math.cube on complex, imaginary part

Percentage Accurate: 82.4% → 99.7%

Time: 2.9s

Alternatives: 14

Speedup: 0.8×

• 55.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x_46re, x_46im)
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_46im) + (((x_46re * x_46im) + (x_46im * x_46re)) * x_46re)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 82.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x_46re, x_46im)
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_46im) + (((x_46re * x_46im) + (x_46im * x_46re)) * x_46re)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.4× speedup

Math

\[\mathsf{copysign}\left(1, x.im\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x.im\right| \leq 2 \cdot 10^{+81}:\\ \;\;\;\;\mathsf{fma}\left(-1, {\left(\left|x.im\right|\right)}^{3}, x.re \cdot \mathsf{fma}\left(\left|x.im\right|, \left|x.im\right| + -1 \cdot \left|x.im\right|, x.re \cdot \left(\left|x.im\right| + 2 \cdot \left|x.im\right|\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - \left|x.im\right|, \left|x.im\right| \cdot \left(\left|x.im\right| + x.re\right), \left|x.im\right| + \left|x.im\right|\right)\\ \end{array} \]

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (*
 (copysign 1.0 x.im)
 (if (<= (fabs x.im) 2e+81)
   (fma
    -1.0
    (pow (fabs x.im) 3.0)
    (*
     x.re
     (fma
      (fabs x.im)
      (+ (fabs x.im) (* -1.0 (fabs x.im)))
      (* x.re (+ (fabs x.im) (* 2.0 (fabs x.im)))))))
   (fma
    (- x.re (fabs x.im))
    (* (fabs x.im) (+ (fabs x.im) x.re))
    (+ (fabs x.im) (fabs x.im))))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (fabs(x_46_im) <= 2e+81) {
		tmp = fma(-1.0, pow(fabs(x_46_im), 3.0), (x_46_re * fma(fabs(x_46_im), (fabs(x_46_im) + (-1.0 * fabs(x_46_im))), (x_46_re * (fabs(x_46_im) + (2.0 * fabs(x_46_im)))))));
	} else {
		tmp = fma((x_46_re - fabs(x_46_im)), (fabs(x_46_im) * (fabs(x_46_im) + x_46_re)), (fabs(x_46_im) + fabs(x_46_im)));
	}
	return copysign(1.0, x_46_im) * tmp;
}

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (abs(x_46_im) <= 2e+81)
		tmp = fma(-1.0, (abs(x_46_im) ^ 3.0), Float64(x_46_re * fma(abs(x_46_im), Float64(abs(x_46_im) + Float64(-1.0 * abs(x_46_im))), Float64(x_46_re * Float64(abs(x_46_im) + Float64(2.0 * abs(x_46_im)))))));
	else
		tmp = fma(Float64(x_46_re - abs(x_46_im)), Float64(abs(x_46_im) * Float64(abs(x_46_im) + x_46_re)), Float64(abs(x_46_im) + abs(x_46_im)));
	end
	return Float64(copysign(1.0, x_46_im) * tmp)
end

Wolfram

code[x$46$re_, x$46$im_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x$46$im], $MachinePrecision], 2e+81], N[(-1.0 * N[Power[N[Abs[x$46$im], $MachinePrecision], 3.0], $MachinePrecision] + N[(x$46$re * N[(N[Abs[x$46$im], $MachinePrecision] * N[(N[Abs[x$46$im], $MachinePrecision] + N[(-1.0 * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(N[Abs[x$46$im], $MachinePrecision] + N[(2.0 * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re - N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[x$46$im], $MachinePrecision] * N[(N[Abs[x$46$im], $MachinePrecision] + x$46$re), $MachinePrecision]), $MachinePrecision] + N[(N[Abs[x$46$im], $MachinePrecision] + N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.9999999999999998e81

   1. Initial program 82.4%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. difference-of-squares N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 91.4%

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

   3. Applied rewrites 91.4%

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

   4. Taylor expanded in x.re around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 86.0%

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

   6. Applied rewrites 86.0%

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

   if 1.9999999999999998e81 < x.im

   1. Initial program 82.4%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. difference-of-squares N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 91.4%

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

   3. Applied rewrites 91.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-+.f64 N/A

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

      7. distribute-rgt-out N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. flip-+ N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. +-inverses N/A

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

      16. +-inverses N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 N/A

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

   5. Applied rewrites 58.7%

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

Alternative 2: 99.7% accurate, 0.8× speedup

Math

\[\mathsf{copysign}\left(1, x.im\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x.im\right| \leq 2 \cdot 10^{+81}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(3 \cdot \left|x.im\right|, x.re, 0\right), x.re, \left(\left|x.im\right| \cdot \left|x.im\right|\right) \cdot \left(-\left|x.im\right|\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - \left|x.im\right|, \left|x.im\right| \cdot \left(\left|x.im\right| + x.re\right), \left|x.im\right| + \left|x.im\right|\right)\\ \end{array} \]

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (*
 (copysign 1.0 x.im)
 (if (<= (fabs x.im) 2e+81)
   (fma
    (fma (* 3.0 (fabs x.im)) x.re 0.0)
    x.re
    (* (* (fabs x.im) (fabs x.im)) (- (fabs x.im))))
   (fma
    (- x.re (fabs x.im))
    (* (fabs x.im) (+ (fabs x.im) x.re))
    (+ (fabs x.im) (fabs x.im))))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (fabs(x_46_im) <= 2e+81) {
		tmp = fma(fma((3.0 * fabs(x_46_im)), x_46_re, 0.0), x_46_re, ((fabs(x_46_im) * fabs(x_46_im)) * -fabs(x_46_im)));
	} else {
		tmp = fma((x_46_re - fabs(x_46_im)), (fabs(x_46_im) * (fabs(x_46_im) + x_46_re)), (fabs(x_46_im) + fabs(x_46_im)));
	}
	return copysign(1.0, x_46_im) * tmp;
}

