math.cube on complex, real part

Percentage Accurate: 82.9% → 99.0%

Time: 12.3s

Alternatives: 14

Speedup: 1.2×

• 43.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 82.9% 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: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ \begin{array}{l} t_0 := x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) - x.im \cdot \left(x.re\_m \cdot x.im + x.re\_m \cdot x.im\right)\\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-44}:\\ \;\;\;\;x.im \cdot \left(x.re\_m \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-126}:\\ \;\;\;\;x.re\_m \cdot \mathsf{fma}\left(x.re\_m, x.re\_m, -3 \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m - x.im, x.re\_m \cdot \left(x.re\_m + x.im\right), x.im + x.im\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (let* ((t_0
         (-
          (* x.re_m (- (* x.re_m x.re_m) (* x.im x.im)))
          (* x.im (+ (* x.re_m x.im) (* x.re_m x.im))))))
   (*
    x.re_s
    (if (<= t_0 -1e-44)
      (* x.im (* x.re_m (* x.im -3.0)))
      (if (<= t_0 2e-126)
        (* x.re_m (fma x.re_m x.re_m (* -3.0 (* x.im x.im))))
        (fma (- x.re_m x.im) (* x.re_m (+ x.re_m x.im)) (+ x.im x.im)))))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double t_0 = (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)));
	double tmp;
	if (t_0 <= -1e-44) {
		tmp = x_46_im * (x_46_re_m * (x_46_im * -3.0));
	} else if (t_0 <= 2e-126) {
		tmp = x_46_re_m * fma(x_46_re_m, x_46_re_m, (-3.0 * (x_46_im * x_46_im)));
	} else {
		tmp = fma((x_46_re_m - x_46_im), (x_46_re_m * (x_46_re_m + x_46_im)), (x_46_im + x_46_im));
	}
	return x_46_re_s * tmp;
}

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	t_0 = Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im))) - Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_re_m * x_46_im))))
	tmp = 0.0
	if (t_0 <= -1e-44)
		tmp = Float64(x_46_im * Float64(x_46_re_m * Float64(x_46_im * -3.0)));
	elseif (t_0 <= 2e-126)
		tmp = Float64(x_46_re_m * fma(x_46_re_m, x_46_re_m, Float64(-3.0 * Float64(x_46_im * x_46_im))));
	else
		tmp = fma(Float64(x_46_re_m - x_46_im), Float64(x_46_re_m * Float64(x_46_re_m + x_46_im)), Float64(x_46_im + x_46_im));
	end
	return Float64(x_46_re_s * tmp)
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := Block[{t$95$0 = N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$re$95$s * If[LessEqual[t$95$0, -1e-44], N[(x$46$im * N[(x$46$re$95$m * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-126], N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$re$95$m + N[(-3.0 * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(x$46$re$95$m * N[(x$46$re$95$m + x$46$im), $MachinePrecision]), $MachinePrecision] + N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -9.99999999999999953e-45

   1. Initial program 94.6%

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

   3. Taylor expanded in x.re around inf

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

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 60.1

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

   5. Applied rewrites 60.1%

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

   6. Taylor expanded in x.re around 0

      \[\leadsto \color{blue}{x.re \cdot \left(-1 \cdot {x.im}^{2} - 2 \cdot {x.im}^{2}\right)} \]
   7. 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. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 35.8

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

   8. Applied rewrites 35.8%

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

   10. Applied rewrites 40.9%

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

   if -9.99999999999999953e-45 < (-.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)) < 1.9999999999999999e-126

   1. Initial program 99.8%

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

   3. Taylor expanded in x.re around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate--l+ N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 99.8

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

   5. Applied rewrites 99.8%

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

   if 1.9999999999999999e-126 < (-.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 63.4%

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

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-neg.f64 65.3

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift--.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. difference-of-squares N/A

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

      21. associate-*r* N/A

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

      22. lower-*.f64 N/A

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

   4. Applied rewrites 85.0%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

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

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. flip-+ N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. +-inverses N/A

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

      18. +-inverses N/A

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

      19. associate-*r/ N/A

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

   6. Applied rewrites 75.1%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -1 \cdot 10^{-44}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{elif}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq 2 \cdot 10^{-126}:\\ \;\;\;\;x.re \cdot \mathsf{fma}\left(x.re, x.re, -3 \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im, x.re \cdot \left(x.re + x.im\right), x.im + x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ \begin{array}{l} t_0 := x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) - x.im \cdot \left(x.re\_m \cdot x.im + x.re\_m \cdot x.im\right)\\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-323}:\\ \;\;\;\;x.im \cdot \left(x.re\_m \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-126}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m - x.im, x.re\_m \cdot \left(x.re\_m + x.im\right), x.im + x.im\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (let* ((t_0
         (-
          (* x.re_m (- (* x.re_m x.re_m) (* x.im x.im)))
          (* x.im (+ (* x.re_m x.im) (* x.re_m x.im))))))
   (*
    x.re_s
    (if (<= t_0 -1e-323)
      (* x.im (* x.re_m (* x.im -3.0)))
      (if (<= t_0 2e-126)
        (* x.re_m (* x.re_m x.re_m))
        (fma (- x.re_m x.im) (* x.re_m (+ x.re_m x.im)) (+ x.im x.im)))))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double t_0 = (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)));
	double tmp;
	if (t_0 <= -1e-323) {
		tmp = x_46_im * (x_46_re_m * (x_46_im * -3.0));
	} else if (t_0 <= 2e-126) {
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	} else {
		tmp = fma((x_46_re_m - x_46_im), (x_46_re_m * (x_46_re_m + x_46_im)), (x_46_im + x_46_im));
	}
	return x_46_re_s * tmp;
}

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	t_0 = Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im))) - Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_re_m * x_46_im))))
	tmp = 0.0
	if (t_0 <= -1e-323)
		tmp = Float64(x_46_im * Float64(x_46_re_m * Float64(x_46_im * -3.0)));
	elseif (t_0 <= 2e-126)
		tmp = Float64(x_46_re_m * Float64(x_46_re_m * x_46_re_m));
	else
		tmp = fma(Float64(x_46_re_m - x_46_im), Float64(x_46_re_m * Float64(x_46_re_m + x_46_im)), Float64(x_46_im + x_46_im));
	end
	return Float64(x_46_re_s * tmp)
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := Block[{t$95$0 = N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$re$95$s * If[LessEqual[t$95$0, -1e-323], N[(x$46$im * N[(x$46$re$95$m * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-126], N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(x$46$re$95$m * N[(x$46$re$95$m + x$46$im), $MachinePrecision]), $MachinePrecision] + N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -9.88131e-324

   1. Initial program 96.0%

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

   3. Taylor expanded in x.re around inf

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

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 60.6

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

   5. Applied rewrites 60.6%

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

   6. Taylor expanded in x.re around 0

      \[\leadsto \color{blue}{x.re \cdot \left(-1 \cdot {x.im}^{2} - 2 \cdot {x.im}^{2}\right)} \]
   7. 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. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 39.1

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

   8. Applied rewrites 39.1%

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

   10. Applied rewrites 42.8%

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

   if -9.88131e-324 < (-.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)) < 1.9999999999999999e-126

   1. Initial program 99.9%

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

   3. Taylor expanded in x.re around inf

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

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 87.0

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

   5. Applied rewrites 87.0%

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

   if 1.9999999999999999e-126 < (-.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 63.4%

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

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-neg.f64 65.3

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift--.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. difference-of-squares N/A

