math.cube on complex, real part

Percentage Accurate: 82.4% → 98.3%

Time: 1.8s

Alternatives: 6

Speedup: 1.8×

• 55.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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)
use fmin_fmax_functions
    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: 82.4% accurate, 1.0× speedup

Math

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

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)
use fmin_fmax_functions
    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: 98.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \left|x.re\right| \cdot \left|x.re\right|\\ \mathsf{copysign}\left(1, x.re\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x.re\right| \leq 2.5 \cdot 10^{-104}:\\ \;\;\;\;\mathsf{fma}\left(\left(x.im \cdot \left|x.re\right|\right) \cdot -3, x.im, t\_0 \cdot \left|x.re\right|\right)\\ \mathbf{elif}\;\left|x.re\right| \leq 5.8 \cdot 10^{+222}:\\ \;\;\;\;\left|x.re\right| \cdot \mathsf{fma}\left(\left|x.re\right|, \left|x.re\right|, -3 \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|x.re\right| \cdot \mathsf{fma}\left(-3 \cdot x.im, x.im, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (let* ((t_0 (* (fabs x.re) (fabs x.re))))
  (*
   (copysign 1.0 x.re)
   (if (<= (fabs x.re) 2.5e-104)
     (fma (* (* x.im (fabs x.re)) -3.0) x.im (* t_0 (fabs x.re)))
     (if (<= (fabs x.re) 5.8e+222)
       (*
        (fabs x.re)
        (fma (fabs x.re) (fabs x.re) (* -3.0 (* x.im x.im))))
       (* (fabs x.re) (fma (* -3.0 x.im) x.im t_0)))))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = fabs(x_46_re) * fabs(x_46_re);
	double tmp;
	if (fabs(x_46_re) <= 2.5e-104) {
		tmp = fma(((x_46_im * fabs(x_46_re)) * -3.0), x_46_im, (t_0 * fabs(x_46_re)));
	} else if (fabs(x_46_re) <= 5.8e+222) {
		tmp = fabs(x_46_re) * fma(fabs(x_46_re), fabs(x_46_re), (-3.0 * (x_46_im * x_46_im)));
	} else {
		tmp = fabs(x_46_re) * fma((-3.0 * x_46_im), x_46_im, t_0);
	}
	return copysign(1.0, x_46_re) * tmp;
}

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(abs(x_46_re) * abs(x_46_re))
	tmp = 0.0
	if (abs(x_46_re) <= 2.5e-104)
		tmp = fma(Float64(Float64(x_46_im * abs(x_46_re)) * -3.0), x_46_im, Float64(t_0 * abs(x_46_re)));
	elseif (abs(x_46_re) <= 5.8e+222)
		tmp = Float64(abs(x_46_re) * fma(abs(x_46_re), abs(x_46_re), Float64(-3.0 * Float64(x_46_im * x_46_im))));
	else
		tmp = Float64(abs(x_46_re) * fma(Float64(-3.0 * x_46_im), x_46_im, t_0));
	end
	return Float64(copysign(1.0, x_46_re) * tmp)
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(N[Abs[x$46$re], $MachinePrecision] * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x$46$re], $MachinePrecision], 2.5e-104], N[(N[(N[(x$46$im * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision] * -3.0), $MachinePrecision] * x$46$im + N[(t$95$0 * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[x$46$re], $MachinePrecision], 5.8e+222], N[(N[Abs[x$46$re], $MachinePrecision] * N[(N[Abs[x$46$re], $MachinePrecision] * N[Abs[x$46$re], $MachinePrecision] + N[(-3.0 * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[x$46$re], $MachinePrecision] * N[(N[(-3.0 * x$46$im), $MachinePrecision] * x$46$im + t$95$0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if x.re < 2.4999999999999999e-104

   1. Initial program 82.4%

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

   2. Taylor expanded in x.im around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 85.3%

         \[\leadsto \mathsf{fma}\left({x.im}^{2}, -1 \cdot x.re - 2 \cdot x.re, {x.re}^{3}\right) \]

