math.cube on complex, real part

Percentage Accurate: 82.5% → 96.1%

Time: 12.6s

Alternatives: 10

Speedup: 1.2×

• 43.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

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_re) - (((x_46_re * x_46_im) + (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_46re) - (x_46im * x_46im)) * x_46re) - (((x_46re * x_46im) + (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_re) - (x_46_im * x_46_im)) * x_46_re) - (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im);
}

Python

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

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_re) - Float64(Float64(Float64(x_46_re * x_46_im) + Float64(x_46_im * x_46_re)) * x_46_im))
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_re) - (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im);
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$46$re), $MachinePrecision] - N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision]

Initial Program: 82.5% accurate, 1.0× speedup

Math

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

FPCore

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

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_re) - (((x_46_re * x_46_im) + (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_46re) - (x_46im * x_46im)) * x_46re) - (((x_46re * x_46im) + (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_re) - (x_46_im * x_46_im)) * x_46_re) - (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im);
}

Python

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

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_re) - Float64(Float64(Float64(x_46_re * x_46_im) + Float64(x_46_im * x_46_re)) * x_46_im))
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_re) - (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im);
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$46$re), $MachinePrecision] - N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision]

Alternative 1: 96.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ \begin{array}{l} \mathbf{if}\;x.im\_m \leq 7.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{\frac{1}{x.re \cdot \left(x.re \cdot x.re + x.im\_m \cdot \left(x.im\_m \cdot -3\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(x.im\_m \cdot \left(x.re \cdot -3\right)\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
(FPCore (x.re x.im_m)
 :precision binary64
 (if (<= x.im_m 7.5e+153)
   (/ 1.0 (/ 1.0 (* x.re (+ (* x.re x.re) (* x.im_m (* x.im_m -3.0))))))
   (* x.im_m (* x.im_m (* x.re -3.0)))))

C

x.im_m = fabs(x_46_im);
double code(double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 7.5e+153) {
		tmp = 1.0 / (1.0 / (x_46_re * ((x_46_re * x_46_re) + (x_46_im_m * (x_46_im_m * -3.0)))));
	} else {
		tmp = x_46_im_m * (x_46_im_m * (x_46_re * -3.0));
	}
	return tmp;
}

Fortran

x.im_m = abs(x_46im)
real(8) function code(x_46re, x_46im_m)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46im_m <= 7.5d+153) then
        tmp = 1.0d0 / (1.0d0 / (x_46re * ((x_46re * x_46re) + (x_46im_m * (x_46im_m * (-3.0d0))))))
    else
        tmp = x_46im_m * (x_46im_m * (x_46re * (-3.0d0)))
    end if
    code = tmp
end function

Java

x.im_m = Math.abs(x_46_im);
public static double code(double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 7.5e+153) {
		tmp = 1.0 / (1.0 / (x_46_re * ((x_46_re * x_46_re) + (x_46_im_m * (x_46_im_m * -3.0)))));
	} else {
		tmp = x_46_im_m * (x_46_im_m * (x_46_re * -3.0));
	}
	return tmp;
}

Python

x.im_m = math.fabs(x_46_im)
def code(x_46_re, x_46_im_m):
	tmp = 0
	if x_46_im_m <= 7.5e+153:
		tmp = 1.0 / (1.0 / (x_46_re * ((x_46_re * x_46_re) + (x_46_im_m * (x_46_im_m * -3.0)))))
	else:
		tmp = x_46_im_m * (x_46_im_m * (x_46_re * -3.0))
	return tmp

Julia

x.im_m = abs(x_46_im)
function code(x_46_re, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 7.5e+153)
		tmp = Float64(1.0 / Float64(1.0 / Float64(x_46_re * Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im_m * Float64(x_46_im_m * -3.0))))));
	else
		tmp = Float64(x_46_im_m * Float64(x_46_im_m * Float64(x_46_re * -3.0)));
	end
	return tmp
end

MATLAB

x.im_m = abs(x_46_im);
function tmp_2 = code(x_46_re, x_46_im_m)
	tmp = 0.0;
	if (x_46_im_m <= 7.5e+153)
		tmp = 1.0 / (1.0 / (x_46_re * ((x_46_re * x_46_re) + (x_46_im_m * (x_46_im_m * -3.0)))));
	else
		tmp = x_46_im_m * (x_46_im_m * (x_46_re * -3.0));
	end
	tmp_2 = tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
code[x$46$re_, x$46$im$95$m_] := If[LessEqual[x$46$im$95$m, 7.5e+153], N[(1.0 / N[(1.0 / N[(x$46$re * N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im$95$m * N[(x$46$im$95$m * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im$95$m * N[(x$46$im$95$m * N[(x$46$re * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 7.50000000000000065e153

   1. Initial program 87.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. *-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-out N/A

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

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

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

      10. +-commutative N/A

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

      11. sub-neg N/A

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

      12. associate-+l+ N/A

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

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

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

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

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

      15. distribute-rgt-neg-out N/A

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

      16. sub-neg N/A

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

      17. count-2 N/A

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

      18. associate-*l* N/A

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

   3. Simplified 92.6%

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

      1. *-commutative N/A

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

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

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

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

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

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

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 92.7%

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

   6. Applied egg-rr 92.7%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

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

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

      6. flip-- N/A

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

      7. associate-*r* N/A

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

      8. swap-sqr N/A

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

      9. metadata-eval N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

   8. Applied egg-rr 92.7%

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

   if 7.50000000000000065e153 < x.im

   1. Initial program 56.6%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. *-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-out N/A