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (abs(x_46_im) <= 2e+81)
		tmp = fma(fma(Float64(3.0 * abs(x_46_im)), x_46_re, 0.0), x_46_re, Float64(Float64(abs(x_46_im) * abs(x_46_im)) * Float64(-abs(x_46_im))));
	else
		tmp = fma(Float64(x_46_re - abs(x_46_im)), Float64(abs(x_46_im) * Float64(abs(x_46_im) + x_46_re)), Float64(abs(x_46_im) + abs(x_46_im)));
	end
	return Float64(copysign(1.0, x_46_im) * tmp)
end

Wolfram

code[x$46$re_, x$46$im_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x$46$im], $MachinePrecision], 2e+81], N[(N[(N[(3.0 * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] * x$46$re + 0.0), $MachinePrecision] * x$46$re + N[(N[(N[Abs[x$46$im], $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] * (-N[Abs[x$46$im], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re - N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[x$46$im], $MachinePrecision] * N[(N[Abs[x$46$im], $MachinePrecision] + x$46$re), $MachinePrecision]), $MachinePrecision] + N[(N[Abs[x$46$im], $MachinePrecision] + N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.9999999999999998e81

   1. Initial program 82.4%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. difference-of-squares N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 91.4%

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

   3. Applied rewrites 91.4%

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

   4. Taylor expanded in x.re around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 86.0%

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

   6. Applied rewrites 86.0%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

   8. Applied rewrites 88.0%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(3 \cdot x.im, x.re, 0\right), \color{blue}{x.re}, \left(x.im \cdot x.im\right) \cdot \left(-x.im\right)\right) \]

   if 1.9999999999999998e81 < x.im

   1. Initial program 82.4%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. difference-of-squares N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 91.4%

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

   3. Applied rewrites 91.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-+.f64 N/A

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

      7. distribute-rgt-out N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. flip-+ N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. +-inverses N/A

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

      16. +-inverses N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 N/A

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

   5. Applied rewrites 58.7%

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

Alternative 3: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left|x.im\right| \cdot x.re\\ t_1 := \left(x.re \cdot x.re - \left|x.im\right| \cdot \left|x.im\right|\right) \cdot \left|x.im\right| + \left(x.re \cdot \left|x.im\right| + t\_0\right) \cdot x.re\\ \mathsf{copysign}\left(1, x.im\right) \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-314}:\\ \;\;\;\;-1 \cdot {\left(\left|x.im\right|\right)}^{3}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_0 \cdot \left(x.re \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - \left|x.im\right|, \left|x.im\right| \cdot \left(\left|x.im\right| + x.re\right), \left|x.im\right| + \left|x.im\right|\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (let* ((t_0 (* (fabs x.im) x.re))
       (t_1
        (+
         (* (- (* x.re x.re) (* (fabs x.im) (fabs x.im))) (fabs x.im))
         (* (+ (* x.re (fabs x.im)) t_0) x.re))))
  (*
   (copysign 1.0 x.im)
   (if (<= t_1 -4e-314)
     (* -1.0 (pow (fabs x.im) 3.0))
     (if (<= t_1 INFINITY)
       (* t_0 (* x.re 3.0))
       (fma
        (- x.re (fabs x.im))
        (* (fabs x.im) (+ (fabs x.im) x.re))
        (+ (fabs x.im) (fabs x.im))))))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = fabs(x_46_im) * x_46_re;
	double t_1 = (((x_46_re * x_46_re) - (fabs(x_46_im) * fabs(x_46_im))) * fabs(x_46_im)) + (((x_46_re * fabs(x_46_im)) + t_0) * x_46_re);
	double tmp;
	if (t_1 <= -4e-314) {
		tmp = -1.0 * pow(fabs(x_46_im), 3.0);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_0 * (x_46_re * 3.0);
	} else {
		tmp = fma((x_46_re - fabs(x_46_im)), (fabs(x_46_im) * (fabs(x_46_im) + x_46_re)), (fabs(x_46_im) + fabs(x_46_im)));
	}
	return copysign(1.0, x_46_im) * tmp;
}

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(abs(x_46_im) * x_46_re)
	t_1 = Float64(Float64(Float64(Float64(x_46_re * x_46_re) - Float64(abs(x_46_im) * abs(x_46_im))) * abs(x_46_im)) + Float64(Float64(Float64(x_46_re * abs(x_46_im)) + t_0) * x_46_re))
	tmp = 0.0
	if (t_1 <= -4e-314)
		tmp = Float64(-1.0 * (abs(x_46_im) ^ 3.0));
	elseif (t_1 <= Inf)
		tmp = Float64(t_0 * Float64(x_46_re * 3.0));
	else
		tmp = fma(Float64(x_46_re - abs(x_46_im)), Float64(abs(x_46_im) * Float64(abs(x_46_im) + x_46_re)), Float64(abs(x_46_im) + abs(x_46_im)));
	end
	return Float64(copysign(1.0, x_46_im) * tmp)
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(N[Abs[x$46$im], $MachinePrecision] * x$46$re), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(N[Abs[x$46$im], $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x$46$re * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$1, -4e-314], N[(-1.0 * N[Power[N[Abs[x$46$im], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(t$95$0 * N[(x$46$re * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re - N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[x$46$im], $MachinePrecision] * N[(N[Abs[x$46$im], $MachinePrecision] + x$46$re), $MachinePrecision]), $MachinePrecision] + N[(N[Abs[x$46$im], $MachinePrecision] + N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < -3.999999999855523e-314

   1. Initial program 82.4%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. difference-of-squares N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 91.4%