   4. Applied rewrites 85.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x.im}^{2}, -1 \cdot x.re - 2 \cdot x.re, {x.re}^{3}\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 80.1%

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

      13. lift-pow.f64 N/A

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

      14. unpow3 N/A

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

      15. lift-*.f64 N/A

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

      16. lower-*.f64 80.0%

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

   6. Applied rewrites 80.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 88.1%

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

   8. Applied rewrites 88.1%

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

   if 2.4999999999999999e-104 < x.re < 5.7999999999999996e222

   1. Initial program 82.4%

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

   2. Taylor expanded in x.im around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 85.3%

         \[\leadsto \mathsf{fma}\left({x.im}^{2}, -1 \cdot x.re - 2 \cdot x.re, {x.re}^{3}\right) \]

   4. Applied rewrites 85.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x.im}^{2}, -1 \cdot x.re - 2 \cdot x.re, {x.re}^{3}\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 80.1%

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

      13. lift-pow.f64 N/A

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

      14. unpow3 N/A

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

      15. lift-*.f64 N/A

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

      16. lower-*.f64 80.0%

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

   6. Applied rewrites 80.0%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-out N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 90.8%

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

   8. Applied rewrites 90.8%

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

   if 5.7999999999999996e222 < x.re

   1. Initial program 82.4%

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

   2. Taylor expanded in x.im around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 85.3%

         \[\leadsto \mathsf{fma}\left({x.im}^{2}, -1 \cdot x.re - 2 \cdot x.re, {x.re}^{3}\right) \]

   4. Applied rewrites 85.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x.im}^{2}, -1 \cdot x.re - 2 \cdot x.re, {x.re}^{3}\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 80.1%

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

      13. lift-pow.f64 N/A

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

      14. unpow3 N/A

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

      15. lift-*.f64 N/A

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

      16. lower-*.f64 80.0%

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

   6. Applied rewrites 80.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-rgt-out N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 87.7%

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

   8. Applied rewrites 87.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

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

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

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

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

      12. sqr-neg-rev N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 90.7%

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

   10. Applied rewrites 90.7%

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

Alternative 2: 98.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \left|x.re\right| \cdot \left|x.re\right|\\ \mathsf{copysign}\left(1, x.re\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x.re\right| \leq 5 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(\left(-3 \cdot x.im\right) \cdot \left|x.re\right|, x.im, t\_0 \cdot \left|x.re\right|\right)\\ \mathbf{elif}\;\left|x.re\right| \leq 5.8 \cdot 10^{+222}:\\ \;\;\;\;\left|x.re\right| \cdot \mathsf{fma}\left(\left|x.re\right|, \left|x.re\right|, -3 \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|x.re\right| \cdot \mathsf{fma}\left(-3 \cdot x.im, x.im, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (let* ((t_0 (* (fabs x.re) (fabs x.re))))
  (*
   (copysign 1.0 x.re)
   (if (<= (fabs x.re) 5e+19)
     (fma (* (* -3.0 x.im) (fabs x.re)) x.im (* t_0 (fabs x.re)))
     (if (<= (fabs x.re) 5.8e+222)
       (*
        (fabs x.re)
        (fma (fabs x.re) (fabs x.re) (* -3.0 (* x.im x.im))))
       (* (fabs x.re) (fma (* -3.0 x.im) x.im t_0)))))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = fabs(x_46_re) * fabs(x_46_re);
	double tmp;
	if (fabs(x_46_re) <= 5e+19) {
		tmp = fma(((-3.0 * x_46_im) * fabs(x_46_re)), x_46_im, (t_0 * fabs(x_46_re)));
	} else if (fabs(x_46_re) <= 5.8e+222) {
		tmp = fabs(x_46_re) * fma(fabs(x_46_re), fabs(x_46_re), (-3.0 * (x_46_im * x_46_im)));
	} else {
		tmp = fabs(x_46_re) * fma((-3.0 * x_46_im), x_46_im, t_0);
	}
	return copysign(1.0, x_46_re) * tmp;
}