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

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

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

      10. +-commutative N/A

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

      11. sub-neg N/A

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

      12. associate-+l+ N/A

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

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

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

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

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

      15. distribute-rgt-neg-out N/A

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

      16. sub-neg N/A

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

      17. count-2 N/A

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

      18. associate-*l* N/A

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

   3. Simplified 53.1%

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

   5. Taylor expanded in x.re around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 75.1%

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

   7. Simplified 75.1%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-lowering-*.f64 92.5%

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

   9. Applied egg-rr 92.5%

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

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

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

       9. *-lowering-*.f64 92.5%

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

   11. Applied egg-rr 92.5%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 7.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{\frac{1}{x.re \cdot \left(x.re \cdot x.re + x.im \cdot \left(x.im \cdot -3\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ \begin{array}{l} \mathbf{if}\;x.im\_m \leq 1.8 \cdot 10^{+153}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re + x.im\_m \cdot \left(x.im\_m \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(x.im\_m \cdot \left(x.re \cdot -3\right)\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
(FPCore (x.re x.im_m)
 :precision binary64
 (if (<= x.im_m 1.8e+153)
   (* x.re (+ (* x.re x.re) (* x.im_m (* x.im_m -3.0))))
   (* x.im_m (* x.im_m (* x.re -3.0)))))

C

x.im_m = fabs(x_46_im);
double code(double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 1.8e+153) {
		tmp = x_46_re * ((x_46_re * x_46_re) + (x_46_im_m * (x_46_im_m * -3.0)));
	} else {
		tmp = x_46_im_m * (x_46_im_m * (x_46_re * -3.0));
	}
	return tmp;
}

Fortran

x.im_m = abs(x_46im)
real(8) function code(x_46re, x_46im_m)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46im_m <= 1.8d+153) then
        tmp = x_46re * ((x_46re * x_46re) + (x_46im_m * (x_46im_m * (-3.0d0))))
    else
        tmp = x_46im_m * (x_46im_m * (x_46re * (-3.0d0)))
    end if
    code = tmp
end function

Java

x.im_m = Math.abs(x_46_im);
public static double code(double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 1.8e+153) {
		tmp = x_46_re * ((x_46_re * x_46_re) + (x_46_im_m * (x_46_im_m * -3.0)));
	} else {
		tmp = x_46_im_m * (x_46_im_m * (x_46_re * -3.0));
	}
	return tmp;
}

Python

x.im_m = math.fabs(x_46_im)
def code(x_46_re, x_46_im_m):
	tmp = 0
	if x_46_im_m <= 1.8e+153:
		tmp = x_46_re * ((x_46_re * x_46_re) + (x_46_im_m * (x_46_im_m * -3.0)))
	else:
		tmp = x_46_im_m * (x_46_im_m * (x_46_re * -3.0))
	return tmp

Julia

x.im_m = abs(x_46_im)
function code(x_46_re, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 1.8e+153)
		tmp = Float64(x_46_re * Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im_m * Float64(x_46_im_m * -3.0))));
	else
		tmp = Float64(x_46_im_m * Float64(x_46_im_m * Float64(x_46_re * -3.0)));
	end
	return tmp
end

MATLAB

x.im_m = abs(x_46_im);
function tmp_2 = code(x_46_re, x_46_im_m)
	tmp = 0.0;
	if (x_46_im_m <= 1.8e+153)
		tmp = x_46_re * ((x_46_re * x_46_re) + (x_46_im_m * (x_46_im_m * -3.0)));
	else
		tmp = x_46_im_m * (x_46_im_m * (x_46_re * -3.0));
	end
	tmp_2 = tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
code[x$46$re_, x$46$im$95$m_] := If[LessEqual[x$46$im$95$m, 1.8e+153], N[(x$46$re * N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im$95$m * N[(x$46$im$95$m * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im$95$m * N[(x$46$im$95$m * N[(x$46$re * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.8e153

   1. Initial program 87.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. *-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-out N/A

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

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

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

      10. +-commutative N/A

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

      11. sub-neg N/A

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

      12. associate-+l+ N/A

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

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

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

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

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

      15. distribute-rgt-neg-out N/A

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

      16. sub-neg N/A

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

      17. count-2 N/A

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

      18. associate-*l* N/A

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

   3. Simplified 92.6%

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

      1. *-commutative N/A

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

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

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

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

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

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

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 92.7%

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

   6. Applied egg-rr 92.7%

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

   if 1.8e153 < x.im

   1. Initial program 56.6%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. *-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-out N/A

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

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

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

      10. +-commutative N/A

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

      11. sub-neg N/A

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

      12. associate-+l+ N/A

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

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

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

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

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

      15. distribute-rgt-neg-out N/A

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

      16. sub-neg N/A

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

      17. count-2 N/A

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

      18. associate-*l* N/A

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

   3. Simplified 53.1%

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

   5. Taylor expanded in x.re around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 75.1%

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

   7. Simplified 75.1%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-lowering-*.f64 92.5%

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

   9. Applied egg-rr 92.5%

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

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

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

       9. *-lowering-*.f64 92.5%

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

   11. Applied egg-rr 92.5%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 1.8 \cdot 10^{+153}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re + x.im \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ \begin{array}{l} \mathbf{if}\;x.im\_m \leq 7.5 \cdot 10^{+153}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re + -3 \cdot \left(x.im\_m \cdot x.im\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(x.im\_m \cdot \left(x.re \cdot -3\right)\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
(FPCore (x.re x.im_m)
 :precision binary64
 (if (<= x.im_m 7.5e+153)
   (* x.re (+ (* x.re x.re) (* -3.0 (* x.im_m x.im_m))))
   (* x.im_m (* x.im_m (* x.re -3.0)))))

