 56.6%

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

   if -2.000000017e-316 < (-.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 77.6%

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

   3. Taylor expanded in x.re around inf

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

      1. cube-mult N/A

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

      2. unpow2 N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 65.1

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

   5. Simplified 65.1%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -2 \cdot 10^{-316}:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) - x.im\_m \cdot \left(x.re\_m \cdot x.im\_m + x.re\_m \cdot x.im\_m\right) \leq -2 \cdot 10^{-316}:\\ \;\;\;\;\left(x.re\_m \cdot x.im\_m\right) \cdot \left(x.im\_m \cdot -3\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<=
       (-
        (* x.re_m (- (* x.re_m x.re_m) (* x.im_m x.im_m)))
        (* x.im_m (+ (* x.re_m x.im_m) (* x.re_m x.im_m))))
       -2e-316)
    (* (* x.re_m x.im_m) (* x.im_m -3.0))
    (* x.re_m (* x.re_m x.re_m)))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)))) <= -2e-316) {
		tmp = (x_46_re_m * x_46_im_m) * (x_46_im_m * -3.0);
	} else {
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	}
	return x_46_re_s * tmp;
}

Fortran

x.im_m = abs(x_46im)
x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (((x_46re_m * ((x_46re_m * x_46re_m) - (x_46im_m * x_46im_m))) - (x_46im_m * ((x_46re_m * x_46im_m) + (x_46re_m * x_46im_m)))) <= (-2d-316)) then
        tmp = (x_46re_m * x_46im_m) * (x_46im_m * (-3.0d0))
    else
        tmp = x_46re_m * (x_46re_m * x_46re_m)
    end if
    code = x_46re_s * tmp
end function

Java

x.im_m = Math.abs(x_46_im);
x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)))) <= -2e-316) {
		tmp = (x_46_re_m * x_46_im_m) * (x_46_im_m * -3.0);
	} else {
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	}
	return x_46_re_s * tmp;
}

Python

x.im_m = math.fabs(x_46_im)
x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if ((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)))) <= -2e-316:
		tmp = (x_46_re_m * x_46_im_m) * (x_46_im_m * -3.0)
	else:
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m)
	return x_46_re_s * tmp

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m))) - Float64(x_46_im_m * Float64(Float64(x_46_re_m * x_46_im_m) + Float64(x_46_re_m * x_46_im_m)))) <= -2e-316)
		tmp = Float64(Float64(x_46_re_m * x_46_im_m) * Float64(x_46_im_m * -3.0));
	else
		tmp = Float64(x_46_re_m * Float64(x_46_re_m * x_46_re_m));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.im_m = abs(x_46_im);
x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)))) <= -2e-316)
		tmp = (x_46_re_m * x_46_im_m) * (x_46_im_m * -3.0);
	else
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * If[LessEqual[N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im$95$m * N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-316], N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m * -3.0), $MachinePrecision]), $MachinePrecision], N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -2.000000017e-316

   1. Initial program 87.4%

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

   3. Taylor expanded in x.re around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

      6. metadata-eval 44.2

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

   5. Simplified 44.2%

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

      1. associate-*l* N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. *-lowering-*.f64 56.6

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

   7. Applied egg-rr 56.6%

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

   if -2.000000017e-316 < (-.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 77.6%

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

   3. Taylor expanded in x.re around inf

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

      1. cube-mult N/A

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

      2. unpow2 N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 65.1

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

   5. Simplified 65.1%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -2 \cdot 10^{-316}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.im \cdot -3\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 96.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) - x.im\_m \cdot \left(x.re\_m \cdot x.im\_m + x.re\_m \cdot x.im\_m\right) \leq -2 \cdot 10^{-316}:\\ \;\;\;\;x.im\_m \cdot \left(\left(x.re\_m \cdot x.im\_m\right) \cdot -3\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<=
       (-
        (* x.re_m (- (* x.re_m x.re_m) (* x.im_m x.im_m)))
        (* x.im_m (+ (* x.re_m x.im_m) (* x.re_m x.im_m))))
       -2e-316)
    (* x.im_m (* (* x.re_m x.im_m) -3.0))
    (* x.re_m (* x.re_m x.re_m)))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)))) <= -2e-316) {
		tmp = x_46_im_m * ((x_46_re_m * x_46_im_m) * -3.0);
	} else {
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	}
	return x_46_re_s * tmp;
}

Fortran

x.im_m = abs(x_46im)
x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (((x_46re_m * ((x_46re_m * x_46re_m) - (x_46im_m * x_46im_m))) - (x_46im_m * ((x_46re_m * x_46im_m) + (x_46re_m * x_46im_m)))) <= (-2d-316)) then
        tmp = x_46im_m * ((x_46re_m * x_46im_m) * (-3.0d0))
    else
        tmp = x_46re_m * (x_46re_m * x_46re_m)
    end if
    code = x_46re_s * tmp
end function

