math.cube on complex, imaginary part

Percentage Accurate: 82.9% → 99.7%

Time: 12.3s

Alternatives: 12

Speedup: 1.1×

• 44.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 82.9% 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: 99.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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, double x_46_im_m) {
	double tmp;
	if (((x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)))) <= ((double) INFINITY)) {
		tmp = fma((x_46_im_m * (x_46_re + x_46_im_m)), (x_46_re - x_46_im_m), (x_46_re * (x_46_re * (x_46_im_m + x_46_im_m))));
	} else {
		tmp = fma((x_46_re + x_46_im_m), (x_46_im_m * (x_46_re - x_46_im_m)), (x_46_im_m + x_46_im_m));
	}
	return x_46_im_s * tmp;
}

Julia

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, x_46_im_m)
	tmp = 0.0
	if (Float64(Float64(x_46_im_m * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m))) + Float64(x_46_re * Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_re * x_46_im_m)))) <= Inf)
		tmp = fma(Float64(x_46_im_m * Float64(x_46_re + x_46_im_m)), Float64(x_46_re - x_46_im_m), Float64(x_46_re * Float64(x_46_re * Float64(x_46_im_m + x_46_im_m))));
	else
		tmp = fma(Float64(x_46_re + x_46_im_m), Float64(x_46_im_m * Float64(x_46_re - x_46_im_m)), Float64(x_46_im_m + x_46_im_m));
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

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_, 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 * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$re * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(x$46$im$95$m * N[(x$46$re + x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$46$re - x$46$im$95$m), $MachinePrecision] + N[(x$46$re * N[(x$46$re * N[(x$46$im$95$m + x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re + x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m * N[(x$46$re - x$46$im$95$m), $MachinePrecision]), $MachinePrecision] + N[(x$46$im$95$m + x$46$im$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 92.1%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Add Preprocessing
   3. 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 34.5

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

   4. Applied egg-rr 34.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.im \cdot \left(x.re + x.im\right), x.re - x.im, x.re \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\right)} \]
   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. difference-of-squares N/A

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

      5. associate-*l* N/A

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

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

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

      7. +-commutative N/A

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

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

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

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

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

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

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

      11. distribute-lft-in N/A

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

      12. flip-+ N/A

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

      13. +-inverses N/A

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

      14. +-inverses N/A

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

      15. associate-*r/ N/A

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

      16. +-inverses N/A

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

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

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

      18. +-inverses N/A

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

      19. +-inverses N/A

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

      20. +-inverses N/A

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

      21. flip-+ N/A

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

   6. Applied egg-rr 100.0%

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

4. Final simplification 99.9%

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

Alternative 2: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := x.im\_m \cdot \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) + x.re \cdot \left(x.re \cdot x.im\_m + x.re \cdot x.im\_m\right)\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+42}:\\ \;\;\;\;x.im\_m \cdot \mathsf{fma}\left(x.re \cdot 3, x.re, x.im\_m \cdot \left(-x.im\_m\right)\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left(x.re \cdot x.im\_m\right) \cdot \left(x.re \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re + x.im\_m, x.im\_m \cdot \left(x.re - x.im\_m\right), x.im\_m + x.im\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0
         (+
          (* x.im_m (- (* x.re x.re) (* x.im_m x.im_m)))
          (* x.re (+ (* x.re x.im_m) (* x.re x.im_m))))))
   (*
    x.im_s
    (if (<= t_0 2e+42)
      (* x.im_m (fma (* x.re 3.0) x.re (* x.im_m (- x.im_m))))
      (if (<= t_0 INFINITY)
        (* (* x.re x.im_m) (* x.re 3.0))
        (fma (+ x.re x.im_m) (* x.im_m (- x.re x.im_m)) (+ x.im_m x.im_m)))))))

C

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, double x_46_im_m) {
	double t_0 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)));
	double tmp;
	if (t_0 <= 2e+42) {
		tmp = x_46_im_m * fma((x_46_re * 3.0), x_46_re, (x_46_im_m * -x_46_im_m));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (x_46_re * x_46_im_m) * (x_46_re * 3.0);
	} else {
		tmp = fma((x_46_re + x_46_im_m), (x_46_im_m * (x_46_re - x_46_im_m)), (x_46_im_m + x_46_im_m));
	}
	return x_46_im_s * tmp;
}

Julia

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, x_46_im_m)
	t_0 = Float64(Float64(x_46_im_m * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m))) + Float64(x_46_re * Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_re * x_46_im_m))))
	tmp = 0.0
	if (t_0 <= 2e+42)
		tmp = Float64(x_46_im_m * fma(Float64(x_46_re * 3.0), x_46_re, Float64(x_46_im_m * Float64(-x_46_im_m))));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(x_46_re * x_46_im_m) * Float64(x_46_re * 3.0));
	else
		tmp = fma(Float64(x_46_re + x_46_im_m), Float64(x_46_im_m * Float64(x_46_re - x_46_im_m)), Float64(x_46_im_m + x_46_im_m));
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

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_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(x$46$im$95$m * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$re * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$0, 2e+42], N[(x$46$im$95$m * N[(N[(x$46$re * 3.0), $MachinePrecision] * x$46$re + N[(x$46$im$95$m * (-x$46$im$95$m)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] * N[(x$46$re * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re + x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m * N[(x$46$re - x$46$im$95$m), $MachinePrecision]), $MachinePrecision] + N[(x$46$im$95$m + x$46$im$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < 2.00000000000000009e42