C

x.im_m = fabs(x_46_im);
double code(double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 7.5e+153) {
		tmp = x_46_re * ((x_46_re * x_46_re) + (-3.0 * (x_46_im_m * x_46_im_m)));
	} else {
		tmp = x_46_im_m * (x_46_im_m * (x_46_re * -3.0));
	}
	return tmp;
}

Fortran

x.im_m = abs(x_46im)
real(8) function code(x_46re, x_46im_m)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46im_m <= 7.5d+153) then
        tmp = x_46re * ((x_46re * x_46re) + ((-3.0d0) * (x_46im_m * x_46im_m)))
    else
        tmp = x_46im_m * (x_46im_m * (x_46re * (-3.0d0)))
    end if
    code = tmp
end function

Java

x.im_m = Math.abs(x_46_im);
public static double code(double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 7.5e+153) {
		tmp = x_46_re * ((x_46_re * x_46_re) + (-3.0 * (x_46_im_m * x_46_im_m)));
	} else {
		tmp = x_46_im_m * (x_46_im_m * (x_46_re * -3.0));
	}
	return tmp;
}

Python

x.im_m = math.fabs(x_46_im)
def code(x_46_re, x_46_im_m):
	tmp = 0
	if x_46_im_m <= 7.5e+153:
		tmp = x_46_re * ((x_46_re * x_46_re) + (-3.0 * (x_46_im_m * x_46_im_m)))
	else:
		tmp = x_46_im_m * (x_46_im_m * (x_46_re * -3.0))
	return tmp

Julia

x.im_m = abs(x_46_im)
function code(x_46_re, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 7.5e+153)
		tmp = Float64(x_46_re * Float64(Float64(x_46_re * x_46_re) + Float64(-3.0 * Float64(x_46_im_m * x_46_im_m))));
	else
		tmp = Float64(x_46_im_m * Float64(x_46_im_m * Float64(x_46_re * -3.0)));
	end
	return tmp
end

MATLAB

x.im_m = abs(x_46_im);
function tmp_2 = code(x_46_re, x_46_im_m)
	tmp = 0.0;
	if (x_46_im_m <= 7.5e+153)
		tmp = x_46_re * ((x_46_re * x_46_re) + (-3.0 * (x_46_im_m * x_46_im_m)));
	else
		tmp = x_46_im_m * (x_46_im_m * (x_46_re * -3.0));
	end
	tmp_2 = tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
code[x$46$re_, x$46$im$95$m_] := If[LessEqual[x$46$im$95$m, 7.5e+153], N[(x$46$re * N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(-3.0 * N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im$95$m * N[(x$46$im$95$m * N[(x$46$re * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 7.50000000000000065e153

   1. Initial program 87.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. *-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-out N/A

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

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

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

      10. +-commutative N/A

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

      11. sub-neg N/A

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

      12. associate-+l+ N/A

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

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

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

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

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

      15. distribute-rgt-neg-out N/A

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

      16. sub-neg N/A

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

      17. count-2 N/A

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

      18. associate-*l* N/A

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

   3. Simplified 92.6%

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

   if 7.50000000000000065e153 < x.im

   1. Initial program 56.6%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. *-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-out N/A

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

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

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

      10. +-commutative N/A

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

      11. sub-neg N/A

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

      12. associate-+l+ N/A

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

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

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

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

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

      15. distribute-rgt-neg-out N/A

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

      16. sub-neg N/A

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

      17. count-2 N/A

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

      18. associate-*l* N/A

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

   3. Simplified 53.1%

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

   5. Taylor expanded in x.re around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 75.1%

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

   7. Simplified 75.1%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-lowering-*.f64 92.5%

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

   9. Applied egg-rr 92.5%

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

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

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

       9. *-lowering-*.f64 92.5%

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

   11. Applied egg-rr 92.5%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 7.5 \cdot 10^{+153}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re + -3 \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ \begin{array}{l} \mathbf{if}\;x.im\_m \leq 31000:\\ \;\;\;\;x.re \cdot \frac{x.re}{\frac{1}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(x.im\_m \cdot \left(x.re \cdot -3\right)\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
(FPCore (x.re x.im_m)
 :precision binary64
 (if (<= x.im_m 31000.0)
   (* x.re (/ x.re (/ 1.0 x.re)))
   (* x.im_m (* x.im_m (* x.re -3.0)))))

C

x.im_m = fabs(x_46_im);
double code(double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 31000.0) {
		tmp = x_46_re * (x_46_re / (1.0 / x_46_re));
	} else {
		tmp = x_46_im_m * (x_46_im_m * (x_46_re * -3.0));
	}
	return tmp;
}

Fortran

x.im_m = abs(x_46im)
real(8) function code(x_46re, x_46im_m)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46im_m <= 31000.0d0) then
        tmp = x_46re * (x_46re / (1.0d0 / x_46re))
    else
        tmp = x_46im_m * (x_46im_m * (x_46re * (-3.0d0)))
    end if
    code = tmp
end function

Java

x.im_m = Math.abs(x_46_im);
public static double code(double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 31000.0) {
		tmp = x_46_re * (x_46_re / (1.0 / x_46_re));
	} else {
		tmp = x_46_im_m * (x_46_im_m * (x_46_re * -3.0));
	}
	return tmp;
}

Python

x.im_m = math.fabs(x_46_im)
def code(x_46_re, x_46_im_m):
	tmp = 0
	if x_46_im_m <= 31000.0:
		tmp = x_46_re * (x_46_re / (1.0 / x_46_re))
	else:
		tmp = x_46_im_m * (x_46_im_m * (x_46_re * -3.0))
	return tmp

