math.cube on complex, real part

Percentage Accurate: 82.6% → 99.8%

Time: 13.6s

Alternatives: 11

Speedup: 1.4×

• 43.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 82.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) - x.im\_m \cdot \left(x.re\_m \cdot x.im\_m + x.re\_m \cdot x.im\_m\right) \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(0 - x.im\_m, x.re\_m \cdot \left(x.im\_m + x.im\_m\right), \left(x.re\_m + x.im\_m\right) \cdot \left(x.re\_m \cdot \left(x.re\_m - x.im\_m\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(x.im\_m \cdot \left(x.re\_m - x.im\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<=
       (-
        (* x.re_m (- (* x.re_m x.re_m) (* x.im_m x.im_m)))
        (* x.im_m (+ (* x.re_m x.im_m) (* x.re_m x.im_m))))
       INFINITY)
    (fma
     (- 0.0 x.im_m)
     (* x.re_m (+ x.im_m x.im_m))
     (* (+ x.re_m x.im_m) (* x.re_m (- x.re_m x.im_m))))
    (* x.re_m (* x.im_m (- x.re_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)))) <= ((double) INFINITY)) {
		tmp = fma((0.0 - x_46_im_m), (x_46_re_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 = x_46_re_m * (x_46_im_m * (x_46_re_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)))) <= Inf)
		tmp = fma(Float64(0.0 - x_46_im_m), Float64(x_46_re_m * Float64(x_46_im_m + 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 = Float64(x_46_re_m * Float64(x_46_im_m * Float64(x_46_re_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], Infinity], N[(N[(0.0 - x$46$im$95$m), $MachinePrecision] * N[(x$46$re$95$m * N[(x$46$im$95$m + x$46$im$95$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]), $MachinePrecision]), $MachinePrecision], N[(x$46$re$95$m * N[(x$46$im$95$m * N[(x$46$re$95$m - 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)) < +inf.0

   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. *-commutative N/A

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

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

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

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

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

      6. neg-sub0 N/A

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

      7. +-inverses N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

      11. +-inverses N/A

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

      12. *-commutative N/A

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

      13. distribute-rgt-out N/A

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

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

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

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

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

      16. difference-of-squares N/A

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

   4. Applied egg-rr 99.8%

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

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

   1. Initial program 0.0%

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

      1. 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. *-commutative N/A

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

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

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

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

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

      6. neg-sub0 N/A

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

      7. +-inverses N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

      11. +-inverses N/A

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

      12. *-commutative N/A

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

      13. distribute-rgt-out N/A

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

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

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

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

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

      16. difference-of-squares N/A

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

   4. Applied egg-rr 31.0%

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

   5. Taylor expanded in x.re around 0

      \[\leadsto \mathsf{fma}\left(0 - x.im, x.re \cdot \left(x.im + x.im\right), \color{blue}{x.im} \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 27.6%

      \[\leadsto \mathsf{fma}\left(0 - x.im, x.re \cdot \left(x.im + x.im\right), \color{blue}{x.im} \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. sub0-neg N/A

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

      4. *-commutative N/A

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

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

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

      6. flip-+ N/A

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

      7. +-inverses N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. +-inverses N/A

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

      12. metadata-eval N/A

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

      13. flip-- N/A

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

      14. metadata-eval N/A

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

      15. mul0-rgt N/A

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

      16. mul0-lft N/A

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

      17. metadata-eval N/A

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

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

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

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

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

      20. --lowering--.f64 69.0

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

   9. Applied egg-rr 69.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.im \cdot \left(x.re - x.im\right), x.re, 0\right)} \]
   10. Step-by-step derivation

       1. +-rgt-identity N/A

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

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

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

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

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

       4. --lowering--.f64 69.0

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

   11. Applied egg-rr 69.0%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(0 - x.im, x.re \cdot \left(x.im + x.im\right), \left(x.re + x.im\right) \cdot \left(x.re \cdot \left(x.re - x.im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ \begin{array}{l} t_0 := x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) - x.im\_m \cdot \left(x.re\_m \cdot x.im\_m + x.re\_m \cdot x.im\_m\right)\\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-317}:\\ \;\;\;\;x.im\_m \cdot \left(\left(x.re\_m \cdot x.im\_m\right) \cdot -3\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m, x.re\_m \cdot x.re\_m, 0\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(x.im\_m \cdot \left(x.re\_m - x.im\_m\right)\right)\\ \end{array} \end{array} \end{array} \]

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.re_m) (* x.im_m x.im_m)))
          (* x.im_m (+ (* x.re_m x.im_m) (* x.re_m x.im_m))))))
   (*
    x.re_s
    (if (<= t_0 -1e-317)
      (* x.im_m (* (* x.re_m x.im_m) -3.0))
      (if (<= t_0 INFINITY)
        (fma x.re_m (* x.re_m x.re_m) 0.0)
        (* x.re_m (* x.im_m (- x.re_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 <= -1e-317) {
		tmp = x_46_im_m * ((x_46_re_m * x_46_im_m) * -3.0);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fma(x_46_re_m, (x_46_re_m * x_46_re_m), 0.0);
	} else {
		tmp = x_46_re_m * (x_46_im_m * (x_46_re_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 <= -1e-317)
		tmp = Float64(x_46_im_m * Float64(Float64(x_46_re_m * x_46_im_m) * -3.0));
	elseif (t_0 <= Inf)
		tmp = fma(x_46_re_m, Float64(x_46_re_m * x_46_re_m), 0.0);
	else
		tmp = Float64(x_46_re_m * Float64(x_46_im_m * Float64(x_46_re_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, -1e-317], N[(x$46$im$95$m * N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] + 0.0), $MachinePrecision], N[(x$46$re$95$m * N[(x$46$im$95$m * N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -1.00000023e-317

   1. Initial program 91.9%

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

   3. Taylor expanded in x.re around 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. +-rgt-identity N/A

         \[\leadsto x.re \cdot \left(\color{blue}{\left({x.im}^{2} + 0\right)} \cdot \left(-1 - 2\right)\right) \]

