math.cube on complex, imaginary part

Percentage Accurate: 82.0% → 96.6%

Time: 9.1s

Alternatives: 5

Speedup: 1.2×

• 44.3% 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: 82.0% 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: 96.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.im\_m \leq 1.7 \cdot 10^{+181}:\\ \;\;\;\;\left(x.im\_m + x.re\right) \cdot \left(x.im\_m \cdot \left(x.re - x.im\_m\right)\right) + x.re \cdot \left(x.im\_m \cdot x.re + x.im\_m \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(0 - x.im\_m \cdot x.im\_m\right)\\ \end{array} \end{array} \]

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 1.7e+181)
    (+
     (* (+ x.im_m x.re) (* x.im_m (- x.re x.im_m)))
     (* x.re (+ (* x.im_m x.re) (* x.im_m x.re))))
    (* x.im_m (- 0.0 (* 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 <= 1.7e+181) {
		tmp = ((x_46_im_m + x_46_re) * (x_46_im_m * (x_46_re - x_46_im_m))) + (x_46_re * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)));
	} else {
		tmp = x_46_im_m * (0.0 - (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 <= 1.7d+181) then
        tmp = ((x_46im_m + x_46re) * (x_46im_m * (x_46re - x_46im_m))) + (x_46re * ((x_46im_m * x_46re) + (x_46im_m * x_46re)))
    else
        tmp = x_46im_m * (0.0d0 - (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 <= 1.7e+181) {
		tmp = ((x_46_im_m + x_46_re) * (x_46_im_m * (x_46_re - x_46_im_m))) + (x_46_re * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)));
	} else {
		tmp = x_46_im_m * (0.0 - (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 <= 1.7e+181:
		tmp = ((x_46_im_m + x_46_re) * (x_46_im_m * (x_46_re - x_46_im_m))) + (x_46_re * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)))
	else:
		tmp = x_46_im_m * (0.0 - (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 <= 1.7e+181)
		tmp = Float64(Float64(Float64(x_46_im_m + x_46_re) * Float64(x_46_im_m * Float64(x_46_re - x_46_im_m))) + Float64(x_46_re * Float64(Float64(x_46_im_m * x_46_re) + Float64(x_46_im_m * x_46_re))));
	else
		tmp = Float64(x_46_im_m * Float64(0.0 - Float64(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 <= 1.7e+181)
		tmp = ((x_46_im_m + x_46_re) * (x_46_im_m * (x_46_re - x_46_im_m))) + (x_46_re * ((x_46_im_m * x_46_re) + (x_46_im_m * x_46_re)));
	else
		tmp = x_46_im_m * (0.0 - (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, 1.7e+181], N[(N[(N[(x$46$im$95$m + x$46$re), $MachinePrecision] * N[(x$46$im$95$m * N[(x$46$re - x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] + N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im$95$m * N[(0.0 - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.70000000000000015e181

   1. Initial program 85.9%

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

      1. difference-of-squares N/A

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

      2. associate-*l* N/A

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

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

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

      4. +-commutative N/A

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

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

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

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

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

      7. --lowering--.f64 95.4%

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

   4. Applied egg-rr 95.4%

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

   if 1.70000000000000015e181 < x.im

   1. Initial program 75.0%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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 85.0%