Java

x.im_m = Math.abs(x_46_im);
x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)))) <= -2e-316) {
		tmp = x_46_im_m * ((x_46_re_m * x_46_im_m) * -3.0);
	} else {
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	}
	return x_46_re_s * tmp;
}

Python

x.im_m = math.fabs(x_46_im)
x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if ((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)))) <= -2e-316:
		tmp = x_46_im_m * ((x_46_re_m * x_46_im_m) * -3.0)
	else:
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m)
	return x_46_re_s * tmp

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m))) - Float64(x_46_im_m * Float64(Float64(x_46_re_m * x_46_im_m) + Float64(x_46_re_m * x_46_im_m)))) <= -2e-316)
		tmp = Float64(x_46_im_m * Float64(Float64(x_46_re_m * x_46_im_m) * -3.0));
	else
		tmp = Float64(x_46_re_m * Float64(x_46_re_m * x_46_re_m));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.im_m = abs(x_46_im);
x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)))) <= -2e-316)
		tmp = x_46_im_m * ((x_46_re_m * x_46_im_m) * -3.0);
	else
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * If[LessEqual[N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im$95$m * N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-316], N[(x$46$im$95$m * N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision], N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -2.000000017e-316

   1. Initial program 87.4%

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

   3. Taylor expanded in x.re around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

      6. metadata-eval 44.2

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

   5. Simplified 44.2%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 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. *-lowering-*.f64 56.6

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

   7. Applied egg-rr 56.6%

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

   if -2.000000017e-316 < (-.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 77.6%

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

   3. Taylor expanded in x.re around inf

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

      1. cube-mult N/A

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

      2. unpow2 N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 65.1

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

   5. Simplified 65.1%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -2 \cdot 10^{-316}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.im\right) \cdot -3\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 90.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) - x.im\_m \cdot \left(x.re\_m \cdot x.im\_m + x.re\_m \cdot x.im\_m\right) \leq -2 \cdot 10^{-316}:\\ \;\;\;\;x.re\_m \cdot \left(\left(x.im\_m \cdot x.im\_m\right) \cdot -3\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<=
       (-
        (* x.re_m (- (* x.re_m x.re_m) (* x.im_m x.im_m)))
        (* x.im_m (+ (* x.re_m x.im_m) (* x.re_m x.im_m))))
       -2e-316)
    (* x.re_m (* (* x.im_m x.im_m) -3.0))
    (* x.re_m (* x.re_m x.re_m)))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)))) <= -2e-316) {
		tmp = x_46_re_m * ((x_46_im_m * x_46_im_m) * -3.0);
	} else {
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	}
	return x_46_re_s * tmp;
}

Fortran

x.im_m = abs(x_46im)
x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (((x_46re_m * ((x_46re_m * x_46re_m) - (x_46im_m * x_46im_m))) - (x_46im_m * ((x_46re_m * x_46im_m) + (x_46re_m * x_46im_m)))) <= (-2d-316)) then
        tmp = x_46re_m * ((x_46im_m * x_46im_m) * (-3.0d0))
    else
        tmp = x_46re_m * (x_46re_m * x_46re_m)
    end if
    code = x_46re_s * tmp
end function

Java

x.im_m = Math.abs(x_46_im);
x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)))) <= -2e-316) {
		tmp = x_46_re_m * ((x_46_im_m * x_46_im_m) * -3.0);
	} else {
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	}
	return x_46_re_s * tmp;
}

Python

x.im_m = math.fabs(x_46_im)
x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if ((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)))) <= -2e-316:
		tmp = x_46_re_m * ((x_46_im_m * x_46_im_m) * -3.0)
	else:
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m)
	return x_46_re_s * tmp

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m))) - Float64(x_46_im_m * Float64(Float64(x_46_re_m * x_46_im_m) + Float64(x_46_re_m * x_46_im_m)))) <= -2e-316)
		tmp = Float64(x_46_re_m * Float64(Float64(x_46_im_m * x_46_im_m) * -3.0));
	else
		tmp = Float64(x_46_re_m * Float64(x_46_re_m * x_46_re_m));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.im_m = abs(x_46_im);
x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)))) <= -2e-316)
		tmp = x_46_re_m * ((x_46_im_m * x_46_im_m) * -3.0);
	else
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * If[LessEqual[N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im$95$m * N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-316], N[(x$46$re$95$m * N[(N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision], N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -2.000000017e-316

   1. Initial program 87.4%

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

   3. Taylor expanded in x.re around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

      6. metadata-eval 44.2

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

   5. Simplified 44.2%

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

   if -2.000000017e-316 < (-.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 77.6%

