math.cube on complex, real part

Percentage Accurate: 81.6% → 99.5%

Time: 9.6s

Alternatives: 10

Speedup: 1.2×

• 44.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 81.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) - x.im \cdot \left(x.re\_m \cdot x.im + x.re\_m \cdot x.im\right) \leq 10^{-73}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m \cdot \left(x.im + x.im\right), -x.im, \left(x.re\_m \cdot \left(x.re\_m + x.im\right)\right) \cdot \left(x.re\_m - x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m + x.im, x.re\_m \cdot \left(x.re\_m - x.im\right), x.im + x.im\right)\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<=
       (-
        (* x.re_m (- (* x.re_m x.re_m) (* x.im x.im)))
        (* x.im (+ (* x.re_m x.im) (* x.re_m x.im))))
       1e-73)
    (fma
     (* x.re_m (+ x.im x.im))
     (- x.im)
     (* (* x.re_m (+ x.re_m x.im)) (- x.re_m x.im)))
    (fma (+ x.re_m x.im) (* x.re_m (- x.re_m x.im)) (+ x.im x.im)))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= 1e-73) {
		tmp = fma((x_46_re_m * (x_46_im + x_46_im)), -x_46_im, ((x_46_re_m * (x_46_re_m + x_46_im)) * (x_46_re_m - x_46_im)));
	} else {
		tmp = fma((x_46_re_m + x_46_im), (x_46_re_m * (x_46_re_m - x_46_im)), (x_46_im + x_46_im));
	}
	return x_46_re_s * tmp;
}

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im))) - Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_re_m * x_46_im)))) <= 1e-73)
		tmp = fma(Float64(x_46_re_m * Float64(x_46_im + x_46_im)), Float64(-x_46_im), Float64(Float64(x_46_re_m * Float64(x_46_re_m + x_46_im)) * Float64(x_46_re_m - x_46_im)));
	else
		tmp = fma(Float64(x_46_re_m + x_46_im), Float64(x_46_re_m * Float64(x_46_re_m - x_46_im)), Float64(x_46_im + x_46_im));
	end
	return Float64(x_46_re_s * tmp)
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-73], N[(N[(x$46$re$95$m * N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision] * (-x$46$im) + N[(N[(x$46$re$95$m * N[(x$46$re$95$m + x$46$im), $MachinePrecision]), $MachinePrecision] * N[(x$46$re$95$m - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m + x$46$im), $MachinePrecision] * N[(x$46$re$95$m * N[(x$46$re$95$m - x$46$im), $MachinePrecision]), $MachinePrecision] + N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < 9.99999999999999997e-74

   1. Initial program 96.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-neg.f64 96.1

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift--.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. difference-of-squares N/A

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

      21. associate-*r* N/A

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

      22. lower-*.f64 N/A

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

   4. Applied rewrites 99.8%

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

   if 9.99999999999999997e-74 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im))

   1. Initial program 72.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-neg.f64 78.3

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift--.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. difference-of-squares N/A

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

      21. associate-*r* N/A

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

      22. lower-*.f64 N/A

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

   4. Applied rewrites 88.0%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. difference-of-squares N/A

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

      9. *-commutative N/A

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

      10. lift-neg.f64 N/A

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

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

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

      12. lift-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. distribute-rgt-in N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. lift-*.f64 N/A