Julia

x.im_m = abs(x_46_im)
function code(x_46_re, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 31000.0)
		tmp = Float64(x_46_re * Float64(x_46_re / Float64(1.0 / x_46_re)));
	else
		tmp = Float64(x_46_im_m * Float64(x_46_im_m * Float64(x_46_re * -3.0)));
	end
	return tmp
end

MATLAB

x.im_m = abs(x_46_im);
function tmp_2 = code(x_46_re, x_46_im_m)
	tmp = 0.0;
	if (x_46_im_m <= 31000.0)
		tmp = x_46_re * (x_46_re / (1.0 / x_46_re));
	else
		tmp = x_46_im_m * (x_46_im_m * (x_46_re * -3.0));
	end
	tmp_2 = tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
code[x$46$re_, x$46$im$95$m_] := If[LessEqual[x$46$im$95$m, 31000.0], N[(x$46$re * N[(x$46$re / N[(1.0 / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im$95$m * N[(x$46$im$95$m * N[(x$46$re * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 31000

   1. Initial program 88.7%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. *-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-out N/A

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

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

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

      10. +-commutative N/A

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

      11. sub-neg N/A

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

      12. associate-+l+ N/A

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

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

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

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

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

      15. distribute-rgt-neg-out N/A

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

      16. sub-neg N/A

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

      17. count-2 N/A

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

      18. associate-*l* N/A

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

   3. Simplified 91.6%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

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

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

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

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

   6. Applied egg-rr 22.3%

      \[\leadsto x.re \cdot \color{blue}{\frac{1}{\frac{x.re \cdot \left(x.re \cdot \left(x.re \cdot x.re\right)\right) + \left(x.im \cdot \left(x.im \cdot -3\right)\right) \cdot \left(x.im \cdot \left(x.im \cdot -3\right) - x.re \cdot x.re\right)}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot \left(x.re \cdot \left(x.re \cdot x.re\right)\right)\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot \left(x.im \cdot \left(x.im \cdot x.im\right)\right)\right)\right) \cdot -27}}} \]

   7. Taylor expanded in x.re around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 73.0%

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

   9. Simplified 73.0%

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

       1. clear-num N/A

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

       2. associate-/l* N/A

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

       3. /-rgt-identity N/A

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

       4. clear-num N/A

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

       5. times-frac N/A

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

       6. *-rgt-identity N/A

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

       7. div-inv N/A

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

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

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

       9. /-lowering-/.f64 73.0%

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

   11. Applied egg-rr 73.0%

       \[\leadsto x.re \cdot \color{blue}{\frac{x.re}{\frac{1}{x.re}}} \]

   if 31000 < x.im

   1. Initial program 65.8%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. *-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-out N/A

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

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

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

      10. +-commutative N/A

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

      11. sub-neg N/A

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

      12. associate-+l+ N/A

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

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

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

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

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

      15. distribute-rgt-neg-out N/A

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

      16. sub-neg N/A

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

      17. count-2 N/A

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

      18. associate-*l* N/A

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

   3. Simplified 76.8%

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

   5. Taylor expanded in x.re around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 69.4%

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

   7. Simplified 69.4%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-lowering-*.f64 77.9%

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

   9. Applied egg-rr 77.9%

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

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

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

       9. *-lowering-*.f64 78.0%

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

   11. Applied egg-rr 78.0%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 31000:\\ \;\;\;\;x.re \cdot \frac{x.re}{\frac{1}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ \begin{array}{l} \mathbf{if}\;x.im\_m \leq 420:\\ \;\;\;\;x.re \cdot \frac{x.re}{\frac{1}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;\left(x.im\_m \cdot -3\right) \cdot \left(x.im\_m \cdot x.re\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
(FPCore (x.re x.im_m)
 :precision binary64
 (if (<= x.im_m 420.0)
   (* x.re (/ x.re (/ 1.0 x.re)))
   (* (* x.im_m -3.0) (* x.im_m x.re))))

C

x.im_m = fabs(x_46_im);
double code(double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 420.0) {
		tmp = x_46_re * (x_46_re / (1.0 / x_46_re));
	} else {
		tmp = (x_46_im_m * -3.0) * (x_46_im_m * x_46_re);
	}
	return tmp;
}

Fortran

x.im_m = abs(x_46im)
real(8) function code(x_46re, x_46im_m)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46im_m <= 420.0d0) then
        tmp = x_46re * (x_46re / (1.0d0 / x_46re))
    else
        tmp = (x_46im_m * (-3.0d0)) * (x_46im_m * x_46re)
    end if
    code = tmp
end function

Java

x.im_m = Math.abs(x_46_im);
public static double code(double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 420.0) {
		tmp = x_46_re * (x_46_re / (1.0 / x_46_re));
	} else {
		tmp = (x_46_im_m * -3.0) * (x_46_im_m * x_46_re);
	}
	return tmp;
}

Python

x.im_m = math.fabs(x_46_im)
def code(x_46_re, x_46_im_m):
	tmp = 0
	if x_46_im_m <= 420.0:
		tmp = x_46_re * (x_46_re / (1.0 / x_46_re))
	else:
		tmp = (x_46_im_m * -3.0) * (x_46_im_m * x_46_re)
	return tmp

Julia

x.im_m = abs(x_46_im)
function code(x_46_re, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 420.0)
		tmp = Float64(x_46_re * Float64(x_46_re / Float64(1.0 / x_46_re)));
	else
		tmp = Float64(Float64(x_46_im_m * -3.0) * Float64(x_46_im_m * x_46_re));
	end
	return tmp
end