      5. unpow2 N/A

         \[\leadsto x.re \cdot \left(\left(\color{blue}{x.im \cdot x.im} + 0\right) \cdot \left(-1 - 2\right)\right) \]

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

         \[\leadsto x.re \cdot \left(\color{blue}{\mathsf{fma}\left(x.im, x.im, 0\right)} \cdot \left(-1 - 2\right)\right) \]

      7. metadata-eval 43.6

         \[\leadsto x.re \cdot \left(\mathsf{fma}\left(x.im, x.im, 0\right) \cdot \color{blue}{-3}\right) \]

   5. Simplified 43.6%

      \[\leadsto \color{blue}{x.re \cdot \left(\mathsf{fma}\left(x.im, x.im, 0\right) \cdot -3\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. associate-*r* N/A

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

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

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

      4. +-rgt-identity N/A

         \[\leadsto \left(x.re \cdot \color{blue}{\left(x.im \cdot x.im + 0\right)}\right) \cdot -3 \]

      5. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left(x.re \cdot \left(x.im \cdot x.im\right) + x.re \cdot 0\right)} \cdot -3 \]

      6. mul0-rgt N/A

         \[\leadsto \left(x.re \cdot \left(x.im \cdot x.im\right) + \color{blue}{0}\right) \cdot -3 \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(x.re, x.im \cdot x.im, 0\right)} \cdot -3 \]

      8. +-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(x.re, \color{blue}{x.im \cdot x.im + 0}, 0\right) \cdot -3 \]

      9. accelerator-lowering-fma.f64 43.6

         \[\leadsto \mathsf{fma}\left(x.re, \color{blue}{\mathsf{fma}\left(x.im, x.im, 0\right)}, 0\right) \cdot -3 \]

   7. Applied egg-rr 43.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.re, \mathsf{fma}\left(x.im, x.im, 0\right), 0\right) \cdot -3} \]

   8. Taylor expanded in x.re around 0

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

      1. *-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 43.6

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

   10. Simplified 43.6%

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. associate-*r* N/A

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

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

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

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

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

       6. *-commutative N/A

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

       7. *-lowering-*.f64 51.6

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

   12. Applied egg-rr 51.6%

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

   if -1.00000023e-317 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < +inf.0

   1. Initial program 92.1%

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

   3. Taylor expanded in x.re around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{{x.re}^{3} + 0} \]

      2. cube-mult N/A

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

      3. unpow2 N/A

         \[\leadsto x.re \cdot \color{blue}{{x.re}^{2}} + 0 \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(x.re, {x.re}^{2}, 0\right)} \]

      5. +-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(x.re, \color{blue}{{x.re}^{2} + 0}, 0\right) \]

      6. unpow2 N/A

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

      7. accelerator-lowering-fma.f64 66.9

         \[\leadsto \mathsf{fma}\left(x.re, \color{blue}{\mathsf{fma}\left(x.re, x.re, 0\right)}, 0\right) \]

   5. Simplified 66.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.re, \mathsf{fma}\left(x.re, x.re, 0\right), 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-lowering-*.f64 66.9

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

   7. Applied egg-rr 66.9%

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

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

   1. Initial program 0.0%

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

      1. 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. *-commutative N/A

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

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

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

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

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

      6. neg-sub0 N/A

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

      7. +-inverses N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

      11. +-inverses N/A

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

      12. *-commutative N/A

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

      13. distribute-rgt-out N/A

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

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

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

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

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

      16. difference-of-squares N/A

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

   4. Applied egg-rr 31.0%

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

   5. Taylor expanded in x.re around 0

      \[\leadsto \mathsf{fma}\left(0 - x.im, x.re \cdot \left(x.im + x.im\right), \color{blue}{x.im} \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 27.6%

      \[\leadsto \mathsf{fma}\left(0 - x.im, x.re \cdot \left(x.im + x.im\right), \color{blue}{x.im} \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. sub0-neg N/A

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

      4. *-commutative N/A

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

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

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

      6. flip-+ N/A

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

      7. +-inverses N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. +-inverses N/A

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

      12. metadata-eval N/A

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

      13. flip-- N/A

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

      14. metadata-eval N/A

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

      15. mul0-rgt N/A

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

      16. mul0-lft N/A

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

      17. metadata-eval N/A

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

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

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

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

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

      20. --lowering--.f64 69.0

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

   9. Applied egg-rr 69.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.im \cdot \left(x.re - x.im\right), x.re, 0\right)} \]
   10. Step-by-step derivation

       1. +-rgt-identity N/A

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

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

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

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

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

       4. --lowering--.f64 69.0

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

   11. Applied egg-rr 69.0%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -1 \cdot 10^{-317}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.im\right) \cdot -3\right)\\ \mathbf{elif}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(x.re, x.re \cdot x.re, 0\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ \begin{array}{l} t_0 := x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) - x.im\_m \cdot \left(x.re\_m \cdot x.im\_m + x.re\_m \cdot x.im\_m\right)\\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-317}:\\ \;\;\;\;x.re\_m \cdot \left(\left(x.im\_m \cdot x.im\_m\right) \cdot -3\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m, x.re\_m \cdot x.re\_m, 0\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(x.im\_m \cdot \left(x.re\_m - x.im\_m\right)\right)\\ \end{array} \end{array} \end{array} \]

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.re_m) (* x.im_m x.im_m)))
          (* x.im_m (+ (* x.re_m x.im_m) (* x.re_m x.im_m))))))
   (*
    x.re_s
    (if (<= t_0 -1e-317)
      (* x.re_m (* (* x.im_m x.im_m) -3.0))
      (if (<= t_0 INFINITY)
        (fma x.re_m (* x.re_m x.re_m) 0.0)
        (* x.re_m (* x.im_m (- x.re_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 <= -1e-317) {
		tmp = x_46_re_m * ((x_46_im_m * x_46_im_m) * -3.0);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fma(x_46_re_m, (x_46_re_m * x_46_re_m), 0.0);
	} else {
		tmp = x_46_re_m * (x_46_im_m * (x_46_re_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 <= -1e-317)
		tmp = Float64(x_46_re_m * Float64(Float64(x_46_im_m * x_46_im_m) * -3.0));
	elseif (t_0 <= Inf)
		tmp = fma(x_46_re_m, Float64(x_46_re_m * x_46_re_m), 0.0);
	else
		tmp = Float64(x_46_re_m * Float64(x_46_im_m * Float64(x_46_re_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, -1e-317], N[(x$46$re$95$m * N[(N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] + 0.0), $MachinePrecision], N[(x$46$re$95$m * N[(x$46$im$95$m * N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -1.00000023e-317