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

   6. Applied rewrites 85.4%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq 10^{-73}:\\ \;\;\;\;\mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), -x.im, \left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re + x.im, x.re \cdot \left(x.re - x.im\right), x.im + x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ \begin{array}{l} t_0 := x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) - x.im \cdot \left(x.re\_m \cdot x.im + x.re\_m \cdot x.im\right)\\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+147}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re\_m \cdot -3\right)\right)\\ \mathbf{elif}\;t\_0 \leq 10^{-73}:\\ \;\;\;\;x.re\_m \cdot \mathsf{fma}\left(x.im \cdot x.im, -3, x.re\_m \cdot x.re\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m + x.im, x.re\_m \cdot \left(x.re\_m - x.im\right), x.im + x.im\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (let* ((t_0
         (-
          (* x.re_m (- (* x.re_m x.re_m) (* x.im x.im)))
          (* x.im (+ (* x.re_m x.im) (* x.re_m x.im))))))
   (*
    x.re_s
    (if (<= t_0 -4e+147)
      (* x.im (* x.im (* x.re_m -3.0)))
      (if (<= t_0 1e-73)
        (* x.re_m (fma (* x.im x.im) -3.0 (* x.re_m x.re_m)))
        (fma (+ x.re_m x.im) (* x.re_m (- x.re_m x.im)) (+ x.im x.im)))))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double t_0 = (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)));
	double tmp;
	if (t_0 <= -4e+147) {
		tmp = x_46_im * (x_46_im * (x_46_re_m * -3.0));
	} else if (t_0 <= 1e-73) {
		tmp = x_46_re_m * fma((x_46_im * x_46_im), -3.0, (x_46_re_m * x_46_re_m));
	} else {
		tmp = fma((x_46_re_m + x_46_im), (x_46_re_m * (x_46_re_m - x_46_im)), (x_46_im + x_46_im));
	}
	return x_46_re_s * tmp;
}

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	t_0 = Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im))) - Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_re_m * x_46_im))))
	tmp = 0.0
	if (t_0 <= -4e+147)
		tmp = Float64(x_46_im * Float64(x_46_im * Float64(x_46_re_m * -3.0)));
	elseif (t_0 <= 1e-73)
		tmp = Float64(x_46_re_m * fma(Float64(x_46_im * x_46_im), -3.0, Float64(x_46_re_m * x_46_re_m)));
	else
		tmp = fma(Float64(x_46_re_m + x_46_im), Float64(x_46_re_m * Float64(x_46_re_m - x_46_im)), Float64(x_46_im + x_46_im));
	end
	return Float64(x_46_re_s * tmp)
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := Block[{t$95$0 = N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$re$95$s * If[LessEqual[t$95$0, -4e+147], N[(x$46$im * N[(x$46$im * N[(x$46$re$95$m * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-73], N[(x$46$re$95$m * N[(N[(x$46$im * x$46$im), $MachinePrecision] * -3.0 + N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m + x$46$im), $MachinePrecision] * N[(x$46$re$95$m * N[(x$46$re$95$m - x$46$im), $MachinePrecision]), $MachinePrecision] + N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -3.9999999999999999e147

   1. Initial program 90.9%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. difference-of-squares N/A

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

      9. lift--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 18.4

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

      13. lift--.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. difference-of-squares N/A

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

      17. lower-*.f64 N/A

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

      18. lower-+.f64 N/A

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

      19. lower--.f64 18.4

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

   4. Applied rewrites 18.4%

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

   5. Taylor expanded in x.im around inf

      \[\leadsto \color{blue}{{x.im}^{2} \cdot \left(\left(-1 \cdot x.re + \left(\frac{x.re \cdot \left(x.re + -1 \cdot x.re\right)}{x.im} + \frac{{x.re}^{3}}{{x.im}^{2}}\right)\right) - 2 \cdot x.re\right)} \]
   6. Step-by-step derivation

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

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

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

      8. distribute-lft-in N/A

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

   7. Applied rewrites 96.8%

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

   8. Taylor expanded in x.re around 0

      \[\leadsto x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 51.9%

       \[\leadsto x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right) \]

   if -3.9999999999999999e147 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < 9.99999999999999997e-74

   1. Initial program 99.8%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. difference-of-squares N/A

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

      9. lift--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 72.0

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

      13. lift--.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. difference-of-squares N/A

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

      17. lower-*.f64 N/A

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

      18. lower-+.f64 N/A

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

      19. lower--.f64 72.0

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

   4. Applied rewrites 72.0%

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

   5. Taylor expanded in x.re around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

   7. Applied rewrites 99.8%

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

   if 9.99999999999999997e-74 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im))

   1. Initial program 72.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-neg.f64 78.3

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift--.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. difference-of-squares N/A

