math.cube on complex, imaginary part

Percentage Accurate: 82.2% → 99.5%

Time: 13.6s

Alternatives: 13

Speedup: 1.1×

• 44.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 82.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 1.56 \cdot 10^{+98}:\\ \;\;\;\;\mathsf{fma}\left(x.im\_m + x.re, x.im\_m \cdot \left(x.re - x.im\_m\right), x.re \cdot \left(x.re \cdot \left(x.im\_m + x.im\_m\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\mathsf{fma}\left(x.re - x.im\_m, x.re \cdot \mathsf{fma}\left(x.im\_m, \frac{x.im\_m}{x.re}, x.im\_m\right), x.im\_m + x.im\_m\right)}}\\ \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.im_m 1.56e+98)
    (fma
     (+ x.im_m x.re)
     (* x.im_m (- x.re x.im_m))
     (* x.re (* x.re (+ x.im_m x.im_m))))
    (/
     1.0
     (/
      1.0
      (fma
       (- x.re x.im_m)
       (* x.re (fma x.im_m (/ x.im_m x.re) x.im_m))
       (+ x.im_m x.im_m)))))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 1.56e+98) {
		tmp = fma((x_46_im_m + x_46_re), (x_46_im_m * (x_46_re - x_46_im_m)), (x_46_re * (x_46_re * (x_46_im_m + x_46_im_m))));
	} else {
		tmp = 1.0 / (1.0 / fma((x_46_re - x_46_im_m), (x_46_re * fma(x_46_im_m, (x_46_im_m / x_46_re), x_46_im_m)), (x_46_im_m + x_46_im_m)));
	}
	return x_46_im_s * tmp;
}

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 1.56e+98)
		tmp = fma(Float64(x_46_im_m + x_46_re), Float64(x_46_im_m * Float64(x_46_re - x_46_im_m)), Float64(x_46_re * Float64(x_46_re * Float64(x_46_im_m + x_46_im_m))));
	else
		tmp = Float64(1.0 / Float64(1.0 / fma(Float64(x_46_re - x_46_im_m), Float64(x_46_re * fma(x_46_im_m, Float64(x_46_im_m / x_46_re), x_46_im_m)), Float64(x_46_im_m + x_46_im_m))));
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 1.56e+98], N[(N[(x$46$im$95$m + x$46$re), $MachinePrecision] * N[(x$46$im$95$m * N[(x$46$re - x$46$im$95$m), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(x$46$re * N[(x$46$im$95$m + x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 / N[(N[(x$46$re - x$46$im$95$m), $MachinePrecision] * N[(x$46$re * N[(x$46$im$95$m * N[(x$46$im$95$m / x$46$re), $MachinePrecision] + x$46$im$95$m), $MachinePrecision]), $MachinePrecision] + N[(x$46$im$95$m + x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.56e98

   1. Initial program 86.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. difference-of-squares N/A

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

      12. associate-*l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 94.9

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 94.9

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

   4. Applied rewrites 94.9%

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

   if 1.56e98 < x.im

   1. Initial program 68.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. difference-of-squares N/A

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

      12. associate-*l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 76.0

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 76.0

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

   4. Applied rewrites 76.0%

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in x.re around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 100.0

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

   9. Applied rewrites 100.0%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 1.56 \cdot 10^{+98}:\\ \;\;\;\;\mathsf{fma}\left(x.im + x.re, x.im \cdot \left(x.re - x.im\right), x.re \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\mathsf{fma}\left(x.re - x.im, x.re \cdot \mathsf{fma}\left(x.im, \frac{x.im}{x.re}, x.im\right), x.im + x.im\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := x.im\_m \cdot \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) + x.re \cdot \left(x.im\_m \cdot x.re + x.im\_m \cdot x.re\right)\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 4 \cdot 10^{+59}:\\ \;\;\;\;x.im\_m \cdot \mathsf{fma}\left(x.re \cdot x.re, 3, x.im\_m \cdot \left(-x.im\_m\right)\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left(x.re \cdot 3\right) \cdot \left(x.im\_m \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im\_m, x.im\_m \cdot \left(x.im\_m + x.re\right), 0\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0
         (+
          (* x.im_m (- (* x.re x.re) (* x.im_m x.im_m)))
          (* x.re (+ (* x.im_m x.re) (* x.im_m x.re))))))
   (*
    x.im_s
    (if (<= t_0 4e+59)
      (* x.im_m (fma (* x.re x.re) 3.0 (* x.im_m (- x.im_m))))
      (if (<= t_0 INFINITY)
        (* (* x.re 3.0) (* x.im_m x.re))
        (fma (- x.re x.im_m) (* x.im_m (+ x.im_m x.re)) 0.0))))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)));
	double tmp;
	if (t_0 <= 4e+59) {
		tmp = x_46_im_m * fma((x_46_re * x_46_re), 3.0, (x_46_im_m * -x_46_im_m));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (x_46_re * 3.0) * (x_46_im_m * x_46_re);
	} else {
		tmp = fma((x_46_re - x_46_im_m), (x_46_im_m * (x_46_im_m + x_46_re)), 0.0);
	}
	return x_46_im_s * tmp;
}