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(abs(x_46_re) * abs(x_46_re))
	tmp = 0.0
	if (abs(x_46_re) <= 5e+19)
		tmp = fma(Float64(Float64(-3.0 * x_46_im) * abs(x_46_re)), x_46_im, Float64(t_0 * abs(x_46_re)));
	elseif (abs(x_46_re) <= 5.8e+222)
		tmp = Float64(abs(x_46_re) * fma(abs(x_46_re), abs(x_46_re), Float64(-3.0 * Float64(x_46_im * x_46_im))));
	else
		tmp = Float64(abs(x_46_re) * fma(Float64(-3.0 * x_46_im), x_46_im, t_0));
	end
	return Float64(copysign(1.0, x_46_re) * tmp)
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(N[Abs[x$46$re], $MachinePrecision] * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x$46$re], $MachinePrecision], 5e+19], N[(N[(N[(-3.0 * x$46$im), $MachinePrecision] * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision] * x$46$im + N[(t$95$0 * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[x$46$re], $MachinePrecision], 5.8e+222], N[(N[Abs[x$46$re], $MachinePrecision] * N[(N[Abs[x$46$re], $MachinePrecision] * N[Abs[x$46$re], $MachinePrecision] + N[(-3.0 * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[x$46$re], $MachinePrecision] * N[(N[(-3.0 * x$46$im), $MachinePrecision] * x$46$im + t$95$0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if x.re < 5e19

   1. Initial program 82.4%

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

   2. Taylor expanded in x.im around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 85.3%

         \[\leadsto \mathsf{fma}\left({x.im}^{2}, -1 \cdot x.re - 2 \cdot x.re, {x.re}^{3}\right) \]

   4. Applied rewrites 85.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x.im}^{2}, -1 \cdot x.re - 2 \cdot x.re, {x.re}^{3}\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 80.1%

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

      13. lift-pow.f64 N/A

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

      14. unpow3 N/A

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

      15. lift-*.f64 N/A

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

      16. lower-*.f64 80.0%

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

   6. Applied rewrites 80.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-rgt-out N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 87.7%

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

   8. Applied rewrites 87.7%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 88.1%

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

   10. Applied rewrites 88.1%

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

   if 5e19 < x.re < 5.7999999999999996e222

   1. Initial program 82.4%

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

   2. Taylor expanded in x.im around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 85.3%

         \[\leadsto \mathsf{fma}\left({x.im}^{2}, -1 \cdot x.re - 2 \cdot x.re, {x.re}^{3}\right) \]

   4. Applied rewrites 85.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x.im}^{2}, -1 \cdot x.re - 2 \cdot x.re, {x.re}^{3}\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 80.1%

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

      13. lift-pow.f64 N/A

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

      14. unpow3 N/A

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

      15. lift-*.f64 N/A

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

      16. lower-*.f64 80.0%

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

   6. Applied rewrites 80.0%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-out N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 90.8%

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

   8. Applied rewrites 90.8%

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

   if 5.7999999999999996e222 < x.re

   1. Initial program 82.4%

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

   2. Taylor expanded in x.im around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 85.3%

         \[\leadsto \mathsf{fma}\left({x.im}^{2}, -1 \cdot x.re - 2 \cdot x.re, {x.re}^{3}\right) \]

   4. Applied rewrites 85.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x.im}^{2}, -1 \cdot x.re - 2 \cdot x.re, {x.re}^{3}\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 80.1%

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

      13. lift-pow.f64 N/A

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

      14. unpow3 N/A

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

      15. lift-*.f64 N/A

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

      16. lower-*.f64 80.0%

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

   6. Applied rewrites 80.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-rgt-out N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 87.7%

