math.cube on complex, real part

Percentage Accurate: 82.4% → 96.9%

Time: 7.4s

Alternatives: 11

Speedup: 1.3×

• 44.4% 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: 82.4% 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: 96.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.re \cdot \mathsf{fma}\left(x.re, x.re, x.im \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{if}\;x.re \leq -6.4 \cdot 10^{+113}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.re \leq 10^{+68}:\\ \;\;\;\;{x.re}^{3} + x.im \cdot \left(x.re \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{elif}\;x.re \leq 5 \cdot 10^{+184}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (* x.re (fma x.re x.re (* x.im (* x.im -3.0))))))
   (if (<= x.re -6.4e+113)
     t_0
     (if (<= x.re 1e+68)
       (+ (pow x.re 3.0) (* x.im (* x.re (* x.im -3.0))))
       (if (<= x.re 5e+184) t_0 (* x.re (* x.re x.re)))))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_re * fma(x_46_re, x_46_re, (x_46_im * (x_46_im * -3.0)));
	double tmp;
	if (x_46_re <= -6.4e+113) {
		tmp = t_0;
	} else if (x_46_re <= 1e+68) {
		tmp = pow(x_46_re, 3.0) + (x_46_im * (x_46_re * (x_46_im * -3.0)));
	} else if (x_46_re <= 5e+184) {
		tmp = t_0;
	} else {
		tmp = x_46_re * (x_46_re * x_46_re);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(x_46_re * fma(x_46_re, x_46_re, Float64(x_46_im * Float64(x_46_im * -3.0))))
	tmp = 0.0
	if (x_46_re <= -6.4e+113)
		tmp = t_0;
	elseif (x_46_re <= 1e+68)
		tmp = Float64((x_46_re ^ 3.0) + Float64(x_46_im * Float64(x_46_re * Float64(x_46_im * -3.0))));
	elseif (x_46_re <= 5e+184)
		tmp = t_0;
	else
		tmp = Float64(x_46_re * Float64(x_46_re * x_46_re));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(x$46$re * N[(x$46$re * x$46$re + N[(x$46$im * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$re, -6.4e+113], t$95$0, If[LessEqual[x$46$re, 1e+68], N[(N[Power[x$46$re, 3.0], $MachinePrecision] + N[(x$46$im * N[(x$46$re * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 5e+184], t$95$0, N[(x$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -6.3999999999999996e113 or 9.99999999999999953e67 < x.re < 4.9999999999999999e184