   1. Initial program 91.9%

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

   3. Taylor expanded in x.re around 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. +-rgt-identity N/A

         \[\leadsto x.re \cdot \left(\color{blue}{\left({x.im}^{2} + 0\right)} \cdot \left(-1 - 2\right)\right) \]

      5. unpow2 N/A

         \[\leadsto x.re \cdot \left(\left(\color{blue}{x.im \cdot x.im} + 0\right) \cdot \left(-1 - 2\right)\right) \]

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

         \[\leadsto x.re \cdot \left(\color{blue}{\mathsf{fma}\left(x.im, x.im, 0\right)} \cdot \left(-1 - 2\right)\right) \]

      7. metadata-eval 43.6

         \[\leadsto x.re \cdot \left(\mathsf{fma}\left(x.im, x.im, 0\right) \cdot \color{blue}{-3}\right) \]

   5. Simplified 43.6%

      \[\leadsto \color{blue}{x.re \cdot \left(\mathsf{fma}\left(x.im, x.im, 0\right) \cdot -3\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-lowering-*.f64 43.6

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

   7. Applied egg-rr 43.6%

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

   if -1.00000023e-317 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < +inf.0

   1. Initial program 92.1%

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

   3. Taylor expanded in x.re around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{{x.re}^{3} + 0} \]

      2. cube-mult N/A

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

      3. unpow2 N/A

         \[\leadsto x.re \cdot \color{blue}{{x.re}^{2}} + 0 \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(x.re, {x.re}^{2}, 0\right)} \]

      5. +-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(x.re, \color{blue}{{x.re}^{2} + 0}, 0\right) \]

      6. unpow2 N/A

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

      7. accelerator-lowering-fma.f64 66.9

         \[\leadsto \mathsf{fma}\left(x.re, \color{blue}{\mathsf{fma}\left(x.re, x.re, 0\right)}, 0\right) \]

   5. Simplified 66.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.re, \mathsf{fma}\left(x.re, x.re, 0\right), 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-lowering-*.f64 66.9

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

   7. Applied egg-rr 66.9%

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

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

   1. Initial program 0.0%

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

      1. 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. *-commutative N/A

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

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

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

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

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

      6. neg-sub0 N/A

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

      7. +-inverses N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

      11. +-inverses N/A

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

      12. *-commutative N/A

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

      13. distribute-rgt-out N/A

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

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

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

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

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

      16. difference-of-squares N/A

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

   4. Applied egg-rr 31.0%

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

   5. Taylor expanded in x.re around 0

      \[\leadsto \mathsf{fma}\left(0 - x.im, x.re \cdot \left(x.im + x.im\right), \color{blue}{x.im} \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 27.6%

      \[\leadsto \mathsf{fma}\left(0 - x.im, x.re \cdot \left(x.im + x.im\right), \color{blue}{x.im} \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. sub0-neg N/A

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

      4. *-commutative N/A

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

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

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

      6. flip-+ N/A

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

      7. +-inverses N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. +-inverses N/A

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

      12. metadata-eval N/A

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

      13. flip-- N/A

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

      14. metadata-eval N/A

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

      15. mul0-rgt N/A

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

      16. mul0-lft N/A

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

      17. metadata-eval N/A

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

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

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

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

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

      20. --lowering--.f64 69.0

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

   9. Applied egg-rr 69.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.im \cdot \left(x.re - x.im\right), x.re, 0\right)} \]
   10. Step-by-step derivation

       1. +-rgt-identity N/A

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

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

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

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

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

       4. --lowering--.f64 69.0

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

   11. Applied egg-rr 69.0%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -1 \cdot 10^{-317}:\\ \;\;\;\;x.re \cdot \left(\left(x.im \cdot x.im\right) \cdot -3\right)\\ \mathbf{elif}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(x.re, x.re \cdot x.re, 0\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ \begin{array}{l} t_0 := x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) - x.im\_m \cdot \left(x.re\_m \cdot x.im\_m + x.re\_m \cdot x.im\_m\right)\\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-317}:\\ \;\;\;\;x.re\_m \cdot \left(x.im\_m \cdot \left(x.im\_m \cdot -3\right)\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m, x.re\_m \cdot x.re\_m, 0\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(x.im\_m \cdot \left(x.re\_m - x.im\_m\right)\right)\\ \end{array} \end{array} \end{array} \]

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.re_m) (* x.im_m x.im_m)))
          (* x.im_m (+ (* x.re_m x.im_m) (* x.re_m x.im_m))))))
   (*
    x.re_s
    (if (<= t_0 -1e-317)
      (* x.re_m (* x.im_m (* x.im_m -3.0)))
      (if (<= t_0 INFINITY)
        (fma x.re_m (* x.re_m x.re_m) 0.0)
        (* x.re_m (* x.im_m (- x.re_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 <= -1e-317) {
		tmp = x_46_re_m * (x_46_im_m * (x_46_im_m * -3.0));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fma(x_46_re_m, (x_46_re_m * x_46_re_m), 0.0);
	} else {
		tmp = x_46_re_m * (x_46_im_m * (x_46_re_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 <= -1e-317)
		tmp = Float64(x_46_re_m * Float64(x_46_im_m * Float64(x_46_im_m * -3.0)));
	elseif (t_0 <= Inf)
		tmp = fma(x_46_re_m, Float64(x_46_re_m * x_46_re_m), 0.0);
	else
		tmp = Float64(x_46_re_m * Float64(x_46_im_m * Float64(x_46_re_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, -1e-317], N[(x$46$re$95$m * N[(x$46$im$95$m * N[(x$46$im$95$m * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] + 0.0), $MachinePrecision], N[(x$46$re$95$m * N[(x$46$im$95$m * N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -1.00000023e-317