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

   3. Simplified 85.0%

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

      1. flip-- N/A

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

      2. clear-num N/A

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

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

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

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

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\left(x.re \cdot x.re\right) \cdot 3 + x.im \cdot x.im\right), \color{blue}{\left(\left(\left(x.re \cdot x.re\right) \cdot 3\right) \cdot \left(\left(x.re \cdot x.re\right) \cdot 3\right) - \left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\left(x.re \cdot x.re\right) \cdot 3\right), \left(x.im \cdot x.im\right)\right), \left(\color{blue}{\left(\left(x.re \cdot x.re\right) \cdot 3\right) \cdot \left(\left(x.re \cdot x.re\right) \cdot 3\right)} - \left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right)\right)\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(x.re \cdot \left(x.re \cdot 3\right)\right), \left(x.im \cdot x.im\right)\right), \left(\color{blue}{\left(\left(x.re \cdot x.re\right) \cdot 3\right)} \cdot \left(\left(x.re \cdot x.re\right) \cdot 3\right) - \left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, \left(x.re \cdot 3\right)\right), \left(x.im \cdot x.im\right)\right), \left(\color{blue}{\left(\left(x.re \cdot x.re\right) \cdot 3\right)} \cdot \left(\left(x.re \cdot x.re\right) \cdot 3\right) - \left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \left(x.im \cdot x.im\right)\right), \left(\left(\left(x.re \cdot x.re\right) \cdot \color{blue}{3}\right) \cdot \left(\left(x.re \cdot x.re\right) \cdot 3\right) - \left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \left(\left(\left(x.re \cdot x.re\right) \cdot 3\right) \cdot \color{blue}{\left(\left(x.re \cdot x.re\right) \cdot 3\right)} - \left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{\_.f64}\left(\left(\left(\left(x.re \cdot x.re\right) \cdot 3\right) \cdot \left(\left(x.re \cdot x.re\right) \cdot 3\right)\right), \color{blue}{\left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right)\right)}\right)\right)\right)\right) \]

      11. swap-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{\_.f64}\left(\left(\left(\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right)\right) \cdot \left(3 \cdot 3\right)\right), \left(\color{blue}{\left(x.im \cdot x.im\right)} \cdot \left(x.im \cdot x.im\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right)\right), \left(3 \cdot 3\right)\right), \left(\color{blue}{\left(x.im \cdot x.im\right)} \cdot \left(x.im \cdot x.im\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x.re \cdot x.re\right), \left(x.re \cdot x.re\right)\right), \left(3 \cdot 3\right)\right), \left(\left(\color{blue}{x.im} \cdot x.im\right) \cdot \left(x.im \cdot x.im\right)\right)\right)\right)\right)\right) \]

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

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

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

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

      16. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.re, x.re\right)\right), 9\right), \left(\left(x.im \cdot \color{blue}{x.im}\right) \cdot \left(x.im \cdot x.im\right)\right)\right)\right)\right)\right) \]

      17. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.re, x.re\right)\right), 9\right), \left(x.im \cdot \color{blue}{\left(x.im \cdot \left(x.im \cdot x.im\right)\right)}\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 0.0%

      \[\leadsto x.im \cdot \color{blue}{\frac{1}{\frac{x.re \cdot \left(x.re \cdot 3\right) + x.im \cdot x.im}{\left(\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right)\right) \cdot 9 - x.im \cdot \left(x.im \cdot \left(x.im \cdot x.im\right)\right)}}} \]

   7. Taylor expanded in x.re around 0

      \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{-1}{{x.im}^{2}}\right)}\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(-1, \color{blue}{\left({x.im}^{2}\right)}\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(-1, \left(x.im \cdot \color{blue}{x.im}\right)\right)\right)\right) \]

      3. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right)\right) \]

   9. Simplified 100.0%

      \[\leadsto x.im \cdot \frac{1}{\color{blue}{\frac{-1}{x.im \cdot x.im}}} \]
   10. Step-by-step derivation

       1. associate-/r/ N/A

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

       2. metadata-eval N/A

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

       3. neg-mul-1 N/A

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

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

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

       5. *-lowering-*.f64 100.0%

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

   11. Applied egg-rr 100.0%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 1.7 \cdot 10^{+181}:\\ \;\;\;\;\left(x.im + x.re\right) \cdot \left(x.im \cdot \left(x.re - x.im\right)\right) + x.re \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(0 - x.im \cdot x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.re \leq 4 \cdot 10^{+146}:\\ \;\;\;\;x.im\_m \cdot \left(\left(x.re \cdot x.re\right) \cdot 3 - x.im\_m \cdot x.im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im\_m \cdot x.re\right) \cdot \left(x.re \cdot 3\right)\\ \end{array} \end{array} \]