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

   3. Applied rewrites 91.4%

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

   4. Taylor expanded in x.re around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 86.0%

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

   6. Applied rewrites 86.0%

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

   7. Taylor expanded in x.re around 0

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

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot {x.im}^{\color{blue}{3}} \]

      2. lower-pow.f64 59.4%

         \[\leadsto -1 \cdot {x.im}^{3} \]

   9. Applied rewrites 59.4%

      \[\leadsto -1 \cdot \color{blue}{{x.im}^{3}} \]

   if -3.999999999855523e-314 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   1. Initial program 82.4%

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

   2. Taylor expanded in x.re around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 49.7%

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

   4. Applied rewrites 49.7%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 49.7%

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

         \[\leadsto \left(x.im + 2 \cdot x.im\right) \cdot \left(x.re \cdot x.re\right) \]

      8. distribute-rgt1-in N/A

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

      9. metadata-eval N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lower-*.f64 55.6%

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 55.6%

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

   6. Applied rewrites 55.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 55.6%

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

   8. Applied rewrites 55.6%

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

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

   1. Initial program 82.4%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. difference-of-squares N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 91.4%

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

   3. Applied rewrites 91.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-+.f64 N/A

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

      7. distribute-rgt-out N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. flip-+ N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. +-inverses N/A

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

      16. +-inverses N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 N/A

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

   5. Applied rewrites 58.7%

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

Alternative 4: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left|x.im\right| \cdot x.re\\ t_1 := \left|x.im\right| + \left|x.im\right|\\ t_2 := \left(x.re \cdot x.re - \left|x.im\right| \cdot \left|x.im\right|\right) \cdot \left|x.im\right| + \left(x.re \cdot \left|x.im\right| + t\_0\right) \cdot x.re\\ \mathsf{copysign}\left(1, x.im\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -4 \cdot 10^{-314}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, x.re \cdot x.re, \left(\left(-\left|x.im\right|\right) \cdot \left|x.im\right|\right) \cdot \left|x.im\right|\right)\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_0 \cdot \left(x.re \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - \left|x.im\right|, \left|x.im\right| \cdot \left(\left|x.im\right| + x.re\right), t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (let* ((t_0 (* (fabs x.im) x.re))
       (t_1 (+ (fabs x.im) (fabs x.im)))
       (t_2
        (+
         (* (- (* x.re x.re) (* (fabs x.im) (fabs x.im))) (fabs x.im))
         (* (+ (* x.re (fabs x.im)) t_0) x.re))))
  (*
   (copysign 1.0 x.im)
   (if (<= t_2 -4e-314)
     (fma
      t_1
      (* x.re x.re)
      (* (* (- (fabs x.im)) (fabs x.im)) (fabs x.im)))
     (if (<= t_2 INFINITY)
       (* t_0 (* x.re 3.0))
       (fma
        (- x.re (fabs x.im))
        (* (fabs x.im) (+ (fabs x.im) x.re))
        t_1))))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = fabs(x_46_im) * x_46_re;
	double t_1 = fabs(x_46_im) + fabs(x_46_im);
	double t_2 = (((x_46_re * x_46_re) - (fabs(x_46_im) * fabs(x_46_im))) * fabs(x_46_im)) + (((x_46_re * fabs(x_46_im)) + t_0) * x_46_re);
	double tmp;
	if (t_2 <= -4e-314) {
		tmp = fma(t_1, (x_46_re * x_46_re), ((-fabs(x_46_im) * fabs(x_46_im)) * fabs(x_46_im)));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_0 * (x_46_re * 3.0);
	} else {
		tmp = fma((x_46_re - fabs(x_46_im)), (fabs(x_46_im) * (fabs(x_46_im) + x_46_re)), t_1);
	}
	return copysign(1.0, x_46_im) * tmp;
}

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(abs(x_46_im) * x_46_re)
	t_1 = Float64(abs(x_46_im) + abs(x_46_im))
	t_2 = Float64(Float64(Float64(Float64(x_46_re * x_46_re) - Float64(abs(x_46_im) * abs(x_46_im))) * abs(x_46_im)) + Float64(Float64(Float64(x_46_re * abs(x_46_im)) + t_0) * x_46_re))
	tmp = 0.0
	if (t_2 <= -4e-314)
		tmp = fma(t_1, Float64(x_46_re * x_46_re), Float64(Float64(Float64(-abs(x_46_im)) * abs(x_46_im)) * abs(x_46_im)));
	elseif (t_2 <= Inf)
		tmp = Float64(t_0 * Float64(x_46_re * 3.0));
	else
		tmp = fma(Float64(x_46_re - abs(x_46_im)), Float64(abs(x_46_im) * Float64(abs(x_46_im) + x_46_re)), t_1);
	end
	return Float64(copysign(1.0, x_46_im) * tmp)
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(N[Abs[x$46$im], $MachinePrecision] * x$46$re), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[x$46$im], $MachinePrecision] + N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(N[Abs[x$46$im], $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x$46$re * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$2, -4e-314], N[(t$95$1 * N[(x$46$re * x$46$re), $MachinePrecision] + N[(N[((-N[Abs[x$46$im], $MachinePrecision]) * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(t$95$0 * N[(x$46$re * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re - N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[x$46$im], $MachinePrecision] * N[(N[Abs[x$46$im], $MachinePrecision] + x$46$re), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < -3.999999999855523e-314

   1. Initial program 82.4%

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

   2. Taylor expanded in x.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 65.9%

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

   4. Applied rewrites 65.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. +-inverses N/A

         \[\leadsto -1 \cdot {x.im}^{3} + x.re \cdot \frac{\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.im \cdot x.re\right) \cdot \left(x.im \cdot x.re\right)}{\color{blue}{0}} \]

      9. +-inverses N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. remove-sound-/ N/A

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

      17. remove-sound-/ N/A

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

   6. Applied rewrites 39.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

   8. Applied rewrites 70.2%

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

   if -3.999999999855523e-314 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   1. Initial program 82.4%