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

      21. associate-*r* N/A

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

      22. lower-*.f64 N/A

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

   4. Applied rewrites 85.0%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

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

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. flip-+ N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. +-inverses N/A

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

      18. +-inverses N/A

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

      19. associate-*r/ N/A

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

   6. Applied rewrites 75.1%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -1 \cdot 10^{-323}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{elif}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq 2 \cdot 10^{-126}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im, x.re \cdot \left(x.re + x.im\right), x.im + x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 3 \cdot 10^{+93}:\\ \;\;\;\;\mathsf{fma}\left(x.im, x.re\_m \cdot \left(x.im \cdot -3\right), x.re\_m \cdot \left(x.re\_m \cdot x.re\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x.im \cdot \mathsf{fma}\left(x.re\_m, \frac{x.re\_m}{x.im}, x.re\_m\right)}{x.re\_m + x.im}, \frac{\left(x.re\_m - x.im\right) \cdot \left(x.re\_m + x.im\right)}{\frac{x.re\_m - x.im}{x.re\_m - x.im}}, x.im + x.im\right)\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 3e+93)
    (fma x.im (* x.re_m (* x.im -3.0)) (* x.re_m (* x.re_m x.re_m)))
    (fma
     (/ (* x.im (fma x.re_m (/ x.re_m x.im) x.re_m)) (+ x.re_m x.im))
     (/
      (* (- x.re_m x.im) (+ x.re_m x.im))
      (/ (- x.re_m x.im) (- x.re_m x.im)))
     (+ x.im x.im)))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 3e+93) {
		tmp = fma(x_46_im, (x_46_re_m * (x_46_im * -3.0)), (x_46_re_m * (x_46_re_m * x_46_re_m)));
	} else {
		tmp = fma(((x_46_im * fma(x_46_re_m, (x_46_re_m / x_46_im), x_46_re_m)) / (x_46_re_m + x_46_im)), (((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im)) / ((x_46_re_m - x_46_im) / (x_46_re_m - x_46_im))), (x_46_im + x_46_im));
	}
	return x_46_re_s * tmp;
}

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_re_m <= 3e+93)
		tmp = fma(x_46_im, Float64(x_46_re_m * Float64(x_46_im * -3.0)), Float64(x_46_re_m * Float64(x_46_re_m * x_46_re_m)));
	else
		tmp = fma(Float64(Float64(x_46_im * fma(x_46_re_m, Float64(x_46_re_m / x_46_im), x_46_re_m)) / Float64(x_46_re_m + x_46_im)), Float64(Float64(Float64(x_46_re_m - x_46_im) * Float64(x_46_re_m + x_46_im)) / Float64(Float64(x_46_re_m - x_46_im) / Float64(x_46_re_m - x_46_im))), Float64(x_46_im + x_46_im));
	end
	return Float64(x_46_re_s * tmp)
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 3e+93], N[(x$46$im * N[(x$46$re$95$m * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision] + N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$46$im * N[(x$46$re$95$m * N[(x$46$re$95$m / x$46$im), $MachinePrecision] + x$46$re$95$m), $MachinePrecision]), $MachinePrecision] / N[(x$46$re$95$m + x$46$im), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(x$46$re$95$m + x$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] / N[(x$46$re$95$m - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 2.99999999999999978e93

   1. Initial program 87.6%

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

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

      2. +-commutative N/A

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

      3. associate--l+ N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 92.7

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

   5. Applied rewrites 92.7%

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

   7. Applied rewrites 92.8%

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

   if 2.99999999999999978e93 < x.re

   1. Initial program 58.5%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-neg.f64 58.5

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift--.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. difference-of-squares N/A

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

      21. associate-*r* N/A

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

      22. lower-*.f64 N/A

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

   4. Applied rewrites 70.7%

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

   5. Taylor expanded in x.im around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 70.7

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

   7. Applied rewrites 70.7%

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

   9. Applied rewrites 100.0%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 3 \cdot 10^{+93}:\\ \;\;\;\;\mathsf{fma}\left(x.im, x.re \cdot \left(x.im \cdot -3\right), x.re \cdot \left(x.re \cdot x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x.im \cdot \mathsf{fma}\left(x.re, \frac{x.re}{x.im}, x.re\right)}{x.re + x.im}, \frac{\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)}{\frac{x.re - x.im}{x.re - x.im}}, x.im + x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) - x.im \cdot \left(x.re\_m \cdot x.im + x.re\_m \cdot x.im\right) \leq 5 \cdot 10^{-319}:\\ \;\;\;\;x.im \cdot \left(x.re\_m \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.im + x.im, x.re\_m, \left(x.re\_m + x.im\right) \cdot \left(x.re\_m \cdot \left(x.re\_m - x.im\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<=
       (-
        (* x.re_m (- (* x.re_m x.re_m) (* x.im x.im)))
        (* x.im (+ (* x.re_m x.im) (* x.re_m x.im))))
       5e-319)
    (* x.im (* x.re_m (* x.im -3.0)))
    (fma
     (+ x.im x.im)
     x.re_m
     (* (+ x.re_m x.im) (* x.re_m (- x.re_m x.im)))))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= 5e-319) {
		tmp = x_46_im * (x_46_re_m * (x_46_im * -3.0));
	} else {
		tmp = fma((x_46_im + x_46_im), x_46_re_m, ((x_46_re_m + x_46_im) * (x_46_re_m * (x_46_re_m - x_46_im))));
	}
	return x_46_re_s * tmp;
}

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im))) - Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_re_m * x_46_im)))) <= 5e-319)
		tmp = Float64(x_46_im * Float64(x_46_re_m * Float64(x_46_im * -3.0)));
	else
		tmp = fma(Float64(x_46_im + x_46_im), x_46_re_m, Float64(Float64(x_46_re_m + x_46_im) * Float64(x_46_re_m * Float64(x_46_re_m - x_46_im))));
	end
	return Float64(x_46_re_s * tmp)
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-319], N[(x$46$im * N[(x$46$re$95$m * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im + x$46$im), $MachinePrecision] * x$46$re$95$m + N[(N[(x$46$re$95$m + x$46$im), $MachinePrecision] * N[(x$46$re$95$m * N[(x$46$re$95$m - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < 4.9999937e-319

   1. Initial program 97.0%

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

   3. Taylor expanded in x.re around inf

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

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 71.1

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

   5. Applied rewrites 71.1%

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

   6. Taylor expanded in x.re around 0

      \[\leadsto \color{blue}{x.re \cdot \left(-1 \cdot {x.im}^{2} - 2 \cdot {x.im}^{2}\right)} \]
   7. 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. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 55.3

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

   8. Applied rewrites 55.3%

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

   10. Applied rewrites 58.0%

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

   if 4.9999937e-319 < (-.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 67.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-neg.f64 68.9

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift--.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. difference-of-squares N/A