   1. Initial program 92.5%

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

   3. 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 92.4%

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

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

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

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

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

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

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

      8. *-lowering-*.f64 92.4

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

   7. Applied egg-rr 92.4%

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

   if 2.00000000000000009e42 < (+.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 91.3%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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 99.9

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

   4. Applied egg-rr 99.9%

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

   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-lft1-in N/A

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

      2. metadata-eval N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 28.1

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

   7. Simplified 28.1%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

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

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

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

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

      8. *-lowering-*.f64 36.6

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

   9. Applied egg-rr 36.6%

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

   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 34.5

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

   4. Applied egg-rr 34.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.im \cdot \left(x.re + x.im\right), x.re - x.im, x.re \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\right)} \]
   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. difference-of-squares N/A

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

      5. associate-*l* N/A

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

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

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

      7. +-commutative N/A

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

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

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

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

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

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

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

      11. distribute-lft-in N/A

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

      12. flip-+ N/A

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

      13. +-inverses N/A

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

      14. +-inverses N/A

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

      15. associate-*r/ N/A

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

      16. +-inverses N/A

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

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

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

      18. +-inverses N/A

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

      19. +-inverses N/A

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

      20. +-inverses N/A

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

      21. flip-+ N/A

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

   6. Applied egg-rr 100.0%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq 2 \cdot 10^{+42}:\\ \;\;\;\;x.im \cdot \mathsf{fma}\left(x.re \cdot 3, x.re, x.im \cdot \left(-x.im\right)\right)\\ \mathbf{elif}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq \infty:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re + x.im, x.im \cdot \left(x.re - x.im\right), x.im + x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := x.im\_m \cdot \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) + x.re \cdot \left(x.re \cdot x.im\_m + x.re \cdot x.im\_m\right)\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+42}:\\ \;\;\;\;x.im\_m \cdot \mathsf{fma}\left(x.im\_m, -x.im\_m, \left(x.re \cdot x.re\right) \cdot 3\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left(x.re \cdot x.im\_m\right) \cdot \left(x.re \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re + x.im\_m, x.im\_m \cdot \left(x.re - x.im\_m\right), x.im\_m + x.im\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0
         (+
          (* x.im_m (- (* x.re x.re) (* x.im_m x.im_m)))
          (* x.re (+ (* x.re x.im_m) (* x.re x.im_m))))))
   (*
    x.im_s
    (if (<= t_0 2e+42)
      (* x.im_m (fma x.im_m (- x.im_m) (* (* x.re x.re) 3.0)))
      (if (<= t_0 INFINITY)
        (* (* x.re x.im_m) (* x.re 3.0))
        (fma (+ x.re x.im_m) (* x.im_m (- x.re x.im_m)) (+ x.im_m x.im_m)))))))

C

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, double x_46_im_m) {
	double t_0 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)));
	double tmp;
	if (t_0 <= 2e+42) {
		tmp = x_46_im_m * fma(x_46_im_m, -x_46_im_m, ((x_46_re * x_46_re) * 3.0));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (x_46_re * x_46_im_m) * (x_46_re * 3.0);
	} else {
		tmp = fma((x_46_re + x_46_im_m), (x_46_im_m * (x_46_re - x_46_im_m)), (x_46_im_m + x_46_im_m));
	}
	return x_46_im_s * tmp;
}

Julia

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, x_46_im_m)
	t_0 = Float64(Float64(x_46_im_m * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m))) + Float64(x_46_re * Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_re * x_46_im_m))))
	tmp = 0.0
	if (t_0 <= 2e+42)
		tmp = Float64(x_46_im_m * fma(x_46_im_m, Float64(-x_46_im_m), Float64(Float64(x_46_re * x_46_re) * 3.0)));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(x_46_re * x_46_im_m) * Float64(x_46_re * 3.0));
	else
		tmp = fma(Float64(x_46_re + x_46_im_m), Float64(x_46_im_m * Float64(x_46_re - x_46_im_m)), Float64(x_46_im_m + x_46_im_m));
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

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_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(x$46$im$95$m * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$re * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$0, 2e+42], N[(x$46$im$95$m * N[(x$46$im$95$m * (-x$46$im$95$m) + N[(N[(x$46$re * x$46$re), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] * N[(x$46$re * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re + x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m * N[(x$46$re - x$46$im$95$m), $MachinePrecision]), $MachinePrecision] + N[(x$46$im$95$m + x$46$im$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < 2.00000000000000009e42

   1. Initial program 92.5%

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

   3. 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 92.4%

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

   if 2.00000000000000009e42 < (+.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 91.3%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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 99.9

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

   4. Applied egg-rr 99.9%

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

   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-lft1-in N/A

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

      2. metadata-eval N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 28.1

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

   7. Simplified 28.1%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

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

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

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

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

      8. *-lowering-*.f64 36.6

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

   9. Applied egg-rr 36.6%

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

   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 34.5

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

   4. Applied egg-rr 34.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.im \cdot \left(x.re + x.im\right), x.re - x.im, x.re \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\right)} \]
   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. difference-of-squares N/A

