math.cube on complex, imaginary part

Percentage Accurate: 82.2% → 99.8%

Time: 9.9s

Alternatives: 8

Speedup: 1.2×

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

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = (((x_46re * x_46re) - (x_46im * x_46im)) * x_46im) + (((x_46re * x_46im) + (x_46im * x_46re)) * x_46re)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 82.2% 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, 0.2× speedup

Math

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

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.im_m 5e-5)
    (fma
     (* x.im_m x.re)
     x.re
     (- (* x.re (* x.im_m (* x.re 2.0))) (* x.im_m (* x.im_m x.im_m))))
    (*
     (* x.im_m x.im_m)
     (* x.im_m (+ -1.0 (* 3.0 (* x.re (/ (/ x.re x.im_m) x.im_m)))))))))

C

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, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 5e-5) {
		tmp = fma((x_46_im_m * x_46_re), x_46_re, ((x_46_re * (x_46_im_m * (x_46_re * 2.0))) - (x_46_im_m * (x_46_im_m * x_46_im_m))));
	} else {
		tmp = (x_46_im_m * x_46_im_m) * (x_46_im_m * (-1.0 + (3.0 * (x_46_re * ((x_46_re / x_46_im_m) / x_46_im_m)))));
	}
	return x_46_im_s * tmp;
}

Julia

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, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 5e-5)
		tmp = fma(Float64(x_46_im_m * x_46_re), x_46_re, Float64(Float64(x_46_re * Float64(x_46_im_m * Float64(x_46_re * 2.0))) - Float64(x_46_im_m * Float64(x_46_im_m * x_46_im_m))));
	else
		tmp = Float64(Float64(x_46_im_m * x_46_im_m) * Float64(x_46_im_m * Float64(-1.0 + Float64(3.0 * Float64(x_46_re * Float64(Float64(x_46_re / x_46_im_m) / x_46_im_m))))));
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

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_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 5e-5], N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] * x$46$re + N[(N[(x$46$re * N[(x$46$im$95$m * N[(x$46$re * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im$95$m * N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m * N[(-1.0 + N[(3.0 * N[(x$46$re * N[(N[(x$46$re / x$46$im$95$m), $MachinePrecision] / x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 5.00000000000000024e-5

   1. Initial program 80.0%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-out N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \color{blue}{\left(\left(x.re + x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right) \]

      8. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right) - \color{blue}{x.im \cdot x.im}\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right), \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

      10. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(2 \cdot x.re\right) \cdot x.re + x.re \cdot x.re\right), \left(x.im \cdot x.im\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(2 \cdot \left(x.re \cdot x.re\right) + x.re \cdot x.re\right), \left(x.im \cdot x.im\right)\right)\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(2 + 1\right) \cdot \left(x.re \cdot x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(3 \cdot \left(x.re \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re \cdot x.re\right) \cdot 3\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x.re \cdot x.re\right), 3\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3\right), \left(x.im \cdot x.im\right)\right)\right) \]

      17. *-lowering-*.f64 84.7%

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3\right), \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

   3. Simplified 84.7%

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

   6. Applied egg-rr 87.7%

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

   if 5.00000000000000024e-5 < x.im

   1. Initial program 74.1%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-out N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \color{blue}{\left(\left(x.re + x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right) \]

      8. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right) - \color{blue}{x.im \cdot x.im}\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right), \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

      10. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(2 \cdot x.re\right) \cdot x.re + x.re \cdot x.re\right), \left(x.im \cdot x.im\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(2 \cdot \left(x.re \cdot x.re\right) + x.re \cdot x.re\right), \left(x.im \cdot x.im\right)\right)\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(2 + 1\right) \cdot \left(x.re \cdot x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(3 \cdot \left(x.re \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re \cdot x.re\right) \cdot 3\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x.re \cdot x.re\right), 3\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3\right), \left(x.im \cdot x.im\right)\right)\right) \]

      17. *-lowering-*.f64 86.3%

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3\right), \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

   3. Simplified 86.3%

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

   5. Taylor expanded in x.im around inf

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

   7. Simplified 99.9%

      \[\leadsto \color{blue}{\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot \left(-1 + 3 \cdot \left(x.re \cdot \frac{\frac{x.re}{x.im}}{x.im}\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.9%

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

Alternative 2: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 3 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(\left(x.im\_m \cdot x.re\right) \cdot 3, x.re, x.im\_m \cdot \left(x.im\_m \cdot \left(0 - x.im\_m\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im\_m \cdot x.im\_m\right) \cdot \left(x.im\_m \cdot \left(-1 + 3 \cdot \left(x.re \cdot \frac{\frac{x.re}{x.im\_m}}{x.im\_m}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.im_m 3e+17)
    (fma (* (* x.im_m x.re) 3.0) x.re (* x.im_m (* x.im_m (- 0.0 x.im_m))))
    (*
     (* x.im_m x.im_m)
     (* x.im_m (+ -1.0 (* 3.0 (* x.re (/ (/ x.re x.im_m) x.im_m)))))))))

C

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, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 3e+17) {
		tmp = fma(((x_46_im_m * x_46_re) * 3.0), x_46_re, (x_46_im_m * (x_46_im_m * (0.0 - x_46_im_m))));
	} else {
		tmp = (x_46_im_m * x_46_im_m) * (x_46_im_m * (-1.0 + (3.0 * (x_46_re * ((x_46_re / x_46_im_m) / x_46_im_m)))));
	}
	return x_46_im_s * tmp;
}

Julia

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, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 3e+17)
		tmp = fma(Float64(Float64(x_46_im_m * x_46_re) * 3.0), x_46_re, Float64(x_46_im_m * Float64(x_46_im_m * Float64(0.0 - x_46_im_m))));
	else
		tmp = Float64(Float64(x_46_im_m * x_46_im_m) * Float64(x_46_im_m * Float64(-1.0 + Float64(3.0 * Float64(x_46_re * Float64(Float64(x_46_re / x_46_im_m) / x_46_im_m))))));
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

