math.cube on complex, real part

Percentage Accurate: 83.1% → 99.7%

Time: 10.8s

Alternatives: 9

Speedup: 0.7×

• 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: 83.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) \cdot x.re\_m - \left(x.im \cdot x.re\_m + x.im \cdot x.re\_m\right) \cdot x.im \leq 5 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(\left(x.re\_m + x.re\_m\right) \cdot x.im, -x.im, \left(x.re\_m - x.im\right) \cdot \left(\mathsf{fma}\left(x.re\_m, \frac{x.re\_m}{x.im}, x.re\_m\right) \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m - x.im, \left(x.im + x.re\_m\right) \cdot x.re\_m, x.re\_m + x.re\_m\right)\\ \end{array} \end{array} \]

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.im x.im)) x.re_m)
        (* (+ (* x.im x.re_m) (* x.im x.re_m)) x.im))
       5e+20)
    (fma
     (* (+ x.re_m x.re_m) x.im)
     (- x.im)
     (* (- x.re_m x.im) (* (fma x.re_m (/ x.re_m x.im) x.re_m) x.im)))
    (fma (- x.re_m x.im) (* (+ x.im x.re_m) 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_im * x_46_im)) * x_46_re_m) - (((x_46_im * x_46_re_m) + (x_46_im * x_46_re_m)) * x_46_im)) <= 5e+20) {
		tmp = fma(((x_46_re_m + x_46_re_m) * x_46_im), -x_46_im, ((x_46_re_m - x_46_im) * (fma(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_im + x_46_re_m) * 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(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im)) * x_46_re_m) - Float64(Float64(Float64(x_46_im * x_46_re_m) + Float64(x_46_im * x_46_re_m)) * x_46_im)) <= 5e+20)
		tmp = fma(Float64(Float64(x_46_re_m + x_46_re_m) * x_46_im), Float64(-x_46_im), Float64(Float64(x_46_re_m - x_46_im) * Float64(fma(x_46_re_m, Float64(x_46_re_m / x_46_im), x_46_re_m) * x_46_im)));
	else
		tmp = fma(Float64(x_46_re_m - x_46_im), Float64(Float64(x_46_im + x_46_re_m) * x_46_re_m), Float64(x_46_re_m + x_46_re_m));
	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[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] - N[(N[(N[(x$46$im * x$46$re$95$m), $MachinePrecision] + N[(x$46$im * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], 5e+20], N[(N[(N[(x$46$re$95$m + x$46$re$95$m), $MachinePrecision] * x$46$im), $MachinePrecision] * (-x$46$im) + N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(N[(x$46$re$95$m * N[(x$46$re$95$m / x$46$im), $MachinePrecision] + x$46$re$95$m), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(N[(x$46$im + x$46$re$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] + 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)) < 5e20

   1. Initial program 92.9%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   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. 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-lft-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-neg.f64 93.0

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift--.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. difference-of-squares N/A

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

      21. associate-*r* N/A

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

      22. lower-*.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x.im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 97.5

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

   7. Applied rewrites 97.5%

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

   if 5e20 < (-.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 59.0%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   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-lft-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-neg.f64 67.8

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift--.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. difference-of-squares N/A

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

      21. associate-*r* N/A

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

      22. lower-*.f64 N/A

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

   4. Applied rewrites 82.3%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. distribute-lft-in N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-+.f64 N/A

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

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

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

      21. *-commutative N/A

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

      22. lift-+.f64 N/A

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

      23. lift-*.f64 N/A

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

      24. *-commutative N/A

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

      25. lift-*.f64 N/A

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

   6. Applied rewrites 87.3%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.im \leq 5 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(\left(x.re + x.re\right) \cdot x.im, -x.im, \left(x.re - x.im\right) \cdot \left(\mathsf{fma}\left(x.re, \frac{x.re}{x.im}, x.re\right) \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im, \left(x.im + x.re\right) \cdot x.re, x.re + x.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ \begin{array}{l} t_0 := \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) \cdot x.re\_m - \left(x.im \cdot x.re\_m + x.im \cdot x.re\_m\right) \cdot x.im\\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\left(-3 \cdot x.im\right) \cdot x.re\_m\right) \cdot x.im\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(-3 \cdot \left(x.im \cdot x.im\right), x.re\_m, \left(x.re\_m \cdot x.re\_m\right) \cdot x.re\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m - x.im, \left(x.im + x.re\_m\right) \cdot x.re\_m, x.re\_m + x.re\_m\right)\\ \end{array} \end{array} \end{array} \]