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

      21. associate-*r* N/A

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

      22. lower-*.f64 N/A

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

   4. Applied rewrites 86.6%

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

   6. Applied rewrites 74.4%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq 5 \cdot 10^{-319}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.im + x.im, x.re, \left(x.re + x.im\right) \cdot \left(x.re \cdot \left(x.re - x.im\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) - x.im \cdot \left(x.re\_m \cdot x.im + x.re\_m \cdot x.im\right) \leq -1 \cdot 10^{-323}:\\ \;\;\;\;x.im \cdot \left(x.re\_m \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m\right)\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<=
       (-
        (* x.re_m (- (* x.re_m x.re_m) (* x.im x.im)))
        (* x.im (+ (* x.re_m x.im) (* x.re_m x.im))))
       -1e-323)
    (* 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_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= -1e-323) {
		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_46re_m) - (x_46im * x_46im))) - (x_46im * ((x_46re_m * x_46im) + (x_46re_m * x_46im)))) <= (-1d-323)) 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_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= -1e-323) {
		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_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= -1e-323:
		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(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im))) - Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_re_m * x_46_im)))) <= -1e-323)
		tmp = Float64(x_46_im * Float64(x_46_re_m * Float64(x_46_im * -3.0)));
	else
		tmp = Float64(x_46_re_m * Float64(x_46_re_m * x_46_re_m));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= -1e-323)
		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[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-323], N[(x$46$im * N[(x$46$re$95$m * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -9.88131e-324

   1. Initial program 96.0%

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

   3. Taylor expanded in x.re around inf

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

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 60.6

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

   5. Applied rewrites 60.6%

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

   6. Taylor expanded in x.re around 0

      \[\leadsto \color{blue}{x.re \cdot \left(-1 \cdot {x.im}^{2} - 2 \cdot {x.im}^{2}\right)} \]
   7. 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. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 39.1

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

   8. Applied rewrites 39.1%

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

   10. Applied rewrites 42.8%

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

   if -9.88131e-324 < (-.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 74.8%

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

   3. Taylor expanded in x.re around inf

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

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 58.8

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

   5. Applied rewrites 58.8%

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

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -1 \cdot 10^{-323}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) - x.im \cdot \left(x.re\_m \cdot x.im + x.re\_m \cdot x.im\right) \leq -1 \cdot 10^{-323}:\\ \;\;\;\;-x.re\_m \cdot \left(x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m\right)\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<=
       (-
        (* x.re_m (- (* x.re_m x.re_m) (* x.im x.im)))
        (* x.im (+ (* x.re_m x.im) (* x.re_m x.im))))
       -1e-323)
    (- (* 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_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= -1e-323) {
		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_46re_m) - (x_46im * x_46im))) - (x_46im * ((x_46re_m * x_46im) + (x_46re_m * x_46im)))) <= (-1d-323)) 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_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= -1e-323) {
		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_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= -1e-323:
		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(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im))) - Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_re_m * x_46_im)))) <= -1e-323)
		tmp = Float64(-Float64(x_46_re_m * Float64(x_46_im * x_46_im)));
	else
		tmp = Float64(x_46_re_m * Float64(x_46_re_m * x_46_re_m));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= -1e-323)
		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[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-323], (-N[(x$46$re$95$m * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -9.88131e-324

   1. Initial program 96.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-neg.f64 95.9

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift--.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. difference-of-squares N/A

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

      21. associate-*r* N/A

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

      22. lower-*.f64 N/A

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

   4. Applied rewrites 99.7%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lift-neg.f64 N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 99.6

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. associate-*l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 99.7

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. associate-*r* N/A

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

      20. *-commutative N/A

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

      21. lift-+.f64 N/A

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

      22. flip-+ N/A

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

   6. Applied rewrites 67.7%

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

   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. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. unpow2 N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 22.5

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

   9. Applied rewrites 22.5%

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

   if -9.88131e-324 < (-.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 74.8%

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

   3. Taylor expanded in x.re around inf

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

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 58.8

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

   5. Applied rewrites 58.8%

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

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -1 \cdot 10^{-323}:\\ \;\;\;\;-x.re \cdot \left(x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 20.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) - x.im \cdot \left(x.re\_m \cdot x.im + x.re\_m \cdot x.im\right) \leq 2 \cdot 10^{-293}:\\ \;\;\;\;\mathsf{fma}\left(x.im, -2, x.im + x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(x.im + x.im\right)\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<=
       (-
        (* x.re_m (- (* x.re_m x.re_m) (* x.im x.im)))
        (* x.im (+ (* x.re_m x.im) (* x.re_m x.im))))
       2e-293)
    (fma x.im -2.0 (+ x.im x.im))
    (* x.re_m (+ x.im x.im)))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= 2e-293) {
		tmp = fma(x_46_im, -2.0, (x_46_im + x_46_im));
	} else {
		tmp = x_46_re_m * (x_46_im + x_46_im);
	}
	return x_46_re_s * tmp;
}

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im))) - Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_re_m * x_46_im)))) <= 2e-293)
		tmp = fma(x_46_im, -2.0, Float64(x_46_im + x_46_im));
	else
		tmp = Float64(x_46_re_m * Float64(x_46_im + x_46_im));
	end
	return Float64(x_46_re_s * tmp)
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-293], N[(x$46$im * -2.0 + N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision], N[(x$46$re$95$m * N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < 2.0000000000000001e-293

   1. Initial program 97.1%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-neg.f64 97.1

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift--.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. difference-of-squares N/A

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

      21. associate-*r* N/A

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

      22. lower-*.f64 N/A

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

   4. Applied rewrites 99.8%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lift-neg.f64 N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 99.7

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. associate-*l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 99.8

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. associate-*r* N/A

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

      20. *-commutative N/A

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

      21. lift-+.f64 N/A

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

      22. flip-+ N/A

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

   6. Applied rewrites 50.3%

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

   7. Taylor expanded in x.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 4.2

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

   9. Applied rewrites 4.2%

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

   11. Applied rewrites 30.2%

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

   if 2.0000000000000001e-293 < (-.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 66.5%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-neg.f64 68.2

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift--.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. difference-of-squares N/A

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

      21. associate-*r* N/A

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

      22. lower-*.f64 N/A

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

   4. Applied rewrites 86.3%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lift-neg.f64 N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 84.5

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. associate-*l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 84.5

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. associate-*r* N/A

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

      20. *-commutative N/A

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

      21. lift-+.f64 N/A

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

      22. flip-+ N/A

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

   6. Applied rewrites 71.4%

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

   7. Taylor expanded in x.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 3.1

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

   9. Applied rewrites 3.1%

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

   11. Applied rewrites 15.0%

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

4. Final simplification 23.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq 2 \cdot 10^{-293}:\\ \;\;\;\;\mathsf{fma}\left(x.im, -2, x.im + x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.im + x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 18.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) - x.im \cdot \left(x.re\_m \cdot x.im + x.re\_m \cdot x.im\right) \leq -5 \cdot 10^{-218}:\\ \;\;\;\;x.im \cdot -2\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(x.im + x.im\right)\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<=
       (-
        (* x.re_m (- (* x.re_m x.re_m) (* x.im x.im)))
        (* x.im (+ (* x.re_m x.im) (* x.re_m x.im))))
       -5e-218)
    (* x.im -2.0)
    (* x.re_m (+ x.im x.im)))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= -5e-218) {
		tmp = x_46_im * -2.0;
	} else {
		tmp = x_46_re_m * (x_46_im + x_46_im);
	}
	return x_46_re_s * tmp;
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (((x_46re_m * ((x_46re_m * x_46re_m) - (x_46im * x_46im))) - (x_46im * ((x_46re_m * x_46im) + (x_46re_m * x_46im)))) <= (-5d-218)) then
        tmp = x_46im * (-2.0d0)
    else
        tmp = x_46re_m * (x_46im + x_46im)
    end if
    code = x_46re_s * tmp
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= -5e-218) {
		tmp = x_46_im * -2.0;
	} else {
		tmp = x_46_re_m * (x_46_im + x_46_im);
	}
	return x_46_re_s * tmp;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	tmp = 0
	if ((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= -5e-218:
		tmp = x_46_im * -2.0
	else:
		tmp = x_46_re_m * (x_46_im + x_46_im)
	return x_46_re_s * tmp

