math.cube on complex, imaginary part

Percentage Accurate: 83.5% → 99.8%

Time: 9.7s

Alternatives: 10

Speedup: 0.5×

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

Specification

Math

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

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: 83.5% 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: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(x.im\_m \cdot x.re\_m + x.im\_m \cdot x.re\_m\right) \cdot x.re\_m + \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 10^{+189}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m \cdot x.re\_m, 2 \cdot x.im\_m, \left(\left(x.im\_m + x.re\_m\right) \cdot x.im\_m\right) \cdot \left(x.re\_m - x.im\_m\right)\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left(3 \cdot \left(x.im\_m \cdot x.re\_m\right)\right) \cdot x.re\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m - x.im\_m, x.im\_m + x.re\_m, 2\right) \cdot x.im\_m\\ \end{array} \end{array} \end{array} \]

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.im_m x.re_m) (* x.im_m x.re_m)) x.re_m)
          (* (- (* x.re_m x.re_m) (* x.im_m x.im_m)) x.im_m))))
   (*
    x.im_s
    (if (<= t_0 1e+189)
      (fma
       (* x.re_m x.re_m)
       (* 2.0 x.im_m)
       (* (* (+ x.im_m x.re_m) x.im_m) (- x.re_m x.im_m)))
      (if (<= t_0 INFINITY)
        (* (* 3.0 (* x.im_m x.re_m)) x.re_m)
        (* (fma (- x.re_m x.im_m) (+ x.im_m x.re_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_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)) * x_46_re_m) + (((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_im_m);
	double tmp;
	if (t_0 <= 1e+189) {
		tmp = fma((x_46_re_m * x_46_re_m), (2.0 * x_46_im_m), (((x_46_im_m + x_46_re_m) * x_46_im_m) * (x_46_re_m - x_46_im_m)));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (3.0 * (x_46_im_m * x_46_re_m)) * x_46_re_m;
	} else {
		tmp = fma((x_46_re_m - x_46_im_m), (x_46_im_m + x_46_re_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_im_m * x_46_re_m) + Float64(x_46_im_m * x_46_re_m)) * x_46_re_m) + Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m))
	tmp = 0.0
	if (t_0 <= 1e+189)
		tmp = fma(Float64(x_46_re_m * x_46_re_m), Float64(2.0 * x_46_im_m), Float64(Float64(Float64(x_46_im_m + x_46_re_m) * x_46_im_m) * Float64(x_46_re_m - x_46_im_m)));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(3.0 * Float64(x_46_im_m * x_46_re_m)) * x_46_re_m);
	else
		tmp = Float64(fma(Float64(x_46_re_m - x_46_im_m), Float64(x_46_im_m + x_46_re_m), 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$46$im$95$m * x$46$re$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] + 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]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$0, 1e+189], N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] * N[(2.0 * x$46$im$95$m), $MachinePrecision] + N[(N[(N[(x$46$im$95$m + x$46$re$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision] * N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(3.0 * N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$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] + 2.0), $MachinePrecision] * x$46$im$95$m), $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)) < 1e189

   1. Initial program 94.4%

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

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. difference-of-squares N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 99.8

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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 94.5%

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

   if 1e189 < (+.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 94.5%

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

   3. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. *-inverses N/A

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

      4. associate-/l* N/A

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

      5. unpow2 N/A

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

      6. cube-mult N/A

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

      7. associate-/l* N/A

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

      8. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {x.re}^{2} + {x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3}} \]

      9. distribute-lft1-in N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{3} \cdot {x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3} \]

      11. associate-*r/ N/A

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

      12. associate-*l* N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. distribute-lft-neg-in N/A

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

      16. *-commutative N/A

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

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

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

   5. Applied rewrites 35.5%

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

   7. Applied rewrites 40.9%

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

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

   1. Initial program 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. Taylor expanded in x.im around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 29.0

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

   5. Applied rewrites 29.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 29.0

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. flip-+ N/A

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

      11. +-inverses N/A

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

      12. +-inverses N/A

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

      13. +-inverses N/A

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

      14. +-inverses N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. flip-+ N/A

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

      18. distribute-lft-in N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

   7. Applied rewrites 29.0%

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

   8. Taylor expanded in x.re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 29.0

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

   10. Applied rewrites 29.0%

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

   11. Taylor expanded in x.im around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. +-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. cancel-sign-sub N/A

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

       9. unpow2 N/A

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

       10. mul-1-neg N/A

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

       11. difference-of-squares N/A

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

       12. mul-1-neg N/A

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

       13. sub-neg N/A

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

       14. unsub-neg N/A

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

       15. mul-1-neg N/A

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

       16. remove-double-neg N/A

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

       17. lower-fma.f64 N/A

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

       18. lower--.f64 N/A

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

       19. lower-+.f64 100.0

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

   13. Applied rewrites 100.0%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im \leq 10^{+189}:\\ \;\;\;\;\mathsf{fma}\left(x.re \cdot x.re, 2 \cdot x.im, \left(\left(x.im + x.re\right) \cdot x.im\right) \cdot \left(x.re - x.im\right)\right)\\ \mathbf{elif}\;\left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im \leq \infty:\\ \;\;\;\;\left(3 \cdot \left(x.im \cdot x.re\right)\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im, x.im + x.re, 2\right) \cdot x.im\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(x.im\_m \cdot x.re\_m + x.im\_m \cdot x.re\_m\right) \cdot x.re\_m + \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(\left(-x.im\_m\right) \cdot x.im\_m, x.im\_m, \left(3 \cdot \left(x.re\_m \cdot x.re\_m\right)\right) \cdot x.im\_m\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left(\left(x.im\_m \cdot x.re\_m\right) \cdot x.re\_m\right) \cdot 3\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m - x.im\_m, x.im\_m + x.re\_m, 2\right) \cdot x.im\_m\\ \end{array} \end{array} \end{array} \]

