math.cube on complex, imaginary part

Percentage Accurate: 82.7% → 99.8%

Time: 11.0s

Alternatives: 9

Speedup: 0.7×

• 43.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

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_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
}

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_46im) + (((x_46re * x_46im) + (x_46im * x_46re)) * x_46re)
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_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
}

Python

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

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

Initial Program: 82.7% accurate, 1.0× speedup

Math

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

FPCore

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

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_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
}

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_46im) + (((x_46re * x_46im) + (x_46im * x_46re)) * x_46re)
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_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
}

Python

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

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

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

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<=
       (+
        (* x.im_m (- (* x.re_m x.re_m) (* x.im_m x.im_m)))
        (* x.re_m (+ (* x.re_m x.im_m) (* x.re_m x.im_m))))
       INFINITY)
    (fma
     (* x.im_m (+ x.re_m x.im_m))
     (- x.re_m x.im_m)
     (* x.re_m (* x.re_m (+ x.im_m x.im_m))))
    (fma (* (- x.re_m x.im_m) (+ x.re_m x.im_m)) x.im_m 0.0))))

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (((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_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)))) <= ((double) INFINITY)) {
		tmp = fma((x_46_im_m * (x_46_re_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 + x_46_im_m))));
	} else {
		tmp = fma(((x_46_re_m - x_46_im_m) * (x_46_re_m + x_46_im_m)), x_46_im_m, 0.0);
	}
	return x_46_im_s * tmp;
}

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (Float64(Float64(x_46_im_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m))) + Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_im_m) + Float64(x_46_re_m * x_46_im_m)))) <= Inf)
		tmp = fma(Float64(x_46_im_m * Float64(x_46_re_m + x_46_im_m)), Float64(x_46_re_m - x_46_im_m), Float64(x_46_re_m * Float64(x_46_re_m * Float64(x_46_im_m + x_46_im_m))));
	else
		tmp = fma(Float64(Float64(x_46_re_m - x_46_im_m) * Float64(x_46_re_m + x_46_im_m)), x_46_im_m, 0.0);
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

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

   1. Initial program 95.4%

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

      1. *-commutative N/A

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

      2. difference-of-squares N/A

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

      11. distribute-rgt-out N/A

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

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

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

      13. +-lowering-+.f64 99.8

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

   4. Applied egg-rr 99.8%

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

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

   1. Initial program 0.0%

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

      1. *-commutative N/A

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

      2. difference-of-squares N/A

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

      11. distribute-rgt-out N/A

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

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

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

      13. +-lowering-+.f64 42.9

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

   4. Applied egg-rr 42.9%

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

      1. associate-*l* N/A

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

      2. difference-of-squares N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. flip-+ N/A

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

      6. unswap-sqr N/A

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

      7. unswap-sqr N/A

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

      8. +-rgt-identity N/A

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

      9. +-rgt-identity N/A

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

      10. +-inverses N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. +-inverses N/A

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

      15. metadata-eval N/A

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

      16. flip-- N/A

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

      17. metadata-eval N/A

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

      18. +-inverses N/A

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

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

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

      20. +-inverses N/A

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

   6. Applied egg-rr 100.0%

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

4. Final simplification 99.8%

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

Alternative 2: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := x.im\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) + x.re\_m \cdot \left(x.re\_m \cdot x.im\_m + x.re\_m \cdot x.im\_m\right)\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-196}:\\ \;\;\;\;\mathsf{fma}\left(x.im\_m, x.im\_m, \mathsf{fma}\left(\mathsf{fma}\left(x.re\_m, x.re\_m, 0\right), -3, 0\right)\right) \cdot \left(0 - x.im\_m\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left(x.re\_m \cdot x.im\_m\right) \cdot \left(x.re\_m \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(x.re\_m - x.im\_m\right) \cdot \left(x.re\_m + x.im\_m\right), x.im\_m, 0\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (let* ((t_0
         (+
          (* x.im_m (- (* x.re_m x.re_m) (* x.im_m x.im_m)))
          (* x.re_m (+ (* x.re_m x.im_m) (* x.re_m x.im_m))))))
   (*
    x.im_s
    (if (<= t_0 4e-196)
      (*
       (fma x.im_m x.im_m (fma (fma x.re_m x.re_m 0.0) -3.0 0.0))
       (- 0.0 x.im_m))
      (if (<= t_0 INFINITY)
        (* (* x.re_m x.im_m) (* x.re_m 3.0))
        (fma (* (- x.re_m x.im_m) (+ x.re_m x.im_m)) x.im_m 0.0))))))

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double t_0 = (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_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)));
	double tmp;
	if (t_0 <= 4e-196) {
		tmp = fma(x_46_im_m, x_46_im_m, fma(fma(x_46_re_m, x_46_re_m, 0.0), -3.0, 0.0)) * (0.0 - x_46_im_m);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (x_46_re_m * x_46_im_m) * (x_46_re_m * 3.0);
	} else {
		tmp = fma(((x_46_re_m - x_46_im_m) * (x_46_re_m + x_46_im_m)), x_46_im_m, 0.0);
	}
	return x_46_im_s * tmp;
}