   1. Initial program 73.8%

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

      1. *-commutative 73.8%

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

      2. distribute-lft-out 73.8%

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

      3. associate-*l* 73.8%

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

      4. *-commutative 73.8%

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

      5. distribute-rgt-out-- 86.9%

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

      6. associate--l- 86.9%

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

      7. associate--l- 86.9%

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

      8. sub-neg 86.9%

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

      9. associate--l+ 86.9%

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

      10. fma-udef 96.7%

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

      11. neg-mul-1 96.7%

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

      12. count-2 96.7%

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

      13. associate-*l* 96.7%

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

      14. distribute-rgt-out-- 96.7%

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

      15. associate-*r* 96.7%

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

      16. metadata-eval 96.7%

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

   3. Simplified 96.7%

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

   if -6.3999999999999996e113 < x.re < 9.99999999999999953e67

   1. Initial program 89.0%

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

      1. *-commutative 89.0%

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

      2. distribute-lft-out 89.0%

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

      3. associate-*l* 89.0%

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

      4. *-commutative 89.0%

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

      5. distribute-rgt-out-- 89.0%

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

      6. associate--l- 89.0%

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

      7. associate--l- 89.0%

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

      8. sub-neg 89.0%

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

      9. associate--l+ 89.0%

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

      10. fma-udef 89.0%

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

      11. neg-mul-1 89.0%

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

      12. count-2 89.0%

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

      13. associate-*l* 89.0%

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

      14. distribute-rgt-out-- 89.0%

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

      15. associate-*r* 89.0%

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

      16. metadata-eval 89.0%

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

   3. Simplified 89.0%

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

      1. fma-udef 89.0%

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

   5. Applied egg-rr 89.0%

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

      1. distribute-lft-in 89.0%

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

      2. cube-unmult 89.1%

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

      3. *-commutative 89.1%

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

      4. associate-*r* 99.9%

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

   7. Applied egg-rr 99.9%

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

   if 4.9999999999999999e184 < x.re

   1. Initial program 51.7%

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

      1. *-commutative 51.7%

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

      2. distribute-lft-out 51.7%

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

      3. associate-*l* 51.7%

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

      4. *-commutative 51.7%

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

      5. distribute-rgt-out-- 75.9%

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

      6. associate--l- 75.9%

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

      7. associate--l- 75.9%

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

      8. sub-neg 75.9%

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

      9. associate--l+ 75.9%

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

      10. fma-udef 79.3%

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

      11. neg-mul-1 79.3%

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

      12. count-2 79.3%

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

      13. associate-*l* 79.3%

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

      14. distribute-rgt-out-- 79.3%

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

      15. associate-*r* 79.3%

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

      16. metadata-eval 79.3%

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

   3. Simplified 79.3%

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

   4. Taylor expanded in x.re around inf 96.6%

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

      1. unpow2 96.6%

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

   6. Simplified 96.6%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -6.4 \cdot 10^{+113}:\\ \;\;\;\;x.re \cdot \mathsf{fma}\left(x.re, x.re, x.im \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{elif}\;x.re \leq 10^{+68}:\\ \;\;\;\;{x.re}^{3} + x.im \cdot \left(x.re \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{elif}\;x.re \leq 5 \cdot 10^{+184}:\\ \;\;\;\;x.re \cdot \mathsf{fma}\left(x.re, x.re, x.im \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \end{array} \]

Alternative 2: 95.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.re \cdot \mathsf{fma}\left(x.re, x.re, x.im \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{if}\;x.re \leq -6.2 \cdot 10^{-71}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.re \leq 1.86 \cdot 10^{-122}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \mathbf{elif}\;x.re \leq 5 \cdot 10^{+186}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (* x.re (fma x.re x.re (* x.im (* x.im -3.0))))))
   (if (<= x.re -6.2e-71)
     t_0
     (if (<= x.re 1.86e-122)
       (* x.im (* x.im (* x.re -3.0)))
       (if (<= x.re 5e+186) t_0 (* x.re (* x.re x.re)))))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_re * fma(x_46_re, x_46_re, (x_46_im * (x_46_im * -3.0)));
	double tmp;
	if (x_46_re <= -6.2e-71) {
		tmp = t_0;
	} else if (x_46_re <= 1.86e-122) {
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0));
	} else if (x_46_re <= 5e+186) {
		tmp = t_0;
	} else {
		tmp = x_46_re * (x_46_re * x_46_re);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(x_46_re * fma(x_46_re, x_46_re, Float64(x_46_im * Float64(x_46_im * -3.0))))
	tmp = 0.0
	if (x_46_re <= -6.2e-71)
		tmp = t_0;
	elseif (x_46_re <= 1.86e-122)
		tmp = Float64(x_46_im * Float64(x_46_im * Float64(x_46_re * -3.0)));
	elseif (x_46_re <= 5e+186)
		tmp = t_0;
	else
		tmp = Float64(x_46_re * Float64(x_46_re * x_46_re));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(x$46$re * N[(x$46$re * x$46$re + N[(x$46$im * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$re, -6.2e-71], t$95$0, If[LessEqual[x$46$re, 1.86e-122], N[(x$46$im * N[(x$46$im * N[(x$46$re * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 5e+186], t$95$0, N[(x$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -6.20000000000000004e-71 or 1.8600000000000001e-122 < x.re < 4.99999999999999954e186