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

      21. associate-*r* N/A

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

      22. lower-*.f64 N/A

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

   4. Applied rewrites 88.0%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. difference-of-squares N/A

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

      9. *-commutative N/A

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

      10. lift-neg.f64 N/A

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

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

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

      12. lift-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. distribute-rgt-in N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. lift-*.f64 N/A

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

   6. Applied rewrites 85.4%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -4 \cdot 10^{+147}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \mathbf{elif}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq 10^{-73}:\\ \;\;\;\;x.re \cdot \mathsf{fma}\left(x.im \cdot x.im, -3, x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re + x.im, x.re \cdot \left(x.re - x.im\right), x.im + x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ \begin{array}{l} t_0 := x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) - x.im \cdot \left(x.re\_m \cdot x.im + x.re\_m \cdot x.im\right)\\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+147}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re\_m \cdot -3\right)\right)\\ \mathbf{elif}\;t\_0 \leq 10^{-73}:\\ \;\;\;\;x.re\_m \cdot \mathsf{fma}\left(x.re\_m, x.re\_m, \left(x.im \cdot x.im\right) \cdot -3\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m + x.im, x.re\_m \cdot \left(x.re\_m - x.im\right), x.im + x.im\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (let* ((t_0
         (-
          (* x.re_m (- (* x.re_m x.re_m) (* x.im x.im)))
          (* x.im (+ (* x.re_m x.im) (* x.re_m x.im))))))
   (*
    x.re_s
    (if (<= t_0 -4e+147)
      (* x.im (* x.im (* x.re_m -3.0)))
      (if (<= t_0 1e-73)
        (* x.re_m (fma x.re_m x.re_m (* (* x.im x.im) -3.0)))
        (fma (+ x.re_m x.im) (* x.re_m (- x.re_m x.im)) (+ x.im x.im)))))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double t_0 = (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)));
	double tmp;
	if (t_0 <= -4e+147) {
		tmp = x_46_im * (x_46_im * (x_46_re_m * -3.0));
	} else if (t_0 <= 1e-73) {
		tmp = x_46_re_m * fma(x_46_re_m, x_46_re_m, ((x_46_im * x_46_im) * -3.0));
	} else {
		tmp = fma((x_46_re_m + x_46_im), (x_46_re_m * (x_46_re_m - x_46_im)), (x_46_im + x_46_im));
	}
	return x_46_re_s * tmp;
}

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	t_0 = Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im))) - Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_re_m * x_46_im))))
	tmp = 0.0
	if (t_0 <= -4e+147)
		tmp = Float64(x_46_im * Float64(x_46_im * Float64(x_46_re_m * -3.0)));
	elseif (t_0 <= 1e-73)
		tmp = Float64(x_46_re_m * fma(x_46_re_m, x_46_re_m, Float64(Float64(x_46_im * x_46_im) * -3.0)));
	else
		tmp = fma(Float64(x_46_re_m + x_46_im), Float64(x_46_re_m * Float64(x_46_re_m - x_46_im)), Float64(x_46_im + x_46_im));
	end
	return Float64(x_46_re_s * tmp)
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := Block[{t$95$0 = N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$re$95$s * If[LessEqual[t$95$0, -4e+147], N[(x$46$im * N[(x$46$im * N[(x$46$re$95$m * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-73], N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$re$95$m + N[(N[(x$46$im * x$46$im), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m + x$46$im), $MachinePrecision] * N[(x$46$re$95$m * N[(x$46$re$95$m - x$46$im), $MachinePrecision]), $MachinePrecision] + N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -3.9999999999999999e147

   1. Initial program 90.9%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. difference-of-squares N/A

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

      9. lift--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 18.4

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

      13. lift--.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. difference-of-squares N/A

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

      17. lower-*.f64 N/A

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

      18. lower-+.f64 N/A

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

      19. lower--.f64 18.4

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

   4. Applied rewrites 18.4%

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

   5. Taylor expanded in x.im around inf

      \[\leadsto \color{blue}{{x.im}^{2} \cdot \left(\left(-1 \cdot x.re + \left(\frac{x.re \cdot \left(x.re + -1 \cdot x.re\right)}{x.im} + \frac{{x.re}^{3}}{{x.im}^{2}}\right)\right) - 2 \cdot x.re\right)} \]
   6. Step-by-step derivation