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	t_0 = Float64(Float64(x_46_im_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m))) + Float64(x_46_re_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 <= 4e-196)
		tmp = Float64(fma(x_46_im_m, x_46_im_m, fma(fma(x_46_re_m, x_46_re_m, 0.0), -3.0, 0.0)) * Float64(0.0 - x_46_im_m));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(x_46_re_m * x_46_im_m) * Float64(x_46_re_m * 3.0));
	else
		tmp = fma(Float64(Float64(x_46_re_m - x_46_im_m) * Float64(x_46_re_m + x_46_im_m)), x_46_im_m, 0.0);
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re$95$m_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(x$46$im$95$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$46$re$95$m * N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$0, 4e-196], N[(N[(x$46$im$95$m * x$46$im$95$m + N[(N[(x$46$re$95$m * x$46$re$95$m + 0.0), $MachinePrecision] * -3.0 + 0.0), $MachinePrecision]), $MachinePrecision] * N[(0.0 - x$46$im$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] * N[(x$46$re$95$m * 3.0), $MachinePrecision]), $MachinePrecision], N[(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] * x$46$im$95$m + 0.0), $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.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < 4.0000000000000002e-196

   1. Initial program 97.2%

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

   3. Taylor expanded in x.re around 0

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

   5. Simplified 96.5%

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

   if 4.0000000000000002e-196 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   1. Initial program 92.7%

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

      1. *-commutative N/A

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

      2. difference-of-squares N/A

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

      11. distribute-rgt-out N/A

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

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

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

      13. +-lowering-+.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x.im around 0

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

      1. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. +-commutative N/A

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

      6. +-rgt-identity N/A

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

      7. distribute-rgt-in N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-lft-in N/A

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

      11. +-commutative N/A

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

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

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

      13. +-rgt-identity N/A

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

      14. distribute-lft1-in N/A

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

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x.im, \color{blue}{3} \cdot {x.re}^{2} + 0, 0\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x.im, \color{blue}{{x.re}^{2} \cdot 3} + 0, 0\right) \]

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

          \[\leadsto \mathsf{fma}\left(x.im, \color{blue}{\mathsf{fma}\left({x.re}^{2}, 3, 0\right)}, 0\right) \]

      18. +-rgt-identity N/A

          \[\leadsto \mathsf{fma}\left(x.im, \mathsf{fma}\left(\color{blue}{{x.re}^{2} + 0}, 3, 0\right), 0\right) \]

      19. unpow2 N/A

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

      20. accelerator-lowering-fma.f64 41.4

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

   7. Simplified 41.4%

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

      1. +-rgt-identity N/A

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

      2. *-lowering-*.f64 41.4

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

   9. Applied egg-rr 41.4%

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

       1. +-rgt-identity N/A

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

       2. +-rgt-identity N/A

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

       3. associate-*l* N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

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

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

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

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

       8. *-lowering-*.f64 48.5

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

   11. Applied egg-rr 48.5%

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

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

   1. Initial program 0.0%

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

      1. *-commutative N/A

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

      2. difference-of-squares N/A

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

      11. distribute-rgt-out N/A

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

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

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

      13. +-lowering-+.f64 42.9

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

   4. Applied egg-rr 42.9%

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

      1. associate-*l* N/A

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

      2. difference-of-squares N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. flip-+ N/A

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

      6. unswap-sqr N/A

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

      7. unswap-sqr N/A

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

      8. +-rgt-identity N/A

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

      9. +-rgt-identity N/A

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

      10. +-inverses N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. +-inverses N/A

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

      15. metadata-eval N/A

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

      16. flip-- N/A

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

      17. metadata-eval N/A

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

      18. +-inverses N/A

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

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

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

      20. +-inverses N/A

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

   6. Applied egg-rr 100.0%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq 4 \cdot 10^{-196}:\\ \;\;\;\;\mathsf{fma}\left(x.im, x.im, \mathsf{fma}\left(\mathsf{fma}\left(x.re, x.re, 0\right), -3, 0\right)\right) \cdot \left(0 - x.im\right)\\ \mathbf{elif}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq \infty:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right), x.im, 0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := x.im\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) + x.re\_m \cdot \left(x.re\_m \cdot x.im\_m + x.re\_m \cdot x.im\_m\right)\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-303}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m, x.re\_m \cdot -3, x.im\_m \cdot x.im\_m\right) \cdot \left(0 - x.im\_m\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left(x.re\_m \cdot x.im\_m\right) \cdot \left(x.re\_m \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(x.re\_m - x.im\_m\right) \cdot \left(x.re\_m + x.im\_m\right), x.im\_m, 0\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (let* ((t_0
         (+
          (* x.im_m (- (* x.re_m x.re_m) (* x.im_m x.im_m)))
          (* x.re_m (+ (* x.re_m x.im_m) (* x.re_m x.im_m))))))
   (*
    x.im_s
    (if (<= t_0 -2e-303)
      (* (fma x.re_m (* x.re_m -3.0) (* x.im_m x.im_m)) (- 0.0 x.im_m))
      (if (<= t_0 INFINITY)
        (* (* x.re_m x.im_m) (* x.re_m 3.0))
        (fma (* (- x.re_m x.im_m) (+ x.re_m x.im_m)) x.im_m 0.0))))))

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double t_0 = (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_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)));
	double tmp;
	if (t_0 <= -2e-303) {
		tmp = fma(x_46_re_m, (x_46_re_m * -3.0), (x_46_im_m * x_46_im_m)) * (0.0 - x_46_im_m);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (x_46_re_m * x_46_im_m) * (x_46_re_m * 3.0);
	} else {
		tmp = fma(((x_46_re_m - x_46_im_m) * (x_46_re_m + x_46_im_m)), x_46_im_m, 0.0);
	}
	return x_46_im_s * tmp;
}