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

   3. Taylor expanded in x.re around inf

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

      1. cube-mult N/A

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

      2. unpow2 N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 65.1

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

   5. Simplified 65.1%

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

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -2 \cdot 10^{-316}:\\ \;\;\;\;x.re \cdot \left(\left(x.im \cdot x.im\right) \cdot -3\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) - x.im\_m \cdot \left(x.re\_m \cdot x.im\_m + x.re\_m \cdot x.im\_m\right) \leq -1 \cdot 10^{-71}:\\ \;\;\;\;x.im\_m \cdot \left(x.im\_m \cdot \left(-2 - x.re\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<=
       (-
        (* x.re_m (- (* x.re_m x.re_m) (* x.im_m x.im_m)))
        (* x.im_m (+ (* x.re_m x.im_m) (* x.re_m x.im_m))))
       -1e-71)
    (* x.im_m (* x.im_m (- -2.0 x.re_m)))
    (* x.re_m (* x.re_m x.re_m)))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)))) <= -1e-71) {
		tmp = x_46_im_m * (x_46_im_m * (-2.0 - x_46_re_m));
	} else {
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	}
	return x_46_re_s * tmp;
}

Fortran

x.im_m = abs(x_46im)
x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (((x_46re_m * ((x_46re_m * x_46re_m) - (x_46im_m * x_46im_m))) - (x_46im_m * ((x_46re_m * x_46im_m) + (x_46re_m * x_46im_m)))) <= (-1d-71)) then
        tmp = x_46im_m * (x_46im_m * ((-2.0d0) - x_46re_m))
    else
        tmp = x_46re_m * (x_46re_m * x_46re_m)
    end if
    code = x_46re_s * tmp
end function

Java

x.im_m = Math.abs(x_46_im);
x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)))) <= -1e-71) {
		tmp = x_46_im_m * (x_46_im_m * (-2.0 - x_46_re_m));
	} else {
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	}
	return x_46_re_s * tmp;
}

Python

x.im_m = math.fabs(x_46_im)
x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if ((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)))) <= -1e-71:
		tmp = x_46_im_m * (x_46_im_m * (-2.0 - x_46_re_m))
	else:
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m)
	return x_46_re_s * tmp

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m))) - Float64(x_46_im_m * Float64(Float64(x_46_re_m * x_46_im_m) + Float64(x_46_re_m * x_46_im_m)))) <= -1e-71)
		tmp = Float64(x_46_im_m * Float64(x_46_im_m * Float64(-2.0 - x_46_re_m)));
	else
		tmp = Float64(x_46_re_m * Float64(x_46_re_m * x_46_re_m));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.im_m = abs(x_46_im);
x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)))) <= -1e-71)
		tmp = x_46_im_m * (x_46_im_m * (-2.0 - x_46_re_m));
	else
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * If[LessEqual[N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im$95$m * N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-71], N[(x$46$im$95$m * N[(x$46$im$95$m * N[(-2.0 - x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -9.9999999999999992e-72

   1. Initial program 85.5%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

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

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

      5. *-commutative N/A

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

      6. distribute-rgt-out N/A

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

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

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

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

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

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

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

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

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

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

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

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

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

      15. --lowering--.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x.im around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

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

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

      5. /-lowering-/.f64 96.7

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

   7. Simplified 96.7%

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

      1. distribute-lft-in N/A

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

      2. flip-+ N/A

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

      3. +-inverses N/A

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

      4. +-inverses N/A

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

      5. +-inverses N/A

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

      6. +-inverses N/A

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

      7. flip-+ N/A

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

      8. +-lowering-+.f64 60.9

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

   9. Applied egg-rr 60.9%

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

   10. Taylor expanded in x.im around inf

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

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

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

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

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

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

       5. sub-neg N/A

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

       6. metadata-eval N/A

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

       7. +-commutative N/A

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

       8. mul-1-neg N/A

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

       9. unsub-neg N/A

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

       10. --lowering--.f64 20.1

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

   12. Simplified 20.1%

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

   if -9.9999999999999992e-72 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im))

   1. Initial program 79.5%

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

   3. Taylor expanded in x.re around inf

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

      1. cube-mult N/A

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

      2. unpow2 N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 63.0

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

   5. Simplified 63.0%

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

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -1 \cdot 10^{-71}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-2 - x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 72.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) - x.im\_m \cdot \left(x.re\_m \cdot x.im\_m + x.re\_m \cdot x.im\_m\right) \leq -1 \cdot 10^{-71}:\\ \;\;\;\;x.im\_m \cdot \left(x.im\_m \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<=
       (-
        (* x.re_m (- (* x.re_m x.re_m) (* x.im_m x.im_m)))
        (* x.im_m (+ (* x.re_m x.im_m) (* x.re_m x.im_m))))
       -1e-71)
    (* x.im_m (* x.im_m -2.0))
    (* x.re_m (* x.re_m x.re_m)))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)))) <= -1e-71) {
		tmp = x_46_im_m * (x_46_im_m * -2.0);
	} else {
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	}
	return x_46_re_s * tmp;
}