   1. Initial program 88.5%

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

      1. *-commutative 88.5%

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

      2. distribute-lft-out 88.5%

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

      3. associate-*l* 88.4%

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

      4. *-commutative 88.4%

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

      5. distribute-rgt-out-- 93.8%

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

      6. associate--l- 93.8%

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

      7. associate--l- 93.8%

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

      8. sub-neg 93.8%

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

      9. associate--l+ 93.8%

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

      10. fma-udef 97.8%

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

      11. neg-mul-1 97.8%

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

      12. count-2 97.8%

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

      13. associate-*l* 97.8%

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

      14. distribute-rgt-out-- 97.8%

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

      15. associate-*r* 97.9%

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

      16. metadata-eval 97.9%

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

   3. Simplified 97.9%

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

   if -6.20000000000000004e-71 < x.re < 1.8600000000000001e-122

   1. Initial program 78.1%

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

      1. *-commutative 78.1%

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

      2. distribute-lft-out 78.1%

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

      3. associate-*l* 78.1%

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

      4. *-commutative 78.1%

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

      5. distribute-rgt-out-- 78.2%

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

      6. associate--l- 78.2%

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

      7. associate--l- 78.2%

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

      8. sub-neg 78.2%

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

      9. associate--l+ 78.2%

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

      10. fma-udef 78.2%

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

      11. neg-mul-1 78.2%

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

      12. count-2 78.2%

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

      13. associate-*l* 78.2%

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

      14. distribute-rgt-out-- 78.2%

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

      15. associate-*r* 78.2%

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

      16. metadata-eval 78.2%

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

   3. Simplified 78.2%

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

      1. fma-udef 78.2%

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

   5. Applied egg-rr 78.2%

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

   6. Taylor expanded in x.re around 0 78.1%

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

      1. unpow2 78.1%

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

   8. Simplified 78.1%

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

      1. add-sqr-sqrt 63.4%

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

      2. pow2 63.4%

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

      3. associate-*r* 62.1%

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

      4. sqrt-prod 48.0%

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

      5. sqrt-prod 20.4%

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

      6. add-sqr-sqrt 59.9%

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

   10. Applied egg-rr 59.9%

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

       1. unpow2 59.9%

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

       2. *-commutative 59.9%

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

       3. associate-*l* 60.0%

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

       4. associate-*l* 60.0%

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

       5. add-sqr-sqrt 99.8%

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

       6. associate-*r* 99.8%

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

       7. *-commutative 99.8%

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

   12. Applied egg-rr 99.8%

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

   if 4.99999999999999954e186 < x.re

   1. Initial program 51.7%

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

      1. *-commutative 51.7%

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

      2. distribute-lft-out 51.7%

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

      3. associate-*l* 51.7%

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

      4. *-commutative 51.7%

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

      5. distribute-rgt-out-- 75.9%

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

      6. associate--l- 75.9%

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

      7. associate--l- 75.9%

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

      8. sub-neg 75.9%

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

      9. associate--l+ 75.9%

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

      10. fma-udef 79.3%

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

      11. neg-mul-1 79.3%

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

      12. count-2 79.3%

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

      13. associate-*l* 79.3%

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

      14. distribute-rgt-out-- 79.3%

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

      15. associate-*r* 79.3%

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

      16. metadata-eval 79.3%

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

   3. Simplified 79.3%

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

   4. Taylor expanded in x.re around inf 96.6%

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

      1. unpow2 96.6%

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

   6. Simplified 96.6%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -6.2 \cdot 10^{-71}:\\ \;\;\;\;x.re \cdot \mathsf{fma}\left(x.re, x.re, x.im \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{elif}\;x.re \leq 1.86 \cdot 10^{-122}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \mathbf{elif}\;x.re \leq 5 \cdot 10^{+186}:\\ \;\;\;\;x.re \cdot \mathsf{fma}\left(x.re, x.re, x.im \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \end{array} \]