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.im x.im)) x.re_m)
          (* (+ (* x.im x.re_m) (* x.im x.re_m)) x.im))))
   (*
    x.re_s
    (if (<= t_0 (- INFINITY))
      (* (* (* -3.0 x.im) x.re_m) x.im)
      (if (<= t_0 5e+20)
        (fma (* -3.0 (* x.im x.im)) x.re_m (* (* x.re_m x.re_m) x.re_m))
        (fma (- x.re_m x.im) (* (+ x.im x.re_m) 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 t_0 = (((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_im * x_46_re_m) + (x_46_im * x_46_re_m)) * x_46_im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = ((-3.0 * x_46_im) * x_46_re_m) * x_46_im;
	} else if (t_0 <= 5e+20) {
		tmp = fma((-3.0 * (x_46_im * x_46_im)), x_46_re_m, ((x_46_re_m * x_46_re_m) * x_46_re_m));
	} else {
		tmp = fma((x_46_re_m - x_46_im), ((x_46_im + x_46_re_m) * 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)
	t_0 = Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im)) * x_46_re_m) - Float64(Float64(Float64(x_46_im * x_46_re_m) + Float64(x_46_im * x_46_re_m)) * x_46_im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(-3.0 * x_46_im) * x_46_re_m) * x_46_im);
	elseif (t_0 <= 5e+20)
		tmp = fma(Float64(-3.0 * Float64(x_46_im * x_46_im)), x_46_re_m, Float64(Float64(x_46_re_m * x_46_re_m) * x_46_re_m));
	else
		tmp = fma(Float64(x_46_re_m - x_46_im), Float64(Float64(x_46_im + x_46_re_m) * x_46_re_m), Float64(x_46_re_m + x_46_re_m));
	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[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] - N[(N[(N[(x$46$im * x$46$re$95$m), $MachinePrecision] + N[(x$46$im * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision]}, N[(x$46$re$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(-3.0 * x$46$im), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] * x$46$im), $MachinePrecision], If[LessEqual[t$95$0, 5e+20], N[(N[(-3.0 * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m + N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(N[(x$46$im + x$46$re$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] + N[(x$46$re$95$m + x$46$re$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 83.4%

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

      1. 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. lift--.f64 N/A

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

      3. flip3-- N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in x.re around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

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

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

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

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 45.9

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

   7. Applied rewrites 45.9%

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

   9. Applied rewrites 45.9%

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

   if -inf.0 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < 5e20

   1. Initial program 99.6%

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

   3. Taylor expanded in x.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

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

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

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

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. metadata-eval N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 99.6

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

   5. Applied rewrites 99.6%

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

   7. Applied rewrites 99.6%

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

   if 5e20 < (-.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 59.0%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   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-lft-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-neg.f64 67.8

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift--.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. difference-of-squares N/A

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

      21. associate-*r* N/A

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

      22. lower-*.f64 N/A

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

   4. Applied rewrites 82.3%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. distribute-lft-in N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-+.f64 N/A

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

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

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

      21. *-commutative N/A

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

      22. lift-+.f64 N/A

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

      23. lift-*.f64 N/A

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

      24. *-commutative N/A

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

      25. lift-*.f64 N/A

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

   6. Applied rewrites 87.3%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.im \leq -\infty:\\ \;\;\;\;\left(\left(-3 \cdot x.im\right) \cdot x.re\right) \cdot x.im\\ \mathbf{elif}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.im \leq 5 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(-3 \cdot \left(x.im \cdot x.im\right), x.re, \left(x.re \cdot x.re\right) \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im, \left(x.im + x.re\right) \cdot x.re, x.re + x.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ \begin{array}{l} t_0 := \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) \cdot x.re\_m - \left(x.im \cdot x.re\_m + x.im \cdot x.re\_m\right) \cdot x.im\\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\left(-3 \cdot x.im\right) \cdot x.re\_m\right) \cdot x.im\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(-3, x.im \cdot x.im, x.re\_m \cdot x.re\_m\right) \cdot x.re\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m - x.im, \left(x.im + x.re\_m\right) \cdot x.re\_m, x.re\_m + x.re\_m\right)\\ \end{array} \end{array} \end{array} \]