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im))) - Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_re_m * x_46_im)))) <= -5e-218)
		tmp = Float64(x_46_im * -2.0);
	else
		tmp = Float64(x_46_re_m * Float64(x_46_im + x_46_im));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= -5e-218)
		tmp = x_46_im * -2.0;
	else
		tmp = x_46_re_m * (x_46_im + x_46_im);
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-218], N[(x$46$im * -2.0), $MachinePrecision], N[(x$46$re$95$m * N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -5.00000000000000041e-218

   1. Initial program 95.4%

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

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-neg.f64 95.4

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift--.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. difference-of-squares N/A

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

      21. associate-*r* N/A

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

      22. lower-*.f64 N/A

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

   4. Applied rewrites 99.7%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lift-neg.f64 N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 99.6

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. associate-*l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 99.6

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. associate-*r* N/A

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

      20. *-commutative N/A

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

      21. lift-+.f64 N/A

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

      22. flip-+ N/A

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

   6. Applied rewrites 75.2%

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

   7. Taylor expanded in x.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 3.2

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

   9. Applied rewrites 3.2%

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

   if -5.00000000000000041e-218 < (-.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 76.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-neg.f64 77.9

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift--.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. difference-of-squares N/A

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

      21. associate-*r* N/A

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

      22. lower-*.f64 N/A

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

   4. Applied rewrites 90.4%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lift-neg.f64 N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 89.2

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. associate-*l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 89.2

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. associate-*r* N/A

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

      20. *-commutative N/A

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

      21. lift-+.f64 N/A

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

      22. flip-+ N/A

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

   6. Applied rewrites 52.3%

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

   7. Taylor expanded in x.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 3.9

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

   9. Applied rewrites 3.9%

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

   11. Applied rewrites 30.2%

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

4. Final simplification 21.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -5 \cdot 10^{-218}:\\ \;\;\;\;x.im \cdot -2\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.im + x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 14.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) - x.im \cdot \left(x.re\_m \cdot x.im + x.re\_m \cdot x.im\right) \leq -1 \cdot 10^{-264}:\\ \;\;\;\;x.im \cdot -2\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im + x.im\right)\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<=
       (-
        (* x.re_m (- (* x.re_m x.re_m) (* x.im x.im)))
        (* x.im (+ (* x.re_m x.im) (* x.re_m x.im))))
       -1e-264)
    (* x.im -2.0)
    (* x.im (+ x.im x.im)))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= -1e-264) {
		tmp = x_46_im * -2.0;
	} else {
		tmp = x_46_im * (x_46_im + x_46_im);
	}
	return x_46_re_s * tmp;
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (((x_46re_m * ((x_46re_m * x_46re_m) - (x_46im * x_46im))) - (x_46im * ((x_46re_m * x_46im) + (x_46re_m * x_46im)))) <= (-1d-264)) then
        tmp = x_46im * (-2.0d0)
    else
        tmp = x_46im * (x_46im + x_46im)
    end if
    code = x_46re_s * tmp
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= -1e-264) {
		tmp = x_46_im * -2.0;
	} else {
		tmp = x_46_im * (x_46_im + x_46_im);
	}
	return x_46_re_s * tmp;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	tmp = 0
	if ((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= -1e-264:
		tmp = x_46_im * -2.0
	else:
		tmp = x_46_im * (x_46_im + x_46_im)
	return x_46_re_s * tmp

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im))) - Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_re_m * x_46_im)))) <= -1e-264)
		tmp = Float64(x_46_im * -2.0);
	else
		tmp = Float64(x_46_im * Float64(x_46_im + x_46_im));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= -1e-264)
		tmp = x_46_im * -2.0;
	else
		tmp = x_46_im * (x_46_im + x_46_im);
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-264], N[(x$46$im * -2.0), $MachinePrecision], N[(x$46$im * N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < -1e-264

   1. Initial program 95.5%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-neg.f64 95.5

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift--.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. difference-of-squares N/A

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

      21. associate-*r* N/A

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

      22. lower-*.f64 N/A

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

   4. Applied rewrites 99.7%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lift-neg.f64 N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 99.6

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. associate-*l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 99.6

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. associate-*r* N/A

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

      20. *-commutative N/A

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

      21. lift-+.f64 N/A

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

      22. flip-+ N/A

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

   6. Applied rewrites 73.8%

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

   7. Taylor expanded in x.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 3.3

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

   9. Applied rewrites 3.3%

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

   if -1e-264 < (-.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 76.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-neg.f64 77.5

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift--.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. difference-of-squares N/A

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

      21. associate-*r* N/A

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

      22. lower-*.f64 N/A

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

   4. Applied rewrites 90.3%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lift-neg.f64 N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 89.0

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. associate-*l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 89.1

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. associate-*r* N/A

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

      20. *-commutative N/A

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

      21. lift-+.f64 N/A

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

      22. flip-+ N/A

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

   6. Applied rewrites 52.6%

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

   7. Taylor expanded in x.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 3.9

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

   9. Applied rewrites 3.9%

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

   11. Applied rewrites 33.8%

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

4. Final simplification 23.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -1 \cdot 10^{-264}:\\ \;\;\;\;x.im \cdot -2\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im + x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 5 \cdot 10^{+84}:\\ \;\;\;\;\mathsf{fma}\left(x.im, x.re\_m \cdot \left(x.im \cdot -3\right), x.re\_m \cdot \left(x.re\_m \cdot x.re\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m - x.im, x.re\_m \cdot \left(x.re\_m + x.im\right), x.im + x.im\right)\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 5e+84)
    (fma x.im (* x.re_m (* x.im -3.0)) (* x.re_m (* x.re_m x.re_m)))
    (fma (- x.re_m x.im) (* x.re_m (+ x.re_m x.im)) (+ x.im x.im)))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 5e+84) {
		tmp = fma(x_46_im, (x_46_re_m * (x_46_im * -3.0)), (x_46_re_m * (x_46_re_m * x_46_re_m)));
	} else {
		tmp = fma((x_46_re_m - x_46_im), (x_46_re_m * (x_46_re_m + x_46_im)), (x_46_im + x_46_im));
	}
	return x_46_re_s * tmp;
}

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_re_m <= 5e+84)
		tmp = fma(x_46_im, Float64(x_46_re_m * Float64(x_46_im * -3.0)), Float64(x_46_re_m * Float64(x_46_re_m * x_46_re_m)));
	else
		tmp = fma(Float64(x_46_re_m - x_46_im), Float64(x_46_re_m * Float64(x_46_re_m + x_46_im)), Float64(x_46_im + x_46_im));
	end
	return Float64(x_46_re_s * tmp)
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 5e+84], N[(x$46$im * N[(x$46$re$95$m * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision] + N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(x$46$re$95$m * N[(x$46$re$95$m + x$46$im), $MachinePrecision]), $MachinePrecision] + N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 5.0000000000000001e84

   1. Initial program 87.5%

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

   3. Taylor expanded in x.re around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate--l+ N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 92.6

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

   5. Applied rewrites 92.6%

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

   7. Applied rewrites 92.7%

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

   if 5.0000000000000001e84 < x.re

   1. Initial program 61.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-neg.f64 61.3

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift--.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. difference-of-squares N/A

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

      21. associate-*r* N/A

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

      22. lower-*.f64 N/A

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

   4. Applied rewrites 72.7%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

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

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. flip-+ N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. +-inverses N/A

