math.cube on complex, imaginary part

Percentage Accurate: 82.6% → 99.7%

Time: 6.1s

Alternatives: 9

Speedup: 0.5×

• 44.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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)
    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: 82.6% accurate, 1.0× speedup

Math

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

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)
    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, 0.5× speedup

Math

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

FPCore

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

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double tmp;
	if (((((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_re_m)) <= ((double) INFINITY)) {
		tmp = ((x_46_im_m * (x_46_im_m + x_46_re_m)) * (x_46_re_m - x_46_im_m)) + ((x_46_re_m * (x_46_im_m + x_46_im_m)) * x_46_re_m);
	} else {
		tmp = fma((x_46_re_m - x_46_im_m), ((x_46_re_m + x_46_im_m) * x_46_im_m), (2.0 * x_46_im_m));
	}
	return x_46_im_s * tmp;
}

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m) + Float64(Float64(Float64(x_46_re_m * x_46_im_m) + Float64(x_46_im_m * x_46_re_m)) * x_46_re_m)) <= Inf)
		tmp = Float64(Float64(Float64(x_46_im_m * Float64(x_46_im_m + x_46_re_m)) * Float64(x_46_re_m - x_46_im_m)) + Float64(Float64(x_46_re_m * Float64(x_46_im_m + x_46_im_m)) * x_46_re_m));
	else
		tmp = fma(Float64(x_46_re_m - x_46_im_m), Float64(Float64(x_46_re_m + x_46_im_m) * x_46_im_m), Float64(2.0 * x_46_im_m));
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 93.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 99.8

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

   6. Applied rewrites 99.8%

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

   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 0.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 19.0

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

   4. Applied rewrites 19.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 19.0

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

   6. Applied rewrites 19.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 19.0

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 19.0

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 19.0

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. distribute-lft-in N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. flip-+ N/A

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

      19. +-inverses N/A

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

      20. +-inverses N/A

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

      21. associate-*r/ N/A

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

   8. Applied rewrites 100.0%

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

Alternative 2: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m + \left(x.re\_m \cdot x.im\_m + x.im\_m \cdot x.re\_m\right) \cdot x.re\_m\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 10^{+226}:\\ \;\;\;\;\left(-\mathsf{fma}\left(x.im\_m, x.im\_m, -3 \cdot \left(x.re\_m \cdot x.re\_m\right)\right)\right) \cdot x.im\_m\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left(x.re\_m \cdot x.im\_m\right) \cdot \left(3 \cdot x.re\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m - x.im\_m, \left(x.re\_m + x.im\_m\right) \cdot x.im\_m, 2 \cdot x.im\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* x.re_m x.re_m) (* x.im_m x.im_m)) x.im_m)
          (* (+ (* x.re_m x.im_m) (* x.im_m x.re_m)) x.re_m))))
   (*
    x.im_s
    (if (<= t_0 1e+226)
      (* (- (fma x.im_m x.im_m (* -3.0 (* x.re_m x.re_m)))) x.im_m)
      (if (<= t_0 INFINITY)
        (* (* x.re_m x.im_m) (* 3.0 x.re_m))
        (fma
         (- x.re_m x.im_m)
         (* (+ x.re_m x.im_m) x.im_m)
         (* 2.0 x.im_m)))))))

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double t_0 = (((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_re_m);
	double tmp;
	if (t_0 <= 1e+226) {
		tmp = -fma(x_46_im_m, x_46_im_m, (-3.0 * (x_46_re_m * x_46_re_m))) * x_46_im_m;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (x_46_re_m * x_46_im_m) * (3.0 * x_46_re_m);
	} else {
		tmp = fma((x_46_re_m - x_46_im_m), ((x_46_re_m + x_46_im_m) * x_46_im_m), (2.0 * x_46_im_m));
	}
	return x_46_im_s * tmp;
}