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = Float64(Float64(x_46_im_m * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m))) + Float64(x_46_re * Float64(Float64(x_46_im_m * x_46_re) + Float64(x_46_im_m * x_46_re))))
	tmp = 0.0
	if (t_0 <= 4e+59)
		tmp = Float64(x_46_im_m * fma(Float64(x_46_re * x_46_re), 3.0, Float64(x_46_im_m * Float64(-x_46_im_m))));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(x_46_re * 3.0) * Float64(x_46_im_m * x_46_re));
	else
		tmp = fma(Float64(x_46_re - x_46_im_m), Float64(x_46_im_m * Float64(x_46_im_m + x_46_re)), 0.0);
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(x$46$im$95$m * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$0, 4e+59], N[(x$46$im$95$m * N[(N[(x$46$re * x$46$re), $MachinePrecision] * 3.0 + N[(x$46$im$95$m * (-x$46$im$95$m)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(x$46$re * 3.0), $MachinePrecision] * N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re - x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m * N[(x$46$im$95$m + x$46$re), $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < 3.99999999999999989e59

   1. Initial program 98.5%

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

   3. Taylor expanded in x.re around 0

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

   5. Applied rewrites 98.5%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-neg.f64 N/A

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

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

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

      10. lift-*.f64 N/A

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

      11. lower-neg.f64 98.5

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

   7. Applied rewrites 98.5%

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

   if 3.99999999999999989e59 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   1. Initial program 86.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. difference-of-squares N/A

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

      12. associate-*l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 99.9

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x.re around inf

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 35.9

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

   7. Applied rewrites 35.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 49.1

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

   9. Applied rewrites 49.1%

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

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

   1. Initial program 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. difference-of-squares N/A

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

      12. associate-*l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 31.3

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 31.3

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

   4. Applied rewrites 31.3%

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

   6. Applied rewrites 96.9%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. remove-double-div 96.9

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 96.9

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

      10. lift-+.f64 N/A

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

      11. flip-+ N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. +-inverses N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. +-inverses N/A

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

      19. metadata-eval N/A

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

      20. flip-- N/A

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

      21. metadata-eval 96.9

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

   8. Applied rewrites 96.9%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right) \leq 4 \cdot 10^{+59}:\\ \;\;\;\;x.im \cdot \mathsf{fma}\left(x.re \cdot x.re, 3, x.im \cdot \left(-x.im\right)\right)\\ \mathbf{elif}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right) \leq \infty:\\ \;\;\;\;\left(x.re \cdot 3\right) \cdot \left(x.im \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im, x.im \cdot \left(x.im + x.re\right), 0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := x.im\_m \cdot \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) + x.re \cdot \left(x.im\_m \cdot x.re + x.im\_m \cdot x.re\right)\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 10^{+22}:\\ \;\;\;\;x.im\_m \cdot \mathsf{fma}\left(x.im\_m, -x.im\_m, 3 \cdot \left(x.re \cdot x.re\right)\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.im\_m \cdot x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im\_m, x.im\_m \cdot \left(x.im\_m + x.re\right), 0\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0
         (+
          (* x.im_m (- (* x.re x.re) (* x.im_m x.im_m)))
          (* x.re (+ (* x.im_m x.re) (* x.im_m x.re))))))
   (*
    x.im_s
    (if (<= t_0 1e+22)
      (* x.im_m (fma x.im_m (- x.im_m) (* 3.0 (* x.re x.re))))
      (if (<= t_0 INFINITY)
        (* 3.0 (* x.re (* x.im_m x.re)))
        (fma (- x.re x.im_m) (* x.im_m (+ x.im_m x.re)) 0.0))))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)));
	double tmp;
	if (t_0 <= 1e+22) {
		tmp = x_46_im_m * fma(x_46_im_m, -x_46_im_m, (3.0 * (x_46_re * x_46_re)));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = 3.0 * (x_46_re * (x_46_im_m * x_46_re));
	} else {
		tmp = fma((x_46_re - x_46_im_m), (x_46_im_m * (x_46_im_m + x_46_re)), 0.0);
	}
	return x_46_im_s * tmp;
}

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = Float64(Float64(x_46_im_m * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m))) + Float64(x_46_re * Float64(Float64(x_46_im_m * x_46_re) + Float64(x_46_im_m * x_46_re))))
	tmp = 0.0
	if (t_0 <= 1e+22)
		tmp = Float64(x_46_im_m * fma(x_46_im_m, Float64(-x_46_im_m), Float64(3.0 * Float64(x_46_re * x_46_re))));
	elseif (t_0 <= Inf)
		tmp = Float64(3.0 * Float64(x_46_re * Float64(x_46_im_m * x_46_re)));
	else
		tmp = fma(Float64(x_46_re - x_46_im_m), Float64(x_46_im_m * Float64(x_46_im_m + x_46_re)), 0.0);
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(x$46$im$95$m * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$0, 1e+22], N[(x$46$im$95$m * N[(x$46$im$95$m * (-x$46$im$95$m) + N[(3.0 * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(3.0 * N[(x$46$re * N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re - x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m * N[(x$46$im$95$m + x$46$re), $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < 1e22

   1. Initial program 98.5%

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

   3. Taylor expanded in x.re around 0

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

   5. Applied rewrites 98.4%

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

   if 1e22 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   1. Initial program 86.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. difference-of-squares N/A