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 4e+146)
    (* x.im_m (- (* (* x.re x.re) 3.0) (* x.im_m x.im_m)))
    (* (* x.im_m x.re) (* 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 <= 4e+146) {
		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_im_m * x_46_re) * (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 <= 4d+146) then
        tmp = x_46im_m * (((x_46re * x_46re) * 3.0d0) - (x_46im_m * x_46im_m))
    else
        tmp = (x_46im_m * x_46re) * (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 <= 4e+146) {
		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_im_m * x_46_re) * (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 <= 4e+146:
		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_im_m * x_46_re) * (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 <= 4e+146)
		tmp = Float64(x_46_im_m * Float64(Float64(Float64(x_46_re * x_46_re) * 3.0) - Float64(x_46_im_m * x_46_im_m)));
	else
		tmp = Float64(Float64(x_46_im_m * x_46_re) * 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 <= 4e+146)
		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_im_m * x_46_re) * (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, 4e+146], N[(x$46$im$95$m * N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] * 3.0), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] * N[(x$46$re * 3.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 3.99999999999999973e146

   1. Initial program 89.8%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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 94.2%

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

   3. Simplified 94.2%

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

   if 3.99999999999999973e146 < x.re

   1. Initial program 49.5%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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 49.5%

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

   3. Simplified 49.5%

      \[\leadsto \color{blue}{x.im \cdot \left(\left(x.re \cdot x.re\right) \cdot 3 - x.im \cdot x.im\right)} \]
   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. *-lowering-*.f64 N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 59.5%

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

   7. Simplified 59.5%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. *-lowering-*.f64 96.4%

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

   9. Applied egg-rr 96.4%

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

Alternative 3: 69.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.re \leq 1.85 \cdot 10^{+96}:\\ \;\;\;\;x.im\_m \cdot \left(0 - x.im\_m \cdot x.im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im\_m \cdot x.re\right) \cdot \left(x.re \cdot 3\right)\\ \end{array} \end{array} \]

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 1.85e+96)
    (* x.im_m (- 0.0 (* x.im_m x.im_m)))
    (* (* x.im_m x.re) (* 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 <= 1.85e+96) {
		tmp = x_46_im_m * (0.0 - (x_46_im_m * x_46_im_m));
	} else {
		tmp = (x_46_im_m * x_46_re) * (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 <= 1.85d+96) then
        tmp = x_46im_m * (0.0d0 - (x_46im_m * x_46im_m))
    else
        tmp = (x_46im_m * x_46re) * (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 <= 1.85e+96) {
		tmp = x_46_im_m * (0.0 - (x_46_im_m * x_46_im_m));
	} else {
		tmp = (x_46_im_m * x_46_re) * (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 <= 1.85e+96:
		tmp = x_46_im_m * (0.0 - (x_46_im_m * x_46_im_m))
	else:
		tmp = (x_46_im_m * x_46_re) * (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 <= 1.85e+96)
		tmp = Float64(x_46_im_m * Float64(0.0 - Float64(x_46_im_m * x_46_im_m)));
	else
		tmp = Float64(Float64(x_46_im_m * x_46_re) * 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 <= 1.85e+96)
		tmp = x_46_im_m * (0.0 - (x_46_im_m * x_46_im_m));
	else
		tmp = (x_46_im_m * x_46_re) * (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, 1.85e+96], N[(x$46$im$95$m * N[(0.0 - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] * N[(x$46$re * 3.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.84999999999999996e96

   1. Initial program 89.8%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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 94.0%