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.im_m x.re_m) (* x.im_m x.re_m)) x.re_m)
          (* (- (* x.re_m x.re_m) (* x.im_m x.im_m)) x.im_m))))
   (*
    x.im_s
    (if (<= t_0 2e-30)
      (fma (* (- x.im_m) x.im_m) x.im_m (* (* 3.0 (* x.re_m x.re_m)) x.im_m))
      (if (<= t_0 INFINITY)
        (* (* (* x.im_m x.re_m) x.re_m) 3.0)
        (* (fma (- x.re_m x.im_m) (+ x.im_m x.re_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_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)) * x_46_re_m) + (((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_im_m);
	double tmp;
	if (t_0 <= 2e-30) {
		tmp = fma((-x_46_im_m * 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_im_m * x_46_re_m) * x_46_re_m) * 3.0;
	} else {
		tmp = fma((x_46_re_m - x_46_im_m), (x_46_im_m + x_46_re_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_im_m * x_46_re_m) + Float64(x_46_im_m * x_46_re_m)) * x_46_re_m) + Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m))
	tmp = 0.0
	if (t_0 <= 2e-30)
		tmp = fma(Float64(Float64(-x_46_im_m) * x_46_im_m), x_46_im_m, Float64(Float64(3.0 * Float64(x_46_re_m * x_46_re_m)) * x_46_im_m));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(Float64(x_46_im_m * x_46_re_m) * x_46_re_m) * 3.0);
	else
		tmp = Float64(fma(Float64(x_46_re_m - x_46_im_m), Float64(x_46_im_m + x_46_re_m), 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$46$im$95$m * x$46$re$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] + 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]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$0, 2e-30], N[(N[((-x$46$im$95$m) * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m + N[(N[(3.0 * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] * 3.0), $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] + 2.0), $MachinePrecision] * x$46$im$95$m), $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)) < 2e-30

   1. Initial program 94.0%

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

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

      3. flip-- N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow-prod-down N/A

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

      11. pow-prod-up N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval N/A

          \[\leadsto \frac{\left({x.re}^{\color{blue}{4}} - \left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right)\right) \cdot x.im}{x.re \cdot x.re + x.im \cdot x.im} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. pow-prod-down N/A

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

      17. pow-prod-up N/A

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

      18. lower-pow.f64 N/A

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

      19. metadata-eval N/A

          \[\leadsto \frac{\left({x.re}^{4} - {x.im}^{\color{blue}{4}}\right) \cdot x.im}{x.re \cdot x.re + x.im \cdot x.im} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

   4. Applied rewrites 49.2%

      \[\leadsto \color{blue}{\frac{\left({x.re}^{4} - {x.im}^{4}\right) \cdot x.im}{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

   5. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 94.0

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

   7. Applied rewrites 94.0%

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

   9. Applied rewrites 94.0%

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

   if 2e-30 < (+.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 95.5%

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

   3. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. *-inverses N/A

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

      4. associate-/l* N/A

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

      5. unpow2 N/A

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

      6. cube-mult N/A

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

      7. associate-/l* N/A

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

      8. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {x.re}^{2} + {x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3}} \]

      9. distribute-lft1-in N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{3} \cdot {x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3} \]

      11. associate-*r/ N/A

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

      12. associate-*l* N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. distribute-lft-neg-in N/A

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

      16. *-commutative N/A

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

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

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

   5. Applied rewrites 35.3%

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

   7. Applied rewrites 39.7%

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

   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. Taylor expanded in x.im around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 29.0

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

   5. Applied rewrites 29.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 29.0

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. flip-+ N/A

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

      11. +-inverses N/A

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

      12. +-inverses N/A

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

      13. +-inverses N/A

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

      14. +-inverses N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. flip-+ N/A

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

      18. distribute-lft-in N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

   7. Applied rewrites 29.0%

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

   8. Taylor expanded in x.re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 29.0

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

   10. Applied rewrites 29.0%

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

   11. Taylor expanded in x.im around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. +-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. cancel-sign-sub N/A