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

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

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

      8. distribute-lft-in N/A

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

   7. Applied rewrites 96.8%

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

   8. Taylor expanded in x.re around 0

      \[\leadsto x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 51.9%

       \[\leadsto x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right) \]

   if -3.9999999999999999e147 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < 9.99999999999999997e-74

   1. Initial program 99.8%

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

   3. Taylor expanded in x.re around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate--l+ N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 99.8

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

   5. Applied rewrites 99.8%

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

   if 9.99999999999999997e-74 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im))

   1. Initial program 72.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-neg.f64 78.3

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift--.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. difference-of-squares N/A

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

      21. associate-*r* N/A

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

      22. lower-*.f64 N/A

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

   4. Applied rewrites 88.0%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. difference-of-squares N/A

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

      9. *-commutative N/A

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

      10. lift-neg.f64 N/A

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

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

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

      12. lift-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. distribute-rgt-in N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. lift-*.f64 N/A

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

   6. Applied rewrites 85.4%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -4 \cdot 10^{+147}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \mathbf{elif}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq 10^{-73}:\\ \;\;\;\;x.re \cdot \mathsf{fma}\left(x.re, x.re, \left(x.im \cdot x.im\right) \cdot -3\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re + x.im, x.re \cdot \left(x.re - x.im\right), x.im + x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ \begin{array}{l} t_0 := x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) - x.im \cdot \left(x.re\_m \cdot x.im + x.re\_m \cdot x.im\right)\\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-303}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re\_m \cdot -3\right)\right)\\ \mathbf{elif}\;t\_0 \leq 10^{-73}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m + x.im, x.re\_m \cdot \left(x.re\_m - x.im\right), x.im + x.im\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (let* ((t_0
         (-
          (* x.re_m (- (* x.re_m x.re_m) (* x.im x.im)))
          (* x.im (+ (* x.re_m x.im) (* x.re_m x.im))))))
   (*
    x.re_s
    (if (<= t_0 -1e-303)
      (* x.im (* x.im (* x.re_m -3.0)))
      (if (<= t_0 1e-73)
        (* x.re_m (* x.re_m x.re_m))
        (fma (+ x.re_m x.im) (* x.re_m (- x.re_m x.im)) (+ x.im x.im)))))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double t_0 = (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)));
	double tmp;
	if (t_0 <= -1e-303) {
		tmp = x_46_im * (x_46_im * (x_46_re_m * -3.0));
	} else if (t_0 <= 1e-73) {
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	} else {
		tmp = fma((x_46_re_m + x_46_im), (x_46_re_m * (x_46_re_m - x_46_im)), (x_46_im + x_46_im));
	}
	return x_46_re_s * tmp;
}

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	t_0 = Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im))) - Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_re_m * x_46_im))))
	tmp = 0.0
	if (t_0 <= -1e-303)
		tmp = Float64(x_46_im * Float64(x_46_im * Float64(x_46_re_m * -3.0)));
	elseif (t_0 <= 1e-73)
		tmp = Float64(x_46_re_m * Float64(x_46_re_m * x_46_re_m));
	else
		tmp = fma(Float64(x_46_re_m + x_46_im), Float64(x_46_re_m * Float64(x_46_re_m - x_46_im)), Float64(x_46_im + x_46_im));
	end
	return Float64(x_46_re_s * tmp)
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := Block[{t$95$0 = N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$re$95$s * If[LessEqual[t$95$0, -1e-303], N[(x$46$im * N[(x$46$im * N[(x$46$re$95$m * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-73], N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m + x$46$im), $MachinePrecision] * N[(x$46$re$95$m * N[(x$46$re$95$m - x$46$im), $MachinePrecision]), $MachinePrecision] + N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -9.99999999999999931e-304

   1. Initial program 93.6%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. difference-of-squares N/A

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

      9. lift--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 39.8

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

      13. lift--.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. difference-of-squares N/A