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.im x.im)) x.re_m)
          (* (+ (* x.im x.re_m) (* x.im x.re_m)) x.im))))
   (*
    x.re_s
    (if (<= t_0 (- INFINITY))
      (* (* (* -3.0 x.im) x.re_m) x.im)
      (if (<= t_0 5e+20)
        (* (fma -3.0 (* x.im x.im) (* x.re_m x.re_m)) x.re_m)
        (fma (- x.re_m x.im) (* (+ x.im x.re_m) 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 t_0 = (((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_im * x_46_re_m) + (x_46_im * x_46_re_m)) * x_46_im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = ((-3.0 * x_46_im) * x_46_re_m) * x_46_im;
	} else if (t_0 <= 5e+20) {
		tmp = fma(-3.0, (x_46_im * x_46_im), (x_46_re_m * x_46_re_m)) * x_46_re_m;
	} else {
		tmp = fma((x_46_re_m - x_46_im), ((x_46_im + x_46_re_m) * 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)
	t_0 = Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im)) * x_46_re_m) - Float64(Float64(Float64(x_46_im * x_46_re_m) + Float64(x_46_im * x_46_re_m)) * x_46_im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(-3.0 * x_46_im) * x_46_re_m) * x_46_im);
	elseif (t_0 <= 5e+20)
		tmp = Float64(fma(-3.0, Float64(x_46_im * x_46_im), Float64(x_46_re_m * x_46_re_m)) * x_46_re_m);
	else
		tmp = fma(Float64(x_46_re_m - x_46_im), Float64(Float64(x_46_im + x_46_re_m) * x_46_re_m), Float64(x_46_re_m + x_46_re_m));
	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[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] - N[(N[(N[(x$46$im * x$46$re$95$m), $MachinePrecision] + N[(x$46$im * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision]}, N[(x$46$re$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(-3.0 * x$46$im), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] * x$46$im), $MachinePrecision], If[LessEqual[t$95$0, 5e+20], N[(N[(-3.0 * N[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision], N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(N[(x$46$im + x$46$re$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] + N[(x$46$re$95$m + x$46$re$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 83.4%

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

      1. 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. lift--.f64 N/A

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

      3. flip3-- N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in x.re around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

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

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

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

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 45.9

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

   7. Applied rewrites 45.9%

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

   9. Applied rewrites 45.9%

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

   if -inf.0 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < 5e20

   1. Initial program 99.6%

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

   3. Taylor expanded in x.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

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

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

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

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. metadata-eval N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 99.6

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

   5. Applied rewrites 99.6%

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

   if 5e20 < (-.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 59.0%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   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-lft-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-neg.f64 67.8

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift--.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. difference-of-squares N/A

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

      21. associate-*r* N/A

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

      22. lower-*.f64 N/A

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

   4. Applied rewrites 82.3%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. distribute-lft-in N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-+.f64 N/A

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

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

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

      21. *-commutative N/A

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

      22. lift-+.f64 N/A

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

      23. lift-*.f64 N/A

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

      24. *-commutative N/A

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

      25. lift-*.f64 N/A

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

   6. Applied rewrites 87.3%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.im \leq -\infty:\\ \;\;\;\;\left(\left(-3 \cdot x.im\right) \cdot x.re\right) \cdot x.im\\ \mathbf{elif}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.im \leq 5 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(-3, x.im \cdot x.im, x.re \cdot x.re\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im, \left(x.im + x.re\right) \cdot x.re, x.re + x.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ \begin{array}{l} t_0 := \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) \cdot x.re\_m - \left(x.im \cdot x.re\_m + x.im \cdot x.re\_m\right) \cdot x.im\\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-298}:\\ \;\;\;\;\left(-3 \cdot \left(x.im \cdot x.re\_m\right)\right) \cdot x.im\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+20}:\\ \;\;\;\;\left(x.re\_m \cdot x.re\_m\right) \cdot x.re\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m - x.im, \left(x.im + x.re\_m\right) \cdot x.re\_m, x.re\_m + x.re\_m\right)\\ \end{array} \end{array} \end{array} \]