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

   3. Simplified 94.0%

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

      1. flip-- N/A

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

      2. clear-num N/A

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

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

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

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

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\left(x.re \cdot x.re\right) \cdot 3 + x.im \cdot x.im\right), \color{blue}{\left(\left(\left(x.re \cdot x.re\right) \cdot 3\right) \cdot \left(\left(x.re \cdot x.re\right) \cdot 3\right) - \left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\left(x.re \cdot x.re\right) \cdot 3\right), \left(x.im \cdot x.im\right)\right), \left(\color{blue}{\left(\left(x.re \cdot x.re\right) \cdot 3\right) \cdot \left(\left(x.re \cdot x.re\right) \cdot 3\right)} - \left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right)\right)\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(x.re \cdot \left(x.re \cdot 3\right)\right), \left(x.im \cdot x.im\right)\right), \left(\color{blue}{\left(\left(x.re \cdot x.re\right) \cdot 3\right)} \cdot \left(\left(x.re \cdot x.re\right) \cdot 3\right) - \left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, \left(x.re \cdot 3\right)\right), \left(x.im \cdot x.im\right)\right), \left(\color{blue}{\left(\left(x.re \cdot x.re\right) \cdot 3\right)} \cdot \left(\left(x.re \cdot x.re\right) \cdot 3\right) - \left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \left(x.im \cdot x.im\right)\right), \left(\left(\left(x.re \cdot x.re\right) \cdot \color{blue}{3}\right) \cdot \left(\left(x.re \cdot x.re\right) \cdot 3\right) - \left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \left(\left(\left(x.re \cdot x.re\right) \cdot 3\right) \cdot \color{blue}{\left(\left(x.re \cdot x.re\right) \cdot 3\right)} - \left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{\_.f64}\left(\left(\left(\left(x.re \cdot x.re\right) \cdot 3\right) \cdot \left(\left(x.re \cdot x.re\right) \cdot 3\right)\right), \color{blue}{\left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right)\right)}\right)\right)\right)\right) \]

      11. swap-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{\_.f64}\left(\left(\left(\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right)\right) \cdot \left(3 \cdot 3\right)\right), \left(\color{blue}{\left(x.im \cdot x.im\right)} \cdot \left(x.im \cdot x.im\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right)\right), \left(3 \cdot 3\right)\right), \left(\color{blue}{\left(x.im \cdot x.im\right)} \cdot \left(x.im \cdot x.im\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x.re \cdot x.re\right), \left(x.re \cdot x.re\right)\right), \left(3 \cdot 3\right)\right), \left(\left(\color{blue}{x.im} \cdot x.im\right) \cdot \left(x.im \cdot x.im\right)\right)\right)\right)\right)\right) \]

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

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

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

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

      16. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.re, x.re\right)\right), 9\right), \left(\left(x.im \cdot \color{blue}{x.im}\right) \cdot \left(x.im \cdot x.im\right)\right)\right)\right)\right)\right) \]

      17. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.re, x.re\right)\right), 9\right), \left(x.im \cdot \color{blue}{\left(x.im \cdot \left(x.im \cdot x.im\right)\right)}\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 47.5%

      \[\leadsto x.im \cdot \color{blue}{\frac{1}{\frac{x.re \cdot \left(x.re \cdot 3\right) + x.im \cdot x.im}{\left(\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right)\right) \cdot 9 - x.im \cdot \left(x.im \cdot \left(x.im \cdot x.im\right)\right)}}} \]

   7. Taylor expanded in x.re around 0

      \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{-1}{{x.im}^{2}}\right)}\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(-1, \color{blue}{\left({x.im}^{2}\right)}\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(-1, \left(x.im \cdot \color{blue}{x.im}\right)\right)\right)\right) \]

      3. *-lowering-*.f64 66.9%

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right)\right) \]

   9. Simplified 66.9%

      \[\leadsto x.im \cdot \frac{1}{\color{blue}{\frac{-1}{x.im \cdot x.im}}} \]
   10. Step-by-step derivation

       1. associate-/r/ N/A

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

       2. metadata-eval N/A

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

       3. neg-mul-1 N/A

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

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

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

       5. *-lowering-*.f64 67.0%

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

   11. Applied egg-rr 67.0%

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

   if 1.84999999999999996e96 < x.re

   1. Initial program 59.6%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. 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 62.0%

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

   3. Simplified 62.0%

      \[\leadsto \color{blue}{x.im \cdot \left(\left(x.re \cdot x.re\right) \cdot 3 - x.im \cdot x.im\right)} \]
   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. *-lowering-*.f64 N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 67.0%