Fortran

x.im_m = abs(x_46im)
x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (((x_46re_m * ((x_46re_m * x_46re_m) - (x_46im_m * x_46im_m))) - (x_46im_m * ((x_46re_m * x_46im_m) + (x_46re_m * x_46im_m)))) <= (-1d-71)) then
        tmp = x_46im_m * (x_46im_m * (-2.0d0))
    else
        tmp = x_46re_m * (x_46re_m * x_46re_m)
    end if
    code = x_46re_s * tmp
end function

Java

x.im_m = Math.abs(x_46_im);
x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)))) <= -1e-71) {
		tmp = x_46_im_m * (x_46_im_m * -2.0);
	} else {
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	}
	return x_46_re_s * tmp;
}

Python

x.im_m = math.fabs(x_46_im)
x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if ((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)))) <= -1e-71:
		tmp = x_46_im_m * (x_46_im_m * -2.0)
	else:
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m)
	return x_46_re_s * tmp

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m))) - Float64(x_46_im_m * Float64(Float64(x_46_re_m * x_46_im_m) + Float64(x_46_re_m * x_46_im_m)))) <= -1e-71)
		tmp = Float64(x_46_im_m * Float64(x_46_im_m * -2.0));
	else
		tmp = Float64(x_46_re_m * Float64(x_46_re_m * x_46_re_m));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.im_m = abs(x_46_im);
x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)))) <= -1e-71)
		tmp = x_46_im_m * (x_46_im_m * -2.0);
	else
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * If[LessEqual[N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im$95$m * N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-71], N[(x$46$im$95$m * N[(x$46$im$95$m * -2.0), $MachinePrecision]), $MachinePrecision], N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -9.9999999999999992e-72

   1. Initial program 85.5%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

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

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

      5. *-commutative N/A

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

      6. distribute-rgt-out N/A

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

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

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

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

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

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

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

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

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

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

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

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

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

      15. --lowering--.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x.im around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

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

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

      5. /-lowering-/.f64 96.7

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

   7. Simplified 96.7%

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

      1. distribute-lft-in N/A

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

      2. flip-+ N/A

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

      3. +-inverses N/A

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

      4. +-inverses N/A

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

      5. +-inverses N/A

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

      6. +-inverses N/A

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

      7. flip-+ N/A

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

      8. +-lowering-+.f64 60.9

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

   9. Applied egg-rr 60.9%

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

   10. Taylor expanded in x.re around 0

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

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

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

       6. *-commutative N/A

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

       7. *-lowering-*.f64 18.6

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

   12. Simplified 18.6%

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

   if -9.9999999999999992e-72 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im))

   1. Initial program 79.5%

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

   3. Taylor expanded in x.re around inf

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

      1. cube-mult N/A

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

      2. unpow2 N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 63.0

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

   5. Simplified 63.0%

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

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -1 \cdot 10^{-71}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 98.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 6.5 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(x.im\_m, x.im\_m \cdot \left(x.re\_m \cdot -3\right), x.re\_m \cdot \left(x.re\_m \cdot x.re\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m - x.im\_m, x.re\_m \cdot \left(x.re\_m + x.im\_m\right), x.im\_m + x.im\_m\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 6.5e+20)
    (fma x.im_m (* x.im_m (* x.re_m -3.0)) (* x.re_m (* x.re_m x.re_m)))
    (fma (- x.re_m x.im_m) (* x.re_m (+ x.re_m x.im_m)) (+ x.im_m x.im_m)))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_re_m <= 6.5e+20) {
		tmp = fma(x_46_im_m, (x_46_im_m * (x_46_re_m * -3.0)), (x_46_re_m * (x_46_re_m * x_46_re_m)));
	} else {
		tmp = fma((x_46_re_m - x_46_im_m), (x_46_re_m * (x_46_re_m + x_46_im_m)), (x_46_im_m + x_46_im_m));
	}
	return x_46_re_s * tmp;
}

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (x_46_re_m <= 6.5e+20)
		tmp = fma(x_46_im_m, Float64(x_46_im_m * Float64(x_46_re_m * -3.0)), Float64(x_46_re_m * Float64(x_46_re_m * x_46_re_m)));
	else
		tmp = fma(Float64(x_46_re_m - x_46_im_m), Float64(x_46_re_m * Float64(x_46_re_m + x_46_im_m)), Float64(x_46_im_m + x_46_im_m));
	end
	return Float64(x_46_re_s * tmp)
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 6.5e+20], N[(x$46$im$95$m * N[(x$46$im$95$m * N[(x$46$re$95$m * -3.0), $MachinePrecision]), $MachinePrecision] + N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision] * N[(x$46$re$95$m * N[(x$46$re$95$m + x$46$im$95$m), $MachinePrecision]), $MachinePrecision] + N[(x$46$im$95$m + x$46$im$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 6.5e20