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

      5. associate-*l* N/A

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

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

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

      7. +-commutative N/A

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

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

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

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

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

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

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

      11. distribute-lft-in N/A

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

      12. flip-+ N/A

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

      13. +-inverses N/A

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

      14. +-inverses N/A

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

      15. associate-*r/ N/A

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

      16. +-inverses N/A

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

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

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

      18. +-inverses N/A

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

      19. +-inverses N/A

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

      20. +-inverses N/A

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

      21. flip-+ N/A

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

   6. Applied egg-rr 100.0%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq 2 \cdot 10^{+42}:\\ \;\;\;\;x.im \cdot \mathsf{fma}\left(x.im, -x.im, \left(x.re \cdot x.re\right) \cdot 3\right)\\ \mathbf{elif}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq \infty:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re + x.im, x.im \cdot \left(x.re - x.im\right), x.im + x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := x.im\_m \cdot \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) + x.re \cdot \left(x.re \cdot x.im\_m + x.re \cdot x.im\_m\right)\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-305}:\\ \;\;\;\;-x.im\_m \cdot \left(x.im\_m \cdot x.im\_m\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im\_m \cdot 3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re + x.im\_m, x.im\_m \cdot \left(x.re - x.im\_m\right), x.im\_m + x.im\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0
         (+
          (* x.im_m (- (* x.re x.re) (* x.im_m x.im_m)))
          (* x.re (+ (* x.re x.im_m) (* x.re x.im_m))))))
   (*
    x.im_s
    (if (<= t_0 -2e-305)
      (- (* x.im_m (* x.im_m x.im_m)))
      (if (<= t_0 INFINITY)
        (* x.re (* x.re (* x.im_m 3.0)))
        (fma (+ x.re x.im_m) (* x.im_m (- x.re x.im_m)) (+ x.im_m x.im_m)))))))

C

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, double x_46_im_m) {
	double t_0 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)));
	double tmp;
	if (t_0 <= -2e-305) {
		tmp = -(x_46_im_m * (x_46_im_m * x_46_im_m));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = x_46_re * (x_46_re * (x_46_im_m * 3.0));
	} else {
		tmp = fma((x_46_re + x_46_im_m), (x_46_im_m * (x_46_re - x_46_im_m)), (x_46_im_m + x_46_im_m));
	}
	return x_46_im_s * tmp;
}

Julia

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, x_46_im_m)
	t_0 = Float64(Float64(x_46_im_m * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m))) + Float64(x_46_re * Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_re * x_46_im_m))))
	tmp = 0.0
	if (t_0 <= -2e-305)
		tmp = Float64(-Float64(x_46_im_m * Float64(x_46_im_m * x_46_im_m)));
	elseif (t_0 <= Inf)
		tmp = Float64(x_46_re * Float64(x_46_re * Float64(x_46_im_m * 3.0)));
	else
		tmp = fma(Float64(x_46_re + x_46_im_m), Float64(x_46_im_m * Float64(x_46_re - x_46_im_m)), Float64(x_46_im_m + x_46_im_m));
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

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_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(x$46$im$95$m * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$re * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$0, -2e-305], (-N[(x$46$im$95$m * N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), If[LessEqual[t$95$0, Infinity], N[(x$46$re * N[(x$46$re * N[(x$46$im$95$m * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re + x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m * N[(x$46$re - x$46$im$95$m), $MachinePrecision]), $MachinePrecision] + N[(x$46$im$95$m + x$46$im$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < -1.99999999999999999e-305

   1. Initial program 88.6%

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

   3. Taylor expanded in x.re around 0

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left({x.im}^{3}\right)} \]

      2. unpow3 N/A

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

      3. unpow2 N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. neg-lowering-neg.f64 45.1

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

   5. Simplified 45.1%

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

   if -1.99999999999999999e-305 < (+.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 95.3%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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 99.9

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

   4. Applied egg-rr 99.9%

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

   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-lft1-in N/A

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

      2. metadata-eval N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 56.8

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

   7. Simplified 56.8%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. *-lowering-*.f64 61.3

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

   9. Applied egg-rr 61.3%

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

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

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

       5. *-commutative N/A

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

       6. *-lowering-*.f64 61.4

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

   11. Applied egg-rr 61.4%

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

   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 34.5

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

   4. Applied egg-rr 34.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.im \cdot \left(x.re + x.im\right), x.re - x.im, x.re \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\right)} \]
   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. difference-of-squares N/A

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

      5. associate-*l* N/A

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

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

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

      7. +-commutative N/A

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

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

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

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

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

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

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

      11. distribute-lft-in N/A

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

      12. flip-+ N/A

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

      13. +-inverses N/A

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

      14. +-inverses N/A

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

      15. associate-*r/ N/A

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

      16. +-inverses N/A

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

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

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

      18. +-inverses N/A

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

      19. +-inverses N/A

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

      20. +-inverses N/A

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

      21. flip-+ N/A

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

   6. Applied egg-rr 100.0%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -2 \cdot 10^{-305}:\\ \;\;\;\;-x.im \cdot \left(x.im \cdot x.im\right)\\ \mathbf{elif}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq \infty:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re + x.im, x.im \cdot \left(x.re - x.im\right), x.im + x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := -x.im\_m \cdot \left(x.im\_m \cdot x.im\_m\right)\\ t_1 := x.im\_m \cdot \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) + x.re \cdot \left(x.re \cdot x.im\_m + x.re \cdot x.im\_m\right)\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-305}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im\_m \cdot 3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0 (- (* x.im_m (* x.im_m x.im_m))))
        (t_1
         (+
          (* x.im_m (- (* x.re x.re) (* x.im_m x.im_m)))
          (* x.re (+ (* x.re x.im_m) (* x.re x.im_m))))))
   (*
    x.im_s
    (if (<= t_1 -2e-305)
      t_0
      (if (<= t_1 INFINITY) (* x.re (* x.re (* x.im_m 3.0))) t_0)))))