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

       9. unpow2 N/A

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

       10. mul-1-neg N/A

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

       11. difference-of-squares N/A

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

       12. mul-1-neg N/A

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

       13. sub-neg N/A

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

       14. unsub-neg N/A

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

       15. mul-1-neg N/A

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

       16. remove-double-neg N/A

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

       17. lower-fma.f64 N/A

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

       18. lower--.f64 N/A

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

       19. lower-+.f64 100.0

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

   13. Applied rewrites 100.0%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im \leq 2 \cdot 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(\left(-x.im\right) \cdot x.im, x.im, \left(3 \cdot \left(x.re \cdot x.re\right)\right) \cdot x.im\right)\\ \mathbf{elif}\;\left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im \leq \infty:\\ \;\;\;\;\left(\left(x.im \cdot x.re\right) \cdot x.re\right) \cdot 3\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im, x.im + x.re, 2\right) \cdot x.im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(x.im\_m \cdot x.re\_m + x.im\_m \cdot x.re\_m\right) \cdot x.re\_m + \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(-3, x.re\_m \cdot x.re\_m, x.im\_m \cdot x.im\_m\right) \cdot \left(-x.im\_m\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left(\left(x.im\_m \cdot x.re\_m\right) \cdot x.re\_m\right) \cdot 3\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m - x.im\_m, x.im\_m + x.re\_m, 2\right) \cdot x.im\_m\\ \end{array} \end{array} \end{array} \]

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.im_m x.re_m) (* x.im_m x.re_m)) x.re_m)
          (* (- (* x.re_m x.re_m) (* x.im_m x.im_m)) x.im_m))))
   (*
    x.im_s
    (if (<= t_0 2e-30)
      (* (fma -3.0 (* x.re_m x.re_m) (* x.im_m x.im_m)) (- x.im_m))
      (if (<= t_0 INFINITY)
        (* (* (* x.im_m x.re_m) x.re_m) 3.0)
        (* (fma (- x.re_m x.im_m) (+ x.im_m x.re_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_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)) * x_46_re_m) + (((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_im_m);
	double tmp;
	if (t_0 <= 2e-30) {
		tmp = fma(-3.0, (x_46_re_m * x_46_re_m), (x_46_im_m * x_46_im_m)) * -x_46_im_m;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = ((x_46_im_m * x_46_re_m) * x_46_re_m) * 3.0;
	} else {
		tmp = fma((x_46_re_m - x_46_im_m), (x_46_im_m + x_46_re_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_im_m * x_46_re_m) + Float64(x_46_im_m * x_46_re_m)) * x_46_re_m) + Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m))
	tmp = 0.0
	if (t_0 <= 2e-30)
		tmp = Float64(fma(-3.0, Float64(x_46_re_m * x_46_re_m), Float64(x_46_im_m * x_46_im_m)) * Float64(-x_46_im_m));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(Float64(x_46_im_m * x_46_re_m) * x_46_re_m) * 3.0);
	else
		tmp = Float64(fma(Float64(x_46_re_m - x_46_im_m), Float64(x_46_im_m + x_46_re_m), 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$46$im$95$m * x$46$re$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] + 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]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$0, 2e-30], N[(N[(-3.0 * 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], If[LessEqual[t$95$0, Infinity], N[(N[(N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] * 3.0), $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] + 2.0), $MachinePrecision] * x$46$im$95$m), $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)) < 2e-30

   1. Initial program 94.0%

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

   3. Taylor expanded in x.im around 0

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

   5. Applied rewrites 94.0%

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

   if 2e-30 < (+.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 95.5%

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

   3. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. *-inverses N/A

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

      4. associate-/l* N/A

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

      5. unpow2 N/A

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

      6. cube-mult N/A

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

      7. associate-/l* N/A

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

      8. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {x.re}^{2} + {x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3}} \]

      9. distribute-lft1-in N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{3} \cdot {x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3} \]

      11. associate-*r/ N/A

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

      12. associate-*l* N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. distribute-lft-neg-in N/A

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

      16. *-commutative N/A

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

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

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

   5. Applied rewrites 35.3%

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

   7. Applied rewrites 39.7%

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

   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. Taylor expanded in x.im around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 29.0

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

   5. Applied rewrites 29.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 29.0

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. flip-+ N/A

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

      11. +-inverses N/A

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

      12. +-inverses N/A

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

      13. +-inverses N/A

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

      14. +-inverses N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. flip-+ N/A

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

      18. distribute-lft-in N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

   7. Applied rewrites 29.0%

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

   8. Taylor expanded in x.re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 29.0

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

   10. Applied rewrites 29.0%

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

   11. Taylor expanded in x.im around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. +-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. cancel-sign-sub N/A

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

       9. unpow2 N/A

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

       10. mul-1-neg N/A

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

       11. difference-of-squares N/A

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

       12. mul-1-neg N/A

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

       13. sub-neg N/A

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

       14. unsub-neg N/A

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

       15. mul-1-neg N/A

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

       16. remove-double-neg N/A

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

       17. lower-fma.f64 N/A

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

       18. lower--.f64 N/A

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

       19. lower-+.f64 100.0

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

   13. Applied rewrites 100.0%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im \leq 2 \cdot 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(-3, x.re \cdot x.re, x.im \cdot x.im\right) \cdot \left(-x.im\right)\\ \mathbf{elif}\;\left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im \leq \infty:\\ \;\;\;\;\left(\left(x.im \cdot x.re\right) \cdot x.re\right) \cdot 3\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im, x.im + x.re, 2\right) \cdot x.im\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(x.im\_m \cdot x.re\_m + x.im\_m \cdot x.re\_m\right) \cdot x.re\_m + \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-308}:\\ \;\;\;\;\left(\left(-x.im\_m\right) \cdot x.im\_m\right) \cdot x.im\_m\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left(\left(x.im\_m \cdot x.re\_m\right) \cdot x.re\_m\right) \cdot 3\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m - x.im\_m, x.im\_m + x.re\_m, 2\right) \cdot x.im\_m\\ \end{array} \end{array} \end{array} \]