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

   8. Applied rewrites 87.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

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

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

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

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

      12. sqr-neg-rev N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 90.7%

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

   10. Applied rewrites 90.7%

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

Alternative 3: 97.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \left|x.re\right| \cdot \left|x.re\right|\\ \mathsf{copysign}\left(1, x.re\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x.re\right| \leq 5 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(-3 \cdot x.im, x.im \cdot \left|x.re\right|, t\_0 \cdot \left|x.re\right|\right)\\ \mathbf{elif}\;\left|x.re\right| \leq 5.8 \cdot 10^{+222}:\\ \;\;\;\;\left|x.re\right| \cdot \mathsf{fma}\left(\left|x.re\right|, \left|x.re\right|, -3 \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|x.re\right| \cdot \mathsf{fma}\left(-3 \cdot x.im, x.im, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (let* ((t_0 (* (fabs x.re) (fabs x.re))))
  (*
   (copysign 1.0 x.re)
   (if (<= (fabs x.re) 5e+19)
     (fma (* -3.0 x.im) (* x.im (fabs x.re)) (* t_0 (fabs x.re)))
     (if (<= (fabs x.re) 5.8e+222)
       (*
        (fabs x.re)
        (fma (fabs x.re) (fabs x.re) (* -3.0 (* x.im x.im))))
       (* (fabs x.re) (fma (* -3.0 x.im) x.im t_0)))))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = fabs(x_46_re) * fabs(x_46_re);
	double tmp;
	if (fabs(x_46_re) <= 5e+19) {
		tmp = fma((-3.0 * x_46_im), (x_46_im * fabs(x_46_re)), (t_0 * fabs(x_46_re)));
	} else if (fabs(x_46_re) <= 5.8e+222) {
		tmp = fabs(x_46_re) * fma(fabs(x_46_re), fabs(x_46_re), (-3.0 * (x_46_im * x_46_im)));
	} else {
		tmp = fabs(x_46_re) * fma((-3.0 * x_46_im), x_46_im, t_0);
	}
	return copysign(1.0, x_46_re) * tmp;
}

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(abs(x_46_re) * abs(x_46_re))
	tmp = 0.0
	if (abs(x_46_re) <= 5e+19)
		tmp = fma(Float64(-3.0 * x_46_im), Float64(x_46_im * abs(x_46_re)), Float64(t_0 * abs(x_46_re)));
	elseif (abs(x_46_re) <= 5.8e+222)
		tmp = Float64(abs(x_46_re) * fma(abs(x_46_re), abs(x_46_re), Float64(-3.0 * Float64(x_46_im * x_46_im))));
	else
		tmp = Float64(abs(x_46_re) * fma(Float64(-3.0 * x_46_im), x_46_im, t_0));
	end
	return Float64(copysign(1.0, x_46_re) * tmp)
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(N[Abs[x$46$re], $MachinePrecision] * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x$46$re], $MachinePrecision], 5e+19], N[(N[(-3.0 * x$46$im), $MachinePrecision] * N[(x$46$im * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[x$46$re], $MachinePrecision], 5.8e+222], N[(N[Abs[x$46$re], $MachinePrecision] * N[(N[Abs[x$46$re], $MachinePrecision] * N[Abs[x$46$re], $MachinePrecision] + N[(-3.0 * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[x$46$re], $MachinePrecision] * N[(N[(-3.0 * x$46$im), $MachinePrecision] * x$46$im + t$95$0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if x.re < 5e19

   1. Initial program 82.4%

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

   2. Taylor expanded in x.im around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 85.3%

         \[\leadsto \mathsf{fma}\left({x.im}^{2}, -1 \cdot x.re - 2 \cdot x.re, {x.re}^{3}\right) \]

   4. Applied rewrites 85.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x.im}^{2}, -1 \cdot x.re - 2 \cdot x.re, {x.re}^{3}\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 80.1%