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	t_0 = Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m) + Float64(Float64(Float64(x_46_re_m * x_46_im_m) + Float64(x_46_im_m * x_46_re_m)) * x_46_re_m))
	tmp = 0.0
	if (t_0 <= 1e+226)
		tmp = Float64(Float64(-fma(x_46_im_m, x_46_im_m, Float64(-3.0 * Float64(x_46_re_m * x_46_re_m)))) * x_46_im_m);
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(x_46_re_m * x_46_im_m) * Float64(3.0 * x_46_re_m));
	else
		tmp = fma(Float64(x_46_re_m - x_46_im_m), Float64(Float64(x_46_re_m + x_46_im_m) * x_46_im_m), Float64(2.0 * x_46_im_m));
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re$95$m_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] + N[(N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$0, 1e+226], N[((-N[(x$46$im$95$m * x$46$im$95$m + N[(-3.0 * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) * x$46$im$95$m), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] * N[(3.0 * x$46$re$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision] * N[(N[(x$46$re$95$m + x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] + N[(2.0 * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < 9.99999999999999961e225

   1. Initial program 97.1%

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

   3. Taylor expanded in x.re around 0

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

      1. distribute-rgt-in N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-lft-in N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. distribute-rgt-in N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   5. Applied rewrites 97.1%

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

   if 9.99999999999999961e225 < (+.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 83.6%

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

   3. Taylor expanded in x.re around inf

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft1-in N/A

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

      7. lower-*.f64 N/A

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

      8. distribute-lft1-in N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 28.5

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

   5. Applied rewrites 28.5%

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

   7. Applied rewrites 44.8%

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

   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 0.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 19.0

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

   4. Applied rewrites 19.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 19.0

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

   6. Applied rewrites 19.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 19.0

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 19.0

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 19.0

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. distribute-lft-in N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. flip-+ N/A

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

      19. +-inverses N/A

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

      20. +-inverses N/A

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

      21. associate-*r/ N/A

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

   8. Applied rewrites 100.0%

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

Alternative 3: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m + \left(x.re\_m \cdot x.im\_m + x.im\_m \cdot x.re\_m\right) \cdot x.re\_m\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-317}:\\ \;\;\;\;\left(\left(-x.im\_m\right) \cdot x.im\_m\right) \cdot x.im\_m\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left(\left(x.re\_m \cdot x.im\_m\right) \cdot x.re\_m\right) \cdot 3\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m - x.im\_m, \left(x.re\_m + x.im\_m\right) \cdot x.im\_m, 2 \cdot x.im\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* x.re_m x.re_m) (* x.im_m x.im_m)) x.im_m)
          (* (+ (* x.re_m x.im_m) (* x.im_m x.re_m)) x.re_m))))
   (*
    x.im_s
    (if (<= t_0 -5e-317)
      (* (* (- x.im_m) x.im_m) x.im_m)
      (if (<= t_0 INFINITY)
        (* (* (* x.re_m x.im_m) x.re_m) 3.0)
        (fma
         (- x.re_m x.im_m)
         (* (+ x.re_m x.im_m) x.im_m)
         (* 2.0 x.im_m)))))))

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double t_0 = (((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_re_m);
	double tmp;
	if (t_0 <= -5e-317) {
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = ((x_46_re_m * x_46_im_m) * x_46_re_m) * 3.0;
	} else {
		tmp = fma((x_46_re_m - x_46_im_m), ((x_46_re_m + x_46_im_m) * x_46_im_m), (2.0 * x_46_im_m));
	}
	return x_46_im_s * tmp;
}

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	t_0 = Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m) + Float64(Float64(Float64(x_46_re_m * x_46_im_m) + Float64(x_46_im_m * x_46_re_m)) * x_46_re_m))
	tmp = 0.0
	if (t_0 <= -5e-317)
		tmp = Float64(Float64(Float64(-x_46_im_m) * x_46_im_m) * x_46_im_m);
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(Float64(x_46_re_m * x_46_im_m) * x_46_re_m) * 3.0);
	else
		tmp = fma(Float64(x_46_re_m - x_46_im_m), Float64(Float64(x_46_re_m + x_46_im_m) * x_46_im_m), Float64(2.0 * x_46_im_m));
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re$95$m_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] + N[(N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$0, -5e-317], N[(N[((-x$46$im$95$m) * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] * 3.0), $MachinePrecision], N[(N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision] * N[(N[(x$46$re$95$m + x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] + N[(2.0 * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < -5.00000017e-317

   1. Initial program 95.3%

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

   3. Taylor expanded in x.re around 0

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

      1. distribute-rgt-in N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-lft-in N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. distribute-rgt-in N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   5. Applied rewrites 95.3%

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

   6. Taylor expanded in x.re around 0

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

   8. Applied rewrites 50.6%

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

   if -5.00000017e-317 < (+.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 92.0%