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_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 3e+17], N[(N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] * 3.0), $MachinePrecision] * x$46$re + N[(x$46$im$95$m * N[(x$46$im$95$m * N[(0.0 - x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m * N[(-1.0 + N[(3.0 * N[(x$46$re * N[(N[(x$46$re / x$46$im$95$m), $MachinePrecision] / x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 3e17

   1. Initial program 80.1%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-out N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \color{blue}{\left(\left(x.re + x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right) \]

      8. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right) - \color{blue}{x.im \cdot x.im}\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right), \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

      10. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(2 \cdot x.re\right) \cdot x.re + x.re \cdot x.re\right), \left(x.im \cdot x.im\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(2 \cdot \left(x.re \cdot x.re\right) + x.re \cdot x.re\right), \left(x.im \cdot x.im\right)\right)\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(2 + 1\right) \cdot \left(x.re \cdot x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(3 \cdot \left(x.re \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re \cdot x.re\right) \cdot 3\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x.re \cdot x.re\right), 3\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3\right), \left(x.im \cdot x.im\right)\right)\right) \]

      17. *-lowering-*.f64 84.8%

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3\right), \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

   3. Simplified 84.8%

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

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

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

      2. distribute-lft-in N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. fma-define N/A

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

      7. fma-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(x.im \cdot \left(x.re \cdot 3\right)\right), \color{blue}{x.re}, \left(x.im \cdot \left(\left(\mathsf{neg}\left(x.im\right)\right) \cdot x.im\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x.im, \left(x.re \cdot 3\right)\right), x.re, \left(x.im \cdot \left(\left(\mathsf{neg}\left(x.im\right)\right) \cdot x.im\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.re, 3\right)\right), x.re, \left(x.im \cdot \left(\left(\mathsf{neg}\left(x.im\right)\right) \cdot x.im\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.re, 3\right)\right), x.re, \mathsf{*.f64}\left(x.im, \left(\left(\mathsf{neg}\left(x.im\right)\right) \cdot x.im\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.re, 3\right)\right), x.re, \mathsf{*.f64}\left(x.im, \left(x.im \cdot \left(\mathsf{neg}\left(x.im\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.re, 3\right)\right), x.re, \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.im, \left(\mathsf{neg}\left(x.im\right)\right)\right)\right)\right) \]

      13. neg-sub0 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.re, 3\right)\right), x.re, \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.im, \left(0 - x.im\right)\right)\right)\right) \]

      14. +-inverses N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.re, 3\right)\right), x.re, \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.im, \left(\left(x.im \cdot x.im - x.im \cdot x.im\right) - x.im\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.re, 3\right)\right), x.re, \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.im, \left(\left(x.im \cdot x.im + \left(\mathsf{neg}\left(x.im\right)\right) \cdot x.im\right) - x.im\right)\right)\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.re, 3\right)\right), x.re, \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.im, \left(\left(\left(\mathsf{neg}\left(x.im\right)\right) \cdot x.im + x.im \cdot x.im\right) - x.im\right)\right)\right)\right) \]

      17. fma-undefine N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.re, 3\right)\right), x.re, \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.im, \left(\mathsf{fma}\left(\mathsf{neg}\left(x.im\right), x.im, x.im \cdot x.im\right) - x.im\right)\right)\right)\right) \]

      18. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.re, 3\right)\right), x.re, \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\mathsf{fma}\left(\mathsf{neg}\left(x.im\right), x.im, x.im \cdot x.im\right)\right), x.im\right)\right)\right)\right) \]

      19. fma-undefine N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.re, 3\right)\right), x.re, \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(\mathsf{neg}\left(x.im\right)\right) \cdot x.im + x.im \cdot x.im\right), x.im\right)\right)\right)\right) \]

      20. +-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.re, 3\right)\right), x.re, \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(x.im \cdot x.im + \left(\mathsf{neg}\left(x.im\right)\right) \cdot x.im\right), x.im\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.re, 3\right)\right), x.re, \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(x.im \cdot x.im - x.im \cdot x.im\right), x.im\right)\right)\right)\right) \]

      22. +-inverses 89.8%

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.re, 3\right)\right), x.re, \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(0, x.im\right)\right)\right)\right) \]

   6. Applied egg-rr 89.8%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\left(x.im \cdot x.re\right) \cdot 3\right), x.re, \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(0, x.im\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\left(x.im \cdot x.re\right), 3\right), x.re, \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(0, x.im\right)\right)\right)\right) \]

      3. *-lowering-*.f64 89.8%

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.re\right), 3\right), x.re, \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(0, x.im\right)\right)\right)\right) \]

   8. Applied egg-rr 89.8%

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

      1. sub0-neg N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.re\right), 3\right), x.re, \left(x.im \cdot \left(x.im \cdot \left(\mathsf{neg}\left(x.im\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.re\right), 3\right), x.re, \left(x.im \cdot \left(\mathsf{neg}\left(x.im \cdot x.im\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.re\right), 3\right), x.re, \left(\mathsf{neg}\left(x.im \cdot \left(x.im \cdot x.im\right)\right)\right)\right) \]

      4. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.re\right), 3\right), x.re, \mathsf{neg.f64}\left(\left(x.im \cdot \left(x.im \cdot x.im\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.re\right), 3\right), x.re, \mathsf{neg.f64}\left(\mathsf{*.f64}\left(x.im, \left(x.im \cdot x.im\right)\right)\right)\right) \]

      6. *-lowering-*.f64 89.8%

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.re\right), 3\right), x.re, \mathsf{neg.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right) \]

   10. Applied egg-rr 89.8%

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

   if 3e17 < x.im

   1. Initial program 73.8%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-out N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \color{blue}{\left(\left(x.re + x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right) \]

      8. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right) - \color{blue}{x.im \cdot x.im}\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right), \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