Alternative 3: 97.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.re \cdot \mathsf{fma}\left(x.re, x.re, x.im \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{if}\;x.re \leq -1 \cdot 10^{+90}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.re \leq 10^{+68}:\\ \;\;\;\;{x.re}^{3} + x.im \cdot \left(-3 \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{elif}\;x.re \leq 5 \cdot 10^{+184}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (* x.re (fma x.re x.re (* x.im (* x.im -3.0))))))
   (if (<= x.re -1e+90)
     t_0
     (if (<= x.re 1e+68)
       (+ (pow x.re 3.0) (* x.im (* -3.0 (* x.re x.im))))
       (if (<= x.re 5e+184) t_0 (* x.re (* x.re x.re)))))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_re * fma(x_46_re, x_46_re, (x_46_im * (x_46_im * -3.0)));
	double tmp;
	if (x_46_re <= -1e+90) {
		tmp = t_0;
	} else if (x_46_re <= 1e+68) {
		tmp = pow(x_46_re, 3.0) + (x_46_im * (-3.0 * (x_46_re * x_46_im)));
	} else if (x_46_re <= 5e+184) {
		tmp = t_0;
	} else {
		tmp = x_46_re * (x_46_re * x_46_re);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(x_46_re * fma(x_46_re, x_46_re, Float64(x_46_im * Float64(x_46_im * -3.0))))
	tmp = 0.0
	if (x_46_re <= -1e+90)
		tmp = t_0;
	elseif (x_46_re <= 1e+68)
		tmp = Float64((x_46_re ^ 3.0) + Float64(x_46_im * Float64(-3.0 * Float64(x_46_re * x_46_im))));
	elseif (x_46_re <= 5e+184)
		tmp = t_0;
	else
		tmp = Float64(x_46_re * Float64(x_46_re * x_46_re));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(x$46$re * N[(x$46$re * x$46$re + N[(x$46$im * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$re, -1e+90], t$95$0, If[LessEqual[x$46$re, 1e+68], N[(N[Power[x$46$re, 3.0], $MachinePrecision] + N[(x$46$im * N[(-3.0 * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 5e+184], t$95$0, N[(x$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -9.99999999999999966e89 or 9.99999999999999953e67 < x.re < 4.9999999999999999e184