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

   2. Taylor expanded in x.re around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 49.7%

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

   4. Applied rewrites 49.7%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 49.7%

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

         \[\leadsto \left(x.im + 2 \cdot x.im\right) \cdot \left(x.re \cdot x.re\right) \]

      8. distribute-rgt1-in N/A

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

      9. metadata-eval N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lower-*.f64 55.6%

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 55.6%

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

   6. Applied rewrites 55.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 55.6%

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

   8. Applied rewrites 55.6%

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

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

   1. Initial program 82.4%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. difference-of-squares N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 91.4%

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

   3. Applied rewrites 91.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-+.f64 N/A

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

      7. distribute-rgt-out N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. flip-+ N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. +-inverses N/A

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

      16. +-inverses N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 N/A

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

   5. Applied rewrites 58.7%

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

Alternative 5: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left|x.im\right| \cdot x.re\\ t_1 := \left|x.im\right| + \left|x.im\right|\\ t_2 := \left(x.re \cdot x.re - \left|x.im\right| \cdot \left|x.im\right|\right) \cdot \left|x.im\right| + \left(x.re \cdot \left|x.im\right| + t\_0\right) \cdot x.re\\ \mathsf{copysign}\left(1, x.im\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -4 \cdot 10^{-314}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, 0, \left(\left(-\left|x.im\right|\right) \cdot \left|x.im\right|\right) \cdot \left|x.im\right|\right)\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_0 \cdot \left(x.re \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - \left|x.im\right|, \left|x.im\right| \cdot \left(\left|x.im\right| + x.re\right), t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (let* ((t_0 (* (fabs x.im) x.re))
       (t_1 (+ (fabs x.im) (fabs x.im)))
       (t_2
        (+
         (* (- (* x.re x.re) (* (fabs x.im) (fabs x.im))) (fabs x.im))
         (* (+ (* x.re (fabs x.im)) t_0) x.re))))
  (*
   (copysign 1.0 x.im)
   (if (<= t_2 -4e-314)
     (fma t_1 0.0 (* (* (- (fabs x.im)) (fabs x.im)) (fabs x.im)))
     (if (<= t_2 INFINITY)
       (* t_0 (* x.re 3.0))
       (fma
        (- x.re (fabs x.im))
        (* (fabs x.im) (+ (fabs x.im) x.re))
        t_1))))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = fabs(x_46_im) * x_46_re;
	double t_1 = fabs(x_46_im) + fabs(x_46_im);
	double t_2 = (((x_46_re * x_46_re) - (fabs(x_46_im) * fabs(x_46_im))) * fabs(x_46_im)) + (((x_46_re * fabs(x_46_im)) + t_0) * x_46_re);
	double tmp;
	if (t_2 <= -4e-314) {
		tmp = fma(t_1, 0.0, ((-fabs(x_46_im) * fabs(x_46_im)) * fabs(x_46_im)));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_0 * (x_46_re * 3.0);
	} else {
		tmp = fma((x_46_re - fabs(x_46_im)), (fabs(x_46_im) * (fabs(x_46_im) + x_46_re)), t_1);
	}
	return copysign(1.0, x_46_im) * tmp;
}

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(abs(x_46_im) * x_46_re)
	t_1 = Float64(abs(x_46_im) + abs(x_46_im))
	t_2 = Float64(Float64(Float64(Float64(x_46_re * x_46_re) - Float64(abs(x_46_im) * abs(x_46_im))) * abs(x_46_im)) + Float64(Float64(Float64(x_46_re * abs(x_46_im)) + t_0) * x_46_re))
	tmp = 0.0
	if (t_2 <= -4e-314)
		tmp = fma(t_1, 0.0, Float64(Float64(Float64(-abs(x_46_im)) * abs(x_46_im)) * abs(x_46_im)));
	elseif (t_2 <= Inf)
		tmp = Float64(t_0 * Float64(x_46_re * 3.0));
	else
		tmp = fma(Float64(x_46_re - abs(x_46_im)), Float64(abs(x_46_im) * Float64(abs(x_46_im) + x_46_re)), t_1);
	end
	return Float64(copysign(1.0, x_46_im) * tmp)
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(N[Abs[x$46$im], $MachinePrecision] * x$46$re), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[x$46$im], $MachinePrecision] + N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(N[Abs[x$46$im], $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x$46$re * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$2, -4e-314], N[(t$95$1 * 0.0 + N[(N[((-N[Abs[x$46$im], $MachinePrecision]) * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(t$95$0 * N[(x$46$re * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re - N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[x$46$im], $MachinePrecision] * N[(N[Abs[x$46$im], $MachinePrecision] + x$46$re), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < -3.999999999855523e-314

   1. Initial program 82.4%

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

   2. Taylor expanded in x.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 65.9%

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

   4. Applied rewrites 65.9%

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

   6. Applied rewrites 59.3%

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

   if -3.999999999855523e-314 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   1. Initial program 82.4%

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

   2. Taylor expanded in x.re around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 49.7%

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

   4. Applied rewrites 49.7%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 49.7%

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

         \[\leadsto \left(x.im + 2 \cdot x.im\right) \cdot \left(x.re \cdot x.re\right) \]

      8. distribute-rgt1-in N/A

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

      9. metadata-eval N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lower-*.f64 55.6%

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 55.6%

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

   6. Applied rewrites 55.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 55.6%

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

   8. Applied rewrites 55.6%

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

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

   1. Initial program 82.4%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. difference-of-squares N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 91.4%

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

   3. Applied rewrites 91.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-+.f64 N/A

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

      7. distribute-rgt-out N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. flip-+ N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. +-inverses N/A