   1. Initial program 82.3%

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

   3. Taylor expanded in x.re around 0

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

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

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

      2. +-commutative N/A

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

      3. associate--l+ N/A

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

      4. unpow2 N/A

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

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

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

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

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

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

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

      8. unpow2 N/A

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

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

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

      10. metadata-eval 85.3

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

   5. Simplified 85.3%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      10. *-lowering-*.f64 91.7

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

   7. Applied egg-rr 91.7%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 91.7

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

   9. Applied egg-rr 91.7%

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

   if 6.5e20 < x.re

   1. Initial program 79.5%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

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

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

      5. *-commutative N/A

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

      6. distribute-rgt-out N/A

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

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

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

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

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

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

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

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

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

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

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

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

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

      15. --lowering--.f64 88.0

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

   4. Applied egg-rr 88.0%

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

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

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. flip-+ N/A

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

      10. +-inverses N/A

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

      11. +-inverses N/A

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

      12. associate-*r/ N/A

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

      13. +-inverses N/A

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

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

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

      15. +-inverses N/A

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

      16. distribute-neg-frac2 N/A

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

      17. +-inverses N/A

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

      18. metadata-eval N/A

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

      19. +-inverses N/A

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

      20. flip-+ N/A

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

      21. distribute-lft-in N/A

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

   6. Applied egg-rr 99.8%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 6.5 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(x.im, x.im \cdot \left(x.re \cdot -3\right), x.re \cdot \left(x.re \cdot x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im, x.re \cdot \left(x.re + x.im\right), x.im + x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 98.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 6.5 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(x.im\_m, x.re\_m \cdot \left(x.im\_m \cdot -3\right), x.re\_m \cdot \left(x.re\_m \cdot x.re\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m - x.im\_m, x.re\_m \cdot \left(x.re\_m + x.im\_m\right), x.im\_m + x.im\_m\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 6.5e+20)
    (fma x.im_m (* x.re_m (* x.im_m -3.0)) (* x.re_m (* x.re_m x.re_m)))
    (fma (- x.re_m x.im_m) (* x.re_m (+ x.re_m x.im_m)) (+ x.im_m x.im_m)))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_re_m <= 6.5e+20) {
		tmp = fma(x_46_im_m, (x_46_re_m * (x_46_im_m * -3.0)), (x_46_re_m * (x_46_re_m * x_46_re_m)));
	} else {
		tmp = fma((x_46_re_m - x_46_im_m), (x_46_re_m * (x_46_re_m + x_46_im_m)), (x_46_im_m + x_46_im_m));
	}
	return x_46_re_s * tmp;
}

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (x_46_re_m <= 6.5e+20)
		tmp = fma(x_46_im_m, Float64(x_46_re_m * Float64(x_46_im_m * -3.0)), Float64(x_46_re_m * Float64(x_46_re_m * x_46_re_m)));
	else
		tmp = fma(Float64(x_46_re_m - x_46_im_m), Float64(x_46_re_m * Float64(x_46_re_m + x_46_im_m)), Float64(x_46_im_m + x_46_im_m));
	end
	return Float64(x_46_re_s * tmp)
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 6.5e+20], N[(x$46$im$95$m * N[(x$46$re$95$m * N[(x$46$im$95$m * -3.0), $MachinePrecision]), $MachinePrecision] + N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision] * N[(x$46$re$95$m * N[(x$46$re$95$m + x$46$im$95$m), $MachinePrecision]), $MachinePrecision] + N[(x$46$im$95$m + x$46$im$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 6.5e20

   1. Initial program 82.3%

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

   3. Taylor expanded in x.re around 0

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

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

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

      2. +-commutative N/A

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

      3. associate--l+ N/A

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

      4. unpow2 N/A

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

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

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

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

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

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

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

      8. unpow2 N/A

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

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

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

      10. metadata-eval 85.3

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

   5. Simplified 85.3%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      10. *-lowering-*.f64 91.7

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

   7. Applied egg-rr 91.7%

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

   if 6.5e20 < x.re

   1. Initial program 79.5%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

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

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

      5. *-commutative N/A

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

      6. distribute-rgt-out N/A

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

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

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

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

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

      9. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \color{blue}{\mathsf{neg}\left(x.im\right)}, \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re\right) \]

      10. difference-of-squares N/A

          \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), \color{blue}{\left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)} \cdot x.re\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), \color{blue}{\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)}\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), \color{blue}{\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)}\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), \color{blue}{\left(x.re + x.im\right)} \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), \left(x.re + x.im\right) \cdot \color{blue}{\left(\left(x.re - x.im\right) \cdot x.re\right)}\right) \]

      15. --lowering--.f64 88.0

          \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), -x.im, \left(x.re + x.im\right) \cdot \left(\color{blue}{\left(x.re - x.im\right)} \cdot x.re\right)\right) \]