C

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, double x_46_im_m) {
	double t_0 = -(x_46_im_m * (x_46_im_m * x_46_im_m));
	double t_1 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)));
	double tmp;
	if (t_1 <= -2e-305) {
		tmp = t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = x_46_re * (x_46_re * (x_46_im_m * 3.0));
	} else {
		tmp = t_0;
	}
	return x_46_im_s * tmp;
}

Java

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, double x_46_im_m) {
	double t_0 = -(x_46_im_m * (x_46_im_m * x_46_im_m));
	double t_1 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)));
	double tmp;
	if (t_1 <= -2e-305) {
		tmp = t_0;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = x_46_re * (x_46_re * (x_46_im_m * 3.0));
	} else {
		tmp = t_0;
	}
	return x_46_im_s * tmp;
}

Python

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, x_46_im_m):
	t_0 = -(x_46_im_m * (x_46_im_m * x_46_im_m))
	t_1 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)))
	tmp = 0
	if t_1 <= -2e-305:
		tmp = t_0
	elif t_1 <= math.inf:
		tmp = x_46_re * (x_46_re * (x_46_im_m * 3.0))
	else:
		tmp = t_0
	return x_46_im_s * tmp

Julia

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, x_46_im_m)
	t_0 = Float64(-Float64(x_46_im_m * Float64(x_46_im_m * x_46_im_m)))
	t_1 = Float64(Float64(x_46_im_m * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m))) + Float64(x_46_re * Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_re * x_46_im_m))))
	tmp = 0.0
	if (t_1 <= -2e-305)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = Float64(x_46_re * Float64(x_46_re * Float64(x_46_im_m * 3.0)));
	else
		tmp = t_0;
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

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, x_46_im_m)
	t_0 = -(x_46_im_m * (x_46_im_m * x_46_im_m));
	t_1 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)));
	tmp = 0.0;
	if (t_1 <= -2e-305)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = x_46_re * (x_46_re * (x_46_im_m * 3.0));
	else
		tmp = t_0;
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

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_, x$46$im$95$m_] := Block[{t$95$0 = (-N[(x$46$im$95$m * N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision])}, Block[{t$95$1 = N[(N[(x$46$im$95$m * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$re * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$1, -2e-305], t$95$0, If[LessEqual[t$95$1, Infinity], N[(x$46$re * N[(x$46$re * N[(x$46$im$95$m * 3.0), $MachinePrecision]), $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)) < -1.99999999999999999e-305 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 69.7%

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

   3. Taylor expanded in x.re around 0

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left({x.im}^{3}\right)} \]

      2. unpow3 N/A

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

      3. unpow2 N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. neg-lowering-neg.f64 49.5

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

   5. Simplified 49.5%

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

   if -1.99999999999999999e-305 < (+.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 95.3%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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 99.9

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

   4. Applied egg-rr 99.9%

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

   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-lft1-in N/A

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

      2. metadata-eval N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 56.8

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

   7. Simplified 56.8%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. *-lowering-*.f64 61.3

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

   9. Applied egg-rr 61.3%

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

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

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

       5. *-commutative N/A

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

       6. *-lowering-*.f64 61.4

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

   11. Applied egg-rr 61.4%

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

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -2 \cdot 10^{-305}:\\ \;\;\;\;-x.im \cdot \left(x.im \cdot x.im\right)\\ \mathbf{elif}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq \infty:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-x.im \cdot \left(x.im \cdot x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := -x.im\_m \cdot \left(x.im\_m \cdot x.im\_m\right)\\ t_1 := x.im\_m \cdot \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) + x.re \cdot \left(x.re \cdot x.im\_m + x.re \cdot x.im\_m\right)\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-305}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left(x.re \cdot x.im\_m\right) \cdot \left(x.re \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0 (- (* x.im_m (* x.im_m x.im_m))))
        (t_1
         (+
          (* x.im_m (- (* x.re x.re) (* x.im_m x.im_m)))
          (* x.re (+ (* x.re x.im_m) (* x.re x.im_m))))))
   (*
    x.im_s
    (if (<= t_1 -2e-305)
      t_0
      (if (<= t_1 INFINITY) (* (* x.re x.im_m) (* x.re 3.0)) t_0)))))

C

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, double x_46_im_m) {
	double t_0 = -(x_46_im_m * (x_46_im_m * x_46_im_m));
	double t_1 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)));
	double tmp;
	if (t_1 <= -2e-305) {
		tmp = t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (x_46_re * x_46_im_m) * (x_46_re * 3.0);
	} else {
		tmp = t_0;
	}
	return x_46_im_s * tmp;
}

Java

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, double x_46_im_m) {
	double t_0 = -(x_46_im_m * (x_46_im_m * x_46_im_m));
	double t_1 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)));
	double tmp;
	if (t_1 <= -2e-305) {
		tmp = t_0;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (x_46_re * x_46_im_m) * (x_46_re * 3.0);
	} else {
		tmp = t_0;
	}
	return x_46_im_s * tmp;
}