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	t_0 = Float64(Float64(x_46_im_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m))) + Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_im_m) + Float64(x_46_re_m * x_46_im_m))))
	tmp = 0.0
	if (t_0 <= -2e-303)
		tmp = Float64(fma(x_46_re_m, Float64(x_46_re_m * -3.0), Float64(x_46_im_m * x_46_im_m)) * Float64(0.0 - x_46_im_m));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(x_46_re_m * x_46_im_m) * Float64(x_46_re_m * 3.0));
	else
		tmp = fma(Float64(Float64(x_46_re_m - x_46_im_m) * Float64(x_46_re_m + x_46_im_m)), x_46_im_m, 0.0);
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re$95$m_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(x$46$im$95$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$46$re$95$m * N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$0, -2e-303], N[(N[(x$46$re$95$m * N[(x$46$re$95$m * -3.0), $MachinePrecision] + N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * N[(0.0 - x$46$im$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] * N[(x$46$re$95$m * 3.0), $MachinePrecision]), $MachinePrecision], N[(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] * x$46$im$95$m + 0.0), $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.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < -1.99999999999999986e-303

   1. Initial program 95.3%

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

   3. Taylor expanded in x.re around 0

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

   5. Simplified 94.1%

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

   6. Taylor expanded in x.im around 0

      \[\leadsto 0 - x.im \cdot \color{blue}{\left(-3 \cdot {x.re}^{2} + {x.im}^{2}\right)} \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto 0 - x.im \cdot \left(-3 \cdot \color{blue}{\left(x.re \cdot x.re\right)} + {x.im}^{2}\right) \]

      2. associate-*r* N/A

         \[\leadsto 0 - x.im \cdot \left(\color{blue}{\left(-3 \cdot x.re\right) \cdot x.re} + {x.im}^{2}\right) \]

      3. *-commutative N/A

         \[\leadsto 0 - x.im \cdot \left(\color{blue}{x.re \cdot \left(-3 \cdot x.re\right)} + {x.im}^{2}\right) \]

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

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

      5. *-commutative N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 94.1

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

   8. Simplified 94.1%

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

   if -1.99999999999999986e-303 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   1. Initial program 95.5%

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

      1. *-commutative N/A

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

      2. difference-of-squares N/A

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

      11. distribute-rgt-out N/A

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

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

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

      13. +-lowering-+.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x.im around 0

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

      1. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. +-commutative N/A

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

      6. +-rgt-identity N/A

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

      7. distribute-rgt-in N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-lft-in N/A

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

      11. +-commutative N/A

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

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

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

      13. +-rgt-identity N/A

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

      14. distribute-lft1-in N/A

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

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x.im, \color{blue}{3} \cdot {x.re}^{2} + 0, 0\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x.im, \color{blue}{{x.re}^{2} \cdot 3} + 0, 0\right) \]

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

          \[\leadsto \mathsf{fma}\left(x.im, \color{blue}{\mathsf{fma}\left({x.re}^{2}, 3, 0\right)}, 0\right) \]

      18. +-rgt-identity N/A

          \[\leadsto \mathsf{fma}\left(x.im, \mathsf{fma}\left(\color{blue}{{x.re}^{2} + 0}, 3, 0\right), 0\right) \]

      19. unpow2 N/A

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

      20. accelerator-lowering-fma.f64 60.6

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

   7. Simplified 60.6%

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

      1. +-rgt-identity N/A

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

      2. *-lowering-*.f64 60.6

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

   9. Applied egg-rr 60.6%

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

       1. +-rgt-identity N/A

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

       2. +-rgt-identity N/A

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

       3. associate-*l* N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

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

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

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

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

       8. *-lowering-*.f64 64.9

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

   11. Applied egg-rr 64.9%

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

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

   1. Initial program 0.0%

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

      1. *-commutative N/A

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

      2. difference-of-squares N/A

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

      11. distribute-rgt-out N/A

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

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

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

      13. +-lowering-+.f64 42.9

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

   4. Applied egg-rr 42.9%

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

      1. associate-*l* N/A

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

      2. difference-of-squares N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. flip-+ N/A

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

      6. unswap-sqr N/A

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

      7. unswap-sqr N/A

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

      8. +-rgt-identity N/A

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

      9. +-rgt-identity N/A

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

      10. +-inverses N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. +-inverses N/A

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

      15. metadata-eval N/A

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

      16. flip-- N/A

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

      17. metadata-eval N/A

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

      18. +-inverses N/A

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

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

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

      20. +-inverses N/A

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

   6. Applied egg-rr 100.0%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -2 \cdot 10^{-303}:\\ \;\;\;\;\mathsf{fma}\left(x.re, x.re \cdot -3, x.im \cdot x.im\right) \cdot \left(0 - x.im\right)\\ \mathbf{elif}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq \infty:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right), x.im, 0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := x.im\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) + x.re\_m \cdot \left(x.re\_m \cdot x.im\_m + x.re\_m \cdot x.im\_m\right)\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-303}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m + x.im\_m, x.im\_m \cdot \left(x.re\_m - x.im\_m\right), 0\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left(x.re\_m \cdot x.im\_m\right) \cdot \left(x.re\_m \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(x.re\_m - x.im\_m\right) \cdot \left(x.re\_m + x.im\_m\right), x.im\_m, 0\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (let* ((t_0
         (+
          (* x.im_m (- (* x.re_m x.re_m) (* x.im_m x.im_m)))
          (* x.re_m (+ (* x.re_m x.im_m) (* x.re_m x.im_m))))))
   (*
    x.im_s
    (if (<= t_0 -2e-303)
      (fma (+ x.re_m x.im_m) (* x.im_m (- x.re_m x.im_m)) 0.0)
      (if (<= t_0 INFINITY)
        (* (* x.re_m x.im_m) (* x.re_m 3.0))
        (fma (* (- x.re_m x.im_m) (+ x.re_m x.im_m)) x.im_m 0.0))))))