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

   7. Simplified 67.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. *-lowering-*.f64 94.7%

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

   9. Applied egg-rr 94.7%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 1.85 \cdot 10^{+96}:\\ \;\;\;\;x.im \cdot \left(0 - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im \cdot x.re\right) \cdot \left(x.re \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;x.re \leq 1.85 \cdot 10^{+96}:\\ \;\;\;\;x.im\_m \cdot \left(0 - x.im\_m \cdot x.im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x.im\_m \cdot \left(x.re \cdot x.re\right)\right)\\ \end{array} \end{array} \]

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 1.85e+96)
    (* x.im_m (- 0.0 (* x.im_m x.im_m)))
    (* 3.0 (* x.im_m (* 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 <= 1.85e+96) {
		tmp = x_46_im_m * (0.0 - (x_46_im_m * x_46_im_m));
	} else {
		tmp = 3.0 * (x_46_im_m * (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 <= 1.85d+96) then
        tmp = x_46im_m * (0.0d0 - (x_46im_m * x_46im_m))
    else
        tmp = 3.0d0 * (x_46im_m * (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 <= 1.85e+96) {
		tmp = x_46_im_m * (0.0 - (x_46_im_m * x_46_im_m));
	} else {
		tmp = 3.0 * (x_46_im_m * (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 <= 1.85e+96:
		tmp = x_46_im_m * (0.0 - (x_46_im_m * x_46_im_m))
	else:
		tmp = 3.0 * (x_46_im_m * (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 <= 1.85e+96)
		tmp = Float64(x_46_im_m * Float64(0.0 - Float64(x_46_im_m * x_46_im_m)));
	else
		tmp = Float64(3.0 * Float64(x_46_im_m * 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 <= 1.85e+96)
		tmp = x_46_im_m * (0.0 - (x_46_im_m * x_46_im_m));
	else
		tmp = 3.0 * (x_46_im_m * (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, 1.85e+96], N[(x$46$im$95$m * N[(0.0 - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 * N[(x$46$im$95$m * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.84999999999999996e96

   1. Initial program 89.8%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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 94.0%

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

   3. Simplified 94.0%

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

      1. flip-- N/A

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

      2. clear-num N/A

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

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

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

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

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\left(x.re \cdot x.re\right) \cdot 3 + x.im \cdot x.im\right), \color{blue}{\left(\left(\left(x.re \cdot x.re\right) \cdot 3\right) \cdot \left(\left(x.re \cdot x.re\right) \cdot 3\right) - \left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\left(x.re \cdot x.re\right) \cdot 3\right), \left(x.im \cdot x.im\right)\right), \left(\color{blue}{\left(\left(x.re \cdot x.re\right) \cdot 3\right) \cdot \left(\left(x.re \cdot x.re\right) \cdot 3\right)} - \left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right)\right)\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(x.re \cdot \left(x.re \cdot 3\right)\right), \left(x.im \cdot x.im\right)\right), \left(\color{blue}{\left(\left(x.re \cdot x.re\right) \cdot 3\right)} \cdot \left(\left(x.re \cdot x.re\right) \cdot 3\right) - \left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, \left(x.re \cdot 3\right)\right), \left(x.im \cdot x.im\right)\right), \left(\color{blue}{\left(\left(x.re \cdot x.re\right) \cdot 3\right)} \cdot \left(\left(x.re \cdot x.re\right) \cdot 3\right) - \left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \left(x.im \cdot x.im\right)\right), \left(\left(\left(x.re \cdot x.re\right) \cdot \color{blue}{3}\right) \cdot \left(\left(x.re \cdot x.re\right) \cdot 3\right) - \left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \left(\left(\left(x.re \cdot x.re\right) \cdot 3\right) \cdot \color{blue}{\left(\left(x.re \cdot x.re\right) \cdot 3\right)} - \left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{\_.f64}\left(\left(\left(\left(x.re \cdot x.re\right) \cdot 3\right) \cdot \left(\left(x.re \cdot x.re\right) \cdot 3\right)\right), \color{blue}{\left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right)\right)}\right)\right)\right)\right) \]