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

      16. +-inverses N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 N/A

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

   5. Applied rewrites 58.7%

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

Alternative 6: 96.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left|x.im\right| \cdot x.re\\ t_1 := \left(\left(-\left|x.im\right|\right) \cdot \left|x.im\right|\right) \cdot \left|x.im\right|\\ t_2 := \left(x.re \cdot x.re - \left|x.im\right| \cdot \left|x.im\right|\right) \cdot \left|x.im\right| + \left(x.re \cdot \left|x.im\right| + t\_0\right) \cdot x.re\\ \mathsf{copysign}\left(1, x.im\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -4 \cdot 10^{-314}:\\ \;\;\;\;\mathsf{fma}\left(\left|x.im\right| + \left|x.im\right|, 0, t\_1\right)\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_0 \cdot \left(x.re \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left|x.im\right|, 2, t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (let* ((t_0 (* (fabs x.im) x.re))
       (t_1 (* (* (- (fabs x.im)) (fabs x.im)) (fabs x.im)))
       (t_2
        (+
         (* (- (* x.re x.re) (* (fabs x.im) (fabs x.im))) (fabs x.im))
         (* (+ (* x.re (fabs x.im)) t_0) x.re))))
  (*
   (copysign 1.0 x.im)
   (if (<= t_2 -4e-314)
     (fma (+ (fabs x.im) (fabs x.im)) 0.0 t_1)
     (if (<= t_2 INFINITY)
       (* t_0 (* x.re 3.0))
       (fma (fabs x.im) 2.0 t_1))))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = fabs(x_46_im) * x_46_re;
	double t_1 = (-fabs(x_46_im) * fabs(x_46_im)) * fabs(x_46_im);
	double t_2 = (((x_46_re * x_46_re) - (fabs(x_46_im) * fabs(x_46_im))) * fabs(x_46_im)) + (((x_46_re * fabs(x_46_im)) + t_0) * x_46_re);
	double tmp;
	if (t_2 <= -4e-314) {
		tmp = fma((fabs(x_46_im) + fabs(x_46_im)), 0.0, t_1);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_0 * (x_46_re * 3.0);
	} else {
		tmp = fma(fabs(x_46_im), 2.0, t_1);
	}
	return copysign(1.0, x_46_im) * tmp;
}

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(abs(x_46_im) * x_46_re)
	t_1 = Float64(Float64(Float64(-abs(x_46_im)) * abs(x_46_im)) * abs(x_46_im))
	t_2 = Float64(Float64(Float64(Float64(x_46_re * x_46_re) - Float64(abs(x_46_im) * abs(x_46_im))) * abs(x_46_im)) + Float64(Float64(Float64(x_46_re * abs(x_46_im)) + t_0) * x_46_re))
	tmp = 0.0
	if (t_2 <= -4e-314)
		tmp = fma(Float64(abs(x_46_im) + abs(x_46_im)), 0.0, t_1);
	elseif (t_2 <= Inf)
		tmp = Float64(t_0 * Float64(x_46_re * 3.0));
	else
		tmp = fma(abs(x_46_im), 2.0, t_1);
	end
	return Float64(copysign(1.0, x_46_im) * tmp)
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(N[Abs[x$46$im], $MachinePrecision] * x$46$re), $MachinePrecision]}, Block[{t$95$1 = N[(N[((-N[Abs[x$46$im], $MachinePrecision]) * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(N[Abs[x$46$im], $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x$46$re * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$2, -4e-314], N[(N[(N[Abs[x$46$im], $MachinePrecision] + N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] * 0.0 + t$95$1), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(t$95$0 * N[(x$46$re * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[Abs[x$46$im], $MachinePrecision] * 2.0 + t$95$1), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < -3.999999999855523e-314

   1. Initial program 82.4%

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

   2. Taylor expanded in x.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 65.9%

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

   4. Applied rewrites 65.9%

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

   6. Applied rewrites 59.3%

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

   if -3.999999999855523e-314 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   1. Initial program 82.4%

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

   2. Taylor expanded in x.re around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 49.7%

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

   4. Applied rewrites 49.7%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 49.7%

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

         \[\leadsto \left(x.im + 2 \cdot x.im\right) \cdot \left(x.re \cdot x.re\right) \]

      8. distribute-rgt1-in N/A

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

      9. metadata-eval N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lower-*.f64 55.6%

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 55.6%

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

   6. Applied rewrites 55.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 55.6%

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

   8. Applied rewrites 55.6%

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

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

   1. Initial program 82.4%

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

   2. Taylor expanded in x.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 65.9%

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

   4. Applied rewrites 65.9%

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

   6. Applied rewrites 39.7%

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

Alternative 7: 89.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left|x.im\right| \cdot x.re\\ t_1 := \mathsf{fma}\left(\left|x.im\right|, 2, \left(\left(-\left|x.im\right|\right) \cdot \left|x.im\right|\right) \cdot \left|x.im\right|\right)\\ t_2 := \left(x.re \cdot x.re - \left|x.im\right| \cdot \left|x.im\right|\right) \cdot \left|x.im\right| + \left(x.re \cdot \left|x.im\right| + t\_0\right) \cdot x.re\\ \mathsf{copysign}\left(1, x.im\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_0 \cdot \left(x.re \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (let* ((t_0 (* (fabs x.im) x.re))
       (t_1
        (fma
         (fabs x.im)
         2.0
         (* (* (- (fabs x.im)) (fabs x.im)) (fabs x.im))))
       (t_2
        (+
         (* (- (* x.re x.re) (* (fabs x.im) (fabs x.im))) (fabs x.im))
         (* (+ (* x.re (fabs x.im)) t_0) x.re))))
  (*
   (copysign 1.0 x.im)
   (if (<= t_2 -1e+20)
     t_1
     (if (<= t_2 INFINITY) (* t_0 (* x.re 3.0)) t_1)))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = fabs(x_46_im) * x_46_re;
	double t_1 = fma(fabs(x_46_im), 2.0, ((-fabs(x_46_im) * fabs(x_46_im)) * fabs(x_46_im)));
	double t_2 = (((x_46_re * x_46_re) - (fabs(x_46_im) * fabs(x_46_im))) * fabs(x_46_im)) + (((x_46_re * fabs(x_46_im)) + t_0) * x_46_re);
	double tmp;
	if (t_2 <= -1e+20) {
		tmp = t_1;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_0 * (x_46_re * 3.0);
	} else {
		tmp = t_1;
	}
	return copysign(1.0, x_46_im) * tmp;
}