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double t_0 = (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_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)));
	double tmp;
	if (t_0 <= -2e-303) {
		tmp = fma((x_46_re_m + x_46_im_m), (x_46_im_m * (x_46_re_m - x_46_im_m)), 0.0);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (x_46_re_m * x_46_im_m) * (x_46_re_m * 3.0);
	} else {
		tmp = fma(((x_46_re_m - x_46_im_m) * (x_46_re_m + x_46_im_m)), x_46_im_m, 0.0);
	}
	return x_46_im_s * tmp;
}

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	t_0 = Float64(Float64(x_46_im_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m))) + Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_im_m) + Float64(x_46_re_m * x_46_im_m))))
	tmp = 0.0
	if (t_0 <= -2e-303)
		tmp = fma(Float64(x_46_re_m + x_46_im_m), Float64(x_46_im_m * Float64(x_46_re_m - x_46_im_m)), 0.0);
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(x_46_re_m * x_46_im_m) * Float64(x_46_re_m * 3.0));
	else
		tmp = fma(Float64(Float64(x_46_re_m - x_46_im_m) * Float64(x_46_re_m + x_46_im_m)), x_46_im_m, 0.0);
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re$95$m_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(x$46$im$95$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$46$re$95$m * N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$0, -2e-303], N[(N[(x$46$re$95$m + x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m * N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] * N[(x$46$re$95$m * 3.0), $MachinePrecision]), $MachinePrecision], N[(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] * x$46$im$95$m + 0.0), $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.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < -1.99999999999999986e-303

   1. Initial program 95.3%

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

      1. *-commutative N/A

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

      2. difference-of-squares N/A

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

      11. distribute-rgt-out N/A

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

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

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

      13. +-lowering-+.f64 99.8

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

   4. Applied egg-rr 99.8%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. flip-+ N/A

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

      6. unswap-sqr N/A

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

      7. unswap-sqr N/A

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

      8. +-rgt-identity N/A

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

      9. +-rgt-identity N/A

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

      10. +-inverses N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. +-inverses N/A

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

      15. metadata-eval N/A

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

      16. flip-- N/A

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

      17. metadata-eval N/A

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

      18. +-inverses N/A

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

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

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

      20. +-inverses N/A

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

   6. Applied egg-rr 78.7%

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

   if -1.99999999999999986e-303 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   1. Initial program 95.5%

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

      1. *-commutative N/A

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

      2. difference-of-squares N/A

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

      11. distribute-rgt-out N/A

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

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

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

      13. +-lowering-+.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x.im around 0

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

      1. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. +-commutative N/A

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

      6. +-rgt-identity N/A

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

      7. distribute-rgt-in N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-lft-in N/A

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

      11. +-commutative N/A

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

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

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

      13. +-rgt-identity N/A

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

      14. distribute-lft1-in N/A

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

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x.im, \color{blue}{3} \cdot {x.re}^{2} + 0, 0\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x.im, \color{blue}{{x.re}^{2} \cdot 3} + 0, 0\right) \]

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

          \[\leadsto \mathsf{fma}\left(x.im, \color{blue}{\mathsf{fma}\left({x.re}^{2}, 3, 0\right)}, 0\right) \]

      18. +-rgt-identity N/A

          \[\leadsto \mathsf{fma}\left(x.im, \mathsf{fma}\left(\color{blue}{{x.re}^{2} + 0}, 3, 0\right), 0\right) \]

      19. unpow2 N/A

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

      20. accelerator-lowering-fma.f64 60.6

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

   7. Simplified 60.6%

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

      1. +-rgt-identity N/A

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

      2. *-lowering-*.f64 60.6

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

   9. Applied egg-rr 60.6%

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

       1. +-rgt-identity N/A

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

       2. +-rgt-identity N/A

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

       3. associate-*l* N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

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

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

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

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

       8. *-lowering-*.f64 64.9

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

   11. Applied egg-rr 64.9%

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

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

   1. Initial program 0.0%

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

      1. *-commutative N/A

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

      2. difference-of-squares N/A

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

      11. distribute-rgt-out N/A

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

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

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

      13. +-lowering-+.f64 42.9

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

   4. Applied egg-rr 42.9%

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

      1. associate-*l* N/A

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

      2. difference-of-squares N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. flip-+ N/A

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

      6. unswap-sqr N/A

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

      7. unswap-sqr N/A

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

      8. +-rgt-identity N/A

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

      9. +-rgt-identity N/A

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

      10. +-inverses N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. +-inverses N/A

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

      15. metadata-eval N/A

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

      16. flip-- N/A

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

      17. metadata-eval N/A

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

      18. +-inverses N/A

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

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

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

      20. +-inverses N/A

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

   6. Applied egg-rr 100.0%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -2 \cdot 10^{-303}:\\ \;\;\;\;\mathsf{fma}\left(x.re + x.im, x.im \cdot \left(x.re - x.im\right), 0\right)\\ \mathbf{elif}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq \infty:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right), x.im, 0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left(x.re\_m - x.im\_m\right) \cdot \left(x.re\_m + x.im\_m\right), x.im\_m, 0\right)\\ t_1 := x.im\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) + x.re\_m \cdot \left(x.re\_m \cdot x.im\_m + x.re\_m \cdot x.im\_m\right)\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-303}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left(x.re\_m \cdot x.im\_m\right) \cdot \left(x.re\_m \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (let* ((t_0 (fma (* (- x.re_m x.im_m) (+ x.re_m x.im_m)) x.im_m 0.0))
        (t_1
         (+
          (* x.im_m (- (* x.re_m x.re_m) (* x.im_m x.im_m)))
          (* x.re_m (+ (* x.re_m x.im_m) (* x.re_m x.im_m))))))
   (*
    x.im_s
    (if (<= t_1 -2e-303)
      t_0
      (if (<= t_1 INFINITY) (* (* x.re_m x.im_m) (* x.re_m 3.0)) t_0)))))