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

      12. associate-*l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 99.9

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x.re around inf

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 36.3

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

   7. Applied rewrites 36.3%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 49.2

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

   9. Applied rewrites 49.2%

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

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

   1. Initial program 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. difference-of-squares N/A

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

      12. associate-*l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 31.3

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 31.3

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

   4. Applied rewrites 31.3%

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

   6. Applied rewrites 96.9%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. remove-double-div 96.9

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 96.9

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

      10. lift-+.f64 N/A

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

      11. flip-+ N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. +-inverses N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. +-inverses N/A

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

      19. metadata-eval N/A

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

      20. flip-- N/A

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

      21. metadata-eval 96.9

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

   8. Applied rewrites 96.9%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right) \leq 10^{+22}:\\ \;\;\;\;x.im \cdot \mathsf{fma}\left(x.im, -x.im, 3 \cdot \left(x.re \cdot x.re\right)\right)\\ \mathbf{elif}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right) \leq \infty:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.im \cdot x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im, x.im \cdot \left(x.im + x.re\right), 0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x.re - x.im\_m, x.im\_m \cdot \left(x.im\_m + x.re\right), 0\right)\\ t_1 := x.im\_m \cdot \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) + x.re \cdot \left(x.im\_m \cdot x.re + x.im\_m \cdot x.re\right)\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-321}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left(x.re \cdot 3\right) \cdot \left(x.im\_m \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0 (fma (- x.re x.im_m) (* x.im_m (+ x.im_m x.re)) 0.0))
        (t_1
         (+
          (* x.im_m (- (* x.re x.re) (* x.im_m x.im_m)))
          (* x.re (+ (* x.im_m x.re) (* x.im_m x.re))))))
   (*
    x.im_s
    (if (<= t_1 -1e-321)
      t_0
      (if (<= t_1 INFINITY) (* (* x.re 3.0) (* x.im_m x.re)) t_0)))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = fma((x_46_re - x_46_im_m), (x_46_im_m * (x_46_im_m + x_46_re)), 0.0);
	double t_1 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)));
	double tmp;
	if (t_1 <= -1e-321) {
		tmp = t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (x_46_re * 3.0) * (x_46_im_m * x_46_re);
	} else {
		tmp = t_0;
	}
	return x_46_im_s * tmp;
}

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = fma(Float64(x_46_re - x_46_im_m), Float64(x_46_im_m * Float64(x_46_im_m + x_46_re)), 0.0)
	t_1 = Float64(Float64(x_46_im_m * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m))) + Float64(x_46_re * Float64(Float64(x_46_im_m * x_46_re) + Float64(x_46_im_m * x_46_re))))
	tmp = 0.0
	if (t_1 <= -1e-321)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(x_46_re * 3.0) * Float64(x_46_im_m * x_46_re));
	else
		tmp = t_0;
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(x$46$re - x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m * N[(x$46$im$95$m + x$46$re), $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im$95$m * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$1, -1e-321], t$95$0, If[LessEqual[t$95$1, Infinity], N[(N[(x$46$re * 3.0), $MachinePrecision] * N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < -9.98013e-322 or +inf.0 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re))

   1. Initial program 73.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. difference-of-squares N/A

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

      12. associate-*l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 82.5

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 82.5

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

   4. Applied rewrites 82.5%

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

   6. Applied rewrites 72.4%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. remove-double-div 72.4

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 72.4

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

      10. lift-+.f64 N/A

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

      11. flip-+ N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. +-inverses N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. +-inverses N/A

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

      19. metadata-eval N/A

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

      20. flip-- N/A

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

      21. metadata-eval 79.1

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

   8. Applied rewrites 79.1%

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

   if -9.98013e-322 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   1. Initial program 92.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. difference-of-squares N/A

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

      12. associate-*l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 99.8

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x.re around inf

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 55.5

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

   7. Applied rewrites 55.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 63.2

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

   9. Applied rewrites 63.2%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right) \leq -1 \cdot 10^{-321}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im, x.im \cdot \left(x.im + x.re\right), 0\right)\\ \mathbf{elif}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right) \leq \infty:\\ \;\;\;\;\left(x.re \cdot 3\right) \cdot \left(x.im \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im, x.im \cdot \left(x.im + x.re\right), 0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := -x.im\_m \cdot \left(x.im\_m \cdot x.im\_m\right)\\ t_1 := x.im\_m \cdot \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) + x.re \cdot \left(x.im\_m \cdot x.re + x.im\_m \cdot x.re\right)\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-321}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left(x.re \cdot 3\right) \cdot \left(x.im\_m \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0 (- (* x.im_m (* x.im_m x.im_m))))
        (t_1
         (+
          (* x.im_m (- (* x.re x.re) (* x.im_m x.im_m)))
          (* x.re (+ (* x.im_m x.re) (* x.im_m x.re))))))
   (*
    x.im_s
    (if (<= t_1 -1e-321)
      t_0
      (if (<= t_1 INFINITY) (* (* x.re 3.0) (* x.im_m x.re)) t_0)))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = -(x_46_im_m * (x_46_im_m * x_46_im_m));
	double t_1 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)));
	double tmp;
	if (t_1 <= -1e-321) {
		tmp = t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (x_46_re * 3.0) * (x_46_im_m * x_46_re);
	} else {
		tmp = t_0;
	}
	return x_46_im_s * tmp;
}