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

   3. Taylor expanded in x.re around inf

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft1-in N/A

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

      7. lower-*.f64 N/A

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

      8. distribute-lft1-in N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 59.0

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

   5. Applied rewrites 59.0%

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

   7. Applied rewrites 66.8%

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

   9. Applied rewrites 66.9%

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

   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 0.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 19.0

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

   4. Applied rewrites 19.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 19.0

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

   6. Applied rewrites 19.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 19.0

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 19.0

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 19.0

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. distribute-lft-in N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. flip-+ N/A

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

      19. +-inverses N/A

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

      20. +-inverses N/A

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

      21. associate-*r/ N/A

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

   8. Applied rewrites 100.0%

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

Alternative 4: 96.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m + \left(x.re\_m \cdot x.im\_m + x.im\_m \cdot x.re\_m\right) \cdot x.re\_m\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-317} \lor \neg \left(t\_0 \leq \infty\right):\\ \;\;\;\;\left(\left(-x.im\_m\right) \cdot x.im\_m\right) \cdot x.im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x.re\_m \cdot x.im\_m\right) \cdot x.re\_m\right) \cdot 3\\ \end{array} \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* x.re_m x.re_m) (* x.im_m x.im_m)) x.im_m)
          (* (+ (* x.re_m x.im_m) (* x.im_m x.re_m)) x.re_m))))
   (*
    x.im_s
    (if (or (<= t_0 -5e-317) (not (<= t_0 INFINITY)))
      (* (* (- x.im_m) x.im_m) x.im_m)
      (* (* (* x.re_m x.im_m) x.re_m) 3.0)))))

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double t_0 = (((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_re_m);
	double tmp;
	if ((t_0 <= -5e-317) || !(t_0 <= ((double) INFINITY))) {
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	} else {
		tmp = ((x_46_re_m * x_46_im_m) * x_46_re_m) * 3.0;
	}
	return x_46_im_s * tmp;
}

Java

x.re_m = Math.abs(x_46_re);
x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double t_0 = (((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_re_m);
	double tmp;
	if ((t_0 <= -5e-317) || !(t_0 <= Double.POSITIVE_INFINITY)) {
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	} else {
		tmp = ((x_46_re_m * x_46_im_m) * x_46_re_m) * 3.0;
	}
	return x_46_im_s * tmp;
}

Python

x.re_m = math.fabs(x_46_re)
x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re_m, x_46_im_m):
	t_0 = (((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_re_m)
	tmp = 0
	if (t_0 <= -5e-317) or not (t_0 <= math.inf):
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m
	else:
		tmp = ((x_46_re_m * x_46_im_m) * x_46_re_m) * 3.0
	return x_46_im_s * tmp

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	t_0 = Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m) + Float64(Float64(Float64(x_46_re_m * x_46_im_m) + Float64(x_46_im_m * x_46_re_m)) * x_46_re_m))
	tmp = 0.0
	if ((t_0 <= -5e-317) || !(t_0 <= Inf))
		tmp = Float64(Float64(Float64(-x_46_im_m) * x_46_im_m) * x_46_im_m);
	else
		tmp = Float64(Float64(Float64(x_46_re_m * x_46_im_m) * x_46_re_m) * 3.0);
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.re_m = abs(x_46_re);
x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re_m, x_46_im_m)
	t_0 = (((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_re_m);
	tmp = 0.0;
	if ((t_0 <= -5e-317) || ~((t_0 <= Inf)))
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	else
		tmp = ((x_46_re_m * x_46_im_m) * x_46_re_m) * 3.0;
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re$95$m_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] + N[(N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[Or[LessEqual[t$95$0, -5e-317], N[Not[LessEqual[t$95$0, Infinity]], $MachinePrecision]], N[(N[((-x$46$im$95$m) * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision], N[(N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] * 3.0), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < -5.00000017e-317 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 79.3%

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

   3. Taylor expanded in x.re around 0

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

      1. distribute-rgt-in N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-lft-in N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. distribute-rgt-in N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   5. Applied rewrites 88.1%

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

   6. Taylor expanded in x.re around 0

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

   8. Applied rewrites 55.7%

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

   if -5.00000017e-317 < (+.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 92.0%

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

   3. Taylor expanded in x.re around inf

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft1-in N/A

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

      7. lower-*.f64 N/A

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

      8. distribute-lft1-in N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 59.0