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.im_m x.re_m) (* x.im_m x.re_m)) x.re_m)
          (* (- (* x.re_m x.re_m) (* x.im_m x.im_m)) x.im_m))))
   (*
    x.im_s
    (if (<= t_0 -1e-308)
      (* (* (- x.im_m) x.im_m) x.im_m)
      (if (<= t_0 INFINITY)
        (* (* (* x.im_m x.re_m) x.re_m) 3.0)
        (* (fma (- x.re_m x.im_m) (+ x.im_m x.re_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_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)) * x_46_re_m) + (((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_im_m);
	double tmp;
	if (t_0 <= -1e-308) {
		tmp = (-x_46_im_m * x_46_im_m) * x_46_im_m;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = ((x_46_im_m * x_46_re_m) * x_46_re_m) * 3.0;
	} else {
		tmp = fma((x_46_re_m - x_46_im_m), (x_46_im_m + x_46_re_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_im_m * x_46_re_m) + Float64(x_46_im_m * x_46_re_m)) * x_46_re_m) + Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m))
	tmp = 0.0
	if (t_0 <= -1e-308)
		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_im_m * x_46_re_m) * x_46_re_m) * 3.0);
	else
		tmp = Float64(fma(Float64(x_46_re_m - x_46_im_m), Float64(x_46_im_m + x_46_re_m), 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$46$im$95$m * x$46$re$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] + 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]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$0, -1e-308], 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$46$im$95$m * x$46$re$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] * 3.0), $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] + 2.0), $MachinePrecision] * x$46$im$95$m), $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)) < -9.9999999999999991e-309

   1. Initial program 91.1%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow-prod-down N/A

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

      11. pow-prod-up N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval N/A

          \[\leadsto \frac{\left({x.re}^{\color{blue}{4}} - \left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right)\right) \cdot x.im}{x.re \cdot x.re + x.im \cdot x.im} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. pow-prod-down N/A

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

      17. pow-prod-up N/A

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

      18. lower-pow.f64 N/A

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

      19. metadata-eval N/A

          \[\leadsto \frac{\left({x.re}^{4} - {x.im}^{\color{blue}{4}}\right) \cdot x.im}{x.re \cdot x.re + x.im \cdot x.im} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

   4. Applied rewrites 41.2%

      \[\leadsto \color{blue}{\frac{\left({x.re}^{4} - {x.im}^{4}\right) \cdot x.im}{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

   5. Taylor expanded in x.im around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-pow.f64 42.7

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

   7. Applied rewrites 42.7%

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

   9. Applied rewrites 42.7%

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

   if -9.9999999999999991e-309 < (+.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 97.5%

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

   3. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. *-inverses N/A

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

      4. associate-/l* N/A

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

      5. unpow2 N/A

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

      6. cube-mult N/A

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

      7. associate-/l* N/A

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

      8. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {x.re}^{2} + {x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3}} \]

      9. distribute-lft1-in N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{3} \cdot {x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3} \]

      11. associate-*r/ N/A

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

      12. associate-*l* N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. distribute-lft-neg-in N/A

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

      16. *-commutative N/A

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

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

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

   5. Applied rewrites 58.1%

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

   7. Applied rewrites 60.6%

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

   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. Taylor expanded in x.im around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 29.0

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

   5. Applied rewrites 29.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 29.0

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. flip-+ N/A

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

      11. +-inverses N/A

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

      12. +-inverses N/A

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

      13. +-inverses N/A

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

      14. +-inverses N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. flip-+ N/A

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

      18. distribute-lft-in N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

   7. Applied rewrites 29.0%

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

   8. Taylor expanded in x.re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 29.0

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

   10. Applied rewrites 29.0%

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

   11. Taylor expanded in x.im around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. +-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. cancel-sign-sub N/A