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = -(x_46_im_m * (x_46_im_m * x_46_im_m));
	double t_1 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)));
	double tmp;
	if (t_1 <= -1e-321) {
		tmp = t_0;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (x_46_re * 3.0) * (x_46_im_m * x_46_re);
	} else {
		tmp = t_0;
	}
	return x_46_im_s * tmp;
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	t_0 = -(x_46_im_m * (x_46_im_m * x_46_im_m))
	t_1 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)))
	tmp = 0
	if t_1 <= -1e-321:
		tmp = t_0
	elif t_1 <= math.inf:
		tmp = (x_46_re * 3.0) * (x_46_im_m * x_46_re)
	else:
		tmp = t_0
	return x_46_im_s * tmp

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = Float64(-Float64(x_46_im_m * Float64(x_46_im_m * x_46_im_m)))
	t_1 = Float64(Float64(x_46_im_m * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m))) + Float64(x_46_re * Float64(Float64(x_46_im_m * x_46_re) + Float64(x_46_im_m * x_46_re))))
	tmp = 0.0
	if (t_1 <= -1e-321)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(x_46_re * 3.0) * Float64(x_46_im_m * x_46_re));
	else
		tmp = t_0;
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = -(x_46_im_m * (x_46_im_m * x_46_im_m));
	t_1 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)));
	tmp = 0.0;
	if (t_1 <= -1e-321)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = (x_46_re * 3.0) * (x_46_im_m * x_46_re);
	else
		tmp = t_0;
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := Block[{t$95$0 = (-N[(x$46$im$95$m * N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision])}, Block[{t$95$1 = N[(N[(x$46$im$95$m * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$1, -1e-321], t$95$0, If[LessEqual[t$95$1, Infinity], N[(N[(x$46$re * 3.0), $MachinePrecision] * N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < -9.98013e-322 or +inf.0 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re))

   1. Initial program 73.1%

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

   3. Taylor expanded in x.re around 0

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

      1. mul-1-neg N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 52.2

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

   5. Applied rewrites 52.2%

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

   if -9.98013e-322 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   1. Initial program 92.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. difference-of-squares N/A

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

      12. associate-*l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 99.8

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x.re around inf

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 55.5

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

   7. Applied rewrites 55.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 63.2

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

   9. Applied rewrites 63.2%

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

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right) \leq -1 \cdot 10^{-321}:\\ \;\;\;\;-x.im \cdot \left(x.im \cdot x.im\right)\\ \mathbf{elif}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right) \leq \infty:\\ \;\;\;\;\left(x.re \cdot 3\right) \cdot \left(x.im \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;-x.im \cdot \left(x.im \cdot x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := -x.im\_m \cdot \left(x.im\_m \cdot x.im\_m\right)\\ t_1 := x.im\_m \cdot \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) + x.re \cdot \left(x.im\_m \cdot x.re + x.im\_m \cdot x.re\right)\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-321}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;x.re \cdot \left(x.im\_m \cdot \left(x.re \cdot 3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0 (- (* x.im_m (* x.im_m x.im_m))))
        (t_1
         (+
          (* x.im_m (- (* x.re x.re) (* x.im_m x.im_m)))
          (* x.re (+ (* x.im_m x.re) (* x.im_m x.re))))))
   (*
    x.im_s
    (if (<= t_1 -1e-321)
      t_0
      (if (<= t_1 INFINITY) (* x.re (* x.im_m (* x.re 3.0))) t_0)))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = -(x_46_im_m * (x_46_im_m * x_46_im_m));
	double t_1 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)));
	double tmp;
	if (t_1 <= -1e-321) {
		tmp = t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = x_46_re * (x_46_im_m * (x_46_re * 3.0));
	} else {
		tmp = t_0;
	}
	return x_46_im_s * tmp;
}

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = -(x_46_im_m * (x_46_im_m * x_46_im_m));
	double t_1 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)));
	double tmp;
	if (t_1 <= -1e-321) {
		tmp = t_0;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = x_46_re * (x_46_im_m * (x_46_re * 3.0));
	} else {
		tmp = t_0;
	}
	return x_46_im_s * tmp;
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	t_0 = -(x_46_im_m * (x_46_im_m * x_46_im_m))
	t_1 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)))
	tmp = 0
	if t_1 <= -1e-321:
		tmp = t_0
	elif t_1 <= math.inf:
		tmp = x_46_re * (x_46_im_m * (x_46_re * 3.0))
	else:
		tmp = t_0
	return x_46_im_s * tmp

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = Float64(-Float64(x_46_im_m * Float64(x_46_im_m * x_46_im_m)))
	t_1 = Float64(Float64(x_46_im_m * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m))) + Float64(x_46_re * Float64(Float64(x_46_im_m * x_46_re) + Float64(x_46_im_m * x_46_re))))
	tmp = 0.0
	if (t_1 <= -1e-321)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = Float64(x_46_re * Float64(x_46_im_m * Float64(x_46_re * 3.0)));
	else
		tmp = t_0;
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = -(x_46_im_m * (x_46_im_m * x_46_im_m));
	t_1 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)));
	tmp = 0.0;
	if (t_1 <= -1e-321)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = x_46_re * (x_46_im_m * (x_46_re * 3.0));
	else
		tmp = t_0;
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := Block[{t$95$0 = (-N[(x$46$im$95$m * N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision])}, Block[{t$95$1 = N[(N[(x$46$im$95$m * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$1, -1e-321], t$95$0, If[LessEqual[t$95$1, Infinity], N[(x$46$re * N[(x$46$im$95$m * N[(x$46$re * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < -9.98013e-322 or +inf.0 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re))