   1. Initial program 91.9%

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

   3. Taylor expanded in x.re around 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. +-rgt-identity N/A

         \[\leadsto x.re \cdot \left(\color{blue}{\left({x.im}^{2} + 0\right)} \cdot \left(-1 - 2\right)\right) \]

      5. unpow2 N/A

         \[\leadsto x.re \cdot \left(\left(\color{blue}{x.im \cdot x.im} + 0\right) \cdot \left(-1 - 2\right)\right) \]

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

         \[\leadsto x.re \cdot \left(\color{blue}{\mathsf{fma}\left(x.im, x.im, 0\right)} \cdot \left(-1 - 2\right)\right) \]

      7. metadata-eval 43.6

         \[\leadsto x.re \cdot \left(\mathsf{fma}\left(x.im, x.im, 0\right) \cdot \color{blue}{-3}\right) \]

   5. Simplified 43.6%

      \[\leadsto \color{blue}{x.re \cdot \left(\mathsf{fma}\left(x.im, x.im, 0\right) \cdot -3\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

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

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

      5. *-lowering-*.f64 43.6

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

   7. Applied egg-rr 43.6%

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

   if -1.00000023e-317 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < +inf.0

   1. Initial program 92.1%

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

   3. Taylor expanded in x.re around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{{x.re}^{3} + 0} \]

      2. cube-mult N/A

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

      3. unpow2 N/A

         \[\leadsto x.re \cdot \color{blue}{{x.re}^{2}} + 0 \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(x.re, {x.re}^{2}, 0\right)} \]

      5. +-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(x.re, \color{blue}{{x.re}^{2} + 0}, 0\right) \]

      6. unpow2 N/A

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

      7. accelerator-lowering-fma.f64 66.9

         \[\leadsto \mathsf{fma}\left(x.re, \color{blue}{\mathsf{fma}\left(x.re, x.re, 0\right)}, 0\right) \]

   5. Simplified 66.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.re, \mathsf{fma}\left(x.re, x.re, 0\right), 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-lowering-*.f64 66.9

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

   7. Applied egg-rr 66.9%

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

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

   1. Initial program 0.0%

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

      1. 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. *-commutative N/A

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

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

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

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

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

      6. neg-sub0 N/A

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

      7. +-inverses N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

      11. +-inverses N/A

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

      12. *-commutative N/A

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

      13. distribute-rgt-out N/A

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

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

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

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

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

      16. difference-of-squares N/A

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

   4. Applied egg-rr 31.0%

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

   5. Taylor expanded in x.re around 0

      \[\leadsto \mathsf{fma}\left(0 - x.im, x.re \cdot \left(x.im + x.im\right), \color{blue}{x.im} \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 27.6%

      \[\leadsto \mathsf{fma}\left(0 - x.im, x.re \cdot \left(x.im + x.im\right), \color{blue}{x.im} \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. sub0-neg N/A

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

      4. *-commutative N/A

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

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

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

      6. flip-+ N/A

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

      7. +-inverses N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. +-inverses N/A

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

      12. metadata-eval N/A

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

      13. flip-- N/A

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

      14. metadata-eval N/A

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

      15. mul0-rgt N/A

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

      16. mul0-lft N/A

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

      17. metadata-eval N/A

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

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

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

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

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

      20. --lowering--.f64 69.0

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

   9. Applied egg-rr 69.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.im \cdot \left(x.re - x.im\right), x.re, 0\right)} \]
   10. Step-by-step derivation

       1. +-rgt-identity N/A

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

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

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

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

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

       4. --lowering--.f64 69.0

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

   11. Applied egg-rr 69.0%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -1 \cdot 10^{-317}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{elif}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(x.re, x.re \cdot x.re, 0\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.8% accurate, 0.5× speedup

Math

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

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<=
       (-
        (* x.re_m (- (* x.re_m x.re_m) (* x.im_m x.im_m)))
        (* x.im_m (+ (* x.re_m x.im_m) (* x.re_m x.im_m))))
       INFINITY)
    (-
     (* (+ x.re_m x.im_m) (* x.re_m (- x.re_m x.im_m)))
     (* x.im_m (* x.re_m (+ x.im_m x.im_m))))
    (* x.re_m (* x.im_m (- x.re_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)))) <= ((double) INFINITY)) {
		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 = x_46_re_m * (x_46_im_m * (x_46_re_m - x_46_im_m));
	}
	return x_46_re_s * tmp;
}

Java

x.im_m = Math.abs(x_46_im);
x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)))) <= Double.POSITIVE_INFINITY) {
		tmp = ((x_46_re_m + x_46_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 = x_46_re_m * (x_46_im_m * (x_46_re_m - x_46_im_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)))) <= math.inf:
		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 = x_46_re_m * (x_46_im_m * (x_46_re_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)))) <= Inf)
		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 = Float64(x_46_re_m * Float64(x_46_im_m * Float64(x_46_re_m - x_46_im_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)))) <= Inf)
		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 = x_46_re_m * (x_46_im_m * (x_46_re_m - x_46_im_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], Infinity], 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[(x$46$re$95$m * N[(x$46$im$95$m * N[(x$46$re$95$m - 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)) < +inf.0

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

   1. Initial program 0.0%

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

      1. 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. *-commutative N/A

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

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

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

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

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

      6. neg-sub0 N/A

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

      7. +-inverses N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

      11. +-inverses N/A

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

      12. *-commutative N/A

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

      13. distribute-rgt-out N/A

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

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

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

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

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

      16. difference-of-squares N/A

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

   4. Applied egg-rr 31.0%

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

   5. Taylor expanded in x.re around 0

      \[\leadsto \mathsf{fma}\left(0 - x.im, x.re \cdot \left(x.im + x.im\right), \color{blue}{x.im} \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 27.6%

      \[\leadsto \mathsf{fma}\left(0 - x.im, x.re \cdot \left(x.im + x.im\right), \color{blue}{x.im} \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. sub0-neg N/A