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

      13. lift-pow.f64 N/A

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

      14. unpow3 N/A

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

      15. lift-*.f64 N/A

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

      16. lower-*.f64 80.0%

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

   6. Applied rewrites 80.0%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 88.1%

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

   8. Applied rewrites 88.1%

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

   if 5e19 < x.re < 5.7999999999999996e222

   1. Initial program 82.4%

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

   2. Taylor expanded in x.im around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 85.3%

         \[\leadsto \mathsf{fma}\left({x.im}^{2}, -1 \cdot x.re - 2 \cdot x.re, {x.re}^{3}\right) \]

   4. Applied rewrites 85.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x.im}^{2}, -1 \cdot x.re - 2 \cdot x.re, {x.re}^{3}\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 80.1%

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

      13. lift-pow.f64 N/A

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

      14. unpow3 N/A

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

      15. lift-*.f64 N/A

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

      16. lower-*.f64 80.0%

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

   6. Applied rewrites 80.0%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-out N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 90.8%

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

   8. Applied rewrites 90.8%

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

   if 5.7999999999999996e222 < x.re

   1. Initial program 82.4%

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

   2. Taylor expanded in x.im around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 85.3%

         \[\leadsto \mathsf{fma}\left({x.im}^{2}, -1 \cdot x.re - 2 \cdot x.re, {x.re}^{3}\right) \]

   4. Applied rewrites 85.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x.im}^{2}, -1 \cdot x.re - 2 \cdot x.re, {x.re}^{3}\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 80.1%

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

      13. lift-pow.f64 N/A

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

      14. unpow3 N/A

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

      15. lift-*.f64 N/A

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

      16. lower-*.f64 80.0%

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

   6. Applied rewrites 80.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-rgt-out N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 87.7%

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

   8. Applied rewrites 87.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

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

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

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

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

      12. sqr-neg-rev N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 90.7%

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

   10. Applied rewrites 90.7%

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

Alternative 4: 92.4% accurate, 0.9× speedup

Math

\[\mathsf{copysign}\left(1, x.re\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x.re\right| \leq 5.8 \cdot 10^{+222}:\\ \;\;\;\;\left|x.re\right| \cdot \mathsf{fma}\left(\left|x.re\right|, \left|x.re\right|, -3 \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|x.re\right| \cdot \mathsf{fma}\left(-3 \cdot x.im, x.im, \left|x.re\right| \cdot \left|x.re\right|\right)\\ \end{array} \]

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (*
 (copysign 1.0 x.re)
 (if (<= (fabs x.re) 5.8e+222)
   (*
    (fabs x.re)
    (fma (fabs x.re) (fabs x.re) (* -3.0 (* x.im x.im))))
   (*
    (fabs x.re)
    (fma (* -3.0 x.im) x.im (* (fabs x.re) (fabs x.re)))))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (fabs(x_46_re) <= 5.8e+222) {
		tmp = fabs(x_46_re) * fma(fabs(x_46_re), fabs(x_46_re), (-3.0 * (x_46_im * x_46_im)));
	} else {
		tmp = fabs(x_46_re) * fma((-3.0 * x_46_im), x_46_im, (fabs(x_46_re) * fabs(x_46_re)));
	}
	return copysign(1.0, x_46_re) * tmp;
}

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (abs(x_46_re) <= 5.8e+222)
		tmp = Float64(abs(x_46_re) * fma(abs(x_46_re), abs(x_46_re), Float64(-3.0 * Float64(x_46_im * x_46_im))));
	else
		tmp = Float64(abs(x_46_re) * fma(Float64(-3.0 * x_46_im), x_46_im, Float64(abs(x_46_re) * abs(x_46_re))));
	end
	return Float64(copysign(1.0, x_46_re) * tmp)
end