      10. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(2 \cdot x.re\right) \cdot x.re + x.re \cdot x.re\right), \left(x.im \cdot x.im\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(2 \cdot \left(x.re \cdot x.re\right) + x.re \cdot x.re\right), \left(x.im \cdot x.im\right)\right)\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(2 + 1\right) \cdot \left(x.re \cdot x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(3 \cdot \left(x.re \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re \cdot x.re\right) \cdot 3\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x.re \cdot x.re\right), 3\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3\right), \left(x.im \cdot x.im\right)\right)\right) \]

      17. *-lowering-*.f64 86.1%

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3\right), \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

   3. Simplified 86.1%

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

   5. Taylor expanded in x.im around inf

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

   7. Simplified 99.9%

      \[\leadsto \color{blue}{\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot \left(-1 + 3 \cdot \left(x.re \cdot \frac{\frac{x.re}{x.im}}{x.im}\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 3 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(\left(x.im \cdot x.re\right) \cdot 3, x.re, x.im \cdot \left(x.im \cdot \left(0 - x.im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot \left(-1 + 3 \cdot \left(x.re \cdot \frac{\frac{x.re}{x.im}}{x.im}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 1.05 \cdot 10^{-96}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.im\_m \cdot x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im\_m \cdot x.im\_m\right) \cdot \left(x.im\_m \cdot \left(-1 + 3 \cdot \left(x.re \cdot \frac{\frac{x.re}{x.im\_m}}{x.im\_m}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.im_m 1.05e-96)
    (* 3.0 (* x.re (* x.im_m x.re)))
    (*
     (* x.im_m x.im_m)
     (* x.im_m (+ -1.0 (* 3.0 (* x.re (/ (/ x.re x.im_m) x.im_m)))))))))

C

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, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 1.05e-96) {
		tmp = 3.0 * (x_46_re * (x_46_im_m * x_46_re));
	} else {
		tmp = (x_46_im_m * x_46_im_m) * (x_46_im_m * (-1.0 + (3.0 * (x_46_re * ((x_46_re / x_46_im_m) / x_46_im_m)))));
	}
	return x_46_im_s * tmp;
}

Fortran

x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46im_m <= 1.05d-96) then
        tmp = 3.0d0 * (x_46re * (x_46im_m * x_46re))
    else
        tmp = (x_46im_m * x_46im_m) * (x_46im_m * ((-1.0d0) + (3.0d0 * (x_46re * ((x_46re / x_46im_m) / x_46im_m)))))
    end if
    code = x_46im_s * tmp
end function

Java

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, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 1.05e-96) {
		tmp = 3.0 * (x_46_re * (x_46_im_m * x_46_re));
	} else {
		tmp = (x_46_im_m * x_46_im_m) * (x_46_im_m * (-1.0 + (3.0 * (x_46_re * ((x_46_re / x_46_im_m) / x_46_im_m)))));
	}
	return x_46_im_s * tmp;
}

Python

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, x_46_im_m):
	tmp = 0
	if x_46_im_m <= 1.05e-96:
		tmp = 3.0 * (x_46_re * (x_46_im_m * x_46_re))
	else:
		tmp = (x_46_im_m * x_46_im_m) * (x_46_im_m * (-1.0 + (3.0 * (x_46_re * ((x_46_re / x_46_im_m) / x_46_im_m)))))
	return x_46_im_s * tmp

Julia

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, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 1.05e-96)
		tmp = Float64(3.0 * Float64(x_46_re * Float64(x_46_im_m * x_46_re)));
	else
		tmp = Float64(Float64(x_46_im_m * x_46_im_m) * Float64(x_46_im_m * Float64(-1.0 + Float64(3.0 * Float64(x_46_re * Float64(Float64(x_46_re / x_46_im_m) / x_46_im_m))))));
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

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, x_46_im_m)
	tmp = 0.0;
	if (x_46_im_m <= 1.05e-96)
		tmp = 3.0 * (x_46_re * (x_46_im_m * x_46_re));
	else
		tmp = (x_46_im_m * x_46_im_m) * (x_46_im_m * (-1.0 + (3.0 * (x_46_re * ((x_46_re / x_46_im_m) / x_46_im_m)))));
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

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_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$im$95$m, 1.05e-96], N[(3.0 * N[(x$46$re * N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m * N[(-1.0 + N[(3.0 * N[(x$46$re * N[(N[(x$46$re / x$46$im$95$m), $MachinePrecision] / x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.05000000000000001e-96

   1. Initial program 77.7%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-out N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \color{blue}{\left(\left(x.re + x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right) \]

      8. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right) - \color{blue}{x.im \cdot x.im}\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right), \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

      10. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(2 \cdot x.re\right) \cdot x.re + x.re \cdot x.re\right), \left(x.im \cdot x.im\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(2 \cdot \left(x.re \cdot x.re\right) + x.re \cdot x.re\right), \left(x.im \cdot x.im\right)\right)\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(2 + 1\right) \cdot \left(x.re \cdot x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(3 \cdot \left(x.re \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re \cdot x.re\right) \cdot 3\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x.re \cdot x.re\right), 3\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3\right), \left(x.im \cdot x.im\right)\right)\right) \]

      17. *-lowering-*.f64 83.0%

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3\right), \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

   3. Simplified 83.0%

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

   5. Taylor expanded in x.im around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \color{blue}{\left(3 \cdot {x.re}^{2}\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(3, \color{blue}{\left({x.re}^{2}\right)}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(3, \left(x.re \cdot \color{blue}{x.re}\right)\right)\right) \]

      7. *-lowering-*.f64 54.0%

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(x.re, \color{blue}{x.re}\right)\right)\right) \]

   7. Simplified 54.0%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(x.re \cdot x.re\right) \cdot x.im\right), \color{blue}{3}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x.im \cdot \left(x.re \cdot x.re\right)\right), 3\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, \left(x.re \cdot x.re\right)\right), 3\right) \]

      7. *-lowering-*.f64 54.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.re, x.re\right)\right), 3\right) \]

   9. Applied egg-rr 54.0%

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

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(x.im \cdot x.re\right) \cdot x.re\right), 3\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(x.re \cdot x.im\right) \cdot x.re\right), 3\right) \]