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.im x.im)) x.re_m)
          (* (+ (* x.im x.re_m) (* x.im x.re_m)) x.im))))
   (*
    x.re_s
    (if (<= t_0 -4e-298)
      (* (* -3.0 (* x.im x.re_m)) x.im)
      (if (<= t_0 5e+20)
        (* (* x.re_m x.re_m) x.re_m)
        (fma (- x.re_m x.im) (* (+ x.im x.re_m) 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 t_0 = (((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_im * x_46_re_m) + (x_46_im * x_46_re_m)) * x_46_im);
	double tmp;
	if (t_0 <= -4e-298) {
		tmp = (-3.0 * (x_46_im * x_46_re_m)) * x_46_im;
	} else if (t_0 <= 5e+20) {
		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_im + x_46_re_m) * 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)
	t_0 = Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im)) * x_46_re_m) - Float64(Float64(Float64(x_46_im * x_46_re_m) + Float64(x_46_im * x_46_re_m)) * x_46_im))
	tmp = 0.0
	if (t_0 <= -4e-298)
		tmp = Float64(Float64(-3.0 * Float64(x_46_im * x_46_re_m)) * x_46_im);
	elseif (t_0 <= 5e+20)
		tmp = Float64(Float64(x_46_re_m * x_46_re_m) * x_46_re_m);
	else
		tmp = fma(Float64(x_46_re_m - x_46_im), Float64(Float64(x_46_im + x_46_re_m) * x_46_re_m), Float64(x_46_re_m + x_46_re_m));
	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[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] - N[(N[(N[(x$46$im * x$46$re$95$m), $MachinePrecision] + N[(x$46$im * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision]}, N[(x$46$re$95$s * If[LessEqual[t$95$0, -4e-298], N[(N[(-3.0 * N[(x$46$im * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision], If[LessEqual[t$95$0, 5e+20], N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision], N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(N[(x$46$im + x$46$re$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] + N[(x$46$re$95$m + x$46$re$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -3.99999999999999965e-298

   1. Initial program 89.8%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   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. lift--.f64 N/A

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

      3. flip3-- N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 13.9%

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

   5. Taylor expanded in x.re around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

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

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

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

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 50.6

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

   7. Applied rewrites 50.6%

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

   if -3.99999999999999965e-298 < (-.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)) < 5e20

   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. unpow3 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 74.2

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

   5. Applied rewrites 74.2%

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

   if 5e20 < (-.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 59.0%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   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-lft-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-neg.f64 67.8

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift--.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. difference-of-squares N/A

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

      21. associate-*r* N/A

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

      22. lower-*.f64 N/A

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

   4. Applied rewrites 82.3%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. distribute-lft-in N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-+.f64 N/A

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

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

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

      21. *-commutative N/A

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

      22. lift-+.f64 N/A

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

      23. lift-*.f64 N/A

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

      24. *-commutative N/A

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

      25. lift-*.f64 N/A

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

   6. Applied rewrites 87.3%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.im \leq -4 \cdot 10^{-298}:\\ \;\;\;\;\left(-3 \cdot \left(x.im \cdot x.re\right)\right) \cdot x.im\\ \mathbf{elif}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.im \leq 5 \cdot 10^{+20}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im, \left(x.im + x.re\right) \cdot x.re, x.re + x.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ \begin{array}{l} t_0 := \left(x.im + x.re\_m\right) \cdot x.re\_m\\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) \cdot x.re\_m - \left(x.im \cdot x.re\_m + x.im \cdot x.re\_m\right) \cdot x.im \leq 5 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(\left(x.re\_m + x.re\_m\right) \cdot x.im, -x.im, t\_0 \cdot \left(x.re\_m - x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m - x.im, t\_0, x.re\_m + x.re\_m\right)\\ \end{array} \end{array} \end{array} \]