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

      4. *-commutative N/A

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

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

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

      6. flip-+ N/A

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

      7. +-inverses N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. +-inverses N/A

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

      12. metadata-eval N/A

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

      13. flip-- N/A

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

      14. metadata-eval N/A

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

      15. mul0-rgt N/A

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

      16. mul0-lft N/A

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

      17. metadata-eval N/A

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

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

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

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

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

      20. --lowering--.f64 69.0

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

   9. Applied egg-rr 69.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.im \cdot \left(x.re - x.im\right), x.re, 0\right)} \]
   10. Step-by-step derivation

       1. +-rgt-identity N/A

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

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

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

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

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

       4. --lowering--.f64 69.0

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

   11. Applied egg-rr 69.0%

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

4. Final simplification 96.3%

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

Alternative 6: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) - x.im\_m \cdot \left(x.re\_m \cdot x.im\_m + x.re\_m \cdot x.im\_m\right) \leq -1 \cdot 10^{-317}:\\ \;\;\;\;x.im\_m \cdot \left(\left(x.re\_m \cdot x.im\_m\right) \cdot -3\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m - x.im\_m, x.re\_m \cdot \left(x.re\_m + x.im\_m\right), 0\right)\\ \end{array} \end{array} \]

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-317)
    (* x.im_m (* (* x.re_m x.im_m) -3.0))
    (fma (- x.re_m x.im_m) (* x.re_m (+ x.re_m x.im_m)) 0.0))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)))) <= -1e-317) {
		tmp = x_46_im_m * ((x_46_re_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)), 0.0);
	}
	return x_46_re_s * tmp;
}

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m))) - Float64(x_46_im_m * Float64(Float64(x_46_re_m * x_46_im_m) + Float64(x_46_re_m * x_46_im_m)))) <= -1e-317)
		tmp = Float64(x_46_im_m * Float64(Float64(x_46_re_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)), 0.0);
	end
	return Float64(x_46_re_s * tmp)
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * If[LessEqual[N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im$95$m * N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-317], N[(x$46$im$95$m * N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] * -3.0), $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] + 0.0), $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)) < -1.00000023e-317

   1. Initial program 91.9%

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

   3. Taylor expanded in x.re around 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. +-rgt-identity N/A

         \[\leadsto x.re \cdot \left(\color{blue}{\left({x.im}^{2} + 0\right)} \cdot \left(-1 - 2\right)\right) \]

      5. unpow2 N/A

         \[\leadsto x.re \cdot \left(\left(\color{blue}{x.im \cdot x.im} + 0\right) \cdot \left(-1 - 2\right)\right) \]

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

         \[\leadsto x.re \cdot \left(\color{blue}{\mathsf{fma}\left(x.im, x.im, 0\right)} \cdot \left(-1 - 2\right)\right) \]

      7. metadata-eval 43.6

         \[\leadsto x.re \cdot \left(\mathsf{fma}\left(x.im, x.im, 0\right) \cdot \color{blue}{-3}\right) \]

   5. Simplified 43.6%

      \[\leadsto \color{blue}{x.re \cdot \left(\mathsf{fma}\left(x.im, x.im, 0\right) \cdot -3\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. associate-*r* N/A

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

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

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

      4. +-rgt-identity N/A

         \[\leadsto \left(x.re \cdot \color{blue}{\left(x.im \cdot x.im + 0\right)}\right) \cdot -3 \]

      5. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left(x.re \cdot \left(x.im \cdot x.im\right) + x.re \cdot 0\right)} \cdot -3 \]

      6. mul0-rgt N/A

         \[\leadsto \left(x.re \cdot \left(x.im \cdot x.im\right) + \color{blue}{0}\right) \cdot -3 \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(x.re, x.im \cdot x.im, 0\right)} \cdot -3 \]

      8. +-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(x.re, \color{blue}{x.im \cdot x.im + 0}, 0\right) \cdot -3 \]

      9. accelerator-lowering-fma.f64 43.6

         \[\leadsto \mathsf{fma}\left(x.re, \color{blue}{\mathsf{fma}\left(x.im, x.im, 0\right)}, 0\right) \cdot -3 \]

   7. Applied egg-rr 43.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.re, \mathsf{fma}\left(x.im, x.im, 0\right), 0\right) \cdot -3} \]

   8. Taylor expanded in x.re around 0

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

      1. *-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 43.6

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

   10. Simplified 43.6%

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. associate-*r* N/A

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

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

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

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

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

       6. *-commutative N/A

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

       7. *-lowering-*.f64 51.6

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

   12. Applied egg-rr 51.6%

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

   if -1.00000023e-317 < (-.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 76.8%

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

      1. 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. *-commutative N/A

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

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

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

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

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

      6. neg-sub0 N/A

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

      7. +-inverses N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

      11. +-inverses N/A

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

      12. *-commutative N/A

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

      13. distribute-rgt-out N/A

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

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

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

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

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

      16. difference-of-squares N/A

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

   4. Applied egg-rr 88.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0 - x.im, x.re \cdot \left(x.im + x.im\right), \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)\right)} \]
   5. Step-by-step derivation

      1. +-commutative N/A

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

      2. sub0-neg N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

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

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

      6. associate-*r* N/A

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

      7. flip-+ N/A

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

      8. +-inverses N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. +-inverses N/A

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

      13. metadata-eval N/A

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

      14. flip-- N/A

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

      15. metadata-eval N/A

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

      16. associate-*r* N/A

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

      17. mul0-rgt N/A

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

      18. *-commutative N/A

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

      19. mul0-lft N/A

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

      20. metadata-eval N/A

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

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

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

   6. Applied egg-rr 82.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.re - x.im, x.re \cdot \left(x.im + x.re\right), 0\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -1 \cdot 10^{-317}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.im\right) \cdot -3\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im, x.re \cdot \left(x.re + x.im\right), 0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 2.8 \cdot 10^{+105}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m, \mathsf{fma}\left(x.re\_m, x.re\_m, \left(x.im\_m \cdot x.im\_m\right) \cdot -3\right), 0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m, x.im\_m, 0\right) \cdot \mathsf{fma}\left(-3, x.im\_m, x.re\_m\right)\\ \end{array} \end{array} \]