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

   5. Applied rewrites 59.0%

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

   7. Applied rewrites 66.8%

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

   9. Applied rewrites 66.9%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq -5 \cdot 10^{-317} \lor \neg \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq \infty\right):\\ \;\;\;\;\left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x.re \cdot x.im\right) \cdot x.re\right) \cdot 3\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m + \left(x.re\_m \cdot x.im\_m + x.im\_m \cdot x.re\_m\right) \cdot x.re\_m\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-317} \lor \neg \left(t\_0 \leq \infty\right):\\ \;\;\;\;\left(\left(-x.im\_m\right) \cdot x.im\_m\right) \cdot x.im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x.im\_m \cdot x.re\_m\right) \cdot 3\right) \cdot x.re\_m\\ \end{array} \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* x.re_m x.re_m) (* x.im_m x.im_m)) x.im_m)
          (* (+ (* x.re_m x.im_m) (* x.im_m x.re_m)) x.re_m))))
   (*
    x.im_s
    (if (or (<= t_0 -5e-317) (not (<= t_0 INFINITY)))
      (* (* (- x.im_m) x.im_m) x.im_m)
      (* (* (* x.im_m x.re_m) 3.0) x.re_m)))))

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double t_0 = (((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_re_m);
	double tmp;
	if ((t_0 <= -5e-317) || !(t_0 <= ((double) INFINITY))) {
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	} else {
		tmp = ((x_46_im_m * x_46_re_m) * 3.0) * x_46_re_m;
	}
	return x_46_im_s * tmp;
}

Java

x.re_m = Math.abs(x_46_re);
x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double t_0 = (((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_re_m);
	double tmp;
	if ((t_0 <= -5e-317) || !(t_0 <= Double.POSITIVE_INFINITY)) {
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	} else {
		tmp = ((x_46_im_m * x_46_re_m) * 3.0) * x_46_re_m;
	}
	return x_46_im_s * tmp;
}

Python

x.re_m = math.fabs(x_46_re)
x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re_m, x_46_im_m):
	t_0 = (((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_re_m)
	tmp = 0
	if (t_0 <= -5e-317) or not (t_0 <= math.inf):
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m
	else:
		tmp = ((x_46_im_m * x_46_re_m) * 3.0) * x_46_re_m
	return x_46_im_s * tmp

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	t_0 = Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m) + Float64(Float64(Float64(x_46_re_m * x_46_im_m) + Float64(x_46_im_m * x_46_re_m)) * x_46_re_m))
	tmp = 0.0
	if ((t_0 <= -5e-317) || !(t_0 <= Inf))
		tmp = Float64(Float64(Float64(-x_46_im_m) * x_46_im_m) * x_46_im_m);
	else
		tmp = Float64(Float64(Float64(x_46_im_m * x_46_re_m) * 3.0) * x_46_re_m);
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.re_m = abs(x_46_re);
x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re_m, x_46_im_m)
	t_0 = (((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_re_m);
	tmp = 0.0;
	if ((t_0 <= -5e-317) || ~((t_0 <= Inf)))
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	else
		tmp = ((x_46_im_m * x_46_re_m) * 3.0) * x_46_re_m;
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re$95$m_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] + N[(N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[Or[LessEqual[t$95$0, -5e-317], N[Not[LessEqual[t$95$0, Infinity]], $MachinePrecision]], N[(N[((-x$46$im$95$m) * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision], N[(N[(N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision] * 3.0), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < -5.00000017e-317 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 79.3%

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

   3. Taylor expanded in x.re around 0

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

      1. distribute-rgt-in N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-lft-in N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. distribute-rgt-in N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   5. Applied rewrites 88.1%

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

   6. Taylor expanded in x.re around 0

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

   8. Applied rewrites 55.7%

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

   if -5.00000017e-317 < (+.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 92.0%

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

   3. Taylor expanded in x.re around inf

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft1-in N/A

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

      7. lower-*.f64 N/A

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

      8. distribute-lft1-in N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 59.0