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.im x.re_m) x.re_m)))
   (*
    x.re_s
    (if (<=
         (-
          (* (- (* x.re_m x.re_m) (* x.im x.im)) x.re_m)
          (* (+ (* x.im x.re_m) (* x.im x.re_m)) x.im))
         5e+20)
      (fma (* (+ x.re_m x.re_m) x.im) (- x.im) (* t_0 (- x.re_m x.im)))
      (fma (- x.re_m x.im) t_0 (+ 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 t_0 = (x_46_im + x_46_re_m) * x_46_re_m;
	double tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im)) * x_46_re_m) - (((x_46_im * x_46_re_m) + (x_46_im * x_46_re_m)) * x_46_im)) <= 5e+20) {
		tmp = fma(((x_46_re_m + x_46_re_m) * x_46_im), -x_46_im, (t_0 * (x_46_re_m - x_46_im)));
	} else {
		tmp = fma((x_46_re_m - x_46_im), t_0, (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)
	t_0 = Float64(Float64(x_46_im + x_46_re_m) * x_46_re_m)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im)) * x_46_re_m) - Float64(Float64(Float64(x_46_im * x_46_re_m) + Float64(x_46_im * x_46_re_m)) * x_46_im)) <= 5e+20)
		tmp = fma(Float64(Float64(x_46_re_m + x_46_re_m) * x_46_im), Float64(-x_46_im), Float64(t_0 * Float64(x_46_re_m - x_46_im)));
	else
		tmp = fma(Float64(x_46_re_m - x_46_im), t_0, Float64(x_46_re_m + x_46_re_m));
	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$46$im + x$46$re$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]}, N[(x$46$re$95$s * If[LessEqual[N[(N[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] - N[(N[(N[(x$46$im * x$46$re$95$m), $MachinePrecision] + N[(x$46$im * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], 5e+20], N[(N[(N[(x$46$re$95$m + x$46$re$95$m), $MachinePrecision] * x$46$im), $MachinePrecision] * (-x$46$im) + N[(t$95$0 * N[(x$46$re$95$m - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * t$95$0 + 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)) < 5e20

   1. Initial program 92.9%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   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. 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-lft-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-neg.f64 93.0

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift--.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. difference-of-squares N/A

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

      21. associate-*r* N/A

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

      22. lower-*.f64 N/A

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

   4. Applied rewrites 99.8%

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

   if 5e20 < (-.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 59.0%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   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-lft-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-neg.f64 67.8

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift--.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. difference-of-squares N/A

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

      21. associate-*r* N/A

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

      22. lower-*.f64 N/A

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

   4. Applied rewrites 82.3%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. distribute-lft-in N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-+.f64 N/A

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

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

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

      21. *-commutative N/A

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

      22. lift-+.f64 N/A

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

      23. lift-*.f64 N/A

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

      24. *-commutative N/A

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

      25. lift-*.f64 N/A

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

   6. Applied rewrites 87.3%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.im \leq 5 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(\left(x.re + x.re\right) \cdot x.im, -x.im, \left(\left(x.im + x.re\right) \cdot x.re\right) \cdot \left(x.re - x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im, \left(x.im + x.re\right) \cdot x.re, x.re + x.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) \cdot x.re\_m - \left(x.im \cdot x.re\_m + x.im \cdot x.re\_m\right) \cdot x.im \leq -4 \cdot 10^{-298}:\\ \;\;\;\;\left(-3 \cdot \left(x.im \cdot x.re\_m\right)\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\left(x.re\_m \cdot x.re\_m\right) \cdot x.re\_m\\ \end{array} \end{array} \]

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.im x.im)) x.re_m)
        (* (+ (* x.im x.re_m) (* x.im x.re_m)) x.im))
       -4e-298)
    (* (* -3.0 (* x.im x.re_m)) x.im)
    (* (* 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_im * x_46_im)) * x_46_re_m) - (((x_46_im * x_46_re_m) + (x_46_im * x_46_re_m)) * x_46_im)) <= -4e-298) {
		tmp = (-3.0 * (x_46_im * x_46_re_m)) * x_46_im;
	} 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_46im * x_46im)) * x_46re_m) - (((x_46im * x_46re_m) + (x_46im * x_46re_m)) * x_46im)) <= (-4d-298)) then
        tmp = ((-3.0d0) * (x_46im * x_46re_m)) * x_46im
    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_im * x_46_im)) * x_46_re_m) - (((x_46_im * x_46_re_m) + (x_46_im * x_46_re_m)) * x_46_im)) <= -4e-298) {
		tmp = (-3.0 * (x_46_im * x_46_re_m)) * x_46_im;
	} 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_im * x_46_im)) * x_46_re_m) - (((x_46_im * x_46_re_m) + (x_46_im * x_46_re_m)) * x_46_im)) <= -4e-298:
		tmp = (-3.0 * (x_46_im * x_46_re_m)) * x_46_im
	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(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im)) * x_46_re_m) - Float64(Float64(Float64(x_46_im * x_46_re_m) + Float64(x_46_im * x_46_re_m)) * x_46_im)) <= -4e-298)
		tmp = Float64(Float64(-3.0 * Float64(x_46_im * x_46_re_m)) * x_46_im);
	else
		tmp = Float64(Float64(x_46_re_m * 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_im * x_46_im)) * x_46_re_m) - (((x_46_im * x_46_re_m) + (x_46_im * x_46_re_m)) * x_46_im)) <= -4e-298)
		tmp = (-3.0 * (x_46_im * x_46_re_m)) * x_46_im;
	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[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] - N[(N[(N[(x$46$im * x$46$re$95$m), $MachinePrecision] + N[(x$46$im * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], -4e-298], N[(N[(-3.0 * N[(x$46$im * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision], N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] * x$46$re$95$m), $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)) < -3.99999999999999965e-298