       3. /-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{x.re}{1} \cdot x.im\right) \cdot x.re\right), 3\right) \]

       4. associate-/r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{x.re}{\frac{1}{x.im}} \cdot x.re\right), 3\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{x.re}{\frac{1}{x.im}}\right), x.re\right), 3\right) \]

       6. associate-/r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{x.re}{1} \cdot x.im\right), x.re\right), 3\right) \]

       7. /-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x.re \cdot x.im\right), x.re\right), 3\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x.im \cdot x.re\right), x.re\right), 3\right) \]

       9. *-lowering-*.f64 65.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, x.re\right), x.re\right), 3\right) \]

   11. Applied egg-rr 65.5%

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

   if 1.05000000000000001e-96 < x.im

   1. Initial program 80.1%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-out N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \color{blue}{\left(\left(x.re + x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right) \]

      8. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right) - \color{blue}{x.im \cdot x.im}\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right), \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

      10. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(2 \cdot x.re\right) \cdot x.re + x.re \cdot x.re\right), \left(x.im \cdot x.im\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(2 \cdot \left(x.re \cdot x.re\right) + x.re \cdot x.re\right), \left(x.im \cdot x.im\right)\right)\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(2 + 1\right) \cdot \left(x.re \cdot x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(3 \cdot \left(x.re \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re \cdot x.re\right) \cdot 3\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x.re \cdot x.re\right), 3\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3\right), \left(x.im \cdot x.im\right)\right)\right) \]

      17. *-lowering-*.f64 89.4%

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3\right), \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

   3. Simplified 89.4%

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

   5. Taylor expanded in x.im around inf

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

   7. Simplified 95.5%

      \[\leadsto \color{blue}{\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot \left(-1 + 3 \cdot \left(x.re \cdot \frac{\frac{x.re}{x.im}}{x.im}\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 1.05 \cdot 10^{-96}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.im \cdot x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot \left(-1 + 3 \cdot \left(x.re \cdot \frac{\frac{x.re}{x.im}}{x.im}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.re \leq 1.1 \cdot 10^{+152}:\\ \;\;\;\;x.im\_m \cdot \left(x.re \cdot \left(x.re \cdot 3\right) - x.im\_m \cdot x.im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.im\_m \cdot \left(x.re \cdot 3\right)\right)\\ \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.re 1.1e+152)
    (* x.im_m (- (* x.re (* x.re 3.0)) (* x.im_m x.im_m)))
    (* x.re (* x.im_m (* x.re 3.0))))))

C

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, double x_46_im_m) {
	double tmp;
	if (x_46_re <= 1.1e+152) {
		tmp = x_46_im_m * ((x_46_re * (x_46_re * 3.0)) - (x_46_im_m * x_46_im_m));
	} else {
		tmp = x_46_re * (x_46_im_m * (x_46_re * 3.0));
	}
	return x_46_im_s * tmp;
}

Fortran

x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46re <= 1.1d+152) then
        tmp = x_46im_m * ((x_46re * (x_46re * 3.0d0)) - (x_46im_m * x_46im_m))
    else
        tmp = x_46re * (x_46im_m * (x_46re * 3.0d0))
    end if
    code = x_46im_s * tmp
end function

Java

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, double x_46_im_m) {
	double tmp;
	if (x_46_re <= 1.1e+152) {
		tmp = x_46_im_m * ((x_46_re * (x_46_re * 3.0)) - (x_46_im_m * x_46_im_m));
	} else {
		tmp = x_46_re * (x_46_im_m * (x_46_re * 3.0));
	}
	return x_46_im_s * tmp;
}

Python

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, x_46_im_m):
	tmp = 0
	if x_46_re <= 1.1e+152:
		tmp = x_46_im_m * ((x_46_re * (x_46_re * 3.0)) - (x_46_im_m * x_46_im_m))
	else:
		tmp = x_46_re * (x_46_im_m * (x_46_re * 3.0))
	return x_46_im_s * tmp

Julia

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, x_46_im_m)
	tmp = 0.0
	if (x_46_re <= 1.1e+152)
		tmp = Float64(x_46_im_m * Float64(Float64(x_46_re * Float64(x_46_re * 3.0)) - Float64(x_46_im_m * x_46_im_m)));
	else
		tmp = Float64(x_46_re * Float64(x_46_im_m * Float64(x_46_re * 3.0)));
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

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, x_46_im_m)
	tmp = 0.0;
	if (x_46_re <= 1.1e+152)
		tmp = x_46_im_m * ((x_46_re * (x_46_re * 3.0)) - (x_46_im_m * x_46_im_m));
	else
		tmp = x_46_re * (x_46_im_m * (x_46_re * 3.0));
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

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_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$re, 1.1e+152], N[(x$46$im$95$m * N[(N[(x$46$re * N[(x$46$re * 3.0), $MachinePrecision]), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re * N[(x$46$im$95$m * N[(x$46$re * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.0999999999999999e152

   1. Initial program 83.8%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-out N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \color{blue}{\left(\left(x.re + x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right) \]

      8. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right) - \color{blue}{x.im \cdot x.im}\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right), \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

      10. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(2 \cdot x.re\right) \cdot x.re + x.re \cdot x.re\right), \left(x.im \cdot x.im\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(2 \cdot \left(x.re \cdot x.re\right) + x.re \cdot x.re\right), \left(x.im \cdot x.im\right)\right)\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(2 + 1\right) \cdot \left(x.re \cdot x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(3 \cdot \left(x.re \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re \cdot x.re\right) \cdot 3\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x.re \cdot x.re\right), 3\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3\right), \left(x.im \cdot x.im\right)\right)\right) \]

      17. *-lowering-*.f64 91.3%

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3\right), \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