   1. Initial program 75.8%

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

      1. *-commutative 75.8%

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

      2. distribute-lft-out 75.8%

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

      3. associate-*l* 75.7%

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

      4. *-commutative 75.7%

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

      5. distribute-rgt-out-- 87.9%

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

      6. associate--l- 87.9%

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

      7. associate--l- 87.9%

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

      8. sub-neg 87.9%

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

      9. associate--l+ 87.9%

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

      10. fma-udef 96.9%

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

      11. neg-mul-1 96.9%

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

      12. count-2 96.9%

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

      13. associate-*l* 96.9%

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

      14. distribute-rgt-out-- 96.9%

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

      15. associate-*r* 97.0%

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

      16. metadata-eval 97.0%

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

   3. Simplified 97.0%

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

   if -9.99999999999999966e89 < x.re < 9.99999999999999953e67

   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. *-commutative 88.7%

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

      2. distribute-lft-out 88.7%

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

      3. associate-*l* 88.6%

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

      4. *-commutative 88.6%

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

      5. distribute-rgt-out-- 88.7%

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

      6. associate--l- 88.7%

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

      7. associate--l- 88.7%

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

      8. sub-neg 88.7%

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

      9. associate--l+ 88.7%

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

      10. fma-udef 88.7%

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

      11. neg-mul-1 88.7%

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

      12. count-2 88.7%

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

      13. associate-*l* 88.7%

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

      14. distribute-rgt-out-- 88.7%

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

      15. associate-*r* 88.7%

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

      16. metadata-eval 88.7%

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

   3. Simplified 88.7%

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

      1. fma-udef 88.7%

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

   5. Applied egg-rr 88.7%

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

      1. distribute-lft-in 88.7%

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

      2. cube-unmult 88.8%

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

      3. *-commutative 88.8%

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

      4. associate-*r* 99.9%

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

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in x.re around 0 99.8%

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

   if 4.9999999999999999e184 < x.re

   1. Initial program 51.7%

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

      1. *-commutative 51.7%

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

      2. distribute-lft-out 51.7%

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

      3. associate-*l* 51.7%

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

      4. *-commutative 51.7%

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

      5. distribute-rgt-out-- 75.9%

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

      6. associate--l- 75.9%

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

      7. associate--l- 75.9%

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

      8. sub-neg 75.9%

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

      9. associate--l+ 75.9%

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

      10. fma-udef 79.3%

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

      11. neg-mul-1 79.3%

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

      12. count-2 79.3%

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

      13. associate-*l* 79.3%

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

      14. distribute-rgt-out-- 79.3%

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

      15. associate-*r* 79.3%

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

      16. metadata-eval 79.3%

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

   3. Simplified 79.3%

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

   4. Taylor expanded in x.re around inf 96.6%

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

      1. unpow2 96.6%

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

   6. Simplified 96.6%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -1 \cdot 10^{+90}:\\ \;\;\;\;x.re \cdot \mathsf{fma}\left(x.re, x.re, x.im \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{elif}\;x.re \leq 10^{+68}:\\ \;\;\;\;{x.re}^{3} + x.im \cdot \left(-3 \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{elif}\;x.re \leq 5 \cdot 10^{+184}:\\ \;\;\;\;x.re \cdot \mathsf{fma}\left(x.re, x.re, x.im \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \end{array} \]

Alternative 4: 75.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -2.7 \cdot 10^{+79} \lor \neg \left(x.im \leq -18500000000 \lor \neg \left(x.im \leq -9.8 \cdot 10^{-48}\right) \land x.im \leq 7800000\right):\\ \;\;\;\;-3 \cdot \left(x.re \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -2.7e+79)
         (not
          (or (<= x.im -18500000000.0)
              (and (not (<= x.im -9.8e-48)) (<= x.im 7800000.0)))))
   (* -3.0 (* x.re (* x.im x.im)))
   (* x.re (* x.re x.re))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -2.7e+79) || !((x_46_im <= -18500000000.0) || (!(x_46_im <= -9.8e-48) && (x_46_im <= 7800000.0)))) {
		tmp = -3.0 * (x_46_re * (x_46_im * x_46_im));
	} else {
		tmp = x_46_re * (x_46_re * x_46_re);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46im <= (-2.7d+79)) .or. (.not. (x_46im <= (-18500000000.0d0)) .or. (.not. (x_46im <= (-9.8d-48))) .and. (x_46im <= 7800000.0d0))) then
        tmp = (-3.0d0) * (x_46re * (x_46im * x_46im))
    else
        tmp = x_46re * (x_46re * x_46re)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -2.7e+79) || !((x_46_im <= -18500000000.0) || (!(x_46_im <= -9.8e-48) && (x_46_im <= 7800000.0)))) {
		tmp = -3.0 * (x_46_re * (x_46_im * x_46_im));
	} else {
		tmp = x_46_re * (x_46_re * x_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_im <= -2.7e+79) or not ((x_46_im <= -18500000000.0) or (not (x_46_im <= -9.8e-48) and (x_46_im <= 7800000.0))):
		tmp = -3.0 * (x_46_re * (x_46_im * x_46_im))
	else:
		tmp = x_46_re * (x_46_re * x_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_im <= -2.7e+79) || !((x_46_im <= -18500000000.0) || (!(x_46_im <= -9.8e-48) && (x_46_im <= 7800000.0))))
		tmp = Float64(-3.0 * Float64(x_46_re * Float64(x_46_im * x_46_im)));
	else
		tmp = Float64(x_46_re * Float64(x_46_re * x_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_im <= -2.7e+79) || ~(((x_46_im <= -18500000000.0) || (~((x_46_im <= -9.8e-48)) && (x_46_im <= 7800000.0)))))
		tmp = -3.0 * (x_46_re * (x_46_im * x_46_im));
	else
		tmp = x_46_re * (x_46_re * x_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$im, -2.7e+79], N[Not[Or[LessEqual[x$46$im, -18500000000.0], And[N[Not[LessEqual[x$46$im, -9.8e-48]], $MachinePrecision], LessEqual[x$46$im, 7800000.0]]]], $MachinePrecision]], N[(-3.0 * N[(x$46$re * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -2.7e79 or -1.85e10 < x.im < -9.8000000000000005e-48 or 7.8e6 < x.im

