math.cube on complex, real part

Percentage Accurate: 83.1% → 99.8%

Time: 21.7s

Alternatives: 8

Speedup: 1.4×

• 44.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 83.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 92.2%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. neg-sub0 N/A

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

      7. +-inverses N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

      11. +-inverses N/A

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

      12. *-commutative N/A

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

      13. distribute-rgt-out N/A

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

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

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

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

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

      16. difference-of-squares N/A

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

   4. Applied egg-rr 99.8%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 99.8

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

   6. Applied egg-rr 99.8%

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

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

   1. Initial program 0.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. neg-sub0 N/A

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

      7. +-inverses N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

      11. +-inverses N/A

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

      12. *-commutative N/A

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

      13. distribute-rgt-out N/A

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

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

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

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

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

      16. difference-of-squares N/A

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

   4. Applied egg-rr 50.0%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 50.0

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

   6. Applied egg-rr 50.0%

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

   7. Taylor expanded in x.re around 0

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

   9. Simplified 33.3%

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

       1. +-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

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

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

       5. flip-+ N/A

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

       6. +-inverses N/A

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

       7. metadata-eval N/A

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

       8. metadata-eval N/A

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

       9. metadata-eval N/A

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

       10. +-inverses N/A

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

       11. metadata-eval N/A

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

       12. flip-- N/A

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

       13. metadata-eval N/A

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

       14. mul0-rgt N/A

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

       15. mul0-lft N/A

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

       16. metadata-eval N/A

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

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

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

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

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

       19. --lowering--.f64 63.9

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

   11. Applied egg-rr 63.9%

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

4. Final simplification 94.8%

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

Alternative 2: 98.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ \begin{array}{l} t_0 := x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) - x.im\_m \cdot \left(x.re\_m \cdot x.im\_m + x.re\_m \cdot x.im\_m\right)\\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-286}:\\ \;\;\;\;\left(x.re\_m \cdot x.im\_m\right) \cdot \left(x.im\_m \cdot -3\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.im\_m \cdot \left(x.re\_m - x.im\_m\right), x.re\_m, 0\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (let* ((t_0
         (-
          (* x.re_m (- (* x.re_m x.re_m) (* x.im_m x.im_m)))
          (* x.im_m (+ (* x.re_m x.im_m) (* x.re_m x.im_m))))))
   (*
    x.re_s
    (if (<= t_0 -1e-286)
      (* (* x.re_m x.im_m) (* x.im_m -3.0))
      (if (<= t_0 INFINITY)
        (* x.re_m (* x.re_m x.re_m))
        (fma (* x.im_m (- x.re_m x.im_m)) x.re_m 0.0))))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double t_0 = (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m))) - (x_46_im_m * ((x_46_re_m * x_46_im_m) + (x_46_re_m * x_46_im_m)));
	double tmp;
	if (t_0 <= -1e-286) {
		tmp = (x_46_re_m * x_46_im_m) * (x_46_im_m * -3.0);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	} else {
		tmp = fma((x_46_im_m * (x_46_re_m - x_46_im_m)), x_46_re_m, 0.0);
	}
	return x_46_re_s * tmp;
}

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	t_0 = Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m))) - Float64(x_46_im_m * Float64(Float64(x_46_re_m * x_46_im_m) + Float64(x_46_re_m * x_46_im_m))))
	tmp = 0.0
	if (t_0 <= -1e-286)
		tmp = Float64(Float64(x_46_re_m * x_46_im_m) * Float64(x_46_im_m * -3.0));
	elseif (t_0 <= Inf)
		tmp = Float64(x_46_re_m * Float64(x_46_re_m * x_46_re_m));
	else
		tmp = fma(Float64(x_46_im_m * Float64(x_46_re_m - x_46_im_m)), x_46_re_m, 0.0);
	end
	return Float64(x_46_re_s * tmp)
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im$95$m * N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$re$95$s * If[LessEqual[t$95$0, -1e-286], N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m * -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im$95$m * N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m + 0.0), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -1.00000000000000005e-286

   1. Initial program 91.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. neg-sub0 N/A

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

      7. +-inverses N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

      11. +-inverses N/A

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

      12. *-commutative N/A

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

      13. distribute-rgt-out N/A

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

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

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

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

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

      16. difference-of-squares N/A

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x.im around inf

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. +-rgt-identity N/A

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

      5. *-commutative N/A

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

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

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

      7. metadata-eval N/A

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

      8. associate-*r* N/A

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

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

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

      10. +-rgt-identity N/A

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

      11. *-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

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

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

      15. *-lowering-*.f64 43.6

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

   7. Simplified 43.6%

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

      1. +-rgt-identity N/A

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

      2. +-rgt-identity N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. *-lowering-*.f64 52.3