   3. Simplified 91.3%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(x.re \cdot x.re\right) \cdot 3 - x.im \cdot x.im\right), \color{blue}{x.im}\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\left(x.re \cdot x.re\right) \cdot 3\right), \left(x.im \cdot x.im\right)\right), x.im\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(x.re \cdot \left(x.re \cdot 3\right)\right), \left(x.im \cdot x.im\right)\right), x.im\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(x.re \cdot 3\right)\right), \left(x.im \cdot x.im\right)\right), x.im\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \left(x.im \cdot x.im\right)\right), x.im\right) \]

      7. *-lowering-*.f64 91.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), x.im\right) \]

   6. Applied egg-rr 91.3%

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

   if 1.0999999999999999e152 < x.re

   1. Initial program 39.0%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-out N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \color{blue}{\left(\left(x.re + x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right) \]

      8. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right) - \color{blue}{x.im \cdot x.im}\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right), \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

      10. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(2 \cdot x.re\right) \cdot x.re + x.re \cdot x.re\right), \left(x.im \cdot x.im\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(2 \cdot \left(x.re \cdot x.re\right) + x.re \cdot x.re\right), \left(x.im \cdot x.im\right)\right)\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(2 + 1\right) \cdot \left(x.re \cdot x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(3 \cdot \left(x.re \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re \cdot x.re\right) \cdot 3\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x.re \cdot x.re\right), 3\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3\right), \left(x.im \cdot x.im\right)\right)\right) \]

      17. *-lowering-*.f64 39.0%

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3\right), \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

   3. Simplified 39.0%

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

   5. Taylor expanded in x.im around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \color{blue}{\left(3 \cdot {x.re}^{2}\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(3, \color{blue}{\left({x.re}^{2}\right)}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(3, \left(x.re \cdot \color{blue}{x.re}\right)\right)\right) \]

      7. *-lowering-*.f64 62.3%

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(x.re, \color{blue}{x.re}\right)\right)\right) \]

   7. Simplified 62.3%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x.im \cdot \left(x.re \cdot 3\right)\right), \color{blue}{x.re}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, \left(x.re \cdot 3\right)\right), x.re\right) \]

      6. *-lowering-*.f64 93.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.re, 3\right)\right), x.re\right) \]

   9. Applied egg-rr 93.1%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 1.1 \cdot 10^{+152}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.re \cdot 3\right) - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.re \leq 1.1 \cdot 10^{+152}:\\ \;\;\;\;x.im\_m \cdot \left(3 \cdot \left(x.re \cdot x.re\right) - x.im\_m \cdot x.im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.im\_m \cdot \left(x.re \cdot 3\right)\right)\\ \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.re 1.1e+152)
    (* x.im_m (- (* 3.0 (* x.re x.re)) (* x.im_m x.im_m)))
    (* x.re (* x.im_m (* x.re 3.0))))))

C

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, double x_46_im_m) {
	double tmp;
	if (x_46_re <= 1.1e+152) {
		tmp = x_46_im_m * ((3.0 * (x_46_re * x_46_re)) - (x_46_im_m * x_46_im_m));
	} else {
		tmp = x_46_re * (x_46_im_m * (x_46_re * 3.0));
	}
	return x_46_im_s * tmp;
}

Fortran

x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46re <= 1.1d+152) then
        tmp = x_46im_m * ((3.0d0 * (x_46re * x_46re)) - (x_46im_m * x_46im_m))
    else
        tmp = x_46re * (x_46im_m * (x_46re * 3.0d0))
    end if
    code = x_46im_s * tmp
end function

Java

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, double x_46_im_m) {
	double tmp;
	if (x_46_re <= 1.1e+152) {
		tmp = x_46_im_m * ((3.0 * (x_46_re * x_46_re)) - (x_46_im_m * x_46_im_m));
	} else {
		tmp = x_46_re * (x_46_im_m * (x_46_re * 3.0));
	}
	return x_46_im_s * tmp;
}

Python

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, x_46_im_m):
	tmp = 0
	if x_46_re <= 1.1e+152:
		tmp = x_46_im_m * ((3.0 * (x_46_re * x_46_re)) - (x_46_im_m * x_46_im_m))
	else:
		tmp = x_46_re * (x_46_im_m * (x_46_re * 3.0))
	return x_46_im_s * tmp

Julia

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, x_46_im_m)
	tmp = 0.0
	if (x_46_re <= 1.1e+152)
		tmp = Float64(x_46_im_m * Float64(Float64(3.0 * Float64(x_46_re * x_46_re)) - Float64(x_46_im_m * x_46_im_m)));
	else
		tmp = Float64(x_46_re * Float64(x_46_im_m * Float64(x_46_re * 3.0)));
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

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, x_46_im_m)
	tmp = 0.0;
	if (x_46_re <= 1.1e+152)
		tmp = x_46_im_m * ((3.0 * (x_46_re * x_46_re)) - (x_46_im_m * x_46_im_m));
	else
		tmp = x_46_re * (x_46_im_m * (x_46_re * 3.0));
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

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_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$re, 1.1e+152], N[(x$46$im$95$m * N[(N[(3.0 * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re * N[(x$46$im$95$m * N[(x$46$re * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.0999999999999999e152

   1. Initial program 83.8%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-out N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \color{blue}{\left(\left(x.re + x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right) \]

      8. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right) - \color{blue}{x.im \cdot x.im}\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right), \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

      10. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(2 \cdot x.re\right) \cdot x.re + x.re \cdot x.re\right), \left(x.im \cdot x.im\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(2 \cdot \left(x.re \cdot x.re\right) + x.re \cdot x.re\right), \left(x.im \cdot x.im\right)\right)\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(2 + 1\right) \cdot \left(x.re \cdot x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(3 \cdot \left(x.re \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re \cdot x.re\right) \cdot 3\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x.re \cdot x.re\right), 3\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3\right), \left(x.im \cdot x.im\right)\right)\right) \]

      17. *-lowering-*.f64 91.3%

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3\right), \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

   3. Simplified 91.3%

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

   if 1.0999999999999999e152 < x.re

   1. Initial program 39.0%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-out N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \color{blue}{\left(\left(x.re + x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right) \]

      8. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right) - \color{blue}{x.im \cdot x.im}\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right), \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