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(abs(x_46_im) * x_46_re)
	t_1 = fma(abs(x_46_im), 2.0, Float64(Float64(Float64(-abs(x_46_im)) * abs(x_46_im)) * abs(x_46_im)))
	t_2 = Float64(Float64(Float64(Float64(x_46_re * x_46_re) - Float64(abs(x_46_im) * abs(x_46_im))) * abs(x_46_im)) + Float64(Float64(Float64(x_46_re * abs(x_46_im)) + t_0) * x_46_re))
	tmp = 0.0
	if (t_2 <= -1e+20)
		tmp = t_1;
	elseif (t_2 <= Inf)
		tmp = Float64(t_0 * Float64(x_46_re * 3.0));
	else
		tmp = t_1;
	end
	return Float64(copysign(1.0, x_46_im) * tmp)
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(N[Abs[x$46$im], $MachinePrecision] * x$46$re), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[x$46$im], $MachinePrecision] * 2.0 + N[(N[((-N[Abs[x$46$im], $MachinePrecision]) * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(N[Abs[x$46$im], $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x$46$re * N[Abs[x$46$im], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$2, -1e+20], t$95$1, If[LessEqual[t$95$2, Infinity], N[(t$95$0 * N[(x$46$re * 3.0), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < -1e20 or +inf.0 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re))

   1. Initial program 82.4%

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

   2. Taylor expanded in x.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 65.9%

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

   4. Applied rewrites 65.9%

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

   6. Applied rewrites 39.7%

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

   if -1e20 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   1. Initial program 82.4%

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

   2. Taylor expanded in x.re around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 49.7%

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

   4. Applied rewrites 49.7%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 49.7%

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

         \[\leadsto \left(x.im + 2 \cdot x.im\right) \cdot \left(x.re \cdot x.re\right) \]

      8. distribute-rgt1-in N/A

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

      9. metadata-eval N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lower-*.f64 55.6%

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 55.6%

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

   6. Applied rewrites 55.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 55.6%

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

   8. Applied rewrites 55.6%

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

Alternative 8: 55.6% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

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_46im * x_46re) * (x_46re * 3.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.4%

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

2. Taylor expanded in x.re around inf

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

   1. lower-*.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 49.7%

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

4. Applied rewrites 49.7%

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

   1. lift-pow.f64 N/A

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

   2. pow2 N/A

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

   3. lift-*.f64 49.7%

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-+.f64 N/A

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

   7. lift-*.f64 N/A

      \[\leadsto \left(x.im + 2 \cdot x.im\right) \cdot \left(x.re \cdot x.re\right) \]

   8. distribute-rgt1-in N/A

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

   9. metadata-eval N/A

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

   10. associate-*l* N/A

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

   11. lower-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. associate-*r* N/A

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

   14. *-commutative N/A

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

   15. lift-*.f64 N/A

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

   16. lower-*.f64 55.6%

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lift-*.f64 55.6%

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

6. Applied rewrites 55.6%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 55.6%

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

8. Applied rewrites 55.6%

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

Alternative 9: 55.6% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

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_46im * (3.0d0 * x_46re))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.4%

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

2. Taylor expanded in x.re around inf

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

   1. lower-*.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 49.7%

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

4. Applied rewrites 49.7%

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

   1. lift-pow.f64 N/A

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

   2. pow2 N/A

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

   3. lift-*.f64 49.7%

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-+.f64 N/A

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

   7. lift-*.f64 N/A

      \[\leadsto \left(x.im + 2 \cdot x.im\right) \cdot \left(x.re \cdot x.re\right) \]

   8. distribute-rgt1-in N/A

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

   9. metadata-eval N/A

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

   10. associate-*l* N/A

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

   11. lower-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. associate-*r* N/A

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

   14. *-commutative N/A

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

   15. lift-*.f64 N/A

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

   16. lower-*.f64 55.6%

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lift-*.f64 55.6%

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

6. Applied rewrites 55.6%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 55.6%

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

8. Applied rewrites 55.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 55.6%

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 55.6%

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

10. Applied rewrites 55.6%

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

Alternative 10: 55.6% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

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 = 3.0d0 * ((x_46im * x_46re) * x_46re)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.4%

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

2. Taylor expanded in x.re around inf

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

   1. lower-*.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 49.7%

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

4. Applied rewrites 49.7%

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

   1. lift-pow.f64 N/A

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

   2. pow2 N/A

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

   3. lift-*.f64 49.7%

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-+.f64 N/A

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

   7. lift-*.f64 N/A

      \[\leadsto \left(x.im + 2 \cdot x.im\right) \cdot \left(x.re \cdot x.re\right) \]

   8. distribute-rgt1-in N/A

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

   9. metadata-eval N/A

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

   10. associate-*l* N/A

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

   11. lower-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. associate-*r* N/A

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

   14. *-commutative N/A

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

   15. lift-*.f64 N/A

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

   16. lower-*.f64 55.6%

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lift-*.f64 55.6%

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

6. Applied rewrites 55.6%

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

Alternative 11: 35.1% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

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_46im + x_46im) * (x_46re * x_46re)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.4%