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

   5. Applied rewrites 59.0%

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

   7. Applied rewrites 66.8%

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

   9. Applied rewrites 66.8%

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

   11. Applied rewrites 66.8%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq -5 \cdot 10^{-317} \lor \neg \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq \infty\right):\\ \;\;\;\;\left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x.im \cdot x.re\right) \cdot 3\right) \cdot x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m + \left(x.re\_m \cdot x.im\_m + x.im\_m \cdot x.re\_m\right) \cdot x.re\_m\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-317} \lor \neg \left(t\_0 \leq \infty\right):\\ \;\;\;\;\left(\left(-x.im\_m\right) \cdot x.im\_m\right) \cdot x.im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(x.re\_m \cdot x.im\_m\right) \cdot \left(3 \cdot x.re\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* x.re_m x.re_m) (* x.im_m x.im_m)) x.im_m)
          (* (+ (* x.re_m x.im_m) (* x.im_m x.re_m)) x.re_m))))
   (*
    x.im_s
    (if (or (<= t_0 -5e-317) (not (<= t_0 INFINITY)))
      (* (* (- x.im_m) x.im_m) x.im_m)
      (* (* x.re_m x.im_m) (* 3.0 x.re_m))))))

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double t_0 = (((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_re_m);
	double tmp;
	if ((t_0 <= -5e-317) || !(t_0 <= ((double) INFINITY))) {
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	} else {
		tmp = (x_46_re_m * x_46_im_m) * (3.0 * x_46_re_m);
	}
	return x_46_im_s * tmp;
}

Java

x.re_m = Math.abs(x_46_re);
x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double t_0 = (((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_re_m);
	double tmp;
	if ((t_0 <= -5e-317) || !(t_0 <= Double.POSITIVE_INFINITY)) {
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	} else {
		tmp = (x_46_re_m * x_46_im_m) * (3.0 * x_46_re_m);
	}
	return x_46_im_s * tmp;
}

Python

x.re_m = math.fabs(x_46_re)
x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re_m, x_46_im_m):
	t_0 = (((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_re_m)
	tmp = 0
	if (t_0 <= -5e-317) or not (t_0 <= math.inf):
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m
	else:
		tmp = (x_46_re_m * x_46_im_m) * (3.0 * x_46_re_m)
	return x_46_im_s * tmp

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	t_0 = Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m) + Float64(Float64(Float64(x_46_re_m * x_46_im_m) + Float64(x_46_im_m * x_46_re_m)) * x_46_re_m))
	tmp = 0.0
	if ((t_0 <= -5e-317) || !(t_0 <= Inf))
		tmp = Float64(Float64(Float64(-x_46_im_m) * x_46_im_m) * x_46_im_m);
	else
		tmp = Float64(Float64(x_46_re_m * x_46_im_m) * Float64(3.0 * x_46_re_m));
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.re_m = abs(x_46_re);
x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re_m, x_46_im_m)
	t_0 = (((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_re_m);
	tmp = 0.0;
	if ((t_0 <= -5e-317) || ~((t_0 <= Inf)))
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	else
		tmp = (x_46_re_m * x_46_im_m) * (3.0 * x_46_re_m);
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re$95$m_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] + N[(N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[Or[LessEqual[t$95$0, -5e-317], N[Not[LessEqual[t$95$0, Infinity]], $MachinePrecision]], N[(N[((-x$46$im$95$m) * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision], N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] * N[(3.0 * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < -5.00000017e-317 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 79.3%

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

   3. Taylor expanded in x.re around 0

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

      1. distribute-rgt-in N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-lft-in N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. distribute-rgt-in N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   5. Applied rewrites 88.1%

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

   6. Taylor expanded in x.re around 0

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

   8. Applied rewrites 55.7%

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

   if -5.00000017e-317 < (+.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 92.0%

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

   3. Taylor expanded in x.re around inf

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft1-in N/A

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

      7. lower-*.f64 N/A

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

      8. distribute-lft1-in N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 59.0

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

   5. Applied rewrites 59.0%

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

   7. Applied rewrites 66.8%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq -5 \cdot 10^{-317} \lor \neg \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq \infty\right):\\ \;\;\;\;\left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(3 \cdot x.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m + \left(x.re\_m \cdot x.im\_m + x.im\_m \cdot x.re\_m\right) \cdot x.re\_m\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-317} \lor \neg \left(t\_0 \leq \infty\right):\\ \;\;\;\;\left(\left(-x.im\_m\right) \cdot x.im\_m\right) \cdot x.im\_m\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(\left(3 \cdot x.re\_m\right) \cdot x.im\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* x.re_m x.re_m) (* x.im_m x.im_m)) x.im_m)
          (* (+ (* x.re_m x.im_m) (* x.im_m x.re_m)) x.re_m))))
   (*
    x.im_s
    (if (or (<= t_0 -5e-317) (not (<= t_0 INFINITY)))
      (* (* (- x.im_m) x.im_m) x.im_m)
      (* x.re_m (* (* 3.0 x.re_m) x.im_m))))))