   4. Applied egg-rr 88.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), -x.im, \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)\right)} \]
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) + \left(x.re \cdot \left(x.im + x.im\right)\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right) \cdot x.re} + \left(x.re \cdot \left(x.im + x.im\right)\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right)} \cdot x.re + \left(x.re \cdot \left(x.im + x.im\right)\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\left(x.re - x.im\right) \cdot \left(\left(x.re + x.im\right) \cdot x.re\right)} + \left(x.re \cdot \left(x.im + x.im\right)\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right) \]

      5. distribute-rgt-neg-out N/A

         \[\leadsto \left(x.re - x.im\right) \cdot \left(\left(x.re + x.im\right) \cdot x.re\right) + \color{blue}{\left(\mathsf{neg}\left(\left(x.re \cdot \left(x.im + x.im\right)\right) \cdot x.im\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto \left(x.re - x.im\right) \cdot \left(\left(x.re + x.im\right) \cdot x.re\right) + \left(\mathsf{neg}\left(\color{blue}{x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)}\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \left(x.re - x.im\right) \cdot \left(\left(x.re + x.im\right) \cdot x.re\right) + \left(\mathsf{neg}\left(\color{blue}{\left(x.im \cdot x.re\right) \cdot \left(x.im + x.im\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left(x.re - x.im\right) \cdot \left(\left(x.re + x.im\right) \cdot x.re\right) + \left(\mathsf{neg}\left(\color{blue}{\left(x.re \cdot x.im\right)} \cdot \left(x.im + x.im\right)\right)\right) \]

      9. flip-+ N/A

         \[\leadsto \left(x.re - x.im\right) \cdot \left(\left(x.re + x.im\right) \cdot x.re\right) + \left(\mathsf{neg}\left(\left(x.re \cdot x.im\right) \cdot \color{blue}{\frac{x.im \cdot x.im - x.im \cdot x.im}{x.im - x.im}}\right)\right) \]

      10. +-inverses N/A

          \[\leadsto \left(x.re - x.im\right) \cdot \left(\left(x.re + x.im\right) \cdot x.re\right) + \left(\mathsf{neg}\left(\left(x.re \cdot x.im\right) \cdot \frac{\color{blue}{0}}{x.im - x.im}\right)\right) \]

      11. +-inverses N/A

          \[\leadsto \left(x.re - x.im\right) \cdot \left(\left(x.re + x.im\right) \cdot x.re\right) + \left(\mathsf{neg}\left(\left(x.re \cdot x.im\right) \cdot \frac{0}{\color{blue}{0}}\right)\right) \]

      12. associate-*r/ N/A

          \[\leadsto \left(x.re - x.im\right) \cdot \left(\left(x.re + x.im\right) \cdot x.re\right) + \left(\mathsf{neg}\left(\color{blue}{\frac{\left(x.re \cdot x.im\right) \cdot 0}{0}}\right)\right) \]

      13. +-inverses N/A

          \[\leadsto \left(x.re - x.im\right) \cdot \left(\left(x.re + x.im\right) \cdot x.re\right) + \left(\mathsf{neg}\left(\frac{\left(x.re \cdot x.im\right) \cdot \color{blue}{\left(x.re \cdot x.im - x.re \cdot x.im\right)}}{0}\right)\right) \]

      14. distribute-lft-out-- N/A

          \[\leadsto \left(x.re - x.im\right) \cdot \left(\left(x.re + x.im\right) \cdot x.re\right) + \left(\mathsf{neg}\left(\frac{\color{blue}{\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)}}{0}\right)\right) \]

      15. +-inverses N/A

          \[\leadsto \left(x.re - x.im\right) \cdot \left(\left(x.re + x.im\right) \cdot x.re\right) + \left(\mathsf{neg}\left(\frac{\color{blue}{0}}{0}\right)\right) \]

      16. distribute-neg-frac2 N/A

          \[\leadsto \left(x.re - x.im\right) \cdot \left(\left(x.re + x.im\right) \cdot x.re\right) + \color{blue}{\frac{0}{\mathsf{neg}\left(0\right)}} \]

      17. +-inverses N/A

          \[\leadsto \left(x.re - x.im\right) \cdot \left(\left(x.re + x.im\right) \cdot x.re\right) + \frac{\color{blue}{\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)}}{\mathsf{neg}\left(0\right)} \]

      18. metadata-eval N/A

          \[\leadsto \left(x.re - x.im\right) \cdot \left(\left(x.re + x.im\right) \cdot x.re\right) + \frac{\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)}{\color{blue}{0}} \]

      19. +-inverses N/A

          \[\leadsto \left(x.re - x.im\right) \cdot \left(\left(x.re + x.im\right) \cdot x.re\right) + \frac{\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)}{\color{blue}{x.re \cdot x.im - x.re \cdot x.im}} \]

      20. flip-+ N/A

          \[\leadsto \left(x.re - x.im\right) \cdot \left(\left(x.re + x.im\right) \cdot x.re\right) + \color{blue}{\left(x.re \cdot x.im + x.re \cdot x.im\right)} \]