MATLAB

x.im_m = abs(x_46_im);
function tmp_2 = code(x_46_re, x_46_im_m)
	tmp = 0.0;
	if (x_46_im_m <= 420.0)
		tmp = x_46_re * (x_46_re / (1.0 / x_46_re));
	else
		tmp = (x_46_im_m * -3.0) * (x_46_im_m * x_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
code[x$46$re_, x$46$im$95$m_] := If[LessEqual[x$46$im$95$m, 420.0], N[(x$46$re * N[(x$46$re / N[(1.0 / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im$95$m * -3.0), $MachinePrecision] * N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 420

   1. Initial program 88.7%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. *-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-out N/A

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

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

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

      10. +-commutative N/A

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

      11. sub-neg N/A

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

      12. associate-+l+ N/A

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

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

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

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

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

      15. distribute-rgt-neg-out N/A

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

      16. sub-neg N/A

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

      17. count-2 N/A

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

      18. associate-*l* N/A

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

   3. Simplified 91.6%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

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

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

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

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

   6. Applied egg-rr 22.3%

      \[\leadsto x.re \cdot \color{blue}{\frac{1}{\frac{x.re \cdot \left(x.re \cdot \left(x.re \cdot x.re\right)\right) + \left(x.im \cdot \left(x.im \cdot -3\right)\right) \cdot \left(x.im \cdot \left(x.im \cdot -3\right) - x.re \cdot x.re\right)}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot \left(x.re \cdot \left(x.re \cdot x.re\right)\right)\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot \left(x.im \cdot \left(x.im \cdot x.im\right)\right)\right)\right) \cdot -27}}} \]

   7. Taylor expanded in x.re around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 73.0%

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

   9. Simplified 73.0%

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

       1. clear-num N/A

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

       2. associate-/l* N/A

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

       3. /-rgt-identity N/A

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

       4. clear-num N/A

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

       5. times-frac N/A

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

       6. *-rgt-identity N/A

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

       7. div-inv N/A

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

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

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

       9. /-lowering-/.f64 73.0%

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

   11. Applied egg-rr 73.0%

       \[\leadsto x.re \cdot \color{blue}{\frac{x.re}{\frac{1}{x.re}}} \]

   if 420 < x.im

   1. Initial program 65.8%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. *-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-out N/A

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

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

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

      10. +-commutative N/A

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

      11. sub-neg N/A

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

      12. associate-+l+ N/A

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

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

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

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

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

      15. distribute-rgt-neg-out N/A

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

      16. sub-neg N/A

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

      17. count-2 N/A

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

      18. associate-*l* N/A

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

   3. Simplified 76.8%

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

   5. Taylor expanded in x.re around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 69.4%

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

   7. Simplified 69.4%

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

      1. associate-*l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

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

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

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

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

      7. *-lowering-*.f64 77.9%

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

   9. Applied egg-rr 77.9%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 420:\\ \;\;\;\;x.re \cdot \frac{x.re}{\frac{1}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;\left(x.im \cdot -3\right) \cdot \left(x.im \cdot x.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ \begin{array}{l} \mathbf{if}\;x.im\_m \leq 9000000:\\ \;\;\;\;x.re \cdot \frac{x.re}{\frac{1}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(-3 \cdot \left(x.im\_m \cdot x.re\right)\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
(FPCore (x.re x.im_m)
 :precision binary64
 (if (<= x.im_m 9000000.0)
   (* x.re (/ x.re (/ 1.0 x.re)))
   (* x.im_m (* -3.0 (* x.im_m x.re)))))

C

x.im_m = fabs(x_46_im);
double code(double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 9000000.0) {
		tmp = x_46_re * (x_46_re / (1.0 / x_46_re));
	} else {
		tmp = x_46_im_m * (-3.0 * (x_46_im_m * x_46_re));
	}
	return tmp;
}

Fortran

x.im_m = abs(x_46im)
real(8) function code(x_46re, x_46im_m)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46im_m <= 9000000.0d0) then
        tmp = x_46re * (x_46re / (1.0d0 / x_46re))
    else
        tmp = x_46im_m * ((-3.0d0) * (x_46im_m * x_46re))
    end if
    code = tmp
end function

Java

x.im_m = Math.abs(x_46_im);
public static double code(double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 9000000.0) {
		tmp = x_46_re * (x_46_re / (1.0 / x_46_re));
	} else {
		tmp = x_46_im_m * (-3.0 * (x_46_im_m * x_46_re));
	}
	return tmp;
}

Python

x.im_m = math.fabs(x_46_im)
def code(x_46_re, x_46_im_m):
	tmp = 0
	if x_46_im_m <= 9000000.0:
		tmp = x_46_re * (x_46_re / (1.0 / x_46_re))
	else:
		tmp = x_46_im_m * (-3.0 * (x_46_im_m * x_46_re))
	return tmp

Julia

x.im_m = abs(x_46_im)
function code(x_46_re, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 9000000.0)
		tmp = Float64(x_46_re * Float64(x_46_re / Float64(1.0 / x_46_re)));
	else
		tmp = Float64(x_46_im_m * Float64(-3.0 * Float64(x_46_im_m * x_46_re)));
	end
	return tmp
end

MATLAB

x.im_m = abs(x_46_im);
function tmp_2 = code(x_46_re, x_46_im_m)
	tmp = 0.0;
	if (x_46_im_m <= 9000000.0)
		tmp = x_46_re * (x_46_re / (1.0 / x_46_re));
	else
		tmp = x_46_im_m * (-3.0 * (x_46_im_m * x_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
code[x$46$re_, x$46$im$95$m_] := If[LessEqual[x$46$im$95$m, 9000000.0], N[(x$46$re * N[(x$46$re / N[(1.0 / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im$95$m * N[(-3.0 * N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 9e6