      11. swap-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{\_.f64}\left(\left(\left(\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right)\right) \cdot \left(3 \cdot 3\right)\right), \left(\color{blue}{\left(x.im \cdot x.im\right)} \cdot \left(x.im \cdot x.im\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right)\right), \left(3 \cdot 3\right)\right), \left(\color{blue}{\left(x.im \cdot x.im\right)} \cdot \left(x.im \cdot x.im\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x.re \cdot x.re\right), \left(x.re \cdot x.re\right)\right), \left(3 \cdot 3\right)\right), \left(\left(\color{blue}{x.im} \cdot x.im\right) \cdot \left(x.im \cdot x.im\right)\right)\right)\right)\right)\right) \]

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

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

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

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

      16. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.re, x.re\right)\right), 9\right), \left(\left(x.im \cdot \color{blue}{x.im}\right) \cdot \left(x.im \cdot x.im\right)\right)\right)\right)\right)\right) \]

      17. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.re, x.re\right)\right), 9\right), \left(x.im \cdot \color{blue}{\left(x.im \cdot \left(x.im \cdot x.im\right)\right)}\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 47.5%

      \[\leadsto x.im \cdot \color{blue}{\frac{1}{\frac{x.re \cdot \left(x.re \cdot 3\right) + x.im \cdot x.im}{\left(\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right)\right) \cdot 9 - x.im \cdot \left(x.im \cdot \left(x.im \cdot x.im\right)\right)}}} \]

   7. Taylor expanded in x.re around 0

      \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{-1}{{x.im}^{2}}\right)}\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(-1, \color{blue}{\left({x.im}^{2}\right)}\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(-1, \left(x.im \cdot \color{blue}{x.im}\right)\right)\right)\right) \]

      3. *-lowering-*.f64 66.9%

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right)\right) \]

   9. Simplified 66.9%

      \[\leadsto x.im \cdot \frac{1}{\color{blue}{\frac{-1}{x.im \cdot x.im}}} \]
   10. Step-by-step derivation

       1. associate-/r/ N/A

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

       2. metadata-eval N/A

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

       3. neg-mul-1 N/A

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

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

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

       5. *-lowering-*.f64 67.0%

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

   11. Applied egg-rr 67.0%

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

   if 1.84999999999999996e96 < x.re

   1. Initial program 59.6%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. 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 62.0%

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

   3. Simplified 62.0%

      \[\leadsto \color{blue}{x.im \cdot \left(\left(x.re \cdot x.re\right) \cdot 3 - x.im \cdot x.im\right)} \]
   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. *-lowering-*.f64 N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 67.0%

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

   7. Simplified 67.0%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 1.85 \cdot 10^{+96}:\\ \;\;\;\;x.im \cdot \left(0 - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x.im \cdot \left(x.re \cdot x.re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 58.4% accurate, 2.7× speedup

Math

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

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 (- 0.0 (* 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) {
	return x_46_im_s * (x_46_im_m * (0.0 - (x_46_im_m * 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 * (0.0d0 - (x_46im_m * 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 * (0.0 - (x_46_im_m * 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 * (0.0 - (x_46_im_m * 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(0.0 - Float64(x_46_im_m * 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 * (0.0 - (x_46_im_m * 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[(0.0 - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.1%

   \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
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 89.0%