      10. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(2 \cdot x.re\right) \cdot x.re + x.re \cdot x.re\right), \left(x.im \cdot x.im\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(2 \cdot \left(x.re \cdot x.re\right) + x.re \cdot x.re\right), \left(x.im \cdot x.im\right)\right)\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(2 + 1\right) \cdot \left(x.re \cdot x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(3 \cdot \left(x.re \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re \cdot x.re\right) \cdot 3\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x.re \cdot x.re\right), 3\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3\right), \left(x.im \cdot x.im\right)\right)\right) \]

      17. *-lowering-*.f64 39.0%

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3\right), \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

   3. Simplified 39.0%

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

   5. Taylor expanded in x.im around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \color{blue}{\left(3 \cdot {x.re}^{2}\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(3, \color{blue}{\left({x.re}^{2}\right)}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(3, \left(x.re \cdot \color{blue}{x.re}\right)\right)\right) \]

      7. *-lowering-*.f64 62.3%

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(x.re, \color{blue}{x.re}\right)\right)\right) \]

   7. Simplified 62.3%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x.im \cdot \left(x.re \cdot 3\right)\right), \color{blue}{x.re}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, \left(x.re \cdot 3\right)\right), x.re\right) \]

      6. *-lowering-*.f64 93.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.re, 3\right)\right), x.re\right) \]

   9. Applied egg-rr 93.1%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 1.1 \cdot 10^{+152}:\\ \;\;\;\;x.im \cdot \left(3 \cdot \left(x.re \cdot x.re\right) - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.re \leq 3.4 \cdot 10^{+110}:\\ \;\;\;\;x.im\_m \cdot \left(x.im\_m \cdot \left(0 - x.im\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.im\_m \cdot \left(x.re \cdot 3\right)\right)\\ \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.re 3.4e+110)
    (* x.im_m (* x.im_m (- 0.0 x.im_m)))
    (* x.re (* x.im_m (* x.re 3.0))))))

C

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, double x_46_im_m) {
	double tmp;
	if (x_46_re <= 3.4e+110) {
		tmp = x_46_im_m * (x_46_im_m * (0.0 - x_46_im_m));
	} else {
		tmp = x_46_re * (x_46_im_m * (x_46_re * 3.0));
	}
	return x_46_im_s * tmp;
}

Fortran

x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46re <= 3.4d+110) then
        tmp = x_46im_m * (x_46im_m * (0.0d0 - x_46im_m))
    else
        tmp = x_46re * (x_46im_m * (x_46re * 3.0d0))
    end if
    code = x_46im_s * tmp
end function

Java

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, double x_46_im_m) {
	double tmp;
	if (x_46_re <= 3.4e+110) {
		tmp = x_46_im_m * (x_46_im_m * (0.0 - x_46_im_m));
	} else {
		tmp = x_46_re * (x_46_im_m * (x_46_re * 3.0));
	}
	return x_46_im_s * tmp;
}

Python

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, x_46_im_m):
	tmp = 0
	if x_46_re <= 3.4e+110:
		tmp = x_46_im_m * (x_46_im_m * (0.0 - x_46_im_m))
	else:
		tmp = x_46_re * (x_46_im_m * (x_46_re * 3.0))
	return x_46_im_s * tmp

Julia

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, x_46_im_m)
	tmp = 0.0
	if (x_46_re <= 3.4e+110)
		tmp = Float64(x_46_im_m * Float64(x_46_im_m * Float64(0.0 - x_46_im_m)));
	else
		tmp = Float64(x_46_re * Float64(x_46_im_m * Float64(x_46_re * 3.0)));
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

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, x_46_im_m)
	tmp = 0.0;
	if (x_46_re <= 3.4e+110)
		tmp = x_46_im_m * (x_46_im_m * (0.0 - x_46_im_m));
	else
		tmp = x_46_re * (x_46_im_m * (x_46_re * 3.0));
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

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_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$re, 3.4e+110], N[(x$46$im$95$m * N[(x$46$im$95$m * N[(0.0 - x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re * N[(x$46$im$95$m * N[(x$46$re * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 3.4000000000000001e110

   1. Initial program 85.0%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-out N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \color{blue}{\left(\left(x.re + x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right) \]

      8. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right) - \color{blue}{x.im \cdot x.im}\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right), \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

      10. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(2 \cdot x.re\right) \cdot x.re + x.re \cdot x.re\right), \left(x.im \cdot x.im\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(2 \cdot \left(x.re \cdot x.re\right) + x.re \cdot x.re\right), \left(x.im \cdot x.im\right)\right)\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(2 + 1\right) \cdot \left(x.re \cdot x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(3 \cdot \left(x.re \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re \cdot x.re\right) \cdot 3\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x.re \cdot x.re\right), 3\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3\right), \left(x.im \cdot x.im\right)\right)\right) \]

      17. *-lowering-*.f64 91.0%

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3\right), \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

   3. Simplified 91.0%

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

   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

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

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{{x.im}^{3}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left({x.im}^{3}\right)}\right) \]

      4. cube-mult N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \left(x.im \cdot \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \left(x.im \cdot {x.im}^{\color{blue}{2}}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x.im, \color{blue}{\left({x.im}^{2}\right)}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x.im, \left(x.im \cdot \color{blue}{x.im}\right)\right)\right) \]

      8. *-lowering-*.f64 67.3%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

   7. Simplified 67.3%

      \[\leadsto \color{blue}{0 - x.im \cdot \left(x.im \cdot x.im\right)} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(x.im \cdot \left(x.im \cdot x.im\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(x.im, \left(x.im \cdot x.im\right)\right)\right) \]

      4. *-lowering-*.f64 67.3%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.im, x.im\right)\right)\right) \]

   9. Applied egg-rr 67.3%

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

   if 3.4000000000000001e110 < x.re

   1. Initial program 41.3%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-out N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \color{blue}{\left(\left(x.re + x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right) \]

      8. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right) - \color{blue}{x.im \cdot x.im}\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right), \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