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

      17. lower-*.f64 N/A

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

      18. lower-+.f64 N/A

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

      19. lower--.f64 39.8

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

   4. Applied rewrites 39.8%

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

   5. Taylor expanded in x.im around inf

      \[\leadsto \color{blue}{{x.im}^{2} \cdot \left(\left(-1 \cdot x.re + \left(\frac{x.re \cdot \left(x.re + -1 \cdot x.re\right)}{x.im} + \frac{{x.re}^{3}}{{x.im}^{2}}\right)\right) - 2 \cdot x.re\right)} \]
   6. Step-by-step derivation

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

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

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

      8. distribute-lft-in N/A

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

   7. Applied rewrites 87.8%

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

   8. Taylor expanded in x.re around 0

      \[\leadsto x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 51.5%

       \[\leadsto x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right) \]

   if -9.99999999999999931e-304 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < 9.99999999999999997e-74

   1. Initial program 99.8%

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

   3. Taylor expanded in x.re around inf

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

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 77.2

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

   5. Applied rewrites 77.2%

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

   if 9.99999999999999997e-74 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im))

   1. Initial program 72.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-neg.f64 78.3

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift--.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. difference-of-squares N/A

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

      21. associate-*r* N/A

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

      22. lower-*.f64 N/A

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

   4. Applied rewrites 88.0%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. difference-of-squares N/A

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

      9. *-commutative N/A

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

      10. lift-neg.f64 N/A

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

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

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

      12. lift-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. distribute-rgt-in N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. lift-*.f64 N/A

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

   6. Applied rewrites 85.4%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -1 \cdot 10^{-303}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \mathbf{elif}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq 10^{-73}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re + x.im, x.re \cdot \left(x.re - x.im\right), x.im + x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) - x.im \cdot \left(x.re\_m \cdot x.im + x.re\_m \cdot x.im\right) \leq 2 \cdot 10^{-304}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re\_m \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.im + x.im, x.re\_m, \left(x.re\_m + x.im\right) \cdot \left(x.re\_m \cdot \left(x.re\_m - x.im\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<=
       (-
        (* x.re_m (- (* x.re_m x.re_m) (* x.im x.im)))
        (* x.im (+ (* x.re_m x.im) (* x.re_m x.im))))
       2e-304)
    (* x.im (* x.im (* x.re_m -3.0)))
    (fma
     (+ x.im x.im)
     x.re_m
     (* (+ x.re_m x.im) (* x.re_m (- x.re_m x.im)))))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= 2e-304) {
		tmp = x_46_im * (x_46_im * (x_46_re_m * -3.0));
	} else {
		tmp = fma((x_46_im + x_46_im), x_46_re_m, ((x_46_re_m + x_46_im) * (x_46_re_m * (x_46_re_m - x_46_im))));
	}
	return x_46_re_s * tmp;
}

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im))) - Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_re_m * x_46_im)))) <= 2e-304)
		tmp = Float64(x_46_im * Float64(x_46_im * Float64(x_46_re_m * -3.0)));
	else
		tmp = fma(Float64(x_46_im + x_46_im), x_46_re_m, Float64(Float64(x_46_re_m + x_46_im) * Float64(x_46_re_m * Float64(x_46_re_m - x_46_im))));
	end
	return Float64(x_46_re_s * tmp)
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-304], N[(x$46$im * N[(x$46$im * N[(x$46$re$95$m * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im + x$46$im), $MachinePrecision] * x$46$re$95$m + N[(N[(x$46$re$95$m + x$46$im), $MachinePrecision] * N[(x$46$re$95$m * N[(x$46$re$95$m - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < 1.99999999999999994e-304

   1. Initial program 95.3%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. difference-of-squares N/A

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

      9. lift--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 43.1

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

      13. lift--.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. difference-of-squares N/A

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

      17. lower-*.f64 N/A

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

      18. lower-+.f64 N/A

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

      19. lower--.f64 43.1

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

   4. Applied rewrites 43.1%

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

   5. Taylor expanded in x.im around inf

      \[\leadsto \color{blue}{{x.im}^{2} \cdot \left(\left(-1 \cdot x.re + \left(\frac{x.re \cdot \left(x.re + -1 \cdot x.re\right)}{x.im} + \frac{{x.re}^{3}}{{x.im}^{2}}\right)\right) - 2 \cdot x.re\right)} \]
   6. Step-by-step derivation