   1. Initial program 89.8%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   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. lift--.f64 N/A

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

      3. flip3-- N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 13.9%

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

   5. Taylor expanded in x.re around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

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

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

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

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 50.6

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

   7. Applied rewrites 50.6%

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

   if -3.99999999999999965e-298 < (-.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 73.8%

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

   3. Taylor expanded in x.re around inf

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 69.1

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

   5. Applied rewrites 69.1%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.im \leq -4 \cdot 10^{-298}:\\ \;\;\;\;\left(-3 \cdot \left(x.im \cdot x.re\right)\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) \cdot x.re\_m - \left(x.im \cdot x.re\_m + x.im \cdot x.re\_m\right) \cdot x.im \leq -4 \cdot 10^{-298}:\\ \;\;\;\;\left(\left(x.im \cdot x.re\_m\right) \cdot x.im\right) \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\left(x.re\_m \cdot x.re\_m\right) \cdot x.re\_m\\ \end{array} \end{array} \]

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.im x.im)) x.re_m)
        (* (+ (* x.im x.re_m) (* x.im x.re_m)) x.im))
       -4e-298)
    (* (* (* x.im x.re_m) 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_im * x_46_im)) * x_46_re_m) - (((x_46_im * x_46_re_m) + (x_46_im * x_46_re_m)) * x_46_im)) <= -4e-298) {
		tmp = ((x_46_im * x_46_re_m) * 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_46im * x_46im)) * x_46re_m) - (((x_46im * x_46re_m) + (x_46im * x_46re_m)) * x_46im)) <= (-4d-298)) then
        tmp = ((x_46im * x_46re_m) * 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_im * x_46_im)) * x_46_re_m) - (((x_46_im * x_46_re_m) + (x_46_im * x_46_re_m)) * x_46_im)) <= -4e-298) {
		tmp = ((x_46_im * x_46_re_m) * 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_im * x_46_im)) * x_46_re_m) - (((x_46_im * x_46_re_m) + (x_46_im * x_46_re_m)) * x_46_im)) <= -4e-298:
		tmp = ((x_46_im * x_46_re_m) * 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(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im)) * x_46_re_m) - Float64(Float64(Float64(x_46_im * x_46_re_m) + Float64(x_46_im * x_46_re_m)) * x_46_im)) <= -4e-298)
		tmp = Float64(Float64(Float64(x_46_im * x_46_re_m) * x_46_im) * -3.0);
	else
		tmp = Float64(Float64(x_46_re_m * 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_im * x_46_im)) * x_46_re_m) - (((x_46_im * x_46_re_m) + (x_46_im * x_46_re_m)) * x_46_im)) <= -4e-298)
		tmp = ((x_46_im * x_46_re_m) * 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[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] - N[(N[(N[(x$46$im * x$46$re$95$m), $MachinePrecision] + N[(x$46$im * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], -4e-298], N[(N[(N[(x$46$im * x$46$re$95$m), $MachinePrecision] * x$46$im), $MachinePrecision] * -3.0), $MachinePrecision], N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] * x$46$re$95$m), $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)) < -3.99999999999999965e-298