      21. distribute-lft-in N/A

          \[\leadsto \left(x.re - x.im\right) \cdot \left(\left(x.re + x.im\right) \cdot x.re\right) + \color{blue}{x.re \cdot \left(x.im + x.im\right)} \]

   6. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.re - x.im, \left(x.re + x.im\right) \cdot x.re, x.im + x.im\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 6.5 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(x.im, x.re \cdot \left(x.im \cdot -3\right), x.re \cdot \left(x.re \cdot x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im, x.re \cdot \left(x.re + x.im\right), x.im + x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 26.7% accurate, 3.6× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \left(x.im\_m \cdot \left(x.im\_m \cdot -2\right)\right) \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (* x.re_s (* x.im_m (* x.im_m -2.0))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	return x_46_re_s * (x_46_im_m * (x_46_im_m * -2.0));
}

Fortran

x.im_m = abs(x_46im)
x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    code = x_46re_s * (x_46im_m * (x_46im_m * (-2.0d0)))
end function

Java

x.im_m = Math.abs(x_46_im);
x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	return x_46_re_s * (x_46_im_m * (x_46_im_m * -2.0));
}

Python

x.im_m = math.fabs(x_46_im)
x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im_m):
	return x_46_re_s * (x_46_im_m * (x_46_im_m * -2.0))

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	return Float64(x_46_re_s * Float64(x_46_im_m * Float64(x_46_im_m * -2.0)))
end

MATLAB

x.im_m = abs(x_46_im);
x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp = code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = x_46_re_s * (x_46_im_m * (x_46_im_m * -2.0));
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * N[(x$46$im$95$m * N[(x$46$im$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.7%

   \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
2. Add Preprocessing
3. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \color{blue}{\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re + \left(\mathsf{neg}\left(\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\right)\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re} \]

   3. distribute-rgt-neg-in N/A

      \[\leadsto \color{blue}{\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right)} + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re \]

   4. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.re \cdot x.im + x.im \cdot x.re, \mathsf{neg}\left(x.im\right), \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re\right)} \]

   5. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{x.im \cdot x.re} + x.im \cdot x.re, \mathsf{neg}\left(x.im\right), \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re\right) \]

   6. distribute-rgt-out N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{x.re \cdot \left(x.im + x.im\right)}, \mathsf{neg}\left(x.im\right), \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{x.re \cdot \left(x.im + x.im\right)}, \mathsf{neg}\left(x.im\right), \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{fma}\left(x.re \cdot \color{blue}{\left(x.im + x.im\right)}, \mathsf{neg}\left(x.im\right), \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re\right) \]

   9. neg-lowering-neg.f64 N/A

      \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \color{blue}{\mathsf{neg}\left(x.im\right)}, \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re\right) \]

   10. difference-of-squares N/A

       \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), \color{blue}{\left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)} \cdot x.re\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), \color{blue}{\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)}\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), \color{blue}{\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)}\right) \]

   13. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), \color{blue}{\left(x.re + x.im\right)} \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), \left(x.re + x.im\right) \cdot \color{blue}{\left(\left(x.re - x.im\right) \cdot x.re\right)}\right) \]

   15. --lowering--.f64 93.2

       \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), -x.im, \left(x.re + x.im\right) \cdot \left(\color{blue}{\left(x.re - x.im\right)} \cdot x.re\right)\right) \]

4. Applied egg-rr 93.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), -x.im, \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)\right)} \]

5. Taylor expanded in x.im around inf

   \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), \color{blue}{\left(x.im \cdot \left(1 + \frac{x.re}{x.im}\right)\right)} \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)\right) \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), \left(x.im \cdot \color{blue}{\left(\frac{x.re}{x.im} + 1\right)}\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)\right) \]

   2. distribute-lft-in N/A

      \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), \color{blue}{\left(x.im \cdot \frac{x.re}{x.im} + x.im \cdot 1\right)} \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)\right) \]

   3. *-rgt-identity N/A

      \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), \left(x.im \cdot \frac{x.re}{x.im} + \color{blue}{x.im}\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)\right) \]

   4. accelerator-lowering-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), \color{blue}{\mathsf{fma}\left(x.im, \frac{x.re}{x.im}, x.im\right)} \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)\right) \]

   5. /-lowering-/.f64 91.0

      \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), -x.im, \mathsf{fma}\left(x.im, \color{blue}{\frac{x.re}{x.im}}, x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)\right) \]

7. Simplified 91.0%

   \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), -x.im, \color{blue}{\mathsf{fma}\left(x.im, \frac{x.re}{x.im}, x.im\right)} \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)\right) \]
8. Step-by-step derivation

   1. distribute-lft-in N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{x.re \cdot x.im + x.re \cdot x.im}, \mathsf{neg}\left(x.im\right), \mathsf{fma}\left(x.im, \frac{x.re}{x.im}, x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)\right) \]