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

      18. +-inverses N/A

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

      19. associate-*r/ N/A

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

   6. Applied rewrites 100.0%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 5 \cdot 10^{+84}:\\ \;\;\;\;\mathsf{fma}\left(x.im, x.re \cdot \left(x.im \cdot -3\right), x.re \cdot \left(x.re \cdot x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im, x.re \cdot \left(x.re + x.im\right), x.im + x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 25.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.im \leq 1.5 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(x.im, -2, x.im + x.im\right)\\ \mathbf{elif}\;x.im \leq 2.3 \cdot 10^{+148}:\\ \;\;\;\;x.re\_m \cdot \left(x.im + x.im\right)\\ \mathbf{else}:\\ \;\;\;\;-x.im \cdot \left(x.im + x.im\right)\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<= x.im 1.5e-11)
    (fma x.im -2.0 (+ x.im x.im))
    (if (<= x.im 2.3e+148)
      (* x.re_m (+ x.im x.im))
      (- (* x.im (+ x.im x.im)))))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_im <= 1.5e-11) {
		tmp = fma(x_46_im, -2.0, (x_46_im + x_46_im));
	} else if (x_46_im <= 2.3e+148) {
		tmp = x_46_re_m * (x_46_im + x_46_im);
	} else {
		tmp = -(x_46_im * (x_46_im + x_46_im));
	}
	return x_46_re_s * tmp;
}

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_im <= 1.5e-11)
		tmp = fma(x_46_im, -2.0, Float64(x_46_im + x_46_im));
	elseif (x_46_im <= 2.3e+148)
		tmp = Float64(x_46_re_m * Float64(x_46_im + x_46_im));
	else
		tmp = Float64(-Float64(x_46_im * Float64(x_46_im + x_46_im)));
	end
	return Float64(x_46_re_s * tmp)
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[x$46$im, 1.5e-11], N[(x$46$im * -2.0 + N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 2.3e+148], N[(x$46$re$95$m * N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision], (-N[(x$46$im * N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision])]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x.im < 1.5e-11

   1. Initial program 90.4%

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

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-neg.f64 91.4

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift--.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. difference-of-squares N/A

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

      21. associate-*r* N/A

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

      22. lower-*.f64 N/A

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

   4. Applied rewrites 97.1%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lift-neg.f64 N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 96.0

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. associate-*l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 96.0

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. associate-*r* N/A

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

      20. *-commutative N/A

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

      21. lift-+.f64 N/A

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

      22. flip-+ N/A

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

   6. Applied rewrites 59.6%

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

   7. Taylor expanded in x.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 4.0

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

   9. Applied rewrites 4.0%

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

   11. Applied rewrites 23.0%

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

   if 1.5e-11 < x.im < 2.3000000000000001e148

   1. Initial program 85.4%

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

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-neg.f64 85.4

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift--.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. difference-of-squares N/A

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

      21. associate-*r* N/A

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

      22. lower-*.f64 N/A

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

   4. Applied rewrites 85.3%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lift-neg.f64 N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 85.2

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

         \[\leadsto \color{blue}{\left(x.re - x.im\right) \cdot \left(x.re \cdot \left(x.re + x.im\right)\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left(x.re - x.im\right) \cdot \color{blue}{\left(x.re \cdot \left(x.re + x.im\right)\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \left(x.re - x.im\right) \cdot \color{blue}{\left(\left(x.re + x.im\right) \cdot x.re\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.re} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \color{blue}{\left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)} \cdot x.re - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \color{blue}{\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      16. lower-*.f64 85.2

          \[\leadsto \left(x.re + x.im\right) \cdot \color{blue}{\left(\left(x.re - x.im\right) \cdot x.re\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) - \color{blue}{x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)} \]

      18. lift-*.f64 N/A

          \[\leadsto \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) - x.im \cdot \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right)} \]

      19. associate-*r* N/A

          \[\leadsto \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) - \color{blue}{\left(x.im \cdot x.re\right) \cdot \left(x.im + x.im\right)} \]

      20. *-commutative N/A

          \[\leadsto \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) - \color{blue}{\left(x.re \cdot x.im\right)} \cdot \left(x.im + x.im\right) \]

      21. lift-+.f64 N/A

          \[\leadsto \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) - \left(x.re \cdot x.im\right) \cdot \color{blue}{\left(x.im + x.im\right)} \]

      22. flip-+ N/A

          \[\leadsto \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) - \left(x.re \cdot x.im\right) \cdot \color{blue}{\frac{x.im \cdot x.im - x.im \cdot x.im}{x.im - x.im}} \]

   6. Applied rewrites 31.9%

      \[\leadsto \color{blue}{\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) - \left(x.im + x.im\right)} \]

   7. Taylor expanded in x.re around 0

      \[\leadsto \color{blue}{-2 \cdot x.im} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{x.im \cdot -2} \]

      2. lower-*.f64 3.0

         \[\leadsto \color{blue}{x.im \cdot -2} \]

   9. Applied rewrites 3.0%

      \[\leadsto \color{blue}{x.im \cdot -2} \]
   10. Step-by-step derivation

   11. Applied rewrites 16.3%

       \[\leadsto \left(x.im + x.im\right) \cdot \color{blue}{x.re} \]

   if 2.3000000000000001e148 < x.im

   1. Initial program 48.6%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im} \]

      2. sub-neg N/A

         \[\leadsto \color{blue}{\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re + \left(\mathsf{neg}\left(\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\right)\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im}\right)\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right)} + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x.re \cdot x.im + x.im \cdot x.re, \mathsf{neg}\left(x.im\right), \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x.re \cdot x.im + x.im \cdot x.re}, \mathsf{neg}\left(x.im\right), \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x.re \cdot x.im} + x.im \cdot x.re, \mathsf{neg}\left(x.im\right), \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x.im \cdot x.re} + x.im \cdot x.re, \mathsf{neg}\left(x.im\right), \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x.im \cdot x.re + \color{blue}{x.im \cdot x.re}, \mathsf{neg}\left(x.im\right), \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re\right) \]

      11. distribute-rgt-out N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{x.re \cdot \left(x.im + x.im\right)}, \mathsf{neg}\left(x.im\right), \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{x.re \cdot \left(x.im + x.im\right)}, \mathsf{neg}\left(x.im\right), \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re\right) \]

      13. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(x.re \cdot \color{blue}{\left(x.im + x.im\right)}, \mathsf{neg}\left(x.im\right), \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re\right) \]

      14. lower-neg.f64 48.6

          \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \color{blue}{-x.im}, \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), \color{blue}{\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re}\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), \color{blue}{x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)}\right) \]

      17. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), x.re \cdot \color{blue}{\left(x.re \cdot x.re - x.im \cdot x.im\right)}\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), x.re \cdot \left(\color{blue}{x.re \cdot x.re} - x.im \cdot x.im\right)\right) \]

      19. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), x.re \cdot \left(x.re \cdot x.re - \color{blue}{x.im \cdot x.im}\right)\right) \]

      20. difference-of-squares N/A

          \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), x.re \cdot \color{blue}{\left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)}\right) \]

      21. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), \color{blue}{\left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), \color{blue}{\left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)}\right) \]

   4. Applied rewrites 83.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), -x.im, \left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right) + \left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right) + \left(x.re \cdot \left(x.im + x.im\right)\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right) + \color{blue}{\left(\mathsf{neg}\left(x.im\right)\right) \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)} \]