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.im_m 2.8e+105)
    (fma x.re_m (fma x.re_m x.re_m (* (* x.im_m x.im_m) -3.0)) 0.0)
    (* (fma x.re_m x.im_m 0.0) (fma -3.0 x.im_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_im_m <= 2.8e+105) {
		tmp = fma(x_46_re_m, fma(x_46_re_m, x_46_re_m, ((x_46_im_m * x_46_im_m) * -3.0)), 0.0);
	} else {
		tmp = fma(x_46_re_m, x_46_im_m, 0.0) * fma(-3.0, x_46_im_m, x_46_re_m);
	}
	return x_46_re_s * tmp;
}

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 2.8e+105)
		tmp = fma(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)), 0.0);
	else
		tmp = Float64(fma(x_46_re_m, x_46_im_m, 0.0) * fma(-3.0, x_46_im_m, x_46_re_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$46$im$95$m, 2.8e+105], 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] + 0.0), $MachinePrecision], N[(N[(x$46$re$95$m * x$46$im$95$m + 0.0), $MachinePrecision] * N[(-3.0 * x$46$im$95$m + x$46$re$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 2.8000000000000001e105

   1. Initial program 89.1%

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

   3. Taylor expanded in x.re around 0

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

   5. Simplified 94.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.re, \mathsf{fma}\left(x.re, x.re, \mathsf{fma}\left(x.im, x.im, 0\right) \cdot -3\right), 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(x.re, \mathsf{fma}\left(x.re, x.re, \color{blue}{\left(x.im \cdot x.im\right)} \cdot -3\right), 0\right) \]

      2. *-lowering-*.f64 94.6

         \[\leadsto \mathsf{fma}\left(x.re, \mathsf{fma}\left(x.re, x.re, \color{blue}{\left(x.im \cdot x.im\right)} \cdot -3\right), 0\right) \]

   7. Applied egg-rr 94.6%

      \[\leadsto \mathsf{fma}\left(x.re, \mathsf{fma}\left(x.re, x.re, \color{blue}{\left(x.im \cdot x.im\right)} \cdot -3\right), 0\right) \]

   if 2.8000000000000001e105 < x.im

   1. Initial program 37.1%

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

      1. 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. *-commutative N/A

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

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

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

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

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

      6. neg-sub0 N/A

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

      7. +-inverses N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

      11. +-inverses N/A

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

      12. *-commutative N/A

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

      13. distribute-rgt-out N/A

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

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

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

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

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

      16. difference-of-squares N/A

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

   4. Applied egg-rr 78.3%

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

   5. Taylor expanded in x.re around 0

      \[\leadsto \mathsf{fma}\left(0 - x.im, x.re \cdot \left(x.im + x.im\right), \color{blue}{x.im} \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 78.3%

      \[\leadsto \mathsf{fma}\left(0 - x.im, x.re \cdot \left(x.im + x.im\right), \color{blue}{x.im} \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)\right) \]
   8. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 78.3

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

   9. Applied egg-rr 78.3%

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

   10. Taylor expanded in x.im around 0

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

       1. distribute-lft-in N/A

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

       2. mul-1-neg N/A

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

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

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

       4. mul-1-neg N/A

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

       5. associate-+l+ N/A

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

       6. distribute-rgt-in N/A

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

       7. associate-*r* N/A

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

       8. associate-*l* N/A

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

       9. *-commutative N/A

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

       10. *-commutative N/A

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

       11. +-commutative N/A

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

       12. unpow2 N/A

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

       13. associate-*r* N/A

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

       14. distribute-rgt-in N/A

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

       15. *-commutative N/A

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

       16. associate-*l* N/A

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

       17. *-commutative N/A

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

       18. distribute-rgt-out N/A

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

   12. Simplified 94.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x.re, x.im, 0\right) \cdot \mathsf{fma}\left(-3, x.im, x.re\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 99.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 1.65 \cdot 10^{+105}:\\ \;\;\;\;x.re\_m \cdot \mathsf{fma}\left(x.im\_m, x.im\_m \cdot -3, x.re\_m \cdot x.re\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m, x.im\_m, 0\right) \cdot \mathsf{fma}\left(-3, x.im\_m, x.re\_m\right)\\ \end{array} \end{array} \]

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.im_m 1.65e+105)
    (* x.re_m (fma x.im_m (* x.im_m -3.0) (* x.re_m x.re_m)))
    (* (fma x.re_m x.im_m 0.0) (fma -3.0 x.im_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_im_m <= 1.65e+105) {
		tmp = x_46_re_m * fma(x_46_im_m, (x_46_im_m * -3.0), (x_46_re_m * x_46_re_m));
	} else {
		tmp = fma(x_46_re_m, x_46_im_m, 0.0) * fma(-3.0, x_46_im_m, x_46_re_m);
	}
	return x_46_re_s * tmp;
}

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 1.65e+105)
		tmp = Float64(x_46_re_m * fma(x_46_im_m, Float64(x_46_im_m * -3.0), Float64(x_46_re_m * x_46_re_m)));
	else
		tmp = Float64(fma(x_46_re_m, x_46_im_m, 0.0) * fma(-3.0, x_46_im_m, x_46_re_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$46$im$95$m, 1.65e+105], N[(x$46$re$95$m * N[(x$46$im$95$m * N[(x$46$im$95$m * -3.0), $MachinePrecision] + 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 + 0.0), $MachinePrecision] * N[(-3.0 * x$46$im$95$m + x$46$re$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.64999999999999999e105