Python

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, x_46_im_m):
	t_0 = -(x_46_im_m * (x_46_im_m * x_46_im_m))
	t_1 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)))
	tmp = 0
	if t_1 <= -2e-305:
		tmp = t_0
	elif t_1 <= math.inf:
		tmp = (x_46_re * x_46_im_m) * (x_46_re * 3.0)
	else:
		tmp = t_0
	return x_46_im_s * tmp

Julia

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, x_46_im_m)
	t_0 = Float64(-Float64(x_46_im_m * Float64(x_46_im_m * x_46_im_m)))
	t_1 = Float64(Float64(x_46_im_m * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m))) + Float64(x_46_re * Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_re * x_46_im_m))))
	tmp = 0.0
	if (t_1 <= -2e-305)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(x_46_re * x_46_im_m) * Float64(x_46_re * 3.0));
	else
		tmp = t_0;
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

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, x_46_im_m)
	t_0 = -(x_46_im_m * (x_46_im_m * x_46_im_m));
	t_1 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)));
	tmp = 0.0;
	if (t_1 <= -2e-305)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = (x_46_re * x_46_im_m) * (x_46_re * 3.0);
	else
		tmp = t_0;
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

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_, x$46$im$95$m_] := Block[{t$95$0 = (-N[(x$46$im$95$m * N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision])}, Block[{t$95$1 = N[(N[(x$46$im$95$m * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$re * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$1, -2e-305], t$95$0, If[LessEqual[t$95$1, Infinity], N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] * N[(x$46$re * 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)) < -1.99999999999999999e-305 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 69.7%

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

   3. Taylor expanded in x.re around 0

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left({x.im}^{3}\right)} \]

      2. unpow3 N/A

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

      3. unpow2 N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. neg-lowering-neg.f64 49.5

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

   5. Simplified 49.5%

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

   if -1.99999999999999999e-305 < (+.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 95.3%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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 99.9

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

   4. Applied egg-rr 99.9%

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

   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-lft1-in N/A

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

      2. metadata-eval N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 56.8

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

   7. Simplified 56.8%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

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

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

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

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

      8. *-lowering-*.f64 61.3

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

   9. Applied egg-rr 61.3%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -2 \cdot 10^{-305}:\\ \;\;\;\;-x.im \cdot \left(x.im \cdot x.im\right)\\ \mathbf{elif}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq \infty:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;-x.im \cdot \left(x.im \cdot x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := -x.im\_m \cdot \left(x.im\_m \cdot x.im\_m\right)\\ t_1 := x.im\_m \cdot \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) + x.re \cdot \left(x.re \cdot x.im\_m + x.re \cdot x.im\_m\right)\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-305}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;x.re \cdot \left(x.im\_m \cdot \left(x.re \cdot 3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0 (- (* x.im_m (* x.im_m x.im_m))))
        (t_1
         (+
          (* x.im_m (- (* x.re x.re) (* x.im_m x.im_m)))
          (* x.re (+ (* x.re x.im_m) (* x.re x.im_m))))))
   (*
    x.im_s
    (if (<= t_1 -2e-305)
      t_0
      (if (<= t_1 INFINITY) (* x.re (* x.im_m (* x.re 3.0))) t_0)))))

C

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, double x_46_im_m) {
	double t_0 = -(x_46_im_m * (x_46_im_m * x_46_im_m));
	double t_1 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)));
	double tmp;
	if (t_1 <= -2e-305) {
		tmp = t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = x_46_re * (x_46_im_m * (x_46_re * 3.0));
	} else {
		tmp = t_0;
	}
	return x_46_im_s * tmp;
}

Java

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, double x_46_im_m) {
	double t_0 = -(x_46_im_m * (x_46_im_m * x_46_im_m));
	double t_1 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)));
	double tmp;
	if (t_1 <= -2e-305) {
		tmp = t_0;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = x_46_re * (x_46_im_m * (x_46_re * 3.0));
	} else {
		tmp = t_0;
	}
	return x_46_im_s * tmp;
}

Python

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, x_46_im_m):
	t_0 = -(x_46_im_m * (x_46_im_m * x_46_im_m))
	t_1 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)))
	tmp = 0
	if t_1 <= -2e-305:
		tmp = t_0
	elif t_1 <= math.inf:
		tmp = x_46_re * (x_46_im_m * (x_46_re * 3.0))
	else:
		tmp = t_0
	return x_46_im_s * tmp

Julia

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, x_46_im_m)
	t_0 = Float64(-Float64(x_46_im_m * Float64(x_46_im_m * x_46_im_m)))
	t_1 = Float64(Float64(x_46_im_m * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m))) + Float64(x_46_re * Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_re * x_46_im_m))))
	tmp = 0.0
	if (t_1 <= -2e-305)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = Float64(x_46_re * Float64(x_46_im_m * Float64(x_46_re * 3.0)));
	else
		tmp = t_0;
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

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, x_46_im_m)
	t_0 = -(x_46_im_m * (x_46_im_m * x_46_im_m));
	t_1 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)));
	tmp = 0.0;
	if (t_1 <= -2e-305)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = x_46_re * (x_46_im_m * (x_46_re * 3.0));
	else
		tmp = t_0;
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