      4. lift-neg.f64 N/A

         \[\leadsto \left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right) + \color{blue}{\left(\mathsf{neg}\left(x.im\right)\right)} \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      5. cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right) - \color{blue}{x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)} \]

      7. lift--.f64 83.2

         \[\leadsto \color{blue}{\left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \color{blue}{\left(x.re - x.im\right) \cdot \left(x.re \cdot \left(x.re + x.im\right)\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left(x.re - x.im\right) \cdot \color{blue}{\left(x.re \cdot \left(x.re + x.im\right)\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \left(x.re - x.im\right) \cdot \color{blue}{\left(\left(x.re + x.im\right) \cdot x.re\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.re} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \color{blue}{\left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)} \cdot x.re - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \color{blue}{\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      16. lower-*.f64 83.2

          \[\leadsto \left(x.re + x.im\right) \cdot \color{blue}{\left(\left(x.re - x.im\right) \cdot x.re\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) - \color{blue}{x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)} \]

      18. lift-*.f64 N/A

          \[\leadsto \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) - x.im \cdot \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right)} \]

      19. associate-*r* N/A

          \[\leadsto \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) - \color{blue}{\left(x.im \cdot x.re\right) \cdot \left(x.im + x.im\right)} \]

      20. *-commutative N/A

          \[\leadsto \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) - \color{blue}{\left(x.re \cdot x.im\right)} \cdot \left(x.im + x.im\right) \]

      21. lift-+.f64 N/A

          \[\leadsto \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) - \left(x.re \cdot x.im\right) \cdot \color{blue}{\left(x.im + x.im\right)} \]

      22. flip-+ N/A

          \[\leadsto \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) - \left(x.re \cdot x.im\right) \cdot \color{blue}{\frac{x.im \cdot x.im - x.im \cdot x.im}{x.im - x.im}} \]

   6. Applied rewrites 80.5%

      \[\leadsto \color{blue}{\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) - \left(x.im + x.im\right)} \]

   7. Taylor expanded in x.re around 0

      \[\leadsto \color{blue}{-2 \cdot x.im} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{x.im \cdot -2} \]

      2. lower-*.f64 3.1

         \[\leadsto \color{blue}{x.im \cdot -2} \]

   9. Applied rewrites 3.1%

      \[\leadsto \color{blue}{x.im \cdot -2} \]
   10. Step-by-step derivation

   11. Applied rewrites 29.6%

       \[\leadsto \left(x.im + x.im\right) \cdot \color{blue}{\left(-x.im\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 23.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 1.5 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(x.im, -2, x.im + x.im\right)\\ \mathbf{elif}\;x.im \leq 2.3 \cdot 10^{+148}:\\ \;\;\;\;x.re \cdot \left(x.im + x.im\right)\\ \mathbf{else}:\\ \;\;\;\;-x.im \cdot \left(x.im + x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 65.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.im \leq 2.3 \cdot 10^{+148}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m\right)\\ \mathbf{else}:\\ \;\;\;\;-x.im \cdot \left(x.im + x.im\right)\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<= x.im 2.3e+148)
    (* x.re_m (* x.re_m x.re_m))
    (- (* x.im (+ x.im x.im))))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_im <= 2.3e+148) {
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	} else {
		tmp = -(x_46_im * (x_46_im + x_46_im));
	}
	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_46im <= 2.3d+148) then
        tmp = x_46re_m * (x_46re_m * x_46re_m)
    else
        tmp = -(x_46im * (x_46im + x_46im))
    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_im <= 2.3e+148) {
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	} else {
		tmp = -(x_46_im * (x_46_im + x_46_im));
	}
	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_im <= 2.3e+148:
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m)
	else:
		tmp = -(x_46_im * (x_46_im + x_46_im))
	return x_46_re_s * tmp

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_im <= 2.3e+148)
		tmp = Float64(x_46_re_m * Float64(x_46_re_m * x_46_re_m));
	else
		tmp = Float64(-Float64(x_46_im * Float64(x_46_im + x_46_im)));
	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_im <= 2.3e+148)
		tmp = x_46_re_m * (x_46_re_m * x_46_re_m);
	else
		tmp = -(x_46_im * (x_46_im + x_46_im));
	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[x$46$im, 2.3e+148], N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision], (-N[(x$46$im * N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision])]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 2.3000000000000001e148

   1. Initial program 89.7%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   2. Add Preprocessing

   3. Taylor expanded in x.re around inf

      \[\leadsto \color{blue}{{x.re}^{3}} \]
   4. Step-by-step derivation

      1. cube-mult N/A

         \[\leadsto \color{blue}{x.re \cdot \left(x.re \cdot x.re\right)} \]

      2. unpow2 N/A

         \[\leadsto x.re \cdot \color{blue}{{x.re}^{2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{x.re \cdot {x.re}^{2}} \]

      4. unpow2 N/A

         \[\leadsto x.re \cdot \color{blue}{\left(x.re \cdot x.re\right)} \]

      5. lower-*.f64 67.8

         \[\leadsto x.re \cdot \color{blue}{\left(x.re \cdot x.re\right)} \]

   5. Applied rewrites 67.8%

      \[\leadsto \color{blue}{x.re \cdot \left(x.re \cdot x.re\right)} \]

   if 2.3000000000000001e148 < x.im

   1. Initial program 48.6%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im} \]

      2. sub-neg N/A

         \[\leadsto \color{blue}{\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re + \left(\mathsf{neg}\left(\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\right)\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im}\right)\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right)} + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x.re \cdot x.im + x.im \cdot x.re, \mathsf{neg}\left(x.im\right), \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x.re \cdot x.im + x.im \cdot x.re}, \mathsf{neg}\left(x.im\right), \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x.re \cdot x.im} + x.im \cdot x.re, \mathsf{neg}\left(x.im\right), \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x.im \cdot x.re} + x.im \cdot x.re, \mathsf{neg}\left(x.im\right), \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x.im \cdot x.re + \color{blue}{x.im \cdot x.re}, \mathsf{neg}\left(x.im\right), \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re\right) \]

      11. distribute-rgt-out N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{x.re \cdot \left(x.im + x.im\right)}, \mathsf{neg}\left(x.im\right), \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{x.re \cdot \left(x.im + x.im\right)}, \mathsf{neg}\left(x.im\right), \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re\right) \]

      13. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(x.re \cdot \color{blue}{\left(x.im + x.im\right)}, \mathsf{neg}\left(x.im\right), \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re\right) \]

      14. lower-neg.f64 48.6

          \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \color{blue}{-x.im}, \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), \color{blue}{\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re}\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), \color{blue}{x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)}\right) \]

      17. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), x.re \cdot \color{blue}{\left(x.re \cdot x.re - x.im \cdot x.im\right)}\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), x.re \cdot \left(\color{blue}{x.re \cdot x.re} - x.im \cdot x.im\right)\right) \]

      19. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), x.re \cdot \left(x.re \cdot x.re - \color{blue}{x.im \cdot x.im}\right)\right) \]

      20. difference-of-squares N/A

          \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), x.re \cdot \color{blue}{\left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)}\right) \]

      21. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), \color{blue}{\left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), \color{blue}{\left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)}\right) \]

   4. Applied rewrites 83.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), -x.im, \left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right) + \left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right) + \left(x.re \cdot \left(x.im + x.im\right)\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right) + \color{blue}{\left(\mathsf{neg}\left(x.im\right)\right) \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)} \]

      4. lift-neg.f64 N/A

         \[\leadsto \left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right) + \color{blue}{\left(\mathsf{neg}\left(x.im\right)\right)} \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      5. cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right) - \color{blue}{x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)} \]