   1. Initial program 88.7%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. *-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-out N/A

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

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

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

      10. +-commutative N/A

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

      11. sub-neg N/A

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

      12. associate-+l+ N/A

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

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

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

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

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

      15. distribute-rgt-neg-out N/A

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

      16. sub-neg N/A

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

      17. count-2 N/A

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

      18. associate-*l* N/A

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

   3. Simplified 91.6%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

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

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

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

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

   6. Applied egg-rr 22.3%

      \[\leadsto x.re \cdot \color{blue}{\frac{1}{\frac{x.re \cdot \left(x.re \cdot \left(x.re \cdot x.re\right)\right) + \left(x.im \cdot \left(x.im \cdot -3\right)\right) \cdot \left(x.im \cdot \left(x.im \cdot -3\right) - x.re \cdot x.re\right)}{\left(x.re \cdot x.re\right) \cdot \left(x.re \cdot \left(x.re \cdot \left(x.re \cdot x.re\right)\right)\right) + \left(\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot \left(x.im \cdot \left(x.im \cdot x.im\right)\right)\right)\right) \cdot -27}}} \]

   7. Taylor expanded in x.re around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 73.0%

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

   9. Simplified 73.0%

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

       1. clear-num N/A

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

       2. associate-/l* N/A

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

       3. /-rgt-identity N/A

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

       4. clear-num N/A

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

       5. times-frac N/A

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

       6. *-rgt-identity N/A

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

       7. div-inv N/A

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

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

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

       9. /-lowering-/.f64 73.0%

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

   11. Applied egg-rr 73.0%

       \[\leadsto x.re \cdot \color{blue}{\frac{x.re}{\frac{1}{x.re}}} \]

   if 9e6 < x.im

   1. Initial program 65.8%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. *-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-out N/A

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

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

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

      10. +-commutative N/A

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

      11. sub-neg N/A

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

      12. associate-+l+ N/A

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

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

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

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

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

      15. distribute-rgt-neg-out N/A

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

      16. sub-neg N/A

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

      17. count-2 N/A

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

      18. associate-*l* N/A

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

   3. Simplified 76.8%

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

   5. Taylor expanded in x.re around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 69.4%

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

   7. Simplified 69.4%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*l* N/A

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

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

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

      5. *-commutative N/A

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

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

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

      7. *-lowering-*.f64 77.9%

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

   9. Applied egg-rr 77.9%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 9000000:\\ \;\;\;\;x.re \cdot \frac{x.re}{\frac{1}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(-3 \cdot \left(x.im \cdot x.re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ \begin{array}{l} \mathbf{if}\;x.im\_m \leq 54000:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.im\_m \cdot \left(-3 \cdot \left(x.im\_m \cdot x.re\right)\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
(FPCore (x.re x.im_m)
 :precision binary64
 (if (<= x.im_m 54000.0)
   (* x.re (* x.re x.re))
   (* x.im_m (* -3.0 (* x.im_m x.re)))))

C

x.im_m = fabs(x_46_im);
double code(double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 54000.0) {
		tmp = x_46_re * (x_46_re * x_46_re);
	} else {
		tmp = x_46_im_m * (-3.0 * (x_46_im_m * x_46_re));
	}
	return tmp;
}

Fortran

x.im_m = abs(x_46im)
real(8) function code(x_46re, x_46im_m)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46im_m <= 54000.0d0) then
        tmp = x_46re * (x_46re * x_46re)
    else
        tmp = x_46im_m * ((-3.0d0) * (x_46im_m * x_46re))
    end if
    code = tmp
end function

Java

x.im_m = Math.abs(x_46_im);
public static double code(double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 54000.0) {
		tmp = x_46_re * (x_46_re * x_46_re);
	} else {
		tmp = x_46_im_m * (-3.0 * (x_46_im_m * x_46_re));
	}
	return tmp;
}

Python

x.im_m = math.fabs(x_46_im)
def code(x_46_re, x_46_im_m):
	tmp = 0
	if x_46_im_m <= 54000.0:
		tmp = x_46_re * (x_46_re * x_46_re)
	else:
		tmp = x_46_im_m * (-3.0 * (x_46_im_m * x_46_re))
	return tmp

Julia

x.im_m = abs(x_46_im)
function code(x_46_re, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 54000.0)
		tmp = Float64(x_46_re * Float64(x_46_re * x_46_re));
	else
		tmp = Float64(x_46_im_m * Float64(-3.0 * Float64(x_46_im_m * x_46_re)));
	end
	return tmp
end

MATLAB

x.im_m = abs(x_46_im);
function tmp_2 = code(x_46_re, x_46_im_m)
	tmp = 0.0;
	if (x_46_im_m <= 54000.0)
		tmp = x_46_re * (x_46_re * x_46_re);
	else
		tmp = x_46_im_m * (-3.0 * (x_46_im_m * x_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
code[x$46$re_, x$46$im$95$m_] := If[LessEqual[x$46$im$95$m, 54000.0], N[(x$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision], N[(x$46$im$95$m * N[(-3.0 * N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 54000

   1. Initial program 88.7%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. *-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-out N/A

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

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

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

      10. +-commutative N/A

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

      11. sub-neg N/A

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

      12. associate-+l+ N/A

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

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

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

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

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

      15. distribute-rgt-neg-out N/A

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

      16. sub-neg N/A

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

      17. count-2 N/A

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

      18. associate-*l* N/A

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

   3. Simplified 91.6%

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

   5. Taylor expanded in x.re around inf

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

      1. cube-mult N/A

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

      2. unpow2 N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 73.0%

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

   7. Simplified 73.0%

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

   if 54000 < x.im

   1. Initial program 65.8%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. *-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-out N/A