   1. Initial program 73.1%

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

   3. Taylor expanded in x.re around 0

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

      1. mul-1-neg N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 52.2

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

   5. Applied rewrites 52.2%

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

   if -9.98013e-322 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   1. Initial program 92.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. difference-of-squares N/A

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

      12. associate-*l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 99.8

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x.re around inf

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 55.5

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

   7. Applied rewrites 55.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 63.2

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

   9. Applied rewrites 63.2%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right) \leq -1 \cdot 10^{-321}:\\ \;\;\;\;-x.im \cdot \left(x.im \cdot x.im\right)\\ \mathbf{elif}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right) \leq \infty:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-x.im \cdot \left(x.im \cdot x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := -x.im\_m \cdot \left(x.im\_m \cdot x.im\_m\right)\\ t_1 := x.im\_m \cdot \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) + x.re \cdot \left(x.im\_m \cdot x.re + x.im\_m \cdot x.re\right)\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-321}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.im\_m \cdot x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0 (- (* x.im_m (* x.im_m x.im_m))))
        (t_1
         (+
          (* x.im_m (- (* x.re x.re) (* x.im_m x.im_m)))
          (* x.re (+ (* x.im_m x.re) (* x.im_m x.re))))))
   (*
    x.im_s
    (if (<= t_1 -1e-321)
      t_0
      (if (<= t_1 INFINITY) (* 3.0 (* x.re (* x.im_m x.re))) t_0)))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = -(x_46_im_m * (x_46_im_m * x_46_im_m));
	double t_1 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)));
	double tmp;
	if (t_1 <= -1e-321) {
		tmp = t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = 3.0 * (x_46_re * (x_46_im_m * x_46_re));
	} else {
		tmp = t_0;
	}
	return x_46_im_s * tmp;
}

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = -(x_46_im_m * (x_46_im_m * x_46_im_m));
	double t_1 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)));
	double tmp;
	if (t_1 <= -1e-321) {
		tmp = t_0;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = 3.0 * (x_46_re * (x_46_im_m * x_46_re));
	} else {
		tmp = t_0;
	}
	return x_46_im_s * tmp;
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	t_0 = -(x_46_im_m * (x_46_im_m * x_46_im_m))
	t_1 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)))
	tmp = 0
	if t_1 <= -1e-321:
		tmp = t_0
	elif t_1 <= math.inf:
		tmp = 3.0 * (x_46_re * (x_46_im_m * x_46_re))
	else:
		tmp = t_0
	return x_46_im_s * tmp

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = Float64(-Float64(x_46_im_m * Float64(x_46_im_m * x_46_im_m)))
	t_1 = Float64(Float64(x_46_im_m * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m))) + Float64(x_46_re * Float64(Float64(x_46_im_m * x_46_re) + Float64(x_46_im_m * x_46_re))))
	tmp = 0.0
	if (t_1 <= -1e-321)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = Float64(3.0 * Float64(x_46_re * Float64(x_46_im_m * x_46_re)));
	else
		tmp = t_0;
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = -(x_46_im_m * (x_46_im_m * x_46_im_m));
	t_1 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)));
	tmp = 0.0;
	if (t_1 <= -1e-321)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = 3.0 * (x_46_re * (x_46_im_m * x_46_re));
	else
		tmp = t_0;
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := Block[{t$95$0 = (-N[(x$46$im$95$m * N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision])}, Block[{t$95$1 = N[(N[(x$46$im$95$m * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$1, -1e-321], t$95$0, If[LessEqual[t$95$1, Infinity], N[(3.0 * N[(x$46$re * N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < -9.98013e-322 or +inf.0 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re))

   1. Initial program 73.1%

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

   3. Taylor expanded in x.re around 0

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

      1. mul-1-neg N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 52.2

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

   5. Applied rewrites 52.2%

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

   if -9.98013e-322 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   1. Initial program 92.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. difference-of-squares N/A

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

      12. associate-*l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 99.8

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x.re around inf

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 55.5

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

   7. Applied rewrites 55.5%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 63.2

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

   9. Applied rewrites 63.2%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right) \leq -1 \cdot 10^{-321}:\\ \;\;\;\;-x.im \cdot \left(x.im \cdot x.im\right)\\ \mathbf{elif}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right) \leq \infty:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.im \cdot x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-x.im \cdot \left(x.im \cdot x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := -x.im\_m \cdot \left(x.im\_m \cdot x.im\_m\right)\\ t_1 := x.im\_m \cdot \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) + x.re \cdot \left(x.im\_m \cdot x.re + x.im\_m \cdot x.re\right)\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-321}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;3 \cdot \left(x.im\_m \cdot \left(x.re \cdot x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0 (- (* x.im_m (* x.im_m x.im_m))))
        (t_1
         (+
          (* x.im_m (- (* x.re x.re) (* x.im_m x.im_m)))
          (* x.re (+ (* x.im_m x.re) (* x.im_m x.re))))))
   (*
    x.im_s
    (if (<= t_1 -1e-321)
      t_0
      (if (<= t_1 INFINITY) (* 3.0 (* x.im_m (* x.re x.re))) t_0)))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = -(x_46_im_m * (x_46_im_m * x_46_im_m));
	double t_1 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)));
	double tmp;
	if (t_1 <= -1e-321) {
		tmp = t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = 3.0 * (x_46_im_m * (x_46_re * x_46_re));
	} else {
		tmp = t_0;
	}
	return x_46_im_s * tmp;
}