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

   9. Applied egg-rr 52.3%

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

   if -1.00000000000000005e-286 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < +inf.0

   1. Initial program 93.1%

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

   3. Taylor expanded in x.re around inf

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

      1. +-rgt-identity N/A

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. +-rgt-identity N/A

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

      6. unpow2 N/A

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

      7. accelerator-lowering-fma.f64 66.1

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

   5. Simplified 66.1%

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

      1. +-rgt-identity N/A

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

      2. *-lowering-*.f64 66.1

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

   7. Applied egg-rr 66.1%

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

      1. +-rgt-identity N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-lowering-*.f64 66.1

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

   9. Applied egg-rr 66.1%

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

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

   1. Initial program 0.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. neg-sub0 N/A

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

      7. +-inverses N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

      11. +-inverses N/A

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

      12. *-commutative N/A

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

      13. distribute-rgt-out N/A

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

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

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

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

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

      16. difference-of-squares N/A

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

   4. Applied egg-rr 50.0%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 50.0

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

   6. Applied egg-rr 50.0%

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

   7. Taylor expanded in x.re around 0

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

   9. Simplified 33.3%

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

       1. +-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

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

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

       5. flip-+ N/A

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

       6. +-inverses N/A

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

       7. metadata-eval N/A

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

       8. metadata-eval N/A

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

       9. metadata-eval N/A

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

       10. +-inverses N/A

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

       11. metadata-eval N/A

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

       12. flip-- N/A

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

       13. metadata-eval N/A

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

       14. mul0-rgt N/A

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

       15. mul0-lft N/A

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

       16. metadata-eval N/A

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

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

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

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

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

       19. --lowering--.f64 63.9

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

   11. Applied egg-rr 63.9%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -1 \cdot 10^{-286}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.im \cdot -3\right)\\ \mathbf{elif}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq \infty:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.im \cdot \left(x.re - x.im\right), x.re, 0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 92.2%

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

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

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

      2. difference-of-squares N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

      11. distribute-rgt-out N/A

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

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

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

      13. +-lowering-+.f64 99.8

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

   4. Applied egg-rr 99.8%

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

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

   1. Initial program 0.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. neg-sub0 N/A

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

      7. +-inverses N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

      11. +-inverses N/A

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

      12. *-commutative N/A

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

      13. distribute-rgt-out N/A

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

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

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

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

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

      16. difference-of-squares N/A

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

   4. Applied egg-rr 50.0%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 50.0

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

   6. Applied egg-rr 50.0%

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

   7. Taylor expanded in x.re around 0

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

   9. Simplified 33.3%

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

       1. +-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

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

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

       5. flip-+ N/A

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

       6. +-inverses N/A

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

       7. metadata-eval N/A

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

       8. metadata-eval N/A

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

       9. metadata-eval N/A

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

       10. +-inverses N/A

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

       11. metadata-eval N/A

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

       12. flip-- N/A

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

       13. metadata-eval N/A

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

       14. mul0-rgt N/A

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

       15. mul0-lft N/A

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

       16. metadata-eval N/A

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

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

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

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

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

       19. --lowering--.f64 63.9

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

   11. Applied egg-rr 63.9%

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

4. Final simplification 94.7%

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

Alternative 4: 99.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 2.9 \cdot 10^{+118}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m, \mathsf{fma}\left(x.re\_m, x.re\_m, \left(x.im\_m \cdot x.im\_m\right) \cdot -3\right), 0\right)\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(x.re\_m \cdot \mathsf{fma}\left(x.im\_m, -3, x.re\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<= x.im_m 2.9e+118)
    (fma x.re_m (fma x.re_m x.re_m (* (* x.im_m x.im_m) -3.0)) 0.0)
    (* x.im_m (* x.re_m (fma x.im_m -3.0 x.re_m))))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 2.9e+118) {
		tmp = fma(x_46_re_m, fma(x_46_re_m, x_46_re_m, ((x_46_im_m * x_46_im_m) * -3.0)), 0.0);
	} else {
		tmp = x_46_im_m * (x_46_re_m * fma(x_46_im_m, -3.0, x_46_re_m));
	}
	return x_46_re_s * tmp;
}