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

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

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

      10. +-commutative N/A

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

      11. sub-neg N/A

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

      12. associate-+l+ N/A

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

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

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

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

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

      15. distribute-rgt-neg-out N/A

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

      16. sub-neg N/A

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

      17. count-2 N/A

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

      18. associate-*l* N/A

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

   3. Simplified 76.8%

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

   5. Taylor expanded in x.re around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 69.4%

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

   7. Simplified 69.4%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*l* N/A

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

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

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

      5. *-commutative N/A

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

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

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

      7. *-lowering-*.f64 77.9%

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

   9. Applied egg-rr 77.9%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 54000:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(-3 \cdot \left(x.im \cdot x.re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ \begin{array}{l} \mathbf{if}\;x.im\_m \leq 4100000:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \left(x.im\_m \cdot \left(x.im\_m \cdot x.re\right)\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
(FPCore (x.re x.im_m)
 :precision binary64
 (if (<= x.im_m 4100000.0)
   (* x.re (* x.re x.re))
   (* -3.0 (* x.im_m (* x.im_m x.re)))))

C

x.im_m = fabs(x_46_im);
double code(double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 4100000.0) {
		tmp = x_46_re * (x_46_re * x_46_re);
	} else {
		tmp = -3.0 * (x_46_im_m * (x_46_im_m * x_46_re));
	}
	return tmp;
}

Fortran

x.im_m = abs(x_46im)
real(8) function code(x_46re, x_46im_m)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46im_m <= 4100000.0d0) then
        tmp = x_46re * (x_46re * x_46re)
    else
        tmp = (-3.0d0) * (x_46im_m * (x_46im_m * x_46re))
    end if
    code = tmp
end function

Java

x.im_m = Math.abs(x_46_im);
public static double code(double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 4100000.0) {
		tmp = x_46_re * (x_46_re * x_46_re);
	} else {
		tmp = -3.0 * (x_46_im_m * (x_46_im_m * x_46_re));
	}
	return tmp;
}

Python

x.im_m = math.fabs(x_46_im)
def code(x_46_re, x_46_im_m):
	tmp = 0
	if x_46_im_m <= 4100000.0:
		tmp = x_46_re * (x_46_re * x_46_re)
	else:
		tmp = -3.0 * (x_46_im_m * (x_46_im_m * x_46_re))
	return tmp

Julia

x.im_m = abs(x_46_im)
function code(x_46_re, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 4100000.0)
		tmp = Float64(x_46_re * Float64(x_46_re * x_46_re));
	else
		tmp = Float64(-3.0 * Float64(x_46_im_m * Float64(x_46_im_m * x_46_re)));
	end
	return tmp
end

MATLAB

x.im_m = abs(x_46_im);
function tmp_2 = code(x_46_re, x_46_im_m)
	tmp = 0.0;
	if (x_46_im_m <= 4100000.0)
		tmp = x_46_re * (x_46_re * x_46_re);
	else
		tmp = -3.0 * (x_46_im_m * (x_46_im_m * x_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
code[x$46$re_, x$46$im$95$m_] := If[LessEqual[x$46$im$95$m, 4100000.0], N[(x$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision], N[(-3.0 * N[(x$46$im$95$m * N[(x$46$im$95$m * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 4.1e6

   1. Initial program 88.7%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. *-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-out N/A

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

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

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

      10. +-commutative N/A

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

      11. sub-neg N/A

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

      12. associate-+l+ N/A

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

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

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

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

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

      15. distribute-rgt-neg-out N/A

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

      16. sub-neg N/A

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

      17. count-2 N/A

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

      18. associate-*l* N/A

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

   3. Simplified 91.6%

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

   5. Taylor expanded in x.re around inf

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

      1. cube-mult N/A

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

      2. unpow2 N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 73.0%

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

   7. Simplified 73.0%

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

   if 4.1e6 < x.im

   1. Initial program 65.8%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. *-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-out N/A

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

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

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

      10. +-commutative N/A

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

      11. sub-neg N/A

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

      12. associate-+l+ N/A

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

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

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

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

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

      15. distribute-rgt-neg-out N/A

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

      16. sub-neg N/A

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

      17. count-2 N/A

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

      18. associate-*l* N/A

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

   3. Simplified 76.8%

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

   5. Taylor expanded in x.re around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 69.4%

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

   7. Simplified 69.4%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-lowering-*.f64 77.9%

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

   9. Applied egg-rr 77.9%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 4100000:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.im \cdot x.re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ \begin{array}{l} \mathbf{if}\;x.im\_m \leq 41:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \left(x.re \cdot \left(x.im\_m \cdot x.im\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
(FPCore (x.re x.im_m)
 :precision binary64
 (if (<= x.im_m 41.0)
   (* x.re (* x.re x.re))
   (* -3.0 (* x.re (* x.im_m x.im_m)))))

C

x.im_m = fabs(x_46_im);
double code(double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 41.0) {
		tmp = x_46_re * (x_46_re * x_46_re);
	} else {
		tmp = -3.0 * (x_46_re * (x_46_im_m * x_46_im_m));
	}
	return tmp;
}

Fortran

x.im_m = abs(x_46im)
real(8) function code(x_46re, x_46im_m)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46im_m <= 41.0d0) then
        tmp = x_46re * (x_46re * x_46re)
    else
        tmp = (-3.0d0) * (x_46re * (x_46im_m * x_46im_m))
    end if
    code = tmp
end function