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_, x$46$im$95$m_] := Block[{t$95$0 = (-N[(x$46$im$95$m * N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision])}, Block[{t$95$1 = N[(N[(x$46$im$95$m * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$re * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$1, -2e-305], t$95$0, If[LessEqual[t$95$1, Infinity], N[(x$46$re * N[(x$46$im$95$m * N[(x$46$re * 3.0), $MachinePrecision]), $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)) < -1.99999999999999999e-305 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 69.7%

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

   3. Taylor expanded in x.re around 0

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left({x.im}^{3}\right)} \]

      2. unpow3 N/A

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

      3. unpow2 N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. neg-lowering-neg.f64 49.5

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

   5. Simplified 49.5%

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

   if -1.99999999999999999e-305 < (+.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 95.3%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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 99.9

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

   4. Applied egg-rr 99.9%

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

   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-lft1-in N/A

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

      2. metadata-eval N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 56.8

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

   7. Simplified 56.8%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. *-lowering-*.f64 61.3

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

   9. Applied egg-rr 61.3%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -2 \cdot 10^{-305}:\\ \;\;\;\;-x.im \cdot \left(x.im \cdot x.im\right)\\ \mathbf{elif}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq \infty:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-x.im \cdot \left(x.im \cdot x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := -x.im\_m \cdot \left(x.im\_m \cdot x.im\_m\right)\\ t_1 := x.im\_m \cdot \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) + x.re \cdot \left(x.re \cdot x.im\_m + x.re \cdot x.im\_m\right)\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-305}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;x.im\_m \cdot \left(\left(x.re \cdot x.re\right) \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0 (- (* x.im_m (* x.im_m x.im_m))))
        (t_1
         (+
          (* x.im_m (- (* x.re x.re) (* x.im_m x.im_m)))
          (* x.re (+ (* x.re x.im_m) (* x.re x.im_m))))))
   (*
    x.im_s
    (if (<= t_1 -2e-305)
      t_0
      (if (<= t_1 INFINITY) (* x.im_m (* (* x.re x.re) 3.0)) t_0)))))

C

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, double x_46_im_m) {
	double t_0 = -(x_46_im_m * (x_46_im_m * x_46_im_m));
	double t_1 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)));
	double tmp;
	if (t_1 <= -2e-305) {
		tmp = t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = x_46_im_m * ((x_46_re * x_46_re) * 3.0);
	} else {
		tmp = t_0;
	}
	return x_46_im_s * tmp;
}

Java

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, double x_46_im_m) {
	double t_0 = -(x_46_im_m * (x_46_im_m * x_46_im_m));
	double t_1 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)));
	double tmp;
	if (t_1 <= -2e-305) {
		tmp = t_0;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = x_46_im_m * ((x_46_re * x_46_re) * 3.0);
	} else {
		tmp = t_0;
	}
	return x_46_im_s * tmp;
}

Python

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, x_46_im_m):
	t_0 = -(x_46_im_m * (x_46_im_m * x_46_im_m))
	t_1 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)))
	tmp = 0
	if t_1 <= -2e-305:
		tmp = t_0
	elif t_1 <= math.inf:
		tmp = x_46_im_m * ((x_46_re * x_46_re) * 3.0)
	else:
		tmp = t_0
	return x_46_im_s * tmp

Julia

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, x_46_im_m)
	t_0 = Float64(-Float64(x_46_im_m * Float64(x_46_im_m * x_46_im_m)))
	t_1 = Float64(Float64(x_46_im_m * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m))) + Float64(x_46_re * Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_re * x_46_im_m))))
	tmp = 0.0
	if (t_1 <= -2e-305)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = Float64(x_46_im_m * Float64(Float64(x_46_re * x_46_re) * 3.0));
	else
		tmp = t_0;
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

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, x_46_im_m)
	t_0 = -(x_46_im_m * (x_46_im_m * x_46_im_m));
	t_1 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)));
	tmp = 0.0;
	if (t_1 <= -2e-305)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = x_46_im_m * ((x_46_re * x_46_re) * 3.0);
	else
		tmp = t_0;
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

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_, x$46$im$95$m_] := Block[{t$95$0 = (-N[(x$46$im$95$m * N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision])}, Block[{t$95$1 = N[(N[(x$46$im$95$m * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$re * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$1, -2e-305], t$95$0, If[LessEqual[t$95$1, Infinity], N[(x$46$im$95$m * N[(N[(x$46$re * x$46$re), $MachinePrecision] * 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)) < -1.99999999999999999e-305 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 69.7%

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

   3. Taylor expanded in x.re around 0

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left({x.im}^{3}\right)} \]

      2. unpow3 N/A

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

      3. unpow2 N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. neg-lowering-neg.f64 49.5

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

   5. Simplified 49.5%

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

   if -1.99999999999999999e-305 < (+.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 95.3%

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

   3. Taylor expanded in x.re around inf

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft1-in N/A

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

      7. *-rgt-identity N/A

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

      8. *-inverses N/A

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

      9. associate-/l* N/A

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

      10. unpow2 N/A

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

      11. cube-mult N/A

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

      12. associate-/l* N/A

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

      13. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{2 \cdot {x.re}^{2} + {x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3}} \]

      14. distribute-lft1-in N/A

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

      15. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{3} \cdot {x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3} \]