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double t_0 = fma(((x_46_re_m - x_46_im_m) * (x_46_re_m + x_46_im_m)), x_46_im_m, 0.0);
	double t_1 = (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_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)));
	double tmp;
	if (t_1 <= -2e-303) {
		tmp = t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (x_46_re_m * x_46_im_m) * (x_46_re_m * 3.0);
	} else {
		tmp = t_0;
	}
	return x_46_im_s * tmp;
}

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	t_0 = fma(Float64(Float64(x_46_re_m - x_46_im_m) * Float64(x_46_re_m + x_46_im_m)), x_46_im_m, 0.0)
	t_1 = Float64(Float64(x_46_im_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m))) + Float64(x_46_re_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_1 <= -2e-303)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(x_46_re_m * x_46_im_m) * Float64(x_46_re_m * 3.0));
	else
		tmp = t_0;
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re$95$m_, x$46$im$95$m_] := Block[{t$95$0 = N[(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] * x$46$im$95$m + 0.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im$95$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$46$re$95$m * N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$1, -2e-303], t$95$0, If[LessEqual[t$95$1, Infinity], N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] * N[(x$46$re$95$m * 3.0), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < -1.99999999999999986e-303 or +inf.0 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re))

   1. Initial program 76.2%

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

      1. *-commutative N/A

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

      2. difference-of-squares N/A

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

      11. distribute-rgt-out N/A

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

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

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

      13. +-lowering-+.f64 88.4

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

   4. Applied egg-rr 88.4%

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

      1. associate-*l* N/A

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

      2. difference-of-squares N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. flip-+ N/A

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

      6. unswap-sqr N/A

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

      7. unswap-sqr N/A

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

      8. +-rgt-identity N/A

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

      9. +-rgt-identity N/A

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

      10. +-inverses N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. +-inverses N/A

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

      15. metadata-eval N/A

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

      16. flip-- N/A

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

      17. metadata-eval N/A

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

      18. +-inverses N/A

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

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

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

      20. +-inverses N/A

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

   6. Applied egg-rr 82.5%

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

   if -1.99999999999999986e-303 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   1. Initial program 95.5%

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

      1. *-commutative N/A

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

      2. difference-of-squares N/A

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

      11. distribute-rgt-out N/A

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

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

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

      13. +-lowering-+.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x.im around 0

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

      1. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. +-commutative N/A

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

      6. +-rgt-identity N/A

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

      7. distribute-rgt-in N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-lft-in N/A

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

      11. +-commutative N/A

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

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

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

      13. +-rgt-identity N/A

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

      14. distribute-lft1-in N/A

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

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x.im, \color{blue}{3} \cdot {x.re}^{2} + 0, 0\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x.im, \color{blue}{{x.re}^{2} \cdot 3} + 0, 0\right) \]

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

          \[\leadsto \mathsf{fma}\left(x.im, \color{blue}{\mathsf{fma}\left({x.re}^{2}, 3, 0\right)}, 0\right) \]

      18. +-rgt-identity N/A

          \[\leadsto \mathsf{fma}\left(x.im, \mathsf{fma}\left(\color{blue}{{x.re}^{2} + 0}, 3, 0\right), 0\right) \]

      19. unpow2 N/A

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

      20. accelerator-lowering-fma.f64 60.6

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

   7. Simplified 60.6%

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

      1. +-rgt-identity N/A

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

      2. *-lowering-*.f64 60.6

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

   9. Applied egg-rr 60.6%

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

       1. +-rgt-identity N/A

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

       2. +-rgt-identity N/A

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

       3. associate-*l* N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

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

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

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

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

       8. *-lowering-*.f64 64.9

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

   11. Applied egg-rr 64.9%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -2 \cdot 10^{-303}:\\ \;\;\;\;\mathsf{fma}\left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right), x.im, 0\right)\\ \mathbf{elif}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq \infty:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right), x.im, 0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) + x.re\_m \cdot \left(x.re\_m \cdot x.im\_m + x.re\_m \cdot x.im\_m\right) \leq -2 \cdot 10^{-303}:\\ \;\;\;\;x.im\_m \cdot \left(0 - x.im\_m \cdot x.im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re\_m \cdot x.im\_m\right) \cdot \left(x.re\_m \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<=
       (+
        (* x.im_m (- (* x.re_m x.re_m) (* x.im_m x.im_m)))
        (* x.re_m (+ (* x.re_m x.im_m) (* x.re_m x.im_m))))
       -2e-303)
    (* x.im_m (- 0.0 (* x.im_m x.im_m)))
    (* (* x.re_m x.im_m) (* x.re_m 3.0)))))

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (((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_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)))) <= -2e-303) {
		tmp = x_46_im_m * (0.0 - (x_46_im_m * x_46_im_m));
	} else {
		tmp = (x_46_re_m * x_46_im_m) * (x_46_re_m * 3.0);
	}
	return x_46_im_s * tmp;
}