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = -(x_46_im_m * (x_46_im_m * x_46_im_m));
	double t_1 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)));
	double tmp;
	if (t_1 <= -1e-321) {
		tmp = t_0;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = 3.0 * (x_46_im_m * (x_46_re * x_46_re));
	} else {
		tmp = t_0;
	}
	return x_46_im_s * tmp;
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	t_0 = -(x_46_im_m * (x_46_im_m * x_46_im_m))
	t_1 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)))
	tmp = 0
	if t_1 <= -1e-321:
		tmp = t_0
	elif t_1 <= math.inf:
		tmp = 3.0 * (x_46_im_m * (x_46_re * x_46_re))
	else:
		tmp = t_0
	return x_46_im_s * tmp

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = Float64(-Float64(x_46_im_m * Float64(x_46_im_m * x_46_im_m)))
	t_1 = Float64(Float64(x_46_im_m * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m))) + Float64(x_46_re * Float64(Float64(x_46_im_m * x_46_re) + Float64(x_46_im_m * x_46_re))))
	tmp = 0.0
	if (t_1 <= -1e-321)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = Float64(3.0 * Float64(x_46_im_m * Float64(x_46_re * x_46_re)));
	else
		tmp = t_0;
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = -(x_46_im_m * (x_46_im_m * x_46_im_m));
	t_1 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)));
	tmp = 0.0;
	if (t_1 <= -1e-321)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = 3.0 * (x_46_im_m * (x_46_re * x_46_re));
	else
		tmp = t_0;
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := Block[{t$95$0 = (-N[(x$46$im$95$m * N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision])}, Block[{t$95$1 = N[(N[(x$46$im$95$m * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$1, -1e-321], t$95$0, If[LessEqual[t$95$1, Infinity], N[(3.0 * N[(x$46$im$95$m * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < -9.98013e-322 or +inf.0 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re))

   1. Initial program 73.1%

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

   3. Taylor expanded in x.re around 0

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

      1. mul-1-neg N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 52.2

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

   5. Applied rewrites 52.2%

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

   if -9.98013e-322 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   1. Initial program 92.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. difference-of-squares N/A

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

      12. associate-*l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 99.8

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x.re around inf

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 55.5

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

   7. Applied rewrites 55.5%

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

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right) \leq -1 \cdot 10^{-321}:\\ \;\;\;\;-x.im \cdot \left(x.im \cdot x.im\right)\\ \mathbf{elif}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right) \leq \infty:\\ \;\;\;\;3 \cdot \left(x.im \cdot \left(x.re \cdot x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-x.im \cdot \left(x.im \cdot x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := -x.im\_m \cdot \left(x.im\_m \cdot x.im\_m\right)\\ t_1 := x.im\_m \cdot \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) + x.re \cdot \left(x.im\_m \cdot x.re + x.im\_m \cdot x.re\right)\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-321}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;x.re \cdot \left(x.im\_m \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0 (- (* x.im_m (* x.im_m x.im_m))))
        (t_1
         (+
          (* x.im_m (- (* x.re x.re) (* x.im_m x.im_m)))
          (* x.re (+ (* x.im_m x.re) (* x.im_m x.re))))))
   (*
    x.im_s
    (if (<= t_1 -1e-321)
      t_0
      (if (<= t_1 INFINITY) (* x.re (* x.im_m x.re)) t_0)))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = -(x_46_im_m * (x_46_im_m * x_46_im_m));
	double t_1 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)));
	double tmp;
	if (t_1 <= -1e-321) {
		tmp = t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = x_46_re * (x_46_im_m * x_46_re);
	} else {
		tmp = t_0;
	}
	return x_46_im_s * tmp;
}

Java

x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = -(x_46_im_m * (x_46_im_m * x_46_im_m));
	double t_1 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)));
	double tmp;
	if (t_1 <= -1e-321) {
		tmp = t_0;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = x_46_re * (x_46_im_m * x_46_re);
	} else {
		tmp = t_0;
	}
	return x_46_im_s * tmp;
}

Python

x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re, x_46_im_m):
	t_0 = -(x_46_im_m * (x_46_im_m * x_46_im_m))
	t_1 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)))
	tmp = 0
	if t_1 <= -1e-321:
		tmp = t_0
	elif t_1 <= math.inf:
		tmp = x_46_re * (x_46_im_m * x_46_re)
	else:
		tmp = t_0
	return x_46_im_s * tmp

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = Float64(-Float64(x_46_im_m * Float64(x_46_im_m * x_46_im_m)))
	t_1 = Float64(Float64(x_46_im_m * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m))) + Float64(x_46_re * Float64(Float64(x_46_im_m * x_46_re) + Float64(x_46_im_m * x_46_re))))
	tmp = 0.0
	if (t_1 <= -1e-321)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = Float64(x_46_re * Float64(x_46_im_m * x_46_re));
	else
		tmp = t_0;
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = -(x_46_im_m * (x_46_im_m * x_46_im_m));
	t_1 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)));
	tmp = 0.0;
	if (t_1 <= -1e-321)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = x_46_re * (x_46_im_m * x_46_re);
	else
		tmp = t_0;
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := Block[{t$95$0 = (-N[(x$46$im$95$m * N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision])}, Block[{t$95$1 = N[(N[(x$46$im$95$m * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$1, -1e-321], t$95$0, If[LessEqual[t$95$1, Infinity], N[(x$46$re * N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < -9.98013e-322 or +inf.0 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re))