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

2. Taylor expanded in x.re around inf

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

   1. lower-*.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 49.7%

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

4. Applied rewrites 49.7%

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

6. Applied rewrites 21.5%

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

7. Taylor expanded in x.re around 0

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

   1. lower-*.f64 3.1%

      \[\leadsto 2 \cdot x.im \]

9. Applied rewrites 3.1%

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

    1. lift-*.f64 N/A

       \[\leadsto 2 \cdot x.im \]

    2. count-2-rev N/A

       \[\leadsto x.im + x.im \]

    3. flip-+ N/A

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

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

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

    5. sub-flip-reverse N/A

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

    6. mul-1-neg N/A

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

    7. lift-*.f64 N/A

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

    8. lift-+.f64 N/A

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

    9. +-inverses N/A

       \[\leadsto \frac{x.im \cdot \left(x.im + -1 \cdot x.im\right)}{0} \]

    10. +-inverses N/A

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

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

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

    12. sub-flip-reverse N/A

        \[\leadsto \frac{x.im \cdot \left(x.im + -1 \cdot x.im\right)}{x.im \cdot \left(x.im + \left(\mathsf{neg}\left(x.im\right)\right)\right)} \]

    13. mul-1-neg N/A

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

    14. lift-*.f64 N/A

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

    15. lift-+.f64 N/A

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

    16. remove-sound-/ N/A

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

    17. remove-sound-/ N/A

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

    18. lift-+.f64 N/A

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

    19. lift-*.f64 N/A

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

    20. mul-1-neg N/A

        \[\leadsto \frac{x.im \cdot \left(x.im + \left(\mathsf{neg}\left(x.im\right)\right)\right)}{x.im \cdot \left(x.im + -1 \cdot x.im\right)} \]

    21. sub-flip-reverse N/A

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

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

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

    23. +-inverses N/A

        \[\leadsto \frac{0}{x.im \cdot \left(\color{blue}{x.im} + -1 \cdot x.im\right)} \]

    24. +-inverses N/A

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

    25. lift-+.f64 N/A

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

    26. lift-*.f64 N/A

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

    27. mul-1-neg N/A

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

    28. sub-flip-reverse N/A

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

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

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

    30. +-inverses N/A

        \[\leadsto \frac{\left(x.im \cdot x.re\right) \cdot \left(x.im \cdot x.re\right) - \left(x.im \cdot x.re\right) \cdot \left(x.im \cdot x.re\right)}{0} \]

11. Applied rewrites 35.1%

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

Alternative 12: 31.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x.re\right| \leq 10.5:\\ \;\;\;\;\left(x.im + x.im\right) \cdot \left|x.re\right|\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \mathsf{fma}\left(\left|x.re\right|, \left|x.re\right|, 2\right)\\ \end{array} \]

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (if (<= (fabs x.re) 10.5)
  (* (+ x.im x.im) (fabs x.re))
  (* x.im (fma (fabs x.re) (fabs x.re) 2.0))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (fabs(x_46_re) <= 10.5) {
		tmp = (x_46_im + x_46_im) * fabs(x_46_re);
	} else {
		tmp = x_46_im * fma(fabs(x_46_re), fabs(x_46_re), 2.0);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (abs(x_46_re) <= 10.5)
		tmp = Float64(Float64(x_46_im + x_46_im) * abs(x_46_re));
	else
		tmp = Float64(x_46_im * fma(abs(x_46_re), abs(x_46_re), 2.0));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[N[Abs[x$46$re], $MachinePrecision], 10.5], N[(N[(x$46$im + x$46$im), $MachinePrecision] * N[Abs[x$46$re], $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(N[Abs[x$46$re], $MachinePrecision] * N[Abs[x$46$re], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 10.5

   1. Initial program 82.4%

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

   2. Taylor expanded in x.re around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 49.7%

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

   4. Applied rewrites 49.7%

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

   6. Applied rewrites 21.5%

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

   7. Taylor expanded in x.re around 0

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

      1. lower-*.f64 3.1%

         \[\leadsto 2 \cdot x.im \]

   9. Applied rewrites 3.1%

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

       1. lift-*.f64 N/A

          \[\leadsto 2 \cdot x.im \]

       2. count-2-rev N/A

          \[\leadsto x.im + x.im \]

       3. flip-+ N/A

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

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

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

       5. sub-flip-reverse N/A

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

       6. mul-1-neg N/A

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

       7. lift-*.f64 N/A

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

       8. lift-+.f64 N/A

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

       9. +-inverses N/A

          \[\leadsto \frac{x.im \cdot \left(x.im + -1 \cdot x.im\right)}{0} \]

       10. +-inverses N/A

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

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

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

       12. sub-flip-reverse N/A

           \[\leadsto \frac{x.im \cdot \left(x.im + -1 \cdot x.im\right)}{x.im \cdot \left(x.im + \left(\mathsf{neg}\left(x.im\right)\right)\right)} \]

       13. mul-1-neg N/A

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

       14. lift-*.f64 N/A

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

       15. lift-+.f64 N/A

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

       16. remove-sound-/ N/A

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

       17. remove-sound-/ N/A

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

       18. lift-+.f64 N/A

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

       19. lift-*.f64 N/A

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

       20. mul-1-neg N/A

           \[\leadsto \frac{x.im \cdot \left(x.im + \left(\mathsf{neg}\left(x.im\right)\right)\right)}{x.im \cdot \left(x.im + -1 \cdot x.im\right)} \]

       21. sub-flip-reverse N/A

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

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

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

       23. +-inverses N/A

           \[\leadsto \frac{0}{x.im \cdot \left(\color{blue}{x.im} + -1 \cdot x.im\right)} \]