Fortran

x.re_m = abs(x_46re)
x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (((x_46im_m * ((x_46re_m * x_46re_m) - (x_46im_m * x_46im_m))) + (x_46re_m * ((x_46re_m * x_46im_m) + (x_46re_m * x_46im_m)))) <= (-2d-303)) then
        tmp = x_46im_m * (0.0d0 - (x_46im_m * x_46im_m))
    else
        tmp = (x_46re_m * x_46im_m) * (x_46re_m * 3.0d0)
    end if
    code = x_46im_s * tmp
end function

Java

x.re_m = Math.abs(x_46_re);
x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (((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_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)))) <= -2e-303) {
		tmp = x_46_im_m * (0.0 - (x_46_im_m * x_46_im_m));
	} else {
		tmp = (x_46_re_m * x_46_im_m) * (x_46_re_m * 3.0);
	}
	return x_46_im_s * tmp;
}

Python

x.re_m = math.fabs(x_46_re)
x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if ((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_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)))) <= -2e-303:
		tmp = x_46_im_m * (0.0 - (x_46_im_m * x_46_im_m))
	else:
		tmp = (x_46_re_m * x_46_im_m) * (x_46_re_m * 3.0)
	return x_46_im_s * tmp

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (Float64(Float64(x_46_im_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m))) + Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_im_m) + Float64(x_46_re_m * x_46_im_m)))) <= -2e-303)
		tmp = Float64(x_46_im_m * Float64(0.0 - Float64(x_46_im_m * x_46_im_m)));
	else
		tmp = Float64(Float64(x_46_re_m * x_46_im_m) * Float64(x_46_re_m * 3.0));
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.re_m = abs(x_46_re);
x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (((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_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)))) <= -2e-303)
		tmp = x_46_im_m * (0.0 - (x_46_im_m * x_46_im_m));
	else
		tmp = (x_46_re_m * x_46_im_m) * (x_46_re_m * 3.0);
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[N[(N[(x$46$im$95$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$46$re$95$m * N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-303], N[(x$46$im$95$m * N[(0.0 - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] * N[(x$46$re$95$m * 3.0), $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.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < -1.99999999999999986e-303

   1. Initial program 95.3%

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

   3. Taylor expanded in x.re around 0

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

   5. Simplified 94.1%

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

   6. Taylor expanded in x.im around inf

      \[\leadsto 0 - x.im \cdot \color{blue}{{x.im}^{2}} \]
   7. Step-by-step derivation

      1. unpow2 N/A

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

      2. *-lowering-*.f64 52.8

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

   8. Simplified 52.8%

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

   if -1.99999999999999986e-303 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re))

   1. Initial program 83.9%

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

      1. *-commutative N/A

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

      2. difference-of-squares N/A

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

      11. distribute-rgt-out N/A

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

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

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

      13. +-lowering-+.f64 92.9

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

   4. Applied egg-rr 92.9%

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

   5. Taylor expanded in x.im around 0

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

      1. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. +-commutative N/A

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

      6. +-rgt-identity N/A

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

      7. distribute-rgt-in N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-lft-in N/A

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

      11. +-commutative N/A

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

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

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

      13. +-rgt-identity N/A

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

      14. distribute-lft1-in N/A

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

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x.im, \color{blue}{3} \cdot {x.re}^{2} + 0, 0\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x.im, \color{blue}{{x.re}^{2} \cdot 3} + 0, 0\right) \]

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

          \[\leadsto \mathsf{fma}\left(x.im, \color{blue}{\mathsf{fma}\left({x.re}^{2}, 3, 0\right)}, 0\right) \]

      18. +-rgt-identity N/A

          \[\leadsto \mathsf{fma}\left(x.im, \mathsf{fma}\left(\color{blue}{{x.re}^{2} + 0}, 3, 0\right), 0\right) \]

      19. unpow2 N/A

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

      20. accelerator-lowering-fma.f64 58.5

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

   7. Simplified 58.5%

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

      1. +-rgt-identity N/A

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

      2. *-lowering-*.f64 58.5

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

   9. Applied egg-rr 58.5%

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

       1. +-rgt-identity N/A

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

       2. +-rgt-identity N/A

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

       3. associate-*l* N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

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

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

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

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

       8. *-lowering-*.f64 62.2

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

   11. Applied egg-rr 62.2%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -2 \cdot 10^{-303}:\\ \;\;\;\;x.im \cdot \left(0 - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) + x.re\_m \cdot \left(x.re\_m \cdot x.im\_m + x.re\_m \cdot x.im\_m\right) \leq -2 \cdot 10^{-303}:\\ \;\;\;\;x.im\_m \cdot \left(0 - x.im\_m \cdot x.im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(x.im\_m \cdot \left(x.re\_m \cdot 3\right)\right)\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<=
       (+
        (* x.im_m (- (* x.re_m x.re_m) (* x.im_m x.im_m)))
        (* x.re_m (+ (* x.re_m x.im_m) (* x.re_m x.im_m))))
       -2e-303)
    (* x.im_m (- 0.0 (* x.im_m x.im_m)))
    (* x.re_m (* x.im_m (* x.re_m 3.0))))))

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (((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_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)))) <= -2e-303) {
		tmp = x_46_im_m * (0.0 - (x_46_im_m * x_46_im_m));
	} else {
		tmp = x_46_re_m * (x_46_im_m * (x_46_re_m * 3.0));
	}
	return x_46_im_s * tmp;
}