Wolfram

code[x$46$re_, x$46$im_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x$46$re], $MachinePrecision], 5.8e+222], N[(N[Abs[x$46$re], $MachinePrecision] * N[(N[Abs[x$46$re], $MachinePrecision] * N[Abs[x$46$re], $MachinePrecision] + N[(-3.0 * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[x$46$re], $MachinePrecision] * N[(N[(-3.0 * x$46$im), $MachinePrecision] * x$46$im + N[(N[Abs[x$46$re], $MachinePrecision] * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 5.7999999999999996e222

   1. Initial program 82.4%

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

   2. Taylor expanded in x.im around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 85.3%

         \[\leadsto \mathsf{fma}\left({x.im}^{2}, -1 \cdot x.re - 2 \cdot x.re, {x.re}^{3}\right) \]

   4. Applied rewrites 85.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x.im}^{2}, -1 \cdot x.re - 2 \cdot x.re, {x.re}^{3}\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 80.1%

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

      13. lift-pow.f64 N/A

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

      14. unpow3 N/A

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

      15. lift-*.f64 N/A

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

      16. lower-*.f64 80.0%

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

   6. Applied rewrites 80.0%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-out N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 90.8%

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

   8. Applied rewrites 90.8%

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

   if 5.7999999999999996e222 < x.re

   1. Initial program 82.4%

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

   2. Taylor expanded in x.im around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 85.3%

         \[\leadsto \mathsf{fma}\left({x.im}^{2}, -1 \cdot x.re - 2 \cdot x.re, {x.re}^{3}\right) \]

   4. Applied rewrites 85.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x.im}^{2}, -1 \cdot x.re - 2 \cdot x.re, {x.re}^{3}\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 80.1%

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

      13. lift-pow.f64 N/A

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

      14. unpow3 N/A

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

      15. lift-*.f64 N/A

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

      16. lower-*.f64 80.0%

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

   6. Applied rewrites 80.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-rgt-out N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 87.7%

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

   8. Applied rewrites 87.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

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

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

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

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

      12. sqr-neg-rev N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 90.7%

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

   10. Applied rewrites 90.7%

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

Alternative 5: 90.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 82.4%

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

2. Taylor expanded in x.im around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-pow.f64 85.3%

      \[\leadsto \mathsf{fma}\left({x.im}^{2}, -1 \cdot x.re - 2 \cdot x.re, {x.re}^{3}\right) \]

4. Applied rewrites 85.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x.im}^{2}, -1 \cdot x.re - 2 \cdot x.re, {x.re}^{3}\right)} \]
5. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

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

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

   9. associate-*r* N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. metadata-eval 80.1%

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

   13. lift-pow.f64 N/A

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

   14. unpow3 N/A

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

   15. lift-*.f64 N/A

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

   16. lower-*.f64 80.0%

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

6. Applied rewrites 80.0%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. distribute-rgt-out N/A

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

   9. lower-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lower-fma.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 90.8%

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

8. Applied rewrites 90.8%

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

Alternative 6: 87.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 82.4%

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

2. Taylor expanded in x.im around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-pow.f64 85.3%

      \[\leadsto \mathsf{fma}\left({x.im}^{2}, -1 \cdot x.re - 2 \cdot x.re, {x.re}^{3}\right) \]

4. Applied rewrites 85.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x.im}^{2}, -1 \cdot x.re - 2 \cdot x.re, {x.re}^{3}\right)} \]
5. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

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

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

   9. associate-*r* N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. metadata-eval 80.1%

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

   13. lift-pow.f64 N/A

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

   14. unpow3 N/A

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

   15. lift-*.f64 N/A

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

   16. lower-*.f64 80.0%

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

6. Applied rewrites 80.0%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. distribute-rgt-out N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 87.7%

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

8. Applied rewrites 87.7%

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

Developer Target 1: 87.7% accurate, 1.1× speedup

Math

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

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)
use fmin_fmax_functions
    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 2025313 -o setup:search
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64

  :alt
  (! :herbie-platform c (+ (* (* 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)))