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

       9. unpow2 N/A

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

       10. mul-1-neg N/A

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

       11. difference-of-squares N/A

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

       12. mul-1-neg N/A

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

       13. sub-neg N/A

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

       14. unsub-neg N/A

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

       15. mul-1-neg N/A

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

       16. remove-double-neg N/A

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

       17. lower-fma.f64 N/A

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

       18. lower--.f64 N/A

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

       19. lower-+.f64 100.0

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

   13. Applied rewrites 100.0%

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

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im \leq -1 \cdot 10^{-308}:\\ \;\;\;\;\left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \mathbf{elif}\;\left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im \leq \infty:\\ \;\;\;\;\left(\left(x.im \cdot x.re\right) \cdot x.re\right) \cdot 3\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im, x.im + x.re, 2\right) \cdot x.im\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(\left(-x.im\_m\right) \cdot x.im\_m\right) \cdot x.im\_m\\ t_1 := \left(x.im\_m \cdot x.re\_m + x.im\_m \cdot x.re\_m\right) \cdot x.re\_m + \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-308}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left(\left(x.im\_m \cdot x.re\_m\right) \cdot x.re\_m\right) \cdot 3\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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.im_m) x.im_m) x.im_m))
        (t_1
         (+
          (* (+ (* x.im_m x.re_m) (* x.im_m x.re_m)) x.re_m)
          (* (- (* x.re_m x.re_m) (* x.im_m x.im_m)) x.im_m))))
   (*
    x.im_s
    (if (<= t_1 -1e-308)
      t_0
      (if (<= t_1 INFINITY) (* (* (* x.im_m x.re_m) x.re_m) 3.0) t_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_im_m * x_46_im_m) * x_46_im_m;
	double t_1 = (((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)) * x_46_re_m) + (((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_im_m);
	double tmp;
	if (t_1 <= -1e-308) {
		tmp = t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = ((x_46_im_m * x_46_re_m) * x_46_re_m) * 3.0;
	} else {
		tmp = t_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_im_m * x_46_im_m) * x_46_im_m;
	double t_1 = (((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)) * x_46_re_m) + (((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_im_m);
	double tmp;
	if (t_1 <= -1e-308) {
		tmp = t_0;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = ((x_46_im_m * x_46_re_m) * x_46_re_m) * 3.0;
	} else {
		tmp = t_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_im_m * x_46_im_m) * x_46_im_m
	t_1 = (((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)) * x_46_re_m) + (((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_im_m)
	tmp = 0
	if t_1 <= -1e-308:
		tmp = t_0
	elif t_1 <= math.inf:
		tmp = ((x_46_im_m * x_46_re_m) * x_46_re_m) * 3.0
	else:
		tmp = t_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(-x_46_im_m) * x_46_im_m) * x_46_im_m)
	t_1 = Float64(Float64(Float64(Float64(x_46_im_m * x_46_re_m) + Float64(x_46_im_m * x_46_re_m)) * x_46_re_m) + Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m))
	tmp = 0.0
	if (t_1 <= -1e-308)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(x_46_im_m * x_46_re_m) * x_46_re_m) * 3.0);
	else
		tmp = t_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_im_m * x_46_im_m) * x_46_im_m;
	t_1 = (((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)) * x_46_re_m) + (((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_im_m);
	tmp = 0.0;
	if (t_1 <= -1e-308)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = ((x_46_im_m * x_46_re_m) * x_46_re_m) * 3.0;
	else
		tmp = t_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[((-x$46$im$95$m) * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] + 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]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$1, -1e-308], t$95$0, If[LessEqual[t$95$1, Infinity], N[(N[(N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] * 3.0), $MachinePrecision], t$95$0]]), $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)) < -9.9999999999999991e-309 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 70.8%

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

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

      3. flip-- N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow-prod-down N/A

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

      11. pow-prod-up N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval N/A

          \[\leadsto \frac{\left({x.re}^{\color{blue}{4}} - \left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right)\right) \cdot x.im}{x.re \cdot x.re + x.im \cdot x.im} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. pow-prod-down N/A

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

      17. pow-prod-up N/A

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

      18. lower-pow.f64 N/A

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

      19. metadata-eval N/A

          \[\leadsto \frac{\left({x.re}^{4} - {x.im}^{\color{blue}{4}}\right) \cdot x.im}{x.re \cdot x.re + x.im \cdot x.im} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

   4. Applied rewrites 32.0%

      \[\leadsto \color{blue}{\frac{\left({x.re}^{4} - {x.im}^{4}\right) \cdot x.im}{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

   5. Taylor expanded in x.im around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-pow.f64 49.0

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

   7. Applied rewrites 49.0%

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

   9. Applied rewrites 49.0%

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

   if -9.9999999999999991e-309 < (+.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 97.5%

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

   3. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. *-inverses N/A

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

      4. associate-/l* N/A

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

      5. unpow2 N/A

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

      6. cube-mult N/A

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

      7. associate-/l* N/A

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

      8. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {x.re}^{2} + {x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3}} \]

      9. distribute-lft1-in N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{3} \cdot {x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3} \]

      11. associate-*r/ N/A

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

      12. associate-*l* N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. distribute-lft-neg-in N/A