Fortran

x.re_m = abs(x_46re)
x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (((x_46im_m * ((x_46re_m * x_46re_m) - (x_46im_m * x_46im_m))) + (x_46re_m * ((x_46re_m * x_46im_m) + (x_46re_m * x_46im_m)))) <= (-2d-303)) then
        tmp = x_46im_m * (0.0d0 - (x_46im_m * x_46im_m))
    else
        tmp = x_46re_m * (x_46im_m * (x_46re_m * 3.0d0))
    end if
    code = x_46im_s * tmp
end function

Java

x.re_m = Math.abs(x_46_re);
x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (((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_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)))) <= -2e-303) {
		tmp = x_46_im_m * (0.0 - (x_46_im_m * x_46_im_m));
	} else {
		tmp = x_46_re_m * (x_46_im_m * (x_46_re_m * 3.0));
	}
	return x_46_im_s * tmp;
}

Python

x.re_m = math.fabs(x_46_re)
x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if ((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_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)))) <= -2e-303:
		tmp = x_46_im_m * (0.0 - (x_46_im_m * x_46_im_m))
	else:
		tmp = x_46_re_m * (x_46_im_m * (x_46_re_m * 3.0))
	return x_46_im_s * tmp

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (Float64(Float64(x_46_im_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m))) + Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_im_m) + Float64(x_46_re_m * x_46_im_m)))) <= -2e-303)
		tmp = Float64(x_46_im_m * Float64(0.0 - 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 * 3.0)));
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.re_m = abs(x_46_re);
x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (((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_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)))) <= -2e-303)
		tmp = x_46_im_m * (0.0 - (x_46_im_m * x_46_im_m));
	else
		tmp = x_46_re_m * (x_46_im_m * (x_46_re_m * 3.0));
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[N[(N[(x$46$im$95$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$46$re$95$m * N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-303], N[(x$46$im$95$m * N[(0.0 - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re$95$m * N[(x$46$im$95$m * N[(x$46$re$95$m * 3.0), $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.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < -1.99999999999999986e-303

   1. Initial program 95.3%

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

   3. Taylor expanded in x.re around 0

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

   5. Simplified 94.1%

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

   6. Taylor expanded in x.im around inf

      \[\leadsto 0 - x.im \cdot \color{blue}{{x.im}^{2}} \]
   7. Step-by-step derivation

      1. unpow2 N/A

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

      2. *-lowering-*.f64 52.8

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

   8. Simplified 52.8%

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

   if -1.99999999999999986e-303 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re))

   1. Initial program 83.9%

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

      1. *-commutative N/A

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

      2. difference-of-squares N/A

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

      11. distribute-rgt-out N/A

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

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

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

      13. +-lowering-+.f64 92.9

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

   4. Applied egg-rr 92.9%

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

   5. Taylor expanded in x.im around 0

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

      1. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. +-commutative N/A

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

      6. +-rgt-identity N/A

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

      7. distribute-rgt-in N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-lft-in N/A

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

      11. +-commutative N/A

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

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

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

      13. +-rgt-identity N/A

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

      14. distribute-lft1-in N/A

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

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x.im, \color{blue}{3} \cdot {x.re}^{2} + 0, 0\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x.im, \color{blue}{{x.re}^{2} \cdot 3} + 0, 0\right) \]

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

          \[\leadsto \mathsf{fma}\left(x.im, \color{blue}{\mathsf{fma}\left({x.re}^{2}, 3, 0\right)}, 0\right) \]

      18. +-rgt-identity N/A

          \[\leadsto \mathsf{fma}\left(x.im, \mathsf{fma}\left(\color{blue}{{x.re}^{2} + 0}, 3, 0\right), 0\right) \]

      19. unpow2 N/A

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

      20. accelerator-lowering-fma.f64 58.5

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

   7. Simplified 58.5%

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

      1. +-rgt-identity N/A

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

      2. *-lowering-*.f64 58.5

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

   9. Applied egg-rr 58.5%

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

       1. +-rgt-identity N/A

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

       2. +-rgt-identity N/A

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

       3. *-commutative N/A

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

       4. associate-*l* N/A

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

       5. associate-*l* N/A

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

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

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

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

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

       8. *-lowering-*.f64 62.2

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

   11. Applied egg-rr 62.2%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -2 \cdot 10^{-303}:\\ \;\;\;\;x.im \cdot \left(0 - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 4.1 \cdot 10^{+201}:\\ \;\;\;\;x.im\_m \cdot \left(0 - x.im\_m \cdot x.im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(x.im\_m \cdot x.im\_m\right)\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.re_m 4.1e+201)
    (* x.im_m (- 0.0 (* x.im_m x.im_m)))
    (* x.im_m (* x.im_m x.im_m)))))

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_re_m <= 4.1e+201) {
		tmp = x_46_im_m * (0.0 - (x_46_im_m * x_46_im_m));
	} else {
		tmp = x_46_im_m * (x_46_im_m * x_46_im_m);
	}
	return x_46_im_s * tmp;
}

Fortran

x.re_m = abs(x_46re)
x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46re_m <= 4.1d+201) then
        tmp = x_46im_m * (0.0d0 - (x_46im_m * x_46im_m))
    else
        tmp = x_46im_m * (x_46im_m * x_46im_m)
    end if
    code = x_46im_s * tmp
end function

Java

x.re_m = Math.abs(x_46_re);
x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_re_m <= 4.1e+201) {
		tmp = x_46_im_m * (0.0 - (x_46_im_m * x_46_im_m));
	} else {
		tmp = x_46_im_m * (x_46_im_m * x_46_im_m);
	}
	return x_46_im_s * tmp;
}