Java

x.im_m = Math.abs(x_46_im);
public static double code(double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 41.0) {
		tmp = x_46_re * (x_46_re * x_46_re);
	} else {
		tmp = -3.0 * (x_46_re * (x_46_im_m * x_46_im_m));
	}
	return tmp;
}

Python

x.im_m = math.fabs(x_46_im)
def code(x_46_re, x_46_im_m):
	tmp = 0
	if x_46_im_m <= 41.0:
		tmp = x_46_re * (x_46_re * x_46_re)
	else:
		tmp = -3.0 * (x_46_re * (x_46_im_m * x_46_im_m))
	return tmp

Julia

x.im_m = abs(x_46_im)
function code(x_46_re, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 41.0)
		tmp = Float64(x_46_re * Float64(x_46_re * x_46_re));
	else
		tmp = Float64(-3.0 * Float64(x_46_re * Float64(x_46_im_m * x_46_im_m)));
	end
	return tmp
end

MATLAB

x.im_m = abs(x_46_im);
function tmp_2 = code(x_46_re, x_46_im_m)
	tmp = 0.0;
	if (x_46_im_m <= 41.0)
		tmp = x_46_re * (x_46_re * x_46_re);
	else
		tmp = -3.0 * (x_46_re * (x_46_im_m * x_46_im_m));
	end
	tmp_2 = tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
code[x$46$re_, x$46$im$95$m_] := If[LessEqual[x$46$im$95$m, 41.0], N[(x$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision], N[(-3.0 * N[(x$46$re * N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 41

   1. Initial program 88.7%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. *-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-out N/A

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

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

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

      10. +-commutative N/A

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

      11. sub-neg N/A

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

      12. associate-+l+ N/A

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

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

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

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

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

      15. distribute-rgt-neg-out N/A

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

      16. sub-neg N/A

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

      17. count-2 N/A

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

      18. associate-*l* N/A

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

   3. Simplified 91.6%

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

   5. Taylor expanded in x.re around inf

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

      1. cube-mult N/A

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

      2. unpow2 N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 73.0%

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

   7. Simplified 73.0%

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

   if 41 < x.im

   1. Initial program 65.8%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. *-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-out N/A

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

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

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

      10. +-commutative N/A

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

      11. sub-neg N/A

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

      12. associate-+l+ N/A

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

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

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

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

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

      15. distribute-rgt-neg-out N/A

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

      16. sub-neg N/A

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

      17. count-2 N/A

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

      18. associate-*l* N/A

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

   3. Simplified 76.8%

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

   5. Taylor expanded in x.re around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 69.4%

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

   7. Simplified 69.4%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 41:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \left(x.re \cdot \left(x.im \cdot x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.3% accurate, 3.8× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ x.re \cdot \left(x.re \cdot x.re\right) \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
(FPCore (x.re x.im_m) :precision binary64 (* x.re (* x.re x.re)))

C

x.im_m = fabs(x_46_im);
double code(double x_46_re, double x_46_im_m) {
	return x_46_re * (x_46_re * x_46_re);
}

Fortran

x.im_m = abs(x_46im)
real(8) function code(x_46re, x_46im_m)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    code = x_46re * (x_46re * x_46re)
end function

Java

x.im_m = Math.abs(x_46_im);
public static double code(double x_46_re, double x_46_im_m) {
	return x_46_re * (x_46_re * x_46_re);
}

Python

x.im_m = math.fabs(x_46_im)
def code(x_46_re, x_46_im_m):
	return x_46_re * (x_46_re * x_46_re)

Julia

x.im_m = abs(x_46_im)
function code(x_46_re, x_46_im_m)
	return Float64(x_46_re * Float64(x_46_re * x_46_re))
end

MATLAB

x.im_m = abs(x_46_im);
function tmp = code(x_46_re, x_46_im_m)
	tmp = x_46_re * (x_46_re * x_46_re);
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
code[x$46$re_, x$46$im$95$m_] := N[(x$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.8%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

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

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

   4. *-commutative N/A

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

   5. distribute-rgt-out N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. distribute-lft-out N/A

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

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

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

   10. +-commutative N/A

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

   11. sub-neg N/A

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

   12. associate-+l+ N/A

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

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

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

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

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

   15. distribute-rgt-neg-out N/A

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

   16. sub-neg N/A

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

   17. count-2 N/A

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

   18. associate-*l* N/A

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

3. Simplified 88.5%

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

5. Taylor expanded in x.re around inf

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

   1. cube-mult N/A

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

   2. unpow2 N/A

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

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

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

   4. unpow2 N/A

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

   5. *-lowering-*.f64 62.3%

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

7. Simplified 62.3%

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

Developer Target 1: 87.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(x.re \cdot x.re\right) \cdot \left(x.re - x.im\right) + \left(x.re \cdot x.im\right) \cdot \left(x.re - 3 \cdot x.im\right) \end{array} \]

FPCore

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

C

double code(double x_46_re, double x_46_im) {
	return ((x_46_re * x_46_re) * (x_46_re - x_46_im)) + ((x_46_re * x_46_im) * (x_46_re - (3.0 * 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_46re) * (x_46re - x_46im)) + ((x_46re * x_46im) * (x_46re - (3.0d0 * x_46im)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024156 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (* (* x.re x.re) (- x.re x.im)) (* (* x.re x.im) (- x.re (* 3 x.im)))))

  (- (* (- (* x.re x.re) (* x.im x.im)) x.re) (* (+ (* x.re x.im) (* x.im x.re)) x.im)))