   1. Initial program 73.1%

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

   3. Taylor expanded in x.re around 0

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

      1. mul-1-neg N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 52.2

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

   5. Applied rewrites 52.2%

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

   if -9.98013e-322 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   1. Initial program 92.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. difference-of-squares N/A

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

      12. associate-*l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 99.8

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 52.9%

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

   7. Taylor expanded in x.re around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 44.7

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

   9. Applied rewrites 44.7%

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

       1. associate-*r* N/A

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

       2. lift-*.f64 N/A

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

       3. lower-*.f64 45.5

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 45.5

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

   11. Applied rewrites 45.5%

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

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right) \leq -1 \cdot 10^{-321}:\\ \;\;\;\;-x.im \cdot \left(x.im \cdot x.im\right)\\ \mathbf{elif}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right) \leq \infty:\\ \;\;\;\;x.re \cdot \left(x.im \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;-x.im \cdot \left(x.im \cdot x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 1.25 \cdot 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(x.im\_m + x.re, x.im\_m \cdot \left(x.re - x.im\_m\right), x.re \cdot \left(x.re \cdot \left(x.im\_m + x.im\_m\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im\_m, x.im\_m \cdot \left(x.im\_m + x.re\right), 0\right)\\ \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.im_m 1.25e+54)
    (fma
     (+ x.im_m x.re)
     (* x.im_m (- x.re x.im_m))
     (* x.re (* x.re (+ x.im_m x.im_m))))
    (fma (- x.re x.im_m) (* x.im_m (+ x.im_m x.re)) 0.0))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 1.25e+54) {
		tmp = fma((x_46_im_m + x_46_re), (x_46_im_m * (x_46_re - x_46_im_m)), (x_46_re * (x_46_re * (x_46_im_m + x_46_im_m))));
	} else {
		tmp = fma((x_46_re - x_46_im_m), (x_46_im_m * (x_46_im_m + x_46_re)), 0.0);
	}
	return x_46_im_s * tmp;
}

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 1.25e+54)
		tmp = fma(Float64(x_46_im_m + x_46_re), Float64(x_46_im_m * Float64(x_46_re - x_46_im_m)), Float64(x_46_re * Float64(x_46_re * Float64(x_46_im_m + x_46_im_m))));
	else
		tmp = fma(Float64(x_46_re - x_46_im_m), Float64(x_46_im_m * Float64(x_46_im_m + x_46_re)), 0.0);
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 1.25e+54], N[(N[(x$46$im$95$m + x$46$re), $MachinePrecision] * N[(x$46$im$95$m * N[(x$46$re - x$46$im$95$m), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(x$46$re * N[(x$46$im$95$m + x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re - x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m * N[(x$46$im$95$m + x$46$re), $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.25000000000000001e54

   1. Initial program 85.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. difference-of-squares N/A

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

      12. associate-*l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 94.8

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 94.8

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

   4. Applied rewrites 94.8%

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

   if 1.25000000000000001e54 < x.im

   1. Initial program 70.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. difference-of-squares N/A

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

      12. associate-*l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 78.1

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 78.1

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

   4. Applied rewrites 78.1%

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

   6. Applied rewrites 99.9%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. remove-double-div 99.9

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 99.9

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

      10. lift-+.f64 N/A

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

      11. flip-+ N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. +-inverses N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. +-inverses N/A

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

      19. metadata-eval N/A

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

      20. flip-- N/A

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

      21. metadata-eval 99.9

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

   8. Applied rewrites 99.9%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 1.25 \cdot 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(x.im + x.re, x.im \cdot \left(x.re - x.im\right), x.re \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im, x.im \cdot \left(x.im + x.re\right), 0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 99.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 1.25 \cdot 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(x.im\_m \cdot \left(x.re \cdot 3\right), x.re, -x.im\_m \cdot \left(x.im\_m \cdot x.im\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im\_m, x.im\_m \cdot \left(x.im\_m + x.re\right), 0\right)\\ \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.im_m 1.25e+54)
    (fma (* x.im_m (* x.re 3.0)) x.re (- (* x.im_m (* x.im_m x.im_m))))
    (fma (- x.re x.im_m) (* x.im_m (+ x.im_m x.re)) 0.0))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 1.25e+54) {
		tmp = fma((x_46_im_m * (x_46_re * 3.0)), x_46_re, -(x_46_im_m * (x_46_im_m * x_46_im_m)));
	} else {
		tmp = fma((x_46_re - x_46_im_m), (x_46_im_m * (x_46_im_m + x_46_re)), 0.0);
	}
	return x_46_im_s * tmp;
}

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 1.25e+54)
		tmp = fma(Float64(x_46_im_m * Float64(x_46_re * 3.0)), x_46_re, Float64(-Float64(x_46_im_m * Float64(x_46_im_m * x_46_im_m))));
	else
		tmp = fma(Float64(x_46_re - x_46_im_m), Float64(x_46_im_m * Float64(x_46_im_m + x_46_re)), 0.0);
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 1.25e+54], N[(N[(x$46$im$95$m * N[(x$46$re * 3.0), $MachinePrecision]), $MachinePrecision] * x$46$re + (-N[(x$46$im$95$m * N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(N[(x$46$re - x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m * N[(x$46$im$95$m + x$46$re), $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.25000000000000001e54