Python

x.re_m = math.fabs(x_46_re)
x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re_m, x_46_im_m):
	tmp = 0
	if x_46_re_m <= 4.1e+201:
		tmp = x_46_im_m * (0.0 - (x_46_im_m * x_46_im_m))
	else:
		tmp = x_46_im_m * (x_46_im_m * x_46_im_m)
	return x_46_im_s * tmp

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (x_46_re_m <= 4.1e+201)
		tmp = Float64(x_46_im_m * Float64(0.0 - Float64(x_46_im_m * x_46_im_m)));
	else
		tmp = Float64(x_46_im_m * Float64(x_46_im_m * x_46_im_m));
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.re_m = abs(x_46_re);
x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re_m, x_46_im_m)
	tmp = 0.0;
	if (x_46_re_m <= 4.1e+201)
		tmp = x_46_im_m * (0.0 - (x_46_im_m * x_46_im_m));
	else
		tmp = x_46_im_m * (x_46_im_m * x_46_im_m);
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$re$95$m, 4.1e+201], N[(x$46$im$95$m * N[(0.0 - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im$95$m * N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 4.1000000000000002e201

   1. Initial program 90.1%

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

   3. Taylor expanded in x.re around 0

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

   5. Simplified 94.4%

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

   6. Taylor expanded in x.im around inf

      \[\leadsto 0 - x.im \cdot \color{blue}{{x.im}^{2}} \]
   7. Step-by-step derivation

      1. unpow2 N/A

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

      2. *-lowering-*.f64 68.6

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

   8. Simplified 68.6%

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

   if 4.1000000000000002e201 < x.re

   1. Initial program 62.0%

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

   3. Taylor expanded in x.re around 0

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

   5. Simplified 83.7%

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

   6. Taylor expanded in x.im around inf

      \[\leadsto 0 - x.im \cdot \color{blue}{{x.im}^{2}} \]
   7. Step-by-step derivation

      1. unpow2 N/A

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

      2. *-lowering-*.f64 0.7

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

   8. Simplified 0.7%

      \[\leadsto 0 - x.im \cdot \color{blue}{\left(x.im \cdot x.im\right)} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

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

      2. +-rgt-identity N/A

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

      3. flip3-+ N/A

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

      4. distribute-neg-frac N/A

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

   10. Applied egg-rr 40.8%

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

       1. +-rgt-identity N/A

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

       2. *-lowering-*.f64 40.8

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

   12. Applied egg-rr 40.8%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 4.1 \cdot 10^{+201}:\\ \;\;\;\;x.im \cdot \left(0 - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 21.7% accurate, 3.6× speedup

Math

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

FPCore

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

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	return x_46_im_s * (x_46_im_m * (x_46_im_m * x_46_im_m));
}

Fortran

x.re_m = abs(x_46re)
x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    code = x_46im_s * (x_46im_m * (x_46im_m * x_46im_m))
end function

Java

x.re_m = Math.abs(x_46_re);
x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	return x_46_im_s * (x_46_im_m * (x_46_im_m * x_46_im_m));
}

Python

x.re_m = math.fabs(x_46_re)
x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re_m, x_46_im_m):
	return x_46_im_s * (x_46_im_m * (x_46_im_m * x_46_im_m))

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	return Float64(x_46_im_s * Float64(x_46_im_m * Float64(x_46_im_m * x_46_im_m)))
end

MATLAB

x.re_m = abs(x_46_re);
x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp = code(x_46_im_s, x_46_re_m, x_46_im_m)
	tmp = x_46_im_s * (x_46_im_m * (x_46_im_m * x_46_im_m));
end

Wolfram

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

Derivation

1. Initial program 87.6%

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

3. Taylor expanded in x.re around 0

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

5. Simplified 93.5%

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

6. Taylor expanded in x.im around inf

   \[\leadsto 0 - x.im \cdot \color{blue}{{x.im}^{2}} \]
7. Step-by-step derivation

   1. unpow2 N/A

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

   2. *-lowering-*.f64 62.5

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

8. Simplified 62.5%

   \[\leadsto 0 - x.im \cdot \color{blue}{\left(x.im \cdot x.im\right)} \]
9. Step-by-step derivation

   1. sub0-neg N/A

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

   2. +-rgt-identity N/A

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

   3. flip3-+ N/A

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

   4. distribute-neg-frac N/A

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

10. Applied egg-rr 28.1%

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

    1. +-rgt-identity N/A

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

    2. *-lowering-*.f64 28.1

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

12. Applied egg-rr 28.1%

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

13. Final simplification 28.1%

    \[\leadsto x.im \cdot \left(x.im \cdot x.im\right) \]
14. Add Preprocessing

Developer Target 1: 91.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(x.re \cdot x.im\right) \cdot \left(2 \cdot x.re\right) + \left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.re + x.im\right) \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (+ (* (* x.re x.im) (* 2.0 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im))))

C

double code(double x_46_re, double x_46_im) {
	return ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - 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_46im) * (2.0d0 * x_46re)) + ((x_46im * (x_46re - 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_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
}

Python

def code(x_46_re, x_46_im):
	return ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im))

Julia

function code(x_46_re, x_46_im)
	return Float64(Float64(Float64(x_46_re * x_46_im) * Float64(2.0 * x_46_re)) + Float64(Float64(x_46_im * Float64(x_46_re - x_46_im)) * Float64(x_46_re + x_46_im)))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
end

Wolfram

code[x$46$re_, x$46$im_] := N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(2.0 * x$46$re), $MachinePrecision]), $MachinePrecision] + N[(N[(x$46$im * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision] * N[(x$46$re + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024196 
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (* (* x.re x.im) (* 2 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im))))

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