   2. flip-+ N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)}{x.re \cdot x.im - x.re \cdot x.im}}, \mathsf{neg}\left(x.im\right), \mathsf{fma}\left(x.im, \frac{x.re}{x.im}, x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)\right) \]

   3. +-inverses N/A

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0}}{x.re \cdot x.im - x.re \cdot x.im}, \mathsf{neg}\left(x.im\right), \mathsf{fma}\left(x.im, \frac{x.re}{x.im}, x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)\right) \]

   4. +-inverses N/A

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{x.im \cdot x.im - x.im \cdot x.im}}{x.re \cdot x.im - x.re \cdot x.im}, \mathsf{neg}\left(x.im\right), \mathsf{fma}\left(x.im, \frac{x.re}{x.im}, x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)\right) \]

   5. +-inverses N/A

      \[\leadsto \mathsf{fma}\left(\frac{x.im \cdot x.im - x.im \cdot x.im}{\color{blue}{0}}, \mathsf{neg}\left(x.im\right), \mathsf{fma}\left(x.im, \frac{x.re}{x.im}, x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)\right) \]

   6. +-inverses N/A

      \[\leadsto \mathsf{fma}\left(\frac{x.im \cdot x.im - x.im \cdot x.im}{\color{blue}{x.im - x.im}}, \mathsf{neg}\left(x.im\right), \mathsf{fma}\left(x.im, \frac{x.re}{x.im}, x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)\right) \]

   7. flip-+ N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{x.im + x.im}, \mathsf{neg}\left(x.im\right), \mathsf{fma}\left(x.im, \frac{x.re}{x.im}, x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)\right) \]

   8. +-lowering-+.f64 64.1

      \[\leadsto \mathsf{fma}\left(\color{blue}{x.im + x.im}, -x.im, \mathsf{fma}\left(x.im, \frac{x.re}{x.im}, x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)\right) \]

9. Applied egg-rr 64.1%

   \[\leadsto \mathsf{fma}\left(\color{blue}{x.im + x.im}, -x.im, \mathsf{fma}\left(x.im, \frac{x.re}{x.im}, x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)\right) \]

10. Taylor expanded in x.re around 0

    \[\leadsto \color{blue}{-2 \cdot {x.im}^{2}} \]
11. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \color{blue}{{x.im}^{2} \cdot -2} \]

    2. unpow2 N/A

       \[\leadsto \color{blue}{\left(x.im \cdot x.im\right)} \cdot -2 \]

    3. associate-*l* N/A

       \[\leadsto \color{blue}{x.im \cdot \left(x.im \cdot -2\right)} \]

    4. *-commutative N/A

       \[\leadsto x.im \cdot \color{blue}{\left(-2 \cdot x.im\right)} \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \color{blue}{x.im \cdot \left(-2 \cdot x.im\right)} \]

    6. *-commutative N/A

       \[\leadsto x.im \cdot \color{blue}{\left(x.im \cdot -2\right)} \]

    7. *-lowering-*.f64 17.3

       \[\leadsto x.im \cdot \color{blue}{\left(x.im \cdot -2\right)} \]

12. Simplified 17.3%

    \[\leadsto \color{blue}{x.im \cdot \left(x.im \cdot -2\right)} \]
13. Add Preprocessing

Developer Target 1: 99.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(x.re \cdot x.re\right) \cdot \left(x.re - x.im\right) + \left(x.re \cdot x.im\right) \cdot \left(x.re - 3 \cdot x.im\right) \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (+ (* (* x.re x.re) (- x.re x.im)) (* (* x.re x.im) (- x.re (* 3.0 x.im)))))

C

double code(double x_46_re, double x_46_im) {
	return ((x_46_re * x_46_re) * (x_46_re - x_46_im)) + ((x_46_re * x_46_im) * (x_46_re - (3.0 * x_46_im)));
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = ((x_46re * x_46re) * (x_46re - x_46im)) + ((x_46re * x_46im) * (x_46re - (3.0d0 * x_46im)))
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return ((x_46_re * x_46_re) * (x_46_re - x_46_im)) + ((x_46_re * x_46_im) * (x_46_re - (3.0 * x_46_im)));
}

Python

def code(x_46_re, x_46_im):
	return ((x_46_re * x_46_re) * (x_46_re - x_46_im)) + ((x_46_re * x_46_im) * (x_46_re - (3.0 * x_46_im)))

Julia

function code(x_46_re, x_46_im)
	return Float64(Float64(Float64(x_46_re * x_46_re) * Float64(x_46_re - x_46_im)) + Float64(Float64(x_46_re * x_46_im) * Float64(x_46_re - Float64(3.0 * x_46_im))))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = ((x_46_re * x_46_re) * (x_46_re - x_46_im)) + ((x_46_re * x_46_im) * (x_46_re - (3.0 * x_46_im)));
end

Wolfram

code[x$46$re_, x$46$im_] := N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision] + N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(x$46$re - N[(3.0 * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024205 
(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)))