   1. Initial program 64.9%

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

      1. *-commutative 64.9%

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

      2. distribute-lft-out 64.9%

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

      3. associate-*l* 64.8%

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

      4. *-commutative 64.8%

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

      5. distribute-rgt-out-- 71.0%

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

      6. associate--l- 71.0%

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

      7. associate--l- 71.0%

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

      8. sub-neg 71.0%

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

      9. associate--l+ 71.0%

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

      10. fma-udef 77.1%

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

      11. neg-mul-1 77.1%

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

      12. count-2 77.1%

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

      13. associate-*l* 77.1%

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

      14. distribute-rgt-out-- 77.1%

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

      15. associate-*r* 77.2%

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

      16. metadata-eval 77.2%

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

   3. Simplified 77.2%

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

      1. fma-udef 71.0%

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

   5. Applied egg-rr 71.0%

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

   6. Taylor expanded in x.re around 0 63.0%

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

      1. unpow2 63.0%

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

   8. Simplified 63.0%

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

   if -2.7e79 < x.im < -1.85e10 or -9.8000000000000005e-48 < x.im < 7.8e6

   1. Initial program 94.2%

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

      1. *-commutative 94.2%

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

      2. distribute-lft-out 94.2%

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

      3. associate-*l* 94.2%

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

      4. *-commutative 94.2%

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

      5. distribute-rgt-out-- 99.9%

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

      6. associate--l- 99.9%

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

      7. associate--l- 99.9%

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

      8. sub-neg 99.9%

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

      9. associate--l+ 99.9%

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

      10. fma-udef 99.9%

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

      11. neg-mul-1 99.9%

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

      12. count-2 99.9%

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

      13. associate-*l* 99.9%

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

      14. distribute-rgt-out-- 99.9%

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

      15. associate-*r* 99.9%

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

      16. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x.re around inf 92.6%

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

      1. unpow2 92.6%

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

   6. Simplified 92.6%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -2.7 \cdot 10^{+79} \lor \neg \left(x.im \leq -18500000000 \lor \neg \left(x.im \leq -9.8 \cdot 10^{-48}\right) \land x.im \leq 7800000\right):\\ \;\;\;\;-3 \cdot \left(x.re \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \end{array} \]

Alternative 5: 81.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -3.3 \cdot 10^{+78}:\\ \;\;\;\;\left(x.im \cdot -3\right) \cdot \left(x.re \cdot x.im\right)\\ \mathbf{elif}\;x.im \leq -33500000000 \lor \neg \left(x.im \leq -9.8 \cdot 10^{-48}\right) \land x.im \leq 3800000:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im -3.3e+78)
   (* (* x.im -3.0) (* x.re x.im))
   (if (or (<= x.im -33500000000.0)
           (and (not (<= x.im -9.8e-48)) (<= x.im 3800000.0)))
     (* x.re (* x.re x.re))
     (* x.im (* x.im (* x.re -3.0))))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= -3.3e+78) {
		tmp = (x_46_im * -3.0) * (x_46_re * x_46_im);
	} else if ((x_46_im <= -33500000000.0) || (!(x_46_im <= -9.8e-48) && (x_46_im <= 3800000.0))) {
		tmp = x_46_re * (x_46_re * x_46_re);
	} else {
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46im <= (-3.3d+78)) then
        tmp = (x_46im * (-3.0d0)) * (x_46re * x_46im)
    else if ((x_46im <= (-33500000000.0d0)) .or. (.not. (x_46im <= (-9.8d-48))) .and. (x_46im <= 3800000.0d0)) then
        tmp = x_46re * (x_46re * x_46re)
    else
        tmp = x_46im * (x_46im * (x_46re * (-3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= -3.3e+78) {
		tmp = (x_46_im * -3.0) * (x_46_re * x_46_im);
	} else if ((x_46_im <= -33500000000.0) || (!(x_46_im <= -9.8e-48) && (x_46_im <= 3800000.0))) {
		tmp = x_46_re * (x_46_re * x_46_re);
	} else {
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= -3.3e+78:
		tmp = (x_46_im * -3.0) * (x_46_re * x_46_im)
	elif (x_46_im <= -33500000000.0) or (not (x_46_im <= -9.8e-48) and (x_46_im <= 3800000.0)):
		tmp = x_46_re * (x_46_re * x_46_re)
	else:
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0))
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= -3.3e+78)
		tmp = Float64(Float64(x_46_im * -3.0) * Float64(x_46_re * x_46_im));
	elseif ((x_46_im <= -33500000000.0) || (!(x_46_im <= -9.8e-48) && (x_46_im <= 3800000.0)))
		tmp = Float64(x_46_re * Float64(x_46_re * x_46_re));
	else
		tmp = Float64(x_46_im * Float64(x_46_im * Float64(x_46_re * -3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_im <= -3.3e+78)
		tmp = (x_46_im * -3.0) * (x_46_re * x_46_im);
	elseif ((x_46_im <= -33500000000.0) || (~((x_46_im <= -9.8e-48)) && (x_46_im <= 3800000.0)))
		tmp = x_46_re * (x_46_re * x_46_re);
	else
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, -3.3e+78], N[(N[(x$46$im * -3.0), $MachinePrecision] * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x$46$im, -33500000000.0], And[N[Not[LessEqual[x$46$im, -9.8e-48]], $MachinePrecision], LessEqual[x$46$im, 3800000.0]]], N[(x$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(x$46$im * N[(x$46$re * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -3.3e78