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double t_0 = (((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_re_m);
	double tmp;
	if ((t_0 <= -5e-317) || !(t_0 <= ((double) INFINITY))) {
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	} else {
		tmp = x_46_re_m * ((3.0 * x_46_re_m) * x_46_im_m);
	}
	return x_46_im_s * tmp;
}

Java

x.re_m = Math.abs(x_46_re);
x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double t_0 = (((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_re_m);
	double tmp;
	if ((t_0 <= -5e-317) || !(t_0 <= Double.POSITIVE_INFINITY)) {
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	} else {
		tmp = x_46_re_m * ((3.0 * x_46_re_m) * x_46_im_m);
	}
	return x_46_im_s * tmp;
}

Python

x.re_m = math.fabs(x_46_re)
x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re_m, x_46_im_m):
	t_0 = (((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_re_m)
	tmp = 0
	if (t_0 <= -5e-317) or not (t_0 <= math.inf):
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m
	else:
		tmp = x_46_re_m * ((3.0 * x_46_re_m) * x_46_im_m)
	return x_46_im_s * tmp

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	t_0 = Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m) + Float64(Float64(Float64(x_46_re_m * x_46_im_m) + Float64(x_46_im_m * x_46_re_m)) * x_46_re_m))
	tmp = 0.0
	if ((t_0 <= -5e-317) || !(t_0 <= Inf))
		tmp = Float64(Float64(Float64(-x_46_im_m) * x_46_im_m) * x_46_im_m);
	else
		tmp = Float64(x_46_re_m * Float64(Float64(3.0 * x_46_re_m) * x_46_im_m));
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.re_m = abs(x_46_re);
x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re_m, x_46_im_m)
	t_0 = (((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_re_m);
	tmp = 0.0;
	if ((t_0 <= -5e-317) || ~((t_0 <= Inf)))
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	else
		tmp = x_46_re_m * ((3.0 * x_46_re_m) * x_46_im_m);
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re$95$m_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] + N[(N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[Or[LessEqual[t$95$0, -5e-317], N[Not[LessEqual[t$95$0, Infinity]], $MachinePrecision]], N[(N[((-x$46$im$95$m) * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision], N[(x$46$re$95$m * N[(N[(3.0 * x$46$re$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < -5.00000017e-317 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 79.3%

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

   3. Taylor expanded in x.re around 0

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

      1. distribute-rgt-in N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-lft-in N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. distribute-rgt-in N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   5. Applied rewrites 88.1%

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

   6. Taylor expanded in x.re around 0

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

   8. Applied rewrites 55.7%

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

   if -5.00000017e-317 < (+.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 92.0%

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

   3. Taylor expanded in x.re around inf

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft1-in N/A

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

      7. lower-*.f64 N/A

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

      8. distribute-lft1-in N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 59.0

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

   5. Applied rewrites 59.0%

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

   7. Applied rewrites 66.8%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq -5 \cdot 10^{-317} \lor \neg \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq \infty\right):\\ \;\;\;\;\left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(3 \cdot x.re\right) \cdot x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m + \left(x.re\_m \cdot x.im\_m + x.im\_m \cdot x.re\_m\right) \cdot x.re\_m\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-317} \lor \neg \left(t\_0 \leq \infty\right):\\ \;\;\;\;\left(\left(-x.im\_m\right) \cdot x.im\_m\right) \cdot x.im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(x.re\_m \cdot x.re\_m\right) \cdot x.im\_m\\ \end{array} \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re_m x.im_m)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* x.re_m x.re_m) (* x.im_m x.im_m)) x.im_m)
          (* (+ (* x.re_m x.im_m) (* x.im_m x.re_m)) x.re_m))))
   (*
    x.im_s
    (if (or (<= t_0 -5e-317) (not (<= t_0 INFINITY)))
      (* (* (- x.im_m) x.im_m) x.im_m)
      (* (* x.re_m x.re_m) x.im_m)))))