      7. lift--.f64 83.2

         \[\leadsto \color{blue}{\left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \color{blue}{\left(x.re - x.im\right) \cdot \left(x.re \cdot \left(x.re + x.im\right)\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left(x.re - x.im\right) \cdot \color{blue}{\left(x.re \cdot \left(x.re + x.im\right)\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \left(x.re - x.im\right) \cdot \color{blue}{\left(\left(x.re + x.im\right) \cdot x.re\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.re} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \color{blue}{\left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)} \cdot x.re - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \color{blue}{\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      16. lower-*.f64 83.2

          \[\leadsto \left(x.re + x.im\right) \cdot \color{blue}{\left(\left(x.re - x.im\right) \cdot x.re\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

      17. lift-*.f64 N/A

          \[\leadsto \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) - \color{blue}{x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)} \]

      18. lift-*.f64 N/A

          \[\leadsto \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) - x.im \cdot \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right)} \]

      19. associate-*r* N/A

          \[\leadsto \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) - \color{blue}{\left(x.im \cdot x.re\right) \cdot \left(x.im + x.im\right)} \]

      20. *-commutative N/A

          \[\leadsto \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) - \color{blue}{\left(x.re \cdot x.im\right)} \cdot \left(x.im + x.im\right) \]

      21. lift-+.f64 N/A

          \[\leadsto \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) - \left(x.re \cdot x.im\right) \cdot \color{blue}{\left(x.im + x.im\right)} \]

      22. flip-+ N/A

          \[\leadsto \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) - \left(x.re \cdot x.im\right) \cdot \color{blue}{\frac{x.im \cdot x.im - x.im \cdot x.im}{x.im - x.im}} \]

   6. Applied rewrites 80.5%

      \[\leadsto \color{blue}{\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) - \left(x.im + x.im\right)} \]

   7. Taylor expanded in x.re around 0

      \[\leadsto \color{blue}{-2 \cdot x.im} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{x.im \cdot -2} \]

      2. lower-*.f64 3.1

         \[\leadsto \color{blue}{x.im \cdot -2} \]

   9. Applied rewrites 3.1%

      \[\leadsto \color{blue}{x.im \cdot -2} \]
   10. Step-by-step derivation

   11. Applied rewrites 29.6%

       \[\leadsto \left(x.im + x.im\right) \cdot \color{blue}{\left(-x.im\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 2.3 \cdot 10^{+148}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;-x.im \cdot \left(x.im + x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 3.6% accurate, 6.7× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \left(x.im \cdot -2\right) \end{array} \]

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.im -2.0)))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	return x_46_re_s * (x_46_im * -2.0);
}

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_46im * (-2.0d0))
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_im * -2.0);
}

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_im * -2.0)

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	return Float64(x_46_re_s * Float64(x_46_im * -2.0))
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_im * -2.0);
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * N[(x$46$im * -2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.0%

   \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im} \]

   2. sub-neg N/A

      \[\leadsto \color{blue}{\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re + \left(\mathsf{neg}\left(\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\right)\right)} \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\right)\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re} \]

   4. lift-*.f64 N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im}\right)\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re \]

   5. distribute-rgt-neg-in N/A

      \[\leadsto \color{blue}{\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right)} + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re \]

   6. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.re \cdot x.im + x.im \cdot x.re, \mathsf{neg}\left(x.im\right), \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re\right)} \]

   7. lift-+.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{x.re \cdot x.im + x.im \cdot x.re}, \mathsf{neg}\left(x.im\right), \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re\right) \]

   8. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{x.re \cdot x.im} + x.im \cdot x.re, \mathsf{neg}\left(x.im\right), \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{x.im \cdot x.re} + x.im \cdot x.re, \mathsf{neg}\left(x.im\right), \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re\right) \]

   10. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x.im \cdot x.re + \color{blue}{x.im \cdot x.re}, \mathsf{neg}\left(x.im\right), \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re\right) \]

   11. distribute-rgt-out N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{x.re \cdot \left(x.im + x.im\right)}, \mathsf{neg}\left(x.im\right), \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re\right) \]

   12. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{x.re \cdot \left(x.im + x.im\right)}, \mathsf{neg}\left(x.im\right), \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re\right) \]

   13. lower-+.f64 N/A

       \[\leadsto \mathsf{fma}\left(x.re \cdot \color{blue}{\left(x.im + x.im\right)}, \mathsf{neg}\left(x.im\right), \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re\right) \]

   14. lower-neg.f64 83.7

       \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \color{blue}{-x.im}, \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re\right) \]

   15. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), \color{blue}{\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re}\right) \]

   16. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), \color{blue}{x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)}\right) \]

   17. lift--.f64 N/A

       \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), x.re \cdot \color{blue}{\left(x.re \cdot x.re - x.im \cdot x.im\right)}\right) \]

   18. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), x.re \cdot \left(\color{blue}{x.re \cdot x.re} - x.im \cdot x.im\right)\right) \]

   19. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), x.re \cdot \left(x.re \cdot x.re - \color{blue}{x.im \cdot x.im}\right)\right) \]

   20. difference-of-squares N/A

       \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), x.re \cdot \color{blue}{\left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)}\right) \]

   21. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), \color{blue}{\left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)}\right) \]

   22. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), \color{blue}{\left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)}\right) \]

4. Applied rewrites 93.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), -x.im, \left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)\right)} \]
5. Step-by-step derivation

   1. lift-fma.f64 N/A

      \[\leadsto \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right) + \left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right) + \left(x.re \cdot \left(x.im + x.im\right)\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right)} \]

   3. *-commutative N/A

      \[\leadsto \left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right) + \color{blue}{\left(\mathsf{neg}\left(x.im\right)\right) \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)} \]

   4. lift-neg.f64 N/A

      \[\leadsto \left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right) + \color{blue}{\left(\mathsf{neg}\left(x.im\right)\right)} \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

   5. cancel-sign-sub-inv N/A

      \[\leadsto \color{blue}{\left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)} \]

   6. lift-*.f64 N/A

      \[\leadsto \left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right) - \color{blue}{x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)} \]

   7. lift--.f64 92.7

      \[\leadsto \color{blue}{\left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)} \]

   8. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \color{blue}{\left(x.re - x.im\right) \cdot \left(x.re \cdot \left(x.re + x.im\right)\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

   10. lift-*.f64 N/A

       \[\leadsto \left(x.re - x.im\right) \cdot \color{blue}{\left(x.re \cdot \left(x.re + x.im\right)\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \left(x.re - x.im\right) \cdot \color{blue}{\left(\left(x.re + x.im\right) \cdot x.re\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

   12. associate-*r* N/A

       \[\leadsto \color{blue}{\left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.re} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \color{blue}{\left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)} \cdot x.re - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

   14. associate-*l* N/A

       \[\leadsto \color{blue}{\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

   15. lower-*.f64 N/A

       \[\leadsto \color{blue}{\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

   16. lower-*.f64 92.7

       \[\leadsto \left(x.re + x.im\right) \cdot \color{blue}{\left(\left(x.re - x.im\right) \cdot x.re\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

   17. lift-*.f64 N/A

       \[\leadsto \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) - \color{blue}{x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)} \]

   18. lift-*.f64 N/A

       \[\leadsto \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) - x.im \cdot \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right)} \]

   19. associate-*r* N/A

       \[\leadsto \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) - \color{blue}{\left(x.im \cdot x.re\right) \cdot \left(x.im + x.im\right)} \]