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

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

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

      8. distribute-lft-in N/A

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

   7. Applied rewrites 76.6%

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

   8. Taylor expanded in x.re around 0

      \[\leadsto x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 63.9%

       \[\leadsto x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right) \]

   if 1.99999999999999994e-304 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im))

   1. Initial program 77.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-neg.f64 82.3

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift--.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. difference-of-squares N/A

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

      21. associate-*r* N/A

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

      22. lower-*.f64 N/A

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

   4. Applied rewrites 90.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-neg.f64 N/A

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

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

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. distribute-rgt-in N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. neg-mul-1 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. lift-+.f64 N/A

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

      17. lift--.f64 N/A

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

      18. difference-of-squares N/A

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

      19. *-commutative N/A

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

   6. Applied rewrites 79.7%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq 2 \cdot 10^{-304}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.im + x.im, x.re, \left(x.re + x.im\right) \cdot \left(x.re \cdot \left(x.re - x.im\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= -1e-303) {
		tmp = x_46_im * (x_46_im * (x_46_re_m * -3.0));
	} else {
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	}
	return x_46_re_s * tmp;
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (((x_46re_m * ((x_46re_m * x_46re_m) - (x_46im * x_46im))) - (x_46im * ((x_46re_m * x_46im) + (x_46re_m * x_46im)))) <= (-1d-303)) then
        tmp = x_46im * (x_46im * (x_46re_m * (-3.0d0)))
    else
        tmp = x_46re_m * (x_46re_m * x_46re_m)
    end if
    code = x_46re_s * tmp
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= -1e-303) {
		tmp = x_46_im * (x_46_im * (x_46_re_m * -3.0));
	} else {
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	}
	return x_46_re_s * tmp;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	tmp = 0
	if ((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= -1e-303:
		tmp = x_46_im * (x_46_im * (x_46_re_m * -3.0))
	else:
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m)
	return x_46_re_s * tmp

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im))) - Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_re_m * x_46_im)))) <= -1e-303)
		tmp = Float64(x_46_im * Float64(x_46_im * Float64(x_46_re_m * -3.0)));
	else
		tmp = Float64(x_46_re_m * Float64(x_46_re_m * x_46_re_m));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= -1e-303)
		tmp = x_46_im * (x_46_im * (x_46_re_m * -3.0));
	else
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -9.99999999999999931e-304

   1. Initial program 93.6%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. difference-of-squares N/A

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

      9. lift--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 39.8

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

      13. lift--.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. difference-of-squares N/A

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

      17. lower-*.f64 N/A

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

      18. lower-+.f64 N/A

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

      19. lower--.f64 39.8

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

   4. Applied rewrites 39.8%

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

   5. Taylor expanded in x.im around inf

      \[\leadsto \color{blue}{{x.im}^{2} \cdot \left(\left(-1 \cdot x.re + \left(\frac{x.re \cdot \left(x.re + -1 \cdot x.re\right)}{x.im} + \frac{{x.re}^{3}}{{x.im}^{2}}\right)\right) - 2 \cdot x.re\right)} \]
   6. Step-by-step derivation

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

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

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

      8. distribute-lft-in N/A

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

   7. Applied rewrites 87.8%

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

   8. Taylor expanded in x.re around 0

      \[\leadsto x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 51.5%

       \[\leadsto x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right) \]

   if -9.99999999999999931e-304 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im))