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double t_0 = (((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_re_m);
	double tmp;
	if ((t_0 <= -5e-317) || !(t_0 <= ((double) INFINITY))) {
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	} else {
		tmp = (x_46_re_m * x_46_re_m) * x_46_im_m;
	}
	return x_46_im_s * tmp;
}

Java

x.re_m = Math.abs(x_46_re);
x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	double t_0 = (((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_re_m);
	double tmp;
	if ((t_0 <= -5e-317) || !(t_0 <= Double.POSITIVE_INFINITY)) {
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	} else {
		tmp = (x_46_re_m * x_46_re_m) * x_46_im_m;
	}
	return x_46_im_s * tmp;
}

Python

x.re_m = math.fabs(x_46_re)
x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re_m, x_46_im_m):
	t_0 = (((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_re_m)
	tmp = 0
	if (t_0 <= -5e-317) or not (t_0 <= math.inf):
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m
	else:
		tmp = (x_46_re_m * x_46_re_m) * x_46_im_m
	return x_46_im_s * tmp

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	t_0 = Float64(Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m) + Float64(Float64(Float64(x_46_re_m * x_46_im_m) + Float64(x_46_im_m * x_46_re_m)) * x_46_re_m))
	tmp = 0.0
	if ((t_0 <= -5e-317) || !(t_0 <= Inf))
		tmp = Float64(Float64(Float64(-x_46_im_m) * x_46_im_m) * x_46_im_m);
	else
		tmp = Float64(Float64(x_46_re_m * x_46_re_m) * x_46_im_m);
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

x.re_m = abs(x_46_re);
x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp_2 = code(x_46_im_s, x_46_re_m, x_46_im_m)
	t_0 = (((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_im_m) + (((x_46_re_m * x_46_im_m) + (x_46_im_m * x_46_re_m)) * x_46_re_m);
	tmp = 0.0;
	if ((t_0 <= -5e-317) || ~((t_0 <= Inf)))
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	else
		tmp = (x_46_re_m * x_46_re_m) * x_46_im_m;
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re$95$m_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] + N[(N[(N[(x$46$re$95$m * x$46$im$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[Or[LessEqual[t$95$0, -5e-317], N[Not[LessEqual[t$95$0, Infinity]], $MachinePrecision]], N[(N[((-x$46$im$95$m) * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision], N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < -5.00000017e-317 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 79.3%

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

   3. Taylor expanded in x.re around 0

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

      1. distribute-rgt-in N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-lft-in N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. distribute-rgt-in N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   5. Applied rewrites 88.1%

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

   6. Taylor expanded in x.re around 0

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

   8. Applied rewrites 55.7%

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

   if -5.00000017e-317 < (+.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 92.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 99.9

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

   6. Applied rewrites 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 99.9

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.9

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 99.9

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. distribute-lft-in N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. flip-+ N/A

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

      19. +-inverses N/A

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

      20. +-inverses N/A

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

      21. associate-*r/ N/A

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

   8. Applied rewrites 46.8%

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

   9. Taylor expanded in x.re around inf

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

       1. +-commutative N/A

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

       2. distribute-rgt1-in N/A

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

       3. metadata-eval N/A

          \[\leadsto {x.re}^{2} \cdot \left(\frac{x.im \cdot \left(\color{blue}{0} \cdot x.im\right)}{x.re} + x.im\right) \]

       4. mul0-lft N/A

          \[\leadsto {x.re}^{2} \cdot \left(\frac{x.im \cdot \color{blue}{0}}{x.re} + x.im\right) \]

       5. *-commutative N/A

          \[\leadsto {x.re}^{2} \cdot \left(\frac{\color{blue}{0 \cdot x.im}}{x.re} + x.im\right) \]

       6. mul0-lft N/A

          \[\leadsto {x.re}^{2} \cdot \left(\frac{\color{blue}{0}}{x.re} + x.im\right) \]

       7. div0 N/A

          \[\leadsto {x.re}^{2} \cdot \left(\color{blue}{0} + x.im\right) \]

       8. distribute-rgt-in N/A

          \[\leadsto \color{blue}{0 \cdot {x.re}^{2} + x.im \cdot {x.re}^{2}} \]

       9. mul0-lft N/A

          \[\leadsto \color{blue}{0} + x.im \cdot {x.re}^{2} \]