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

      16. *-commutative N/A

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

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

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

   5. Applied rewrites 58.1%

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

   7. Applied rewrites 60.6%

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

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im \leq -1 \cdot 10^{-308}:\\ \;\;\;\;\left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \mathbf{elif}\;\left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im \leq \infty:\\ \;\;\;\;\left(\left(x.im \cdot x.re\right) \cdot x.re\right) \cdot 3\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(\left(-x.im\_m\right) \cdot x.im\_m\right) \cdot x.im\_m\\ t_1 := \left(x.im\_m \cdot x.re\_m + x.im\_m \cdot x.re\_m\right) \cdot x.re\_m + \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-308}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left(3 \cdot \left(x.im\_m \cdot x.re\_m\right)\right) \cdot x.re\_m\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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.im_m) x.im_m) x.im_m))
        (t_1
         (+
          (* (+ (* x.im_m x.re_m) (* x.im_m x.re_m)) x.re_m)
          (* (- (* x.re_m x.re_m) (* x.im_m x.im_m)) x.im_m))))
   (*
    x.im_s
    (if (<= t_1 -1e-308)
      t_0
      (if (<= t_1 INFINITY) (* (* 3.0 (* x.im_m x.re_m)) x.re_m) t_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_im_m * x_46_im_m) * x_46_im_m;
	double t_1 = (((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)) * x_46_re_m) + (((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_im_m);
	double tmp;
	if (t_1 <= -1e-308) {
		tmp = t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (3.0 * (x_46_im_m * x_46_re_m)) * x_46_re_m;
	} else {
		tmp = t_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_im_m * x_46_im_m) * x_46_im_m;
	double t_1 = (((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)) * x_46_re_m) + (((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_im_m);
	double tmp;
	if (t_1 <= -1e-308) {
		tmp = t_0;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (3.0 * (x_46_im_m * x_46_re_m)) * x_46_re_m;
	} else {
		tmp = t_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_im_m * x_46_im_m) * x_46_im_m
	t_1 = (((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)) * x_46_re_m) + (((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_im_m)
	tmp = 0
	if t_1 <= -1e-308:
		tmp = t_0
	elif t_1 <= math.inf:
		tmp = (3.0 * (x_46_im_m * x_46_re_m)) * x_46_re_m
	else:
		tmp = t_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(-x_46_im_m) * x_46_im_m) * x_46_im_m)
	t_1 = Float64(Float64(Float64(Float64(x_46_im_m * x_46_re_m) + Float64(x_46_im_m * x_46_re_m)) * x_46_re_m) + Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m))
	tmp = 0.0
	if (t_1 <= -1e-308)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(3.0 * Float64(x_46_im_m * x_46_re_m)) * x_46_re_m);
	else
		tmp = t_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_im_m * x_46_im_m) * x_46_im_m;
	t_1 = (((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)) * x_46_re_m) + (((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_im_m);
	tmp = 0.0;
	if (t_1 <= -1e-308)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = (3.0 * (x_46_im_m * x_46_re_m)) * x_46_re_m;
	else
		tmp = t_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[((-x$46$im$95$m) * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] + 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]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$1, -1e-308], t$95$0, If[LessEqual[t$95$1, Infinity], N[(N[(3.0 * N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision], t$95$0]]), $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)) < -9.9999999999999991e-309 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 70.8%

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

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

      3. flip-- N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow-prod-down N/A

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

      11. pow-prod-up N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval N/A

          \[\leadsto \frac{\left({x.re}^{\color{blue}{4}} - \left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right)\right) \cdot x.im}{x.re \cdot x.re + x.im \cdot x.im} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. pow-prod-down N/A

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

      17. pow-prod-up N/A

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

      18. lower-pow.f64 N/A

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

      19. metadata-eval N/A

          \[\leadsto \frac{\left({x.re}^{4} - {x.im}^{\color{blue}{4}}\right) \cdot x.im}{x.re \cdot x.re + x.im \cdot x.im} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

   4. Applied rewrites 32.0%

      \[\leadsto \color{blue}{\frac{\left({x.re}^{4} - {x.im}^{4}\right) \cdot x.im}{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

   5. Taylor expanded in x.im around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-pow.f64 49.0

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

   7. Applied rewrites 49.0%

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

   9. Applied rewrites 49.0%

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

   if -9.9999999999999991e-309 < (+.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 97.5%

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

   3. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. *-inverses N/A

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

      4. associate-/l* N/A

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

      5. unpow2 N/A

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

      6. cube-mult N/A

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

      7. associate-/l* N/A

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

      8. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {x.re}^{2} + {x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3}} \]

      9. distribute-lft1-in N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{3} \cdot {x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3} \]

      11. associate-*r/ N/A

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

      12. associate-*l* N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. distribute-lft-neg-in N/A