      10. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(2 \cdot x.re\right) \cdot x.re + x.re \cdot x.re\right), \left(x.im \cdot x.im\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(2 \cdot \left(x.re \cdot x.re\right) + x.re \cdot x.re\right), \left(x.im \cdot x.im\right)\right)\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(2 + 1\right) \cdot \left(x.re \cdot x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(3 \cdot \left(x.re \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re \cdot x.re\right) \cdot 3\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x.re \cdot x.re\right), 3\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3\right), \left(x.im \cdot x.im\right)\right)\right) \]

      17. *-lowering-*.f64 51.8%

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3\right), \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

   3. Simplified 51.8%

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

   5. Taylor expanded in x.im around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \color{blue}{\left(3 \cdot {x.re}^{2}\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(3, \color{blue}{\left({x.re}^{2}\right)}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(3, \left(x.re \cdot \color{blue}{x.re}\right)\right)\right) \]

      7. *-lowering-*.f64 59.7%

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(x.re, \color{blue}{x.re}\right)\right)\right) \]

   7. Simplified 59.7%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x.im \cdot \left(x.re \cdot 3\right)\right), \color{blue}{x.re}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, \left(x.re \cdot 3\right)\right), x.re\right) \]

      6. *-lowering-*.f64 84.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.re, 3\right)\right), x.re\right) \]

   9. Applied egg-rr 84.0%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 3.4 \cdot 10^{+110}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(0 - x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.re \leq 3 \cdot 10^{+110}:\\ \;\;\;\;x.im\_m \cdot \left(x.im\_m \cdot \left(0 - x.im\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(3 \cdot \left(x.re \cdot x.re\right)\right)\\ \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (*
  x.im_s
  (if (<= x.re 3e+110)
    (* x.im_m (* x.im_m (- 0.0 x.im_m)))
    (* x.im_m (* 3.0 (* x.re x.re))))))

C

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, double x_46_im_m) {
	double tmp;
	if (x_46_re <= 3e+110) {
		tmp = x_46_im_m * (x_46_im_m * (0.0 - x_46_im_m));
	} else {
		tmp = x_46_im_m * (3.0 * (x_46_re * x_46_re));
	}
	return x_46_im_s * tmp;
}

Fortran

x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46re <= 3d+110) then
        tmp = x_46im_m * (x_46im_m * (0.0d0 - x_46im_m))
    else
        tmp = x_46im_m * (3.0d0 * (x_46re * x_46re))
    end if
    code = x_46im_s * tmp
end function

Java

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, double x_46_im_m) {
	double tmp;
	if (x_46_re <= 3e+110) {
		tmp = x_46_im_m * (x_46_im_m * (0.0 - x_46_im_m));
	} else {
		tmp = x_46_im_m * (3.0 * (x_46_re * x_46_re));
	}
	return x_46_im_s * tmp;
}

Python

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, x_46_im_m):
	tmp = 0
	if x_46_re <= 3e+110:
		tmp = x_46_im_m * (x_46_im_m * (0.0 - x_46_im_m))
	else:
		tmp = x_46_im_m * (3.0 * (x_46_re * x_46_re))
	return x_46_im_s * tmp

Julia

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, x_46_im_m)
	tmp = 0.0
	if (x_46_re <= 3e+110)
		tmp = Float64(x_46_im_m * Float64(x_46_im_m * Float64(0.0 - x_46_im_m)));
	else
		tmp = Float64(x_46_im_m * Float64(3.0 * Float64(x_46_re * x_46_re)));
	end
	return Float64(x_46_im_s * tmp)
end

MATLAB

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, x_46_im_m)
	tmp = 0.0;
	if (x_46_re <= 3e+110)
		tmp = x_46_im_m * (x_46_im_m * (0.0 - x_46_im_m));
	else
		tmp = x_46_im_m * (3.0 * (x_46_re * x_46_re));
	end
	tmp_2 = x_46_im_s * tmp;
end

Wolfram

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_, x$46$im$95$m_] := N[(x$46$im$95$s * If[LessEqual[x$46$re, 3e+110], N[(x$46$im$95$m * N[(x$46$im$95$m * N[(0.0 - x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im$95$m * N[(3.0 * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 3.00000000000000007e110

   1. Initial program 85.0%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-out N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \color{blue}{\left(\left(x.re + x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right) \]

      8. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right) - \color{blue}{x.im \cdot x.im}\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right), \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

      10. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(2 \cdot x.re\right) \cdot x.re + x.re \cdot x.re\right), \left(x.im \cdot x.im\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(2 \cdot \left(x.re \cdot x.re\right) + x.re \cdot x.re\right), \left(x.im \cdot x.im\right)\right)\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(2 + 1\right) \cdot \left(x.re \cdot x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(3 \cdot \left(x.re \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re \cdot x.re\right) \cdot 3\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x.re \cdot x.re\right), 3\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3\right), \left(x.im \cdot x.im\right)\right)\right) \]

      17. *-lowering-*.f64 91.0%

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3\right), \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

   3. Simplified 91.0%

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

   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

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

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{{x.im}^{3}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left({x.im}^{3}\right)}\right) \]

      4. cube-mult N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \left(x.im \cdot \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \left(x.im \cdot {x.im}^{\color{blue}{2}}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x.im, \color{blue}{\left({x.im}^{2}\right)}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x.im, \left(x.im \cdot \color{blue}{x.im}\right)\right)\right) \]

      8. *-lowering-*.f64 67.3%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

   7. Simplified 67.3%

      \[\leadsto \color{blue}{0 - x.im \cdot \left(x.im \cdot x.im\right)} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(x.im \cdot \left(x.im \cdot x.im\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(x.im, \left(x.im \cdot x.im\right)\right)\right) \]

      4. *-lowering-*.f64 67.3%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.im, x.im\right)\right)\right) \]

   9. Applied egg-rr 67.3%

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

   if 3.00000000000000007e110 < x.re

   1. Initial program 41.3%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-out N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \color{blue}{\left(\left(x.re + x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right) \]