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 2.9e+118)
		tmp = fma(x_46_re_m, fma(x_46_re_m, x_46_re_m, Float64(Float64(x_46_im_m * x_46_im_m) * -3.0)), 0.0);
	else
		tmp = Float64(x_46_im_m * Float64(x_46_re_m * fma(x_46_im_m, -3.0, x_46_re_m)));
	end
	return Float64(x_46_re_s * tmp)
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * If[LessEqual[x$46$im$95$m, 2.9e+118], N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$re$95$m + N[(N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision], N[(x$46$im$95$m * N[(x$46$re$95$m * N[(x$46$im$95$m * -3.0 + x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 2.90000000000000016e118

   1. Initial program 84.7%

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

   3. Taylor expanded in x.re around 0

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

   5. Simplified 94.5%

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

      1. +-rgt-identity N/A

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

      2. *-lowering-*.f64 94.5

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

   7. Applied egg-rr 94.5%

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

   if 2.90000000000000016e118 < x.im

   1. Initial program 51.4%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. neg-sub0 N/A

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

      7. +-inverses N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

      11. +-inverses N/A

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

      12. *-commutative N/A

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

      13. distribute-rgt-out N/A

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

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

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

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

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

      16. difference-of-squares N/A

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

   4. Applied egg-rr 80.8%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 80.8

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

   6. Applied egg-rr 80.8%

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

   7. Taylor expanded in x.re around 0

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

   9. Simplified 80.8%

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

   10. Taylor expanded in x.im around 0

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

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

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

       2. *-commutative N/A

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

       3. distribute-rgt-out N/A

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

       4. metadata-eval N/A

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

       5. associate-*l* N/A

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

       6. unpow2 N/A

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

       7. distribute-lft-out N/A

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

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

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

       9. *-commutative N/A

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

       10. accelerator-lowering-fma.f64 90.3

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

   12. Simplified 90.3%

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

Alternative 5: 99.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 2 \cdot 10^{+121}:\\ \;\;\;\;x.re\_m \cdot \mathsf{fma}\left(x.im\_m, x.im\_m \cdot -3, x.re\_m \cdot x.re\_m\right)\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(x.re\_m \cdot \mathsf{fma}\left(x.im\_m, -3, x.re\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<= x.im_m 2e+121)
    (* x.re_m (fma x.im_m (* x.im_m -3.0) (* x.re_m x.re_m)))
    (* x.im_m (* x.re_m (fma x.im_m -3.0 x.re_m))))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 2e+121) {
		tmp = x_46_re_m * fma(x_46_im_m, (x_46_im_m * -3.0), (x_46_re_m * x_46_re_m));
	} else {
		tmp = x_46_im_m * (x_46_re_m * fma(x_46_im_m, -3.0, x_46_re_m));
	}
	return x_46_re_s * tmp;
}

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 2e+121)
		tmp = Float64(x_46_re_m * fma(x_46_im_m, Float64(x_46_im_m * -3.0), Float64(x_46_re_m * x_46_re_m)));
	else
		tmp = Float64(x_46_im_m * Float64(x_46_re_m * fma(x_46_im_m, -3.0, x_46_re_m)));
	end
	return Float64(x_46_re_s * tmp)
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * If[LessEqual[x$46$im$95$m, 2e+121], N[(x$46$re$95$m * N[(x$46$im$95$m * N[(x$46$im$95$m * -3.0), $MachinePrecision] + N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im$95$m * N[(x$46$re$95$m * N[(x$46$im$95$m * -3.0 + x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 2.00000000000000007e121