   1. Initial program 82.1%

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

   3. Taylor expanded in x.re around inf

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

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 66.0

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

   5. Applied rewrites 66.0%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -1 \cdot 10^{-303}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= -1e-303) {
		tmp = x_46_re_m * ((x_46_im * x_46_im) * -3.0);
	} else {
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	}
	return x_46_re_s * tmp;
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (((x_46re_m * ((x_46re_m * x_46re_m) - (x_46im * x_46im))) - (x_46im * ((x_46re_m * x_46im) + (x_46re_m * x_46im)))) <= (-1d-303)) then
        tmp = x_46re_m * ((x_46im * x_46im) * (-3.0d0))
    else
        tmp = x_46re_m * (x_46re_m * x_46re_m)
    end if
    code = x_46re_s * tmp
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= -1e-303) {
		tmp = x_46_re_m * ((x_46_im * x_46_im) * -3.0);
	} else {
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	}
	return x_46_re_s * tmp;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	tmp = 0
	if ((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= -1e-303:
		tmp = x_46_re_m * ((x_46_im * x_46_im) * -3.0)
	else:
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m)
	return x_46_re_s * tmp

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im))) - Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_re_m * x_46_im)))) <= -1e-303)
		tmp = Float64(x_46_re_m * Float64(Float64(x_46_im * x_46_im) * -3.0));
	else
		tmp = Float64(x_46_re_m * Float64(x_46_re_m * x_46_re_m));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= -1e-303)
		tmp = x_46_re_m * ((x_46_im * x_46_im) * -3.0);
	else
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -9.99999999999999931e-304

   1. Initial program 93.6%

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

   3. Taylor expanded in x.re around 0

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

      1. lower-*.f64 N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval 44.3

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

   5. Applied rewrites 44.3%

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

   if -9.99999999999999931e-304 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im))

   1. Initial program 82.1%

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

   3. Taylor expanded in x.re around inf

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

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 66.0

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

   5. Applied rewrites 66.0%

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

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -1 \cdot 10^{-303}:\\ \;\;\;\;x.re \cdot \left(\left(x.im \cdot x.im\right) \cdot -3\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.im \leq 4.8 \cdot 10^{+51}:\\ \;\;\;\;x.re\_m \cdot \mathsf{fma}\left(x.im \cdot x.im, -3, x.re\_m \cdot x.re\_m\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re\_m \cdot \mathsf{fma}\left(x.re\_m, \frac{x.re\_m}{x.im}, x.im \cdot -3\right)\right)\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<= x.im 4.8e+51)
    (* x.re_m (fma (* x.im x.im) -3.0 (* x.re_m x.re_m)))
    (* x.im (* x.re_m (fma x.re_m (/ x.re_m x.im) (* x.im -3.0)))))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_im <= 4.8e+51) {
		tmp = x_46_re_m * fma((x_46_im * x_46_im), -3.0, (x_46_re_m * x_46_re_m));
	} else {
		tmp = x_46_im * (x_46_re_m * fma(x_46_re_m, (x_46_re_m / x_46_im), (x_46_im * -3.0)));
	}
	return x_46_re_s * tmp;
}

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_im <= 4.8e+51)
		tmp = Float64(x_46_re_m * fma(Float64(x_46_im * x_46_im), -3.0, Float64(x_46_re_m * x_46_re_m)));
	else
		tmp = Float64(x_46_im * Float64(x_46_re_m * fma(x_46_re_m, Float64(x_46_re_m / x_46_im), Float64(x_46_im * -3.0))));
	end
	return Float64(x_46_re_s * tmp)
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[x$46$im, 4.8e+51], N[(x$46$re$95$m * N[(N[(x$46$im * x$46$im), $MachinePrecision] * -3.0 + N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(x$46$re$95$m * N[(x$46$re$95$m * N[(x$46$re$95$m / x$46$im), $MachinePrecision] + N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 4.7999999999999997e51

   1. Initial program 90.0%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. difference-of-squares N/A

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

      9. lift--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 42.9

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

      13. lift--.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. difference-of-squares N/A

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

      17. lower-*.f64 N/A

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

      18. lower-+.f64 N/A

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

      19. lower--.f64 42.9

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

   4. Applied rewrites 42.9%

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

   5. Taylor expanded in x.re around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

   7. Applied rewrites 94.8%

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

   if 4.7999999999999997e51 < x.im

   1. Initial program 68.8%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. difference-of-squares N/A

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

      9. lift--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 20.1

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

      13. lift--.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. difference-of-squares N/A