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

      16. *-commutative N/A

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

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

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

   5. Applied rewrites 58.1%

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

   7. Applied rewrites 60.6%

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

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im \leq -1 \cdot 10^{-308}:\\ \;\;\;\;\left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \mathbf{elif}\;\left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im \leq \infty:\\ \;\;\;\;\left(3 \cdot \left(x.im \cdot x.re\right)\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := \left(\left(-x.im\_m\right) \cdot x.im\_m\right) \cdot x.im\_m\\ t_1 := \left(x.im\_m \cdot x.re\_m + x.im\_m \cdot x.re\_m\right) \cdot x.re\_m + \left(x.re\_m \cdot x.re\_m - x.im\_m \cdot x.im\_m\right) \cdot x.im\_m\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-308}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left(x.im\_m \cdot x.re\_m\right) \cdot x.re\_m\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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.im_m) x.im_m) x.im_m))
        (t_1
         (+
          (* (+ (* x.im_m x.re_m) (* x.im_m x.re_m)) x.re_m)
          (* (- (* x.re_m x.re_m) (* x.im_m x.im_m)) x.im_m))))
   (*
    x.im_s
    (if (<= t_1 -1e-308)
      t_0
      (if (<= t_1 INFINITY) (* (* x.im_m x.re_m) x.re_m) t_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_im_m * x_46_im_m) * x_46_im_m;
	double t_1 = (((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)) * x_46_re_m) + (((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_im_m);
	double tmp;
	if (t_1 <= -1e-308) {
		tmp = t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (x_46_im_m * x_46_re_m) * x_46_re_m;
	} else {
		tmp = t_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_im_m * x_46_im_m) * x_46_im_m;
	double t_1 = (((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)) * x_46_re_m) + (((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_im_m);
	double tmp;
	if (t_1 <= -1e-308) {
		tmp = t_0;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (x_46_im_m * x_46_re_m) * x_46_re_m;
	} else {
		tmp = t_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_im_m * x_46_im_m) * x_46_im_m
	t_1 = (((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)) * x_46_re_m) + (((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_im_m)
	tmp = 0
	if t_1 <= -1e-308:
		tmp = t_0
	elif t_1 <= math.inf:
		tmp = (x_46_im_m * x_46_re_m) * x_46_re_m
	else:
		tmp = t_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(-x_46_im_m) * x_46_im_m) * x_46_im_m)
	t_1 = Float64(Float64(Float64(Float64(x_46_im_m * x_46_re_m) + Float64(x_46_im_m * x_46_re_m)) * x_46_re_m) + Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m))
	tmp = 0.0
	if (t_1 <= -1e-308)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(x_46_im_m * x_46_re_m) * x_46_re_m);
	else
		tmp = t_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_im_m * x_46_im_m) * x_46_im_m;
	t_1 = (((x_46_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)) * x_46_re_m) + (((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_im_m);
	tmp = 0.0;
	if (t_1 <= -1e-308)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = (x_46_im_m * x_46_re_m) * x_46_re_m;
	else
		tmp = t_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[((-x$46$im$95$m) * x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision] + 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]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$1, -1e-308], t$95$0, If[LessEqual[t$95$1, Infinity], N[(N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision], t$95$0]]), $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)) < -9.9999999999999991e-309 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 70.8%

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

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

      3. flip-- N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow-prod-down N/A

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

      11. pow-prod-up N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval N/A

          \[\leadsto \frac{\left({x.re}^{\color{blue}{4}} - \left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right)\right) \cdot x.im}{x.re \cdot x.re + x.im \cdot x.im} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. pow-prod-down N/A

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

      17. pow-prod-up N/A

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

      18. lower-pow.f64 N/A

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

      19. metadata-eval N/A

          \[\leadsto \frac{\left({x.re}^{4} - {x.im}^{\color{blue}{4}}\right) \cdot x.im}{x.re \cdot x.re + x.im \cdot x.im} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

   4. Applied rewrites 32.0%

      \[\leadsto \color{blue}{\frac{\left({x.re}^{4} - {x.im}^{4}\right) \cdot x.im}{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

   5. Taylor expanded in x.im around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-pow.f64 49.0

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

   7. Applied rewrites 49.0%

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

   9. Applied rewrites 49.0%

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

   if -9.9999999999999991e-309 < (+.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 97.5%

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

   3. Taylor expanded in x.im around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 58.1

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

   5. Applied rewrites 58.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 58.2

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. flip-+ N/A

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

      11. +-inverses N/A

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

      12. +-inverses N/A

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

      13. +-inverses N/A

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

      14. +-inverses N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. flip-+ N/A

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

      18. distribute-lft-in N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

   7. Applied rewrites 18.0%

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

   8. Taylor expanded in x.re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 46.4

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

   10. Applied rewrites 46.4%

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

   12. Applied rewrites 46.7%

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

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im \leq -1 \cdot 10^{-308}:\\ \;\;\;\;\left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \mathbf{elif}\;\left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im \leq \infty:\\ \;\;\;\;\left(x.im \cdot x.re\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-x.im\right) \cdot x.im\right) \cdot x.im\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.8% accurate, 0.5× speedup

Math

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

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.im_m x.re_m) (* x.im_m x.re_m)) x.re_m)))
   (*
    x.im_s
    (if (<=
         (+ t_0 (* (- (* x.re_m x.re_m) (* x.im_m x.im_m)) x.im_m))
         INFINITY)
      (+ (* (+ x.im_m x.re_m) (* (- x.re_m x.im_m) x.im_m)) t_0)
      (* (fma (- x.re_m x.im_m) (+ x.im_m x.re_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_im_m * x_46_re_m) + (x_46_im_m * x_46_re_m)) * x_46_re_m;
	double tmp;
	if ((t_0 + (((x_46_re_m * x_46_re_m) - (x_46_im_m * x_46_im_m)) * x_46_im_m)) <= ((double) INFINITY)) {
		tmp = ((x_46_im_m + x_46_re_m) * ((x_46_re_m - x_46_im_m) * x_46_im_m)) + t_0;
	} else {
		tmp = fma((x_46_re_m - x_46_im_m), (x_46_im_m + x_46_re_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(x_46_im_m * x_46_re_m) + Float64(x_46_im_m * x_46_re_m)) * x_46_re_m)
	tmp = 0.0
	if (Float64(t_0 + Float64(Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im_m * x_46_im_m)) * x_46_im_m)) <= Inf)
		tmp = Float64(Float64(Float64(x_46_im_m + x_46_re_m) * Float64(Float64(x_46_re_m - x_46_im_m) * x_46_im_m)) + t_0);
	else
		tmp = Float64(fma(Float64(x_46_re_m - x_46_im_m), Float64(x_46_im_m + x_46_re_m), 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[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision] + N[(x$46$im$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[N[(t$95$0 + 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]), $MachinePrecision], Infinity], N[(N[(N[(x$46$im$95$m + x$46$re$95$m), $MachinePrecision] * N[(N[(x$46$re$95$m - x$46$im$95$m), $MachinePrecision] * x$46$im$95$m), $MachinePrecision]), $MachinePrecision] + t$95$0), $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] + 2.0), $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)) < +inf.0