   1. Initial program 84.7%

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

   3. Taylor expanded in x.re around 0

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

   5. Simplified 94.5%

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

      1. +-rgt-identity N/A

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

      2. *-lowering-*.f64 94.5

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

   7. Applied egg-rr 94.5%

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

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

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

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

      4. +-commutative N/A

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

      5. associate-*l* N/A

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

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

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

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

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

      8. *-lowering-*.f64 93.1

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

   9. Applied egg-rr 93.1%

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

   if 2.00000000000000007e121 < x.im

   1. Initial program 51.4%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. neg-sub0 N/A

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

      7. +-inverses N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

      11. +-inverses N/A

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

      12. *-commutative N/A

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

      13. distribute-rgt-out N/A

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

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

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

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

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

      16. difference-of-squares N/A

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

   4. Applied egg-rr 80.8%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 80.8

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

   6. Applied egg-rr 80.8%

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

   7. Taylor expanded in x.re around 0

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

   9. Simplified 80.8%

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

   10. Taylor expanded in x.im around 0

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

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

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

       2. *-commutative N/A

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

       3. distribute-rgt-out N/A

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

       4. metadata-eval N/A

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

       5. associate-*l* N/A

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

       6. unpow2 N/A

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

       7. distribute-lft-out N/A

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

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

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

       9. *-commutative N/A

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

       10. accelerator-lowering-fma.f64 90.3

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

   12. Simplified 90.3%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 2 \cdot 10^{+121}:\\ \;\;\;\;x.re \cdot \mathsf{fma}\left(x.im, x.im \cdot -3, x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \mathsf{fma}\left(x.im, -3, x.re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 1.26 \cdot 10^{+32}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m - x.im\_m, x.re\_m \cdot x.im\_m, 0\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<= x.im_m 1.26e+32)
    (* x.re_m (* x.re_m x.re_m))
    (fma (- x.re_m x.im_m) (* x.re_m x.im_m) 0.0))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 1.26e+32) {
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	} else {
		tmp = fma((x_46_re_m - x_46_im_m), (x_46_re_m * x_46_im_m), 0.0);
	}
	return x_46_re_s * tmp;
}

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 1.26e+32)
		tmp = Float64(x_46_re_m * Float64(x_46_re_m * x_46_re_m));
	else
		tmp = fma(Float64(x_46_re_m - x_46_im_m), Float64(x_46_re_m * x_46_im_m), 0.0);
	end
	return Float64(x_46_re_s * tmp)
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * If[LessEqual[x$46$im$95$m, 1.26e+32], N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision] * N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] + 0.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.26e32

   1. Initial program 87.3%

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

   3. Taylor expanded in x.re around inf

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

      1. +-rgt-identity N/A

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. +-rgt-identity N/A

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

      6. unpow2 N/A

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

      7. accelerator-lowering-fma.f64 71.2

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

   5. Simplified 71.2%

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

      1. +-rgt-identity N/A

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

      2. *-lowering-*.f64 71.2

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

   7. Applied egg-rr 71.2%

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

      1. +-rgt-identity N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-lowering-*.f64 71.2

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

   9. Applied egg-rr 71.2%

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

   if 1.26e32 < x.im

   1. Initial program 54.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. neg-sub0 N/A

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

      7. +-inverses N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

      11. +-inverses N/A

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

      12. *-commutative N/A

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

      13. distribute-rgt-out N/A

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

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

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

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

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

      16. difference-of-squares N/A

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

   4. Applied egg-rr 80.4%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 80.4

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

   6. Applied egg-rr 80.4%

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

   7. Taylor expanded in x.re around 0

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

   9. Simplified 77.2%

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. *-commutative N/A

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

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

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

       7. flip-+ N/A

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

       8. +-inverses N/A

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

       9. metadata-eval N/A

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

       10. metadata-eval N/A

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

       11. metadata-eval N/A

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

       12. +-inverses N/A

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

       13. metadata-eval N/A

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

       14. flip-- N/A

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

       15. metadata-eval N/A

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

       16. mul0-rgt N/A

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

       17. mul0-lft N/A

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

       18. metadata-eval N/A

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

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

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

       20. --lowering--.f64 N/A

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

       21. *-commutative N/A

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

       22. *-lowering-*.f64 56.9

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

   11. Applied egg-rr 56.9%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 1.26 \cdot 10^{+32}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im, x.re \cdot x.im, 0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 3.4 \cdot 10^{+33}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.im\_m \cdot \left(x.re\_m - x.im\_m\right), x.re\_m, 0\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im_m)
 :precision binary64
 (*
  x.re_s
  (if (<= x.im_m 3.4e+33)
    (* x.re_m (* x.re_m x.re_m))
    (fma (* x.im_m (- x.re_m x.im_m)) x.re_m 0.0))))

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 3.4e+33) {
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	} else {
		tmp = fma((x_46_im_m * (x_46_re_m - x_46_im_m)), x_46_re_m, 0.0);
	}
	return x_46_re_s * tmp;
}