      8. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right) - \color{blue}{x.im \cdot x.im}\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right), \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

      10. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(2 \cdot x.re\right) \cdot x.re + x.re \cdot x.re\right), \left(x.im \cdot x.im\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(2 \cdot \left(x.re \cdot x.re\right) + x.re \cdot x.re\right), \left(x.im \cdot x.im\right)\right)\right) \]

      12. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(2 + 1\right) \cdot \left(x.re \cdot x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(3 \cdot \left(x.re \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re \cdot x.re\right) \cdot 3\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x.re \cdot x.re\right), 3\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3\right), \left(x.im \cdot x.im\right)\right)\right) \]

      17. *-lowering-*.f64 51.8%

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3\right), \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

   3. Simplified 51.8%

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

   5. Taylor expanded in x.im around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \color{blue}{\left(3 \cdot {x.re}^{2}\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(3, \color{blue}{\left({x.re}^{2}\right)}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(3, \left(x.re \cdot \color{blue}{x.re}\right)\right)\right) \]

      7. *-lowering-*.f64 59.7%

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(x.re, \color{blue}{x.re}\right)\right)\right) \]

   7. Simplified 59.7%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 3 \cdot 10^{+110}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(0 - x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(3 \cdot \left(x.re \cdot x.re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 58.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \left(x.im\_m \cdot \left(x.im\_m \cdot \left(0 - x.im\_m\right)\right)\right) \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (* x.im_s (* x.im_m (* x.im_m (- 0.0 x.im_m)))))

C

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, double x_46_im_m) {
	return x_46_im_s * (x_46_im_m * (x_46_im_m * (0.0 - x_46_im_m)));
}

Fortran

x.im\_m = abs(x_46im)
x.im\_s = copysign(1.0d0, x_46im)
real(8) function code(x_46im_s, x_46re, x_46im_m)
    real(8), intent (in) :: x_46im_s
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    code = x_46im_s * (x_46im_m * (x_46im_m * (0.0d0 - x_46im_m)))
end function

Java

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, double x_46_im_m) {
	return x_46_im_s * (x_46_im_m * (x_46_im_m * (0.0 - x_46_im_m)));
}

Python

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, x_46_im_m):
	return x_46_im_s * (x_46_im_m * (x_46_im_m * (0.0 - x_46_im_m)))

Julia

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, x_46_im_m)
	return Float64(x_46_im_s * Float64(x_46_im_m * Float64(x_46_im_m * Float64(0.0 - x_46_im_m))))
end

MATLAB

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, x_46_im_m)
	tmp = x_46_im_s * (x_46_im_m * (x_46_im_m * (0.0 - x_46_im_m)));
end

Wolfram

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_, x$46$im$95$m_] := N[(x$46$im$95$s * N[(x$46$im$95$m * N[(x$46$im$95$m * N[(0.0 - x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.5%

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. distribute-lft-out N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. distribute-lft-out N/A

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

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x.im, \color{blue}{\left(\left(x.re + x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right) \]

   8. associate-+r- N/A

      \[\leadsto \mathsf{*.f64}\left(x.im, \left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right) - \color{blue}{x.im \cdot x.im}\right)\right) \]

   9. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right), \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

   10. count-2 N/A

       \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(2 \cdot x.re\right) \cdot x.re + x.re \cdot x.re\right), \left(x.im \cdot x.im\right)\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(2 \cdot \left(x.re \cdot x.re\right) + x.re \cdot x.re\right), \left(x.im \cdot x.im\right)\right)\right) \]

   12. distribute-lft1-in N/A

       \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(2 + 1\right) \cdot \left(x.re \cdot x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(3 \cdot \left(x.re \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re \cdot x.re\right) \cdot 3\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(x.re \cdot x.re\right), 3\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

   16. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3\right), \left(x.im \cdot x.im\right)\right)\right) \]

   17. *-lowering-*.f64 85.1%

       \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), 3\right), \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

3. Simplified 85.1%

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

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

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

   2. neg-sub0 N/A

      \[\leadsto 0 - \color{blue}{{x.im}^{3}} \]

   3. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left({x.im}^{3}\right)}\right) \]

   4. cube-mult N/A

      \[\leadsto \mathsf{\_.f64}\left(0, \left(x.im \cdot \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{\_.f64}\left(0, \left(x.im \cdot {x.im}^{\color{blue}{2}}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x.im, \color{blue}{\left({x.im}^{2}\right)}\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x.im, \left(x.im \cdot \color{blue}{x.im}\right)\right)\right) \]

   8. *-lowering-*.f64 59.7%

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

7. Simplified 59.7%

   \[\leadsto \color{blue}{0 - x.im \cdot \left(x.im \cdot x.im\right)} \]
8. Step-by-step derivation

   1. sub0-neg N/A

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

   2. neg-lowering-neg.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\left(x.im \cdot \left(x.im \cdot x.im\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(x.im, \left(x.im \cdot x.im\right)\right)\right) \]

   4. *-lowering-*.f64 59.7%

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.im, x.im\right)\right)\right) \]

9. Applied egg-rr 59.7%

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

10. Final simplification 59.7%

    \[\leadsto x.im \cdot \left(x.im \cdot \left(0 - x.im\right)\right) \]
11. Add Preprocessing

Developer Target 1: 91.4% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (+ (* (* x.re x.im) (* 2.0 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im))))

C

double code(double x_46_re, double x_46_im) {
	return ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = ((x_46re * x_46im) * (2.0d0 * x_46re)) + ((x_46im * (x_46re - x_46im)) * (x_46re + x_46im))
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
}

Python

def code(x_46_re, x_46_im):
	return ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im))

Julia

function code(x_46_re, x_46_im)
	return Float64(Float64(Float64(x_46_re * x_46_im) * Float64(2.0 * x_46_re)) + Float64(Float64(x_46_im * Float64(x_46_re - x_46_im)) * Float64(x_46_re + x_46_im)))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
end

Wolfram

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

Reproduce

herbie shell --seed 2024164 
(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)))