   1. Initial program 94.4%

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

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. difference-of-squares N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 99.8

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

   4. Applied rewrites 99.8%

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

   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. Taylor expanded in x.im around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 29.0

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

   5. Applied rewrites 29.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 29.0

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. flip-+ N/A

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

      11. +-inverses N/A

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

      12. +-inverses N/A

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

      13. +-inverses N/A

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

      14. +-inverses N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. flip-+ N/A

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

      18. distribute-lft-in N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

   7. Applied rewrites 29.0%

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

   8. Taylor expanded in x.re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 29.0

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

   10. Applied rewrites 29.0%

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

   11. Taylor expanded in x.im around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. +-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. cancel-sign-sub N/A

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

       9. unpow2 N/A

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

       10. mul-1-neg N/A

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

       11. difference-of-squares N/A

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

       12. mul-1-neg N/A

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

       13. sub-neg N/A

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

       14. unsub-neg N/A

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

       15. mul-1-neg N/A

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

       16. remove-double-neg N/A

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

       17. lower-fma.f64 N/A

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

       18. lower--.f64 N/A

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

       19. lower-+.f64 100.0

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

   13. Applied rewrites 100.0%

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

4. Final simplification 99.8%

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

Alternative 9: 34.6% accurate, 3.6× speedup

Math

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

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.im_m x.re_m) 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) {
	return x_46_im_s * ((x_46_im_m * x_46_re_m) * x_46_re_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_46im_m * x_46re_m) * x_46re_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_im_m * x_46_re_m) * x_46_re_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_im_m * x_46_re_m) * x_46_re_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_im_m * x_46_re_m) * x_46_re_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_im_m * x_46_re_m) * x_46_re_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$46$im$95$m * x$46$re$95$m), $MachinePrecision] * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.0%

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

3. Taylor expanded in x.im around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 51.3

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

5. Applied rewrites 51.3%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lower-fma.f64 51.3

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-+.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. flip-+ N/A

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

   11. +-inverses N/A

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

   12. +-inverses N/A

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

   13. +-inverses N/A

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

   14. +-inverses N/A

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

   15. lift-*.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. flip-+ N/A

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

   18. distribute-lft-in N/A

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

   19. lift-*.f64 N/A

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

   20. lift-*.f64 N/A

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

7. Applied rewrites 21.3%

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

8. Taylor expanded in x.re around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 34.7

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

10. Applied rewrites 34.7%

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

12. Applied rewrites 35.3%

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

Alternative 10: 3.1% accurate, 6.7× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \left(2 \cdot x.im\_m\right) \end{array} \]

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 (* 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) {
	return x_46_im_s * (2.0 * 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 * (2.0d0 * 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 * (2.0 * 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 * (2.0 * 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(2.0 * 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 * (2.0 * 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[(2.0 * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.0%

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

3. Taylor expanded in x.im around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 51.3

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

5. Applied rewrites 51.3%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lower-fma.f64 51.3

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-+.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. flip-+ N/A

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

   11. +-inverses N/A

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

   12. +-inverses N/A

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

   13. +-inverses N/A

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

   14. +-inverses N/A

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

   15. lift-*.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. flip-+ N/A

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

   18. distribute-lft-in N/A

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

   19. lift-*.f64 N/A

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

   20. lift-*.f64 N/A

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

7. Applied rewrites 21.3%

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

8. Taylor expanded in x.re around 0

   \[\leadsto \color{blue}{-1 \cdot {x.im}^{3} + 2 \cdot x.im} \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{2 \cdot x.im + -1 \cdot {x.im}^{3}} \]

   2. unpow3 N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. distribute-rgt-in N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. +-commutative N/A

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

   9. unpow2 N/A

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

   10. associate-*r* N/A

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

   11. lower-fma.f64 N/A

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

   12. mul-1-neg N/A

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

   13. lower-neg.f64 40.8

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

10. Applied rewrites 40.8%

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

11. Taylor expanded in x.im around 0

    \[\leadsto 2 \cdot x.im \]
12. Step-by-step derivation

13. Applied rewrites 3.2%

    \[\leadsto 2 \cdot x.im \]
14. Add Preprocessing

Developer Target 1: 92.0% 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 2024268 
(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)))