   20. *-commutative N/A

       \[\leadsto \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) - \color{blue}{\left(x.re \cdot x.im\right)} \cdot \left(x.im + x.im\right) \]

   21. lift-+.f64 N/A

       \[\leadsto \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) - \left(x.re \cdot x.im\right) \cdot \color{blue}{\left(x.im + x.im\right)} \]

   22. flip-+ N/A

       \[\leadsto \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) - \left(x.re \cdot x.im\right) \cdot \color{blue}{\frac{x.im \cdot x.im - x.im \cdot x.im}{x.im - x.im}} \]

6. Applied rewrites 60.0%

   \[\leadsto \color{blue}{\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) - \left(x.im + x.im\right)} \]

7. Taylor expanded in x.re around 0

   \[\leadsto \color{blue}{-2 \cdot x.im} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{x.im \cdot -2} \]

   2. lower-*.f64 3.7

      \[\leadsto \color{blue}{x.im \cdot -2} \]

9. Applied rewrites 3.7%

   \[\leadsto \color{blue}{x.im \cdot -2} \]
10. Add Preprocessing

Alternative 14: 3.6% accurate, 10.0× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \left(x.im + x.im\right) \end{array} \]

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.im x.im)))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	return x_46_re_s * (x_46_im + x_46_im);
}

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_46im + x_46im)
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_im + x_46_im);
}

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_im + x_46_im)

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	return Float64(x_46_re_s * Float64(x_46_im + x_46_im))
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_im + x_46_im);
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.0%

   \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im} \]

   2. sub-neg N/A

      \[\leadsto \color{blue}{\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re + \left(\mathsf{neg}\left(\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\right)\right)} \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\right)\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re} \]

   4. lift-*.f64 N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im}\right)\right) + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re \]

   5. distribute-rgt-neg-in N/A

      \[\leadsto \color{blue}{\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right)} + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re \]

   6. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.re \cdot x.im + x.im \cdot x.re, \mathsf{neg}\left(x.im\right), \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re\right)} \]

   7. lift-+.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{x.re \cdot x.im + x.im \cdot x.re}, \mathsf{neg}\left(x.im\right), \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re\right) \]

   8. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{x.re \cdot x.im} + x.im \cdot x.re, \mathsf{neg}\left(x.im\right), \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{x.im \cdot x.re} + x.im \cdot x.re, \mathsf{neg}\left(x.im\right), \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re\right) \]

   10. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x.im \cdot x.re + \color{blue}{x.im \cdot x.re}, \mathsf{neg}\left(x.im\right), \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re\right) \]

   11. distribute-rgt-out N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{x.re \cdot \left(x.im + x.im\right)}, \mathsf{neg}\left(x.im\right), \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re\right) \]

   12. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{x.re \cdot \left(x.im + x.im\right)}, \mathsf{neg}\left(x.im\right), \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re\right) \]

   13. lower-+.f64 N/A

       \[\leadsto \mathsf{fma}\left(x.re \cdot \color{blue}{\left(x.im + x.im\right)}, \mathsf{neg}\left(x.im\right), \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re\right) \]

   14. lower-neg.f64 83.7

       \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \color{blue}{-x.im}, \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re\right) \]

   15. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), \color{blue}{\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re}\right) \]

   16. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), \color{blue}{x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)}\right) \]

   17. lift--.f64 N/A

       \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), x.re \cdot \color{blue}{\left(x.re \cdot x.re - x.im \cdot x.im\right)}\right) \]

   18. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), x.re \cdot \left(\color{blue}{x.re \cdot x.re} - x.im \cdot x.im\right)\right) \]

   19. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), x.re \cdot \left(x.re \cdot x.re - \color{blue}{x.im \cdot x.im}\right)\right) \]

   20. difference-of-squares N/A

       \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), x.re \cdot \color{blue}{\left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)}\right) \]

   21. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), \color{blue}{\left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)}\right) \]

   22. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), \mathsf{neg}\left(x.im\right), \color{blue}{\left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)}\right) \]

4. Applied rewrites 93.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), -x.im, \left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)\right)} \]
5. Step-by-step derivation

   1. lift-fma.f64 N/A

      \[\leadsto \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right) + \left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right) + \left(x.re \cdot \left(x.im + x.im\right)\right) \cdot \left(\mathsf{neg}\left(x.im\right)\right)} \]

   3. *-commutative N/A

      \[\leadsto \left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right) + \color{blue}{\left(\mathsf{neg}\left(x.im\right)\right) \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)} \]

   4. lift-neg.f64 N/A

      \[\leadsto \left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right) + \color{blue}{\left(\mathsf{neg}\left(x.im\right)\right)} \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

   5. cancel-sign-sub-inv N/A

      \[\leadsto \color{blue}{\left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)} \]

   6. lift-*.f64 N/A

      \[\leadsto \left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right) - \color{blue}{x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)} \]

   7. lift--.f64 92.7

      \[\leadsto \color{blue}{\left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)} \]

   8. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(x.re \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \color{blue}{\left(x.re - x.im\right) \cdot \left(x.re \cdot \left(x.re + x.im\right)\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

   10. lift-*.f64 N/A

       \[\leadsto \left(x.re - x.im\right) \cdot \color{blue}{\left(x.re \cdot \left(x.re + x.im\right)\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \left(x.re - x.im\right) \cdot \color{blue}{\left(\left(x.re + x.im\right) \cdot x.re\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

   12. associate-*r* N/A

       \[\leadsto \color{blue}{\left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.re} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \color{blue}{\left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)} \cdot x.re - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

   14. associate-*l* N/A

       \[\leadsto \color{blue}{\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

   15. lower-*.f64 N/A

       \[\leadsto \color{blue}{\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

   16. lower-*.f64 92.7

       \[\leadsto \left(x.re + x.im\right) \cdot \color{blue}{\left(\left(x.re - x.im\right) \cdot x.re\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

   17. lift-*.f64 N/A

       \[\leadsto \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) - \color{blue}{x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)} \]

   18. lift-*.f64 N/A

       \[\leadsto \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) - x.im \cdot \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right)} \]

   19. associate-*r* N/A

       \[\leadsto \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) - \color{blue}{\left(x.im \cdot x.re\right) \cdot \left(x.im + x.im\right)} \]

   20. *-commutative N/A

       \[\leadsto \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) - \color{blue}{\left(x.re \cdot x.im\right)} \cdot \left(x.im + x.im\right) \]

   21. lift-+.f64 N/A

       \[\leadsto \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) - \left(x.re \cdot x.im\right) \cdot \color{blue}{\left(x.im + x.im\right)} \]

   22. flip-+ N/A

       \[\leadsto \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) - \left(x.re \cdot x.im\right) \cdot \color{blue}{\frac{x.im \cdot x.im - x.im \cdot x.im}{x.im - x.im}} \]

6. Applied rewrites 60.0%

   \[\leadsto \color{blue}{\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) - \left(x.im + x.im\right)} \]

7. Taylor expanded in x.re around 0

   \[\leadsto \color{blue}{-2 \cdot x.im} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{x.im \cdot -2} \]

   2. lower-*.f64 3.7

      \[\leadsto \color{blue}{x.im \cdot -2} \]

9. Applied rewrites 3.7%

   \[\leadsto \color{blue}{x.im \cdot -2} \]
10. Step-by-step derivation

11. Applied rewrites 3.8%

    \[\leadsto \color{blue}{x.im + x.im} \]
12. Add Preprocessing

Developer Target 1: 87.4% 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 2024233 
(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)))