       10. +-lft-identity N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 43.0

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

   11. Applied rewrites 43.0%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq -5 \cdot 10^{-317} \lor \neg \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \leq \infty\right):\\ \;\;\;\;\left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.im\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 35.3% accurate, 3.6× speedup

Math

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

FPCore

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

C

x.re_m = fabs(x_46_re);
x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	return x_46_im_s * ((x_46_re_m * x_46_re_m) * x_46_im_m);
}

Fortran

x.re_m = abs(x_46re)
x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re_m, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im_m
    code = x_46im_s * ((x_46re_m * x_46re_m) * x_46im_m)
end function

Java

x.re_m = Math.abs(x_46_re);
x.im\_m = Math.abs(x_46_im);
x.im\_s = Math.copySign(1.0, x_46_im);
public static double code(double x_46_im_s, double x_46_re_m, double x_46_im_m) {
	return x_46_im_s * ((x_46_re_m * x_46_re_m) * x_46_im_m);
}

Python

x.re_m = math.fabs(x_46_re)
x.im\_m = math.fabs(x_46_im)
x.im\_s = math.copysign(1.0, x_46_im)
def code(x_46_im_s, x_46_re_m, x_46_im_m):
	return x_46_im_s * ((x_46_re_m * x_46_re_m) * x_46_im_m)

Julia

x.re_m = abs(x_46_re)
x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re_m, x_46_im_m)
	return Float64(x_46_im_s * Float64(Float64(x_46_re_m * x_46_re_m) * x_46_im_m))
end

MATLAB

x.re_m = abs(x_46_re);
x.im\_m = abs(x_46_im);
x.im\_s = sign(x_46_im) * abs(1.0);
function tmp = code(x_46_im_s, x_46_re_m, x_46_im_m)
	tmp = x_46_im_s * ((x_46_re_m * x_46_re_m) * x_46_im_m);
end

Wolfram

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

Derivation

1. Initial program 85.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. difference-of-squares N/A

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

   7. associate-*r* N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 N/A

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

   12. lower--.f64 93.2

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

4. Applied rewrites 93.2%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. distribute-lft-out N/A

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

   6. lower-*.f64 N/A

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

   7. lower-+.f64 93.2

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

6. Applied rewrites 93.2%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 93.2

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 93.2

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 93.2

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. lift-+.f64 N/A

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

   15. distribute-lft-in N/A

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

   16. lift-*.f64 N/A

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

   17. lift-*.f64 N/A

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

   18. flip-+ N/A

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

   19. +-inverses N/A

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

   20. +-inverses N/A

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

   21. associate-*r/ N/A

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

8. Applied rewrites 58.1%

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

9. Taylor expanded in x.re around inf

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

    1. +-commutative N/A

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

    2. distribute-rgt1-in N/A

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

    3. metadata-eval N/A

       \[\leadsto {x.re}^{2} \cdot \left(\frac{x.im \cdot \left(\color{blue}{0} \cdot x.im\right)}{x.re} + x.im\right) \]

    4. mul0-lft N/A

       \[\leadsto {x.re}^{2} \cdot \left(\frac{x.im \cdot \color{blue}{0}}{x.re} + x.im\right) \]

    5. *-commutative N/A

       \[\leadsto {x.re}^{2} \cdot \left(\frac{\color{blue}{0 \cdot x.im}}{x.re} + x.im\right) \]

    6. mul0-lft N/A

       \[\leadsto {x.re}^{2} \cdot \left(\frac{\color{blue}{0}}{x.re} + x.im\right) \]

    7. div0 N/A

       \[\leadsto {x.re}^{2} \cdot \left(\color{blue}{0} + x.im\right) \]

    8. distribute-rgt-in N/A

       \[\leadsto \color{blue}{0 \cdot {x.re}^{2} + x.im \cdot {x.re}^{2}} \]

    9. mul0-lft N/A

       \[\leadsto \color{blue}{0} + x.im \cdot {x.re}^{2} \]

    10. +-lft-identity N/A

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

    11. *-commutative N/A

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

    12. lower-*.f64 N/A

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

    13. unpow2 N/A

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

    14. lower-*.f64 32.9

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

11. Applied rewrites 32.9%

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

Developer Target 1: 91.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(x.re \cdot x.im\right) \cdot \left(2 \cdot x.re\right) + \left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.re + x.im\right) \end{array} \]

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)
    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 2024298 
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"
  :precision binary64

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