   1. Initial program 89.8%

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

   3. Taylor expanded in x.re around 0

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

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

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 50.6

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

   5. Applied rewrites 50.6%

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

   if -3.99999999999999965e-298 < (-.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 73.8%

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

   3. Taylor expanded in x.re around inf

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 69.1

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

   5. Applied rewrites 69.1%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.im \leq -4 \cdot 10^{-298}:\\ \;\;\;\;\left(\left(x.im \cdot x.re\right) \cdot x.im\right) \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) \cdot x.re\_m - \left(x.im \cdot x.re\_m + x.im \cdot x.re\_m\right) \cdot x.im \leq -4 \cdot 10^{-298}:\\ \;\;\;\;\left(\left(-x.re\_m\right) \cdot x.im\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\left(x.re\_m \cdot x.re\_m\right) \cdot x.re\_m\\ \end{array} \end{array} \]

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.im x.im)) x.re_m)
        (* (+ (* x.im x.re_m) (* x.im x.re_m)) x.im))
       -4e-298)
    (* (* (- x.re_m) x.im) x.im)
    (* (* 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_im * x_46_im)) * x_46_re_m) - (((x_46_im * x_46_re_m) + (x_46_im * x_46_re_m)) * x_46_im)) <= -4e-298) {
		tmp = (-x_46_re_m * x_46_im) * x_46_im;
	} 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_46im * x_46im)) * x_46re_m) - (((x_46im * x_46re_m) + (x_46im * x_46re_m)) * x_46im)) <= (-4d-298)) then
        tmp = (-x_46re_m * x_46im) * x_46im
    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_im * x_46_im)) * x_46_re_m) - (((x_46_im * x_46_re_m) + (x_46_im * x_46_re_m)) * x_46_im)) <= -4e-298) {
		tmp = (-x_46_re_m * x_46_im) * x_46_im;
	} 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_im * x_46_im)) * x_46_re_m) - (((x_46_im * x_46_re_m) + (x_46_im * x_46_re_m)) * x_46_im)) <= -4e-298:
		tmp = (-x_46_re_m * x_46_im) * x_46_im
	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(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im)) * x_46_re_m) - Float64(Float64(Float64(x_46_im * x_46_re_m) + Float64(x_46_im * x_46_re_m)) * x_46_im)) <= -4e-298)
		tmp = Float64(Float64(Float64(-x_46_re_m) * x_46_im) * x_46_im);
	else
		tmp = Float64(Float64(x_46_re_m * 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_im * x_46_im)) * x_46_re_m) - (((x_46_im * x_46_re_m) + (x_46_im * x_46_re_m)) * x_46_im)) <= -4e-298)
		tmp = (-x_46_re_m * x_46_im) * x_46_im;
	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[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] - N[(N[(N[(x$46$im * x$46$re$95$m), $MachinePrecision] + N[(x$46$im * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], -4e-298], N[(N[((-x$46$re$95$m) * x$46$im), $MachinePrecision] * x$46$im), $MachinePrecision], N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] * x$46$re$95$m), $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)) < -3.99999999999999965e-298

   1. Initial program 89.8%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   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. 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-lft-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-neg.f64 89.8

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift--.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. difference-of-squares N/A

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

      21. associate-*r* N/A

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

      22. lower-*.f64 N/A

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

   4. Applied rewrites 99.7%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. distribute-lft-in N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-+.f64 N/A

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

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

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

      21. *-commutative N/A

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

      22. lift-+.f64 N/A

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

      23. lift-*.f64 N/A

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

      24. *-commutative N/A

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

      25. lift-*.f64 N/A

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

   6. Applied rewrites 66.0%

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

   7. Taylor expanded in x.im around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 23.5

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

   9. Applied rewrites 23.5%

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

   if -3.99999999999999965e-298 < (-.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 73.8%

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

   3. Taylor expanded in x.re around inf

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 69.1

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

   5. Applied rewrites 69.1%

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

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.im \leq -4 \cdot 10^{-298}:\\ \;\;\;\;\left(\left(-x.re\right) \cdot x.im\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 58.0% accurate, 3.6× speedup

Math

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

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(Float64(x_46_re_m * 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[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.9%

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

3. Taylor expanded in x.re around inf

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

   1. unpow3 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 60.9

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

5. Applied rewrites 60.9%

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

Developer Target 1: 88.0% 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 2024235 
(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)))