       24. +-inverses N/A

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

       25. lift-+.f64 N/A

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

       26. lift-*.f64 N/A

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

       27. mul-1-neg N/A

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

       28. sub-flip-reverse N/A

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

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

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

       30. +-inverses N/A

           \[\leadsto \frac{\left(x.im \cdot x.re\right) \cdot \left(x.im \cdot x.re\right) - \left(x.im \cdot x.re\right) \cdot \left(x.im \cdot x.re\right)}{0} \]

   11. Applied rewrites 20.1%

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

   if 10.5 < x.re

   1. Initial program 82.4%

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

   2. Taylor expanded in x.re around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 49.7%

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

   4. Applied rewrites 49.7%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 49.7%

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

         \[\leadsto \left(x.im + 2 \cdot x.im\right) \cdot \left(x.re \cdot x.re\right) \]

      8. distribute-rgt1-in N/A

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

      9. metadata-eval N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lower-*.f64 55.6%

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 55.6%

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

   6. Applied rewrites 55.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 55.6%

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

   8. Applied rewrites 55.6%

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

   10. Applied rewrites 20.8%

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

Alternative 13: 20.1% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

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_46im + x_46im) * x_46re
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.4%

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

2. Taylor expanded in x.re around inf

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

   1. lower-*.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 49.7%

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

4. Applied rewrites 49.7%

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

6. Applied rewrites 21.5%

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

7. Taylor expanded in x.re around 0

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

   1. lower-*.f64 3.1%

      \[\leadsto 2 \cdot x.im \]

9. Applied rewrites 3.1%

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

    1. lift-*.f64 N/A

       \[\leadsto 2 \cdot x.im \]

    2. count-2-rev N/A

       \[\leadsto x.im + x.im \]

    3. flip-+ N/A

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

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

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

    5. sub-flip-reverse N/A

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

    6. mul-1-neg N/A

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

    7. lift-*.f64 N/A

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

    8. lift-+.f64 N/A

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

    9. +-inverses N/A

       \[\leadsto \frac{x.im \cdot \left(x.im + -1 \cdot x.im\right)}{0} \]

    10. +-inverses N/A

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

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

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

    12. sub-flip-reverse N/A

        \[\leadsto \frac{x.im \cdot \left(x.im + -1 \cdot x.im\right)}{x.im \cdot \left(x.im + \left(\mathsf{neg}\left(x.im\right)\right)\right)} \]

    13. mul-1-neg N/A

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

    14. lift-*.f64 N/A

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

    15. lift-+.f64 N/A

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

    16. remove-sound-/ N/A

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

    17. remove-sound-/ N/A

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

    18. lift-+.f64 N/A

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

    19. lift-*.f64 N/A

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

    20. mul-1-neg N/A

        \[\leadsto \frac{x.im \cdot \left(x.im + \left(\mathsf{neg}\left(x.im\right)\right)\right)}{x.im \cdot \left(x.im + -1 \cdot x.im\right)} \]

    21. sub-flip-reverse N/A

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

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

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

    23. +-inverses N/A

        \[\leadsto \frac{0}{x.im \cdot \left(\color{blue}{x.im} + -1 \cdot x.im\right)} \]

    24. +-inverses N/A

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

    25. lift-+.f64 N/A

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

    26. lift-*.f64 N/A

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

    27. mul-1-neg N/A

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

    28. sub-flip-reverse N/A

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

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

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

    30. +-inverses N/A

        \[\leadsto \frac{\left(x.im \cdot x.re\right) \cdot \left(x.im \cdot x.re\right) - \left(x.im \cdot x.re\right) \cdot \left(x.im \cdot x.re\right)}{0} \]

11. Applied rewrites 20.1%

    \[\leadsto \left(x.im + x.im\right) \cdot x.re \]
12. Add Preprocessing

Alternative 14: 3.1% accurate, 7.4× speedup

Math

\[x.im + x.im \]

FPCore

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

C

double code(double x_46_re, double x_46_im) {
	return x_46_im + 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_46im + x_46im
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.4%

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

2. Taylor expanded in x.re around inf

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

   1. lower-*.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 49.7%

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

4. Applied rewrites 49.7%

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

6. Applied rewrites 21.5%

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

7. Taylor expanded in x.re around 0

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

   1. lower-*.f64 3.1%

      \[\leadsto 2 \cdot x.im \]

9. Applied rewrites 3.1%

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

    1. lift-*.f64 N/A

       \[\leadsto 2 \cdot x.im \]

    2. count-2-rev N/A

       \[\leadsto x.im + x.im \]

    3. lift-+.f64 3.1%

       \[\leadsto x.im + x.im \]

11. Applied rewrites 3.1%

    \[\leadsto x.im + x.im \]
12. Add Preprocessing

Developer Target 1: 91.4% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (x.re x.im)
  :precision binary64
  (+
 (* (* x.re x.im) (* 2.0 x.re))
 (* (* x.im (- x.re x.im)) (+ x.re x.im))))

C

double code(double x_46_re, double x_46_im) {
	return ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
}

Fortran

real(8) function code(x_46re, x_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = ((x_46re * x_46im) * (2.0d0 * x_46re)) + ((x_46im * (x_46re - x_46im)) * (x_46re + x_46im))
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
}

Python

def code(x_46_re, x_46_im):
	return ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im))

Julia

function code(x_46_re, x_46_im)
	return Float64(Float64(Float64(x_46_re * x_46_im) * Float64(2.0 * x_46_re)) + Float64(Float64(x_46_im * Float64(x_46_re - x_46_im)) * Float64(x_46_re + x_46_im)))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
end

Wolfram

code[x$46$re_, x$46$im_] := N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(2.0 * x$46$re), $MachinePrecision]), $MachinePrecision] + N[(N[(x$46$im * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision] * N[(x$46$re + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2025313 -o setup:search
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"
  :precision binary64

  :alt
  (! :herbie-platform c (+ (* (* x.re x.im) (* 2 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im))))

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