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 3.4e+33)
		tmp = Float64(x_46_re_m * Float64(x_46_re_m * x_46_re_m));
	else
		tmp = fma(Float64(x_46_im_m * Float64(x_46_re_m - x_46_im_m)), x_46_re_m, 0.0);
	end
	return Float64(x_46_re_s * tmp)
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im$95$m_] := N[(x$46$re$95$s * If[LessEqual[x$46$im$95$m, 3.4e+33], N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im$95$m * N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m + 0.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 3.3999999999999999e33

   1. Initial program 87.3%

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

   3. Taylor expanded in x.re around inf

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

      1. +-rgt-identity N/A

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. +-rgt-identity N/A

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

      6. unpow2 N/A

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

      7. accelerator-lowering-fma.f64 71.2

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

   5. Simplified 71.2%

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

      1. +-rgt-identity N/A

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

      2. *-lowering-*.f64 71.2

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

   7. Applied egg-rr 71.2%

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

      1. +-rgt-identity N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-lowering-*.f64 71.2

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

   9. Applied egg-rr 71.2%

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

   if 3.3999999999999999e33 < x.im

   1. Initial program 54.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. neg-sub0 N/A

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

      7. +-inverses N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

      11. +-inverses N/A

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

      12. *-commutative N/A

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

      13. distribute-rgt-out N/A

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

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

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

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

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

      16. difference-of-squares N/A

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

   4. Applied egg-rr 80.4%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 80.4

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

   6. Applied egg-rr 80.4%

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

   7. Taylor expanded in x.re around 0

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

   9. Simplified 77.2%

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

       1. +-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

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

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

       5. flip-+ N/A

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

       6. +-inverses N/A

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

       7. metadata-eval N/A

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

       8. metadata-eval N/A

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

       9. metadata-eval N/A

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

       10. +-inverses N/A

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

       11. metadata-eval N/A

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

       12. flip-- N/A

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

       13. metadata-eval N/A

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

       14. mul0-rgt N/A

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

       15. mul0-lft N/A

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

       16. metadata-eval N/A

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

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

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

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

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

       19. --lowering--.f64 55.0

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

   11. Applied egg-rr 55.0%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 3.4 \cdot 10^{+33}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.im \cdot \left(x.re - x.im\right), x.re, 0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 58.8% accurate, 3.6× speedup

Math

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

FPCore

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

C

x.im_m = fabs(x_46_im);
x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	return x_46_re_s * (x_46_re_m * (x_46_re_m * x_46_re_m));
}

Fortran

x.im_m = abs(x_46im)
x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    code = x_46re_s * (x_46re_m * (x_46re_m * x_46re_m))
end function

Java

x.im_m = Math.abs(x_46_im);
x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im_m) {
	return x_46_re_s * (x_46_re_m * (x_46_re_m * x_46_re_m));
}

Python

x.im_m = math.fabs(x_46_im)
x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im_m):
	return x_46_re_s * (x_46_re_m * (x_46_re_m * x_46_re_m))

Julia

x.im_m = abs(x_46_im)
x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im_m)
	return Float64(x_46_re_s * Float64(x_46_re_m * Float64(x_46_re_m * x_46_re_m)))
end

MATLAB

x.im_m = abs(x_46_im);
x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp = code(x_46_re_s, x_46_re_m, x_46_im_m)
	tmp = x_46_re_s * (x_46_re_m * (x_46_re_m * x_46_re_m));
end

Wolfram

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

Derivation

1. Initial program 79.2%

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

3. Taylor expanded in x.re around inf

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

   1. +-rgt-identity N/A

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

   2. cube-mult N/A

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

   3. unpow2 N/A

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

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

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

   5. +-rgt-identity N/A

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

   6. unpow2 N/A

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

   7. accelerator-lowering-fma.f64 60.5

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

5. Simplified 60.5%

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

   1. +-rgt-identity N/A

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

   2. *-lowering-*.f64 60.5

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

7. Applied egg-rr 60.5%

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

   1. +-rgt-identity N/A

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

   2. associate-*r* N/A

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

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

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

   4. *-lowering-*.f64 60.5

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

9. Applied egg-rr 60.5%

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

10. Final simplification 60.5%

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

Developer Target 1: 99.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024198 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (* (* x.re x.re) (- x.re x.im)) (* (* x.re x.im) (- x.re (* 3 x.im)))))

  (- (* (- (* x.re x.re) (* x.im x.im)) x.re) (* (+ (* x.re x.im) (* x.im x.re)) x.im)))