   1. Initial program 89.1%

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

   3. Taylor expanded in x.re around 0

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

   5. Simplified 94.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.re, \mathsf{fma}\left(x.re, x.re, \mathsf{fma}\left(x.im, x.im, 0\right) \cdot -3\right), 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{x.re \cdot \left(x.re \cdot x.re + \left(x.im \cdot x.im + 0\right) \cdot -3\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(x.re \cdot x.re + \left(x.im \cdot x.im + 0\right) \cdot -3\right) \cdot x.re} \]

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

         \[\leadsto \color{blue}{\left(x.re \cdot x.re + \left(x.im \cdot x.im + 0\right) \cdot -3\right) \cdot x.re} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(x.im \cdot x.im + 0\right) \cdot -3 + x.re \cdot x.re\right)} \cdot x.re \]

      5. +-rgt-identity N/A

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

      6. associate-*l* N/A

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

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

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

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

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

      9. +-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(x.im, x.im \cdot -3, \color{blue}{x.re \cdot x.re + 0}\right) \cdot x.re \]

      10. accelerator-lowering-fma.f64 95.5

          \[\leadsto \mathsf{fma}\left(x.im, x.im \cdot -3, \color{blue}{\mathsf{fma}\left(x.re, x.re, 0\right)}\right) \cdot x.re \]

   7. Applied egg-rr 95.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.im, x.im \cdot -3, \mathsf{fma}\left(x.re, x.re, 0\right)\right) \cdot x.re} \]
   8. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-lowering-*.f64 95.5

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

   9. Applied egg-rr 95.5%

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

   if 1.64999999999999999e105 < x.im

   1. Initial program 37.1%

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

      1. 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. *-commutative N/A

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

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

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

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

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

      6. neg-sub0 N/A

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

      7. +-inverses N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

      11. +-inverses N/A

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

      12. *-commutative N/A

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

      13. distribute-rgt-out N/A

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

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

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

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

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

      16. difference-of-squares N/A

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

   4. Applied egg-rr 78.3%

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

   5. Taylor expanded in x.re around 0

      \[\leadsto \mathsf{fma}\left(0 - x.im, x.re \cdot \left(x.im + x.im\right), \color{blue}{x.im} \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 78.3%

      \[\leadsto \mathsf{fma}\left(0 - x.im, x.re \cdot \left(x.im + x.im\right), \color{blue}{x.im} \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)\right) \]
   8. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 78.3

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

   9. Applied egg-rr 78.3%

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

   10. Taylor expanded in x.im around 0

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

       1. distribute-lft-in N/A

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

       2. mul-1-neg N/A

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

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

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

       4. mul-1-neg N/A

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

       5. associate-+l+ N/A

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

       6. distribute-rgt-in N/A

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

       7. associate-*r* N/A

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

       8. associate-*l* N/A

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

       9. *-commutative N/A

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

       10. *-commutative N/A

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

       11. +-commutative N/A

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

       12. unpow2 N/A

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

       13. associate-*r* N/A

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

       14. distribute-rgt-in N/A

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

       15. *-commutative N/A

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

       16. associate-*l* N/A

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

       17. *-commutative N/A

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

       18. distribute-rgt-out N/A

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

   12. Simplified 94.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x.re, x.im, 0\right) \cdot \mathsf{fma}\left(-3, x.im, x.re\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 1.65 \cdot 10^{+105}:\\ \;\;\;\;x.re \cdot \mathsf{fma}\left(x.im, x.im \cdot -3, x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re, x.im, 0\right) \cdot \mathsf{fma}\left(-3, x.im, x.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 4.1 \cdot 10^{+43}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m, x.re\_m \cdot x.re\_m, 0\right)\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(x.re\_m \cdot \left(x.re\_m - x.im\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<= x.im_m 4.1e+43)
    (fma x.re_m (* x.re_m x.re_m) 0.0)
    (* x.im_m (* x.re_m (- x.re_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_im_m <= 4.1e+43) {
		tmp = fma(x_46_re_m, (x_46_re_m * x_46_re_m), 0.0);
	} else {
		tmp = x_46_im_m * (x_46_re_m * (x_46_re_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_im_m <= 4.1e+43)
		tmp = fma(x_46_re_m, Float64(x_46_re_m * x_46_re_m), 0.0);
	else
		tmp = Float64(x_46_im_m * Float64(x_46_re_m * Float64(x_46_re_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$46$im$95$m, 4.1e+43], N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] + 0.0), $MachinePrecision], N[(x$46$im$95$m * N[(x$46$re$95$m * N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 4.1e43

   1. Initial program 90.2%

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

   3. Taylor expanded in x.re around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{{x.re}^{3} + 0} \]

      2. cube-mult N/A

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

      3. unpow2 N/A

         \[\leadsto x.re \cdot \color{blue}{{x.re}^{2}} + 0 \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(x.re, {x.re}^{2}, 0\right)} \]

      5. +-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(x.re, \color{blue}{{x.re}^{2} + 0}, 0\right) \]

      6. unpow2 N/A

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

      7. accelerator-lowering-fma.f64 72.0

         \[\leadsto \mathsf{fma}\left(x.re, \color{blue}{\mathsf{fma}\left(x.re, x.re, 0\right)}, 0\right) \]

   5. Simplified 72.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.re, \mathsf{fma}\left(x.re, x.re, 0\right), 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-lowering-*.f64 72.0

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

   7. Applied egg-rr 72.0%

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

   if 4.1e43 < x.im

   1. Initial program 49.3%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. neg-sub0 N/A

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

      7. +-inverses N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

      11. +-inverses N/A

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

      12. *-commutative N/A

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

      13. distribute-rgt-out N/A

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

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

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

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

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

      16. difference-of-squares N/A

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

   4. Applied egg-rr 81.3%

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

   5. Taylor expanded in x.re around 0

      \[\leadsto \mathsf{fma}\left(0 - x.im, x.re \cdot \left(x.im + x.im\right), \color{blue}{x.im} \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 75.8%

      \[\leadsto \mathsf{fma}\left(0 - x.im, x.re \cdot \left(x.im + x.im\right), \color{blue}{x.im} \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. sub0-neg N/A

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

      4. *-commutative N/A

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

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

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

      6. flip-+ N/A

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

      7. +-inverses N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. +-inverses N/A