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

3. Simplified 89.0%

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

   1. flip-- N/A

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

   2. clear-num N/A

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

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

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

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

      \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\left(x.re \cdot x.re\right) \cdot 3 + x.im \cdot x.im\right), \color{blue}{\left(\left(\left(x.re \cdot x.re\right) \cdot 3\right) \cdot \left(\left(x.re \cdot x.re\right) \cdot 3\right) - \left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right)\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\left(x.re \cdot x.re\right) \cdot 3\right), \left(x.im \cdot x.im\right)\right), \left(\color{blue}{\left(\left(x.re \cdot x.re\right) \cdot 3\right) \cdot \left(\left(x.re \cdot x.re\right) \cdot 3\right)} - \left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right)\right)\right)\right)\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(x.re \cdot \left(x.re \cdot 3\right)\right), \left(x.im \cdot x.im\right)\right), \left(\color{blue}{\left(\left(x.re \cdot x.re\right) \cdot 3\right)} \cdot \left(\left(x.re \cdot x.re\right) \cdot 3\right) - \left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, \left(x.re \cdot 3\right)\right), \left(x.im \cdot x.im\right)\right), \left(\color{blue}{\left(\left(x.re \cdot x.re\right) \cdot 3\right)} \cdot \left(\left(x.re \cdot x.re\right) \cdot 3\right) - \left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \left(x.im \cdot x.im\right)\right), \left(\left(\left(x.re \cdot x.re\right) \cdot \color{blue}{3}\right) \cdot \left(\left(x.re \cdot x.re\right) \cdot 3\right) - \left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \left(\left(\left(x.re \cdot x.re\right) \cdot 3\right) \cdot \color{blue}{\left(\left(x.re \cdot x.re\right) \cdot 3\right)} - \left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{\_.f64}\left(\left(\left(\left(x.re \cdot x.re\right) \cdot 3\right) \cdot \left(\left(x.re \cdot x.re\right) \cdot 3\right)\right), \color{blue}{\left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right)\right)}\right)\right)\right)\right) \]

   11. swap-sqr N/A

       \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{\_.f64}\left(\left(\left(\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right)\right) \cdot \left(3 \cdot 3\right)\right), \left(\color{blue}{\left(x.im \cdot x.im\right)} \cdot \left(x.im \cdot x.im\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right)\right), \left(3 \cdot 3\right)\right), \left(\color{blue}{\left(x.im \cdot x.im\right)} \cdot \left(x.im \cdot x.im\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x.re \cdot x.re\right), \left(x.re \cdot x.re\right)\right), \left(3 \cdot 3\right)\right), \left(\left(\color{blue}{x.im} \cdot x.im\right) \cdot \left(x.im \cdot x.im\right)\right)\right)\right)\right)\right) \]

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

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

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

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

   16. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.re, x.re\right)\right), 9\right), \left(\left(x.im \cdot \color{blue}{x.im}\right) \cdot \left(x.im \cdot x.im\right)\right)\right)\right)\right)\right) \]

   17. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.re, x.re\right)\right), 9\right), \left(x.im \cdot \color{blue}{\left(x.im \cdot \left(x.im \cdot x.im\right)\right)}\right)\right)\right)\right)\right) \]

6. Applied egg-rr 40.3%

   \[\leadsto x.im \cdot \color{blue}{\frac{1}{\frac{x.re \cdot \left(x.re \cdot 3\right) + x.im \cdot x.im}{\left(\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot x.re\right)\right) \cdot 9 - x.im \cdot \left(x.im \cdot \left(x.im \cdot x.im\right)\right)}}} \]

7. Taylor expanded in x.re around 0

   \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{-1}{{x.im}^{2}}\right)}\right)\right) \]
8. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(-1, \color{blue}{\left({x.im}^{2}\right)}\right)\right)\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(-1, \left(x.im \cdot \color{blue}{x.im}\right)\right)\right)\right) \]

   3. *-lowering-*.f64 57.4%

      \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right)\right) \]

9. Simplified 57.4%

   \[\leadsto x.im \cdot \frac{1}{\color{blue}{\frac{-1}{x.im \cdot x.im}}} \]
10. Step-by-step derivation

    1. associate-/r/ N/A

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

    2. metadata-eval N/A

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

    3. neg-mul-1 N/A

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

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

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

    5. *-lowering-*.f64 57.5%

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

11. Applied egg-rr 57.5%

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

12. Final simplification 57.5%

    \[\leadsto x.im \cdot \left(0 - x.im \cdot x.im\right) \]
13. 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 2024139 
(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)))