   1. Initial program 85.9%

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

   3. Taylor expanded in x.re around 0

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

   5. Applied rewrites 91.3%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-in N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-neg.f64 N/A

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

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

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

      16. lift-*.f64 N/A

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

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

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

      18. lift-*.f64 N/A

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

      19. cube-mult N/A

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

      20. lower-neg.f64 N/A

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

      21. cube-mult N/A

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

      22. lift-*.f64 N/A

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

      23. lower-*.f64 89.8

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

   7. Applied rewrites 89.8%

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

   if 1.25000000000000001e54 < x.im

   1. Initial program 70.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. difference-of-squares N/A

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

      12. associate-*l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 78.1

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 78.1

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

   4. Applied rewrites 78.1%

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

   6. Applied rewrites 99.9%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. remove-double-div 99.9

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 99.9

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

      10. lift-+.f64 N/A

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

      11. flip-+ N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. +-inverses N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. +-inverses N/A

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

      19. metadata-eval N/A

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

      20. flip-- N/A

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

      21. metadata-eval 99.9

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

   8. Applied rewrites 99.9%

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

Alternative 12: 35.7% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.7%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. difference-of-squares N/A

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

   12. associate-*l* N/A

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

   13. lower-fma.f64 N/A

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

   14. lower-+.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lower--.f64 91.2

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 91.2

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

4. Applied rewrites 91.2%

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

6. Applied rewrites 62.6%

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

7. Taylor expanded in x.re around inf

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

   1. lower-*.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 36.9

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

9. Applied rewrites 36.9%

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

    1. associate-*r* N/A

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

    2. lift-*.f64 N/A

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

    3. lower-*.f64 37.4

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

    4. lift-*.f64 N/A

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

    5. *-commutative N/A

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

    6. lower-*.f64 37.4

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

11. Applied rewrites 37.4%

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

12. Final simplification 37.4%

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

Alternative 13: 35.0% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.7%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. difference-of-squares N/A

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

   12. associate-*l* N/A

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

   13. lower-fma.f64 N/A

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

   14. lower-+.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{x.re + x.im}, \left(x.re - x.im\right) \cdot x.im, \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\right) \]

   15. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x.re + x.im, \color{blue}{\left(x.re - x.im\right) \cdot x.im}, \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\right) \]

   16. lower--.f64 91.2

       \[\leadsto \mathsf{fma}\left(x.re + x.im, \color{blue}{\left(x.re - x.im\right)} \cdot x.im, \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\right) \]

   17. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x.re + x.im, \left(x.re - x.im\right) \cdot x.im, \color{blue}{\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re}\right) \]

   18. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x.re + x.im, \left(x.re - x.im\right) \cdot x.im, \color{blue}{x.re \cdot \left(x.re \cdot x.im + x.im \cdot x.re\right)}\right) \]

   19. lower-*.f64 91.2

       \[\leadsto \mathsf{fma}\left(x.re + x.im, \left(x.re - x.im\right) \cdot x.im, \color{blue}{x.re \cdot \left(x.re \cdot x.im + x.im \cdot x.re\right)}\right) \]

4. Applied rewrites 91.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x.re + x.im, \left(x.re - x.im\right) \cdot x.im, x.re \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\right)} \]

6. Applied rewrites 62.6%

   \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(x.re - x.im, x.im \cdot \left(x.re + x.im\right), x.im + x.im\right)}}} \]

7. Taylor expanded in x.re around inf

   \[\leadsto \color{blue}{x.im \cdot {x.re}^{2}} \]
8. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{x.im \cdot {x.re}^{2}} \]

   2. unpow2 N/A

      \[\leadsto x.im \cdot \color{blue}{\left(x.re \cdot x.re\right)} \]

   3. lower-*.f64 36.9

      \[\leadsto x.im \cdot \color{blue}{\left(x.re \cdot x.re\right)} \]

9. Applied rewrites 36.9%

   \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot x.re\right)} \]
10. Add Preprocessing

Developer Target 1: 91.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(x.re \cdot x.im\right) \cdot \left(2 \cdot x.re\right) + \left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.re + x.im\right) \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (+ (* (* x.re x.im) (* 2.0 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im))))

C

double code(double x_46_re, double x_46_im) {
	return ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = ((x_46re * x_46im) * (2.0d0 * x_46re)) + ((x_46im * (x_46re - x_46im)) * (x_46re + x_46im))
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
}

Python

def code(x_46_re, x_46_im):
	return ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im))

Julia

function code(x_46_re, x_46_im)
	return Float64(Float64(Float64(x_46_re * x_46_im) * Float64(2.0 * x_46_re)) + Float64(Float64(x_46_im * Float64(x_46_re - x_46_im)) * Float64(x_46_re + x_46_im)))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
end

Wolfram

code[x$46$re_, x$46$im_] := N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(2.0 * x$46$re), $MachinePrecision]), $MachinePrecision] + N[(N[(x$46$im * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision] * N[(x$46$re + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024220 
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (* (* x.re x.im) (* 2 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im))))

  (+ (* (- (* x.re x.re) (* x.im x.im)) x.im) (* (+ (* x.re x.im) (* x.im x.re)) x.re)))