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

      12. metadata-eval N/A

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

      13. flip-- N/A

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

      14. metadata-eval N/A

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

      15. mul0-rgt N/A

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

      16. mul0-lft N/A

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

      17. metadata-eval N/A

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

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

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

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

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

      20. --lowering--.f64 53.6

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

   9. Applied egg-rr 53.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.im \cdot \left(x.re - x.im\right), x.re, 0\right)} \]
   10. Step-by-step derivation

       1. +-rgt-identity N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

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

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

       5. *-commutative N/A

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

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

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

       7. --lowering--.f64 55.9

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

   11. Applied egg-rr 55.9%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 4.1 \cdot 10^{+43}:\\ \;\;\;\;\mathsf{fma}\left(x.re, x.re \cdot x.re, 0\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.re - x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 8.6 \cdot 10^{+42}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m, x.re\_m \cdot x.re\_m, 0\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(x.im\_m \cdot \left(x.re\_m - x.im\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<= x.im_m 8.6e+42)
    (fma x.re_m (* x.re_m x.re_m) 0.0)
    (* x.re_m (* x.im_m (- x.re_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_im_m <= 8.6e+42) {
		tmp = fma(x_46_re_m, (x_46_re_m * x_46_re_m), 0.0);
	} else {
		tmp = x_46_re_m * (x_46_im_m * (x_46_re_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_im_m <= 8.6e+42)
		tmp = fma(x_46_re_m, Float64(x_46_re_m * x_46_re_m), 0.0);
	else
		tmp = Float64(x_46_re_m * Float64(x_46_im_m * Float64(x_46_re_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$46$im$95$m, 8.6e+42], N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] + 0.0), $MachinePrecision], N[(x$46$re$95$m * N[(x$46$im$95$m * N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 8.5999999999999996e42

   1. Initial program 90.2%

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

   3. Taylor expanded in x.re around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{{x.re}^{3} + 0} \]

      2. cube-mult N/A

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

      3. unpow2 N/A

         \[\leadsto x.re \cdot \color{blue}{{x.re}^{2}} + 0 \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(x.re, {x.re}^{2}, 0\right)} \]

      5. +-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(x.re, \color{blue}{{x.re}^{2} + 0}, 0\right) \]

      6. unpow2 N/A

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

      7. accelerator-lowering-fma.f64 72.0

         \[\leadsto \mathsf{fma}\left(x.re, \color{blue}{\mathsf{fma}\left(x.re, x.re, 0\right)}, 0\right) \]

   5. Simplified 72.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.re, \mathsf{fma}\left(x.re, x.re, 0\right), 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-lowering-*.f64 72.0

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

   7. Applied egg-rr 72.0%

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

   if 8.5999999999999996e42 < x.im

   1. Initial program 49.3%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. neg-sub0 N/A

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

      7. +-inverses N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

      11. +-inverses N/A

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

      12. *-commutative N/A

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

      13. distribute-rgt-out N/A

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

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

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

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

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

      16. difference-of-squares N/A

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

   4. Applied egg-rr 81.3%

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

   5. Taylor expanded in x.re around 0

      \[\leadsto \mathsf{fma}\left(0 - x.im, x.re \cdot \left(x.im + x.im\right), \color{blue}{x.im} \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 75.8%

      \[\leadsto \mathsf{fma}\left(0 - x.im, x.re \cdot \left(x.im + x.im\right), \color{blue}{x.im} \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. sub0-neg N/A

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

      4. *-commutative N/A

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

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

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

      6. flip-+ N/A

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

      7. +-inverses N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. +-inverses N/A

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

      12. metadata-eval N/A

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

      13. flip-- N/A

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

      14. metadata-eval N/A

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

      15. mul0-rgt N/A

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

      16. mul0-lft N/A

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

      17. metadata-eval N/A

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

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

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

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

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

      20. --lowering--.f64 53.6

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

   9. Applied egg-rr 53.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.im \cdot \left(x.re - x.im\right), x.re, 0\right)} \]
   10. Step-by-step derivation

       1. +-rgt-identity N/A

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

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

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

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

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

       4. --lowering--.f64 53.6

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

   11. Applied egg-rr 53.6%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 8.6 \cdot 10^{+42}:\\ \;\;\;\;\mathsf{fma}\left(x.re, x.re \cdot x.re, 0\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.7% accurate, 3.3× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \mathsf{fma}\left(x.re\_m, x.re\_m \cdot x.re\_m, 0\right) \end{array} \]

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 (fma x.re_m (* x.re_m x.re_m) 0.0)))

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 * fma(x_46_re_m, (x_46_re_m * x_46_re_m), 0.0);
}

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 * fma(x_46_re_m, Float64(x_46_re_m * x_46_re_m), 0.0))
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$46$re$95$m * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.6%

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

3. Taylor expanded in x.re around inf

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

   1. +-rgt-identity N/A

      \[\leadsto \color{blue}{{x.re}^{3} + 0} \]

   2. cube-mult N/A

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

   3. unpow2 N/A

      \[\leadsto x.re \cdot \color{blue}{{x.re}^{2}} + 0 \]

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

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.re, {x.re}^{2}, 0\right)} \]

   5. +-rgt-identity N/A

      \[\leadsto \mathsf{fma}\left(x.re, \color{blue}{{x.re}^{2} + 0}, 0\right) \]

   6. unpow2 N/A

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

   7. accelerator-lowering-fma.f64 63.2

      \[\leadsto \mathsf{fma}\left(x.re, \color{blue}{\mathsf{fma}\left(x.re, x.re, 0\right)}, 0\right) \]

5. Simplified 63.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x.re, \mathsf{fma}\left(x.re, x.re, 0\right), 0\right)} \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

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

   2. *-lowering-*.f64 63.2

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

7. Applied egg-rr 63.2%

   \[\leadsto \mathsf{fma}\left(x.re, \color{blue}{x.re \cdot x.re}, 0\right) \]
8. Add Preprocessing

Developer Target 1: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024196 
(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)))