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

      17. lower-*.f64 N/A

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

      18. lower-+.f64 N/A

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

      19. lower--.f64 20.1

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

   4. Applied rewrites 20.1%

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

   5. Taylor expanded in x.im around inf

      \[\leadsto \color{blue}{{x.im}^{2} \cdot \left(\left(-1 \cdot x.re + \left(\frac{x.re \cdot \left(x.re + -1 \cdot x.re\right)}{x.im} + \frac{{x.re}^{3}}{{x.im}^{2}}\right)\right) - 2 \cdot x.re\right)} \]
   6. Step-by-step derivation

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

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

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

      8. distribute-lft-in N/A

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

   7. Applied rewrites 97.6%

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

   8. Taylor expanded in x.re around 0

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

   10. Applied rewrites 91.3%

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

   12. Applied rewrites 99.8%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 4.8 \cdot 10^{+51}:\\ \;\;\;\;x.re \cdot \mathsf{fma}\left(x.im \cdot x.im, -3, x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \mathsf{fma}\left(x.re, \frac{x.re}{x.im}, x.im \cdot -3\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 4.8 \cdot 10^{+58}:\\ \;\;\;\;\mathsf{fma}\left(x.im, x.re\_m \cdot \left(x.im \cdot -3\right), x.re\_m \cdot \left(x.re\_m \cdot x.re\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m + x.im, x.re\_m \cdot \left(x.re\_m - x.im\right), x.im + x.im\right)\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 4.8e+58)
    (fma x.im (* x.re_m (* x.im -3.0)) (* x.re_m (* x.re_m x.re_m)))
    (fma (+ x.re_m x.im) (* x.re_m (- x.re_m x.im)) (+ x.im x.im)))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 4.8e+58) {
		tmp = fma(x_46_im, (x_46_re_m * (x_46_im * -3.0)), (x_46_re_m * (x_46_re_m * x_46_re_m)));
	} else {
		tmp = fma((x_46_re_m + x_46_im), (x_46_re_m * (x_46_re_m - x_46_im)), (x_46_im + x_46_im));
	}
	return x_46_re_s * tmp;
}

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_re_m <= 4.8e+58)
		tmp = fma(x_46_im, Float64(x_46_re_m * Float64(x_46_im * -3.0)), Float64(x_46_re_m * Float64(x_46_re_m * x_46_re_m)));
	else
		tmp = fma(Float64(x_46_re_m + x_46_im), Float64(x_46_re_m * Float64(x_46_re_m - x_46_im)), Float64(x_46_im + x_46_im));
	end
	return Float64(x_46_re_s * tmp)
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 4.8e+58], N[(x$46$im * N[(x$46$re$95$m * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision] + N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m + x$46$im), $MachinePrecision] * N[(x$46$re$95$m * N[(x$46$re$95$m - x$46$im), $MachinePrecision]), $MachinePrecision] + N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 4.8e58

   1. Initial program 87.9%

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

   3. Taylor expanded in x.re around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate--l+ N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 92.4

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

   5. Applied rewrites 92.4%

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

   7. Applied rewrites 93.4%

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

   if 4.8e58 < x.re

   1. Initial program 79.2%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-neg.f64 84.9

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift--.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. difference-of-squares N/A

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

      21. associate-*r* N/A

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

      22. lower-*.f64 N/A

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

   4. Applied rewrites 88.7%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. difference-of-squares N/A

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

      9. *-commutative N/A

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

      10. lift-neg.f64 N/A

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

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

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

      12. lift-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. distribute-rgt-in N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. lift-*.f64 N/A

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

   6. Applied rewrites 100.0%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 4.8 \cdot 10^{+58}:\\ \;\;\;\;\mathsf{fma}\left(x.im, x.re \cdot \left(x.im \cdot -3\right), x.re \cdot \left(x.re \cdot x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re + x.im, x.re \cdot \left(x.re - x.im\right), x.im + x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 58.8% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.1%

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

3. Taylor expanded in x.re around inf

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

   1. cube-mult N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 60.2

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

5. Applied rewrites 60.2%

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

Developer Target 1: 87.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024231 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64

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

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