      16. associate-*r/ N/A

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

      17. associate-*l* N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

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

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

   5. Simplified 56.8%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -2 \cdot 10^{-305}:\\ \;\;\;\;-x.im \cdot \left(x.im \cdot x.im\right)\\ \mathbf{elif}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq \infty:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.re\right) \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;-x.im \cdot \left(x.im \cdot x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 5 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(x.re \cdot \left(x.im\_m \cdot 3\right), x.re, -x.im\_m \cdot \left(x.im\_m \cdot x.im\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re + x.im\_m, x.im\_m \cdot \left(x.re - x.im\_m\right), x.im\_m + x.im\_m\right)\\ \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.im_m 5e+102)
    (fma (* x.re (* x.im_m 3.0)) x.re (- (* x.im_m (* x.im_m x.im_m))))
    (fma (+ x.re x.im_m) (* x.im_m (- x.re x.im_m)) (+ x.im_m x.im_m)))))

C

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, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 5e+102) {
		tmp = fma((x_46_re * (x_46_im_m * 3.0)), x_46_re, -(x_46_im_m * (x_46_im_m * x_46_im_m)));
	} else {
		tmp = fma((x_46_re + x_46_im_m), (x_46_im_m * (x_46_re - x_46_im_m)), (x_46_im_m + x_46_im_m));
	}
	return x_46_im_s * tmp;
}

Julia

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, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 5e+102)
		tmp = fma(Float64(x_46_re * Float64(x_46_im_m * 3.0)), x_46_re, Float64(-Float64(x_46_im_m * Float64(x_46_im_m * x_46_im_m))));
	else
		tmp = fma(Float64(x_46_re + x_46_im_m), Float64(x_46_im_m * Float64(x_46_re - x_46_im_m)), Float64(x_46_im_m + x_46_im_m));
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

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_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 5e+102], N[(N[(x$46$re * N[(x$46$im$95$m * 3.0), $MachinePrecision]), $MachinePrecision] * x$46$re + (-N[(x$46$im$95$m * N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(N[(x$46$re + x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m * N[(x$46$re - x$46$im$95$m), $MachinePrecision]), $MachinePrecision] + N[(x$46$im$95$m + x$46$im$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 5e102

   1. Initial program 84.2%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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 89.7%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

      9. associate-*l* N/A

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

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

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

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

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

      12. associate-*l* N/A

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

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

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

      14. *-lowering-*.f64 91.4

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

   7. Applied egg-rr 91.4%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-lowering-*.f64 91.4

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

   9. Applied egg-rr 91.4%

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

   if 5e102 < x.im

   1. Initial program 69.0%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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 76.2

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

   4. Applied egg-rr 76.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.im \cdot \left(x.re + x.im\right), x.re - x.im, x.re \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\right)} \]
   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. difference-of-squares N/A

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

      5. associate-*l* N/A

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

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

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

      7. +-commutative N/A

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

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

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

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

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

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

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

      11. distribute-lft-in N/A

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

      12. flip-+ N/A

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

      13. +-inverses N/A

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

      14. +-inverses N/A

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

      15. associate-*r/ N/A

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

      16. +-inverses N/A

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

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

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

      18. +-inverses N/A

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

      19. +-inverses N/A

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

      20. +-inverses N/A

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

      21. flip-+ N/A

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

   6. Applied egg-rr 100.0%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 5 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(x.re \cdot \left(x.im \cdot 3\right), x.re, -x.im \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re + x.im, x.im \cdot \left(x.re - x.im\right), x.im + x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 5 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(x.im\_m \cdot \left(x.re \cdot 3\right), x.re, -x.im\_m \cdot \left(x.im\_m \cdot x.im\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re + x.im\_m, x.im\_m \cdot \left(x.re - x.im\_m\right), x.im\_m + x.im\_m\right)\\ \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.im_m 5e+102)
    (fma (* x.im_m (* x.re 3.0)) x.re (- (* x.im_m (* x.im_m x.im_m))))
    (fma (+ x.re x.im_m) (* x.im_m (- x.re x.im_m)) (+ x.im_m x.im_m)))))

C

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, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 5e+102) {
		tmp = fma((x_46_im_m * (x_46_re * 3.0)), x_46_re, -(x_46_im_m * (x_46_im_m * x_46_im_m)));
	} else {
		tmp = fma((x_46_re + x_46_im_m), (x_46_im_m * (x_46_re - x_46_im_m)), (x_46_im_m + x_46_im_m));
	}
	return x_46_im_s * tmp;
}

Julia

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, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 5e+102)
		tmp = fma(Float64(x_46_im_m * Float64(x_46_re * 3.0)), x_46_re, Float64(-Float64(x_46_im_m * Float64(x_46_im_m * x_46_im_m))));
	else
		tmp = fma(Float64(x_46_re + x_46_im_m), Float64(x_46_im_m * Float64(x_46_re - x_46_im_m)), Float64(x_46_im_m + x_46_im_m));
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

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_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 5e+102], N[(N[(x$46$im$95$m * N[(x$46$re * 3.0), $MachinePrecision]), $MachinePrecision] * x$46$re + (-N[(x$46$im$95$m * N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(N[(x$46$re + x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m * N[(x$46$re - x$46$im$95$m), $MachinePrecision]), $MachinePrecision] + N[(x$46$im$95$m + x$46$im$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 5e102

   1. Initial program 84.2%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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 89.7%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

      9. associate-*l* N/A

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

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

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

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

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

      12. associate-*l* N/A

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

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

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

      14. *-lowering-*.f64 91.4

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

   7. Applied egg-rr 91.4%

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

   if 5e102 < x.im

   1. Initial program 69.0%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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 76.2

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

   4. Applied egg-rr 76.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.im \cdot \left(x.re + x.im\right), x.re - x.im, x.re \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\right)} \]
   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. difference-of-squares N/A

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

      5. associate-*l* N/A

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

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

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

      7. +-commutative N/A

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

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

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

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

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

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

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

      11. distribute-lft-in N/A

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

      12. flip-+ N/A

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

      13. +-inverses N/A

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

      14. +-inverses N/A

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

      15. associate-*r/ N/A

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

      16. +-inverses N/A

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

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

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

      18. +-inverses N/A

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

      19. +-inverses N/A

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

      20. +-inverses N/A

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

      21. flip-+ N/A

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

   6. Applied egg-rr 100.0%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 5 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(x.im \cdot \left(x.re \cdot 3\right), x.re, -x.im \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re + x.im, x.im \cdot \left(x.re - x.im\right), x.im + x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.re \leq 2.1 \cdot 10^{+163}:\\ \;\;\;\;-x.im\_m \cdot \left(x.im\_m \cdot x.im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(x.re \cdot x.im\_m\right)\\ \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.re 2.1e+163)
    (- (* x.im_m (* x.im_m x.im_m)))
    (* x.im_m (* x.re x.im_m)))))

C

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, double x_46_im_m) {
	double tmp;
	if (x_46_re <= 2.1e+163) {
		tmp = -(x_46_im_m * (x_46_im_m * x_46_im_m));
	} else {
		tmp = x_46_im_m * (x_46_re * x_46_im_m);
	}
	return x_46_im_s * tmp;
}

Fortran

x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46re <= 2.1d+163) then
        tmp = -(x_46im_m * (x_46im_m * x_46im_m))
    else
        tmp = x_46im_m * (x_46re * x_46im_m)
    end if
    code = x_46im_s * tmp
end function

Java

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, double x_46_im_m) {
	double tmp;
	if (x_46_re <= 2.1e+163) {
		tmp = -(x_46_im_m * (x_46_im_m * x_46_im_m));
	} else {
		tmp = x_46_im_m * (x_46_re * x_46_im_m);
	}
	return x_46_im_s * tmp;
}

Python

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, x_46_im_m):
	tmp = 0
	if x_46_re <= 2.1e+163:
		tmp = -(x_46_im_m * (x_46_im_m * x_46_im_m))
	else:
		tmp = x_46_im_m * (x_46_re * x_46_im_m)
	return x_46_im_s * tmp

Julia

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, x_46_im_m)
	tmp = 0.0
	if (x_46_re <= 2.1e+163)
		tmp = Float64(-Float64(x_46_im_m * Float64(x_46_im_m * x_46_im_m)));
	else
		tmp = Float64(x_46_im_m * Float64(x_46_re * x_46_im_m));
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

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, x_46_im_m)
	tmp = 0.0;
	if (x_46_re <= 2.1e+163)
		tmp = -(x_46_im_m * (x_46_im_m * x_46_im_m));
	else
		tmp = x_46_im_m * (x_46_re * x_46_im_m);
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

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_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$re, 2.1e+163], (-N[(x$46$im$95$m * N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), N[(x$46$im$95$m * N[(x$46$re * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 2.1e163

   1. Initial program 84.8%

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

   3. Taylor expanded in x.re around 0

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left({x.im}^{3}\right)} \]

      2. unpow3 N/A

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

      3. unpow2 N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. neg-lowering-neg.f64 67.6

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

   5. Simplified 67.6%

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

   if 2.1e163 < x.re

   1. Initial program 61.8%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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 94.2

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

   4. Applied egg-rr 94.2%

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

   5. Taylor expanded in x.re around inf

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

   7. Simplified 94.2%

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

   8. Taylor expanded in x.im around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

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

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

      4. *-lowering-*.f64 20.8

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

   10. Simplified 20.8%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 2.1 \cdot 10^{+163}:\\ \;\;\;\;-x.im \cdot \left(x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 29.0% accurate, 3.6× speedup

Math

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

FPCore

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

C

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, double x_46_im_m) {
	return x_46_im_s * (x_46_im_m * (x_46_re * x_46_im_m));
}

Fortran

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

Java

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, double x_46_im_m) {
	return x_46_im_s * (x_46_im_m * (x_46_re * x_46_im_m));
}

Python

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, x_46_im_m):
	return x_46_im_s * (x_46_im_m * (x_46_re * x_46_im_m))

Julia

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, x_46_im_m)
	return Float64(x_46_im_s * Float64(x_46_im_m * Float64(x_46_re * x_46_im_m)))
end

MATLAB

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, x_46_im_m)
	tmp = x_46_im_s * (x_46_im_m * (x_46_re * x_46_im_m));
end

Wolfram

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_, x$46$im$95$m_] := N[(x$46$im$95$s * N[(x$46$im$95$m * N[(x$46$re * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.7%

   \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
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 92.4

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

4. Applied egg-rr 92.4%

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

5. Taylor expanded in x.re around inf

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

7. Simplified 64.6%

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

8. Taylor expanded in x.im around inf

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

   1. unpow2 N/A

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

   2. associate-*l* N/A

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

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

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

   4. *-lowering-*.f64 29.2

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

10. Simplified 29.2%

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

11. Final simplification 29.2%

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

Developer Target 1: 91.4% 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 2024204 
(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)))