   1. Initial program 58.5%

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

      1. *-commutative 58.5%

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

      2. distribute-lft-out 58.5%

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

      3. associate-*l* 58.5%

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

      4. *-commutative 58.5%

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

      5. distribute-rgt-out-- 62.8%

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

      6. associate--l- 62.8%

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

      7. associate--l- 62.8%

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

      8. sub-neg 62.8%

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

      9. associate--l+ 62.8%

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

      10. fma-udef 71.3%

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

      11. neg-mul-1 71.3%

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

      12. count-2 71.3%

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

      13. associate-*l* 71.3%

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

      14. distribute-rgt-out-- 71.3%

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

      15. associate-*r* 71.3%

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

      16. metadata-eval 71.3%

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

   3. Simplified 71.3%

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

      1. fma-udef 62.8%

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

   5. Applied egg-rr 62.8%

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

   6. Taylor expanded in x.re around 0 67.0%

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

      1. unpow2 67.0%

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

   8. Simplified 67.0%

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

      1. add-sqr-sqrt 39.0%

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

      2. pow2 39.0%

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

      3. associate-*r* 39.0%

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

      4. sqrt-prod 38.9%

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

      5. sqrt-prod 0.0%

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

      6. add-sqr-sqrt 50.8%

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

   10. Applied egg-rr 50.8%

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

       1. unpow2 50.8%

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

       2. swap-sqr 38.9%

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

       3. add-sqr-sqrt 67.0%

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

       4. associate-*l* 85.0%

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

       5. associate-*r* 84.9%

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

       6. *-commutative 84.9%

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

       7. associate-*l* 85.0%

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

       8. *-commutative 85.0%

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

   12. Applied egg-rr 85.0%

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

   if -3.3e78 < x.im < -3.35e10 or -9.8000000000000005e-48 < x.im < 3.8e6

   1. Initial program 94.2%

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

      1. *-commutative 94.2%

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

      2. distribute-lft-out 94.2%

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

      3. associate-*l* 94.2%

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

      4. *-commutative 94.2%

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

      5. distribute-rgt-out-- 99.9%

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

      6. associate--l- 99.9%

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

      7. associate--l- 99.9%

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

      8. sub-neg 99.9%

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

      9. associate--l+ 99.9%

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

      10. fma-udef 99.9%

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

      11. neg-mul-1 99.9%

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

      12. count-2 99.9%

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

      13. associate-*l* 99.9%

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

      14. distribute-rgt-out-- 99.9%

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

      15. associate-*r* 99.9%

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

      16. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x.re around inf 92.6%

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

      1. unpow2 92.6%

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

   6. Simplified 92.6%

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

   if -3.35e10 < x.im < -9.8000000000000005e-48 or 3.8e6 < x.im

   1. Initial program 69.3%

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

      1. *-commutative 69.3%

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

      2. distribute-lft-out 69.3%

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

      3. associate-*l* 69.3%

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

      4. *-commutative 69.3%

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

      5. distribute-rgt-out-- 76.8%

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

      6. associate--l- 76.8%

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

      7. associate--l- 76.8%

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

      8. sub-neg 76.8%

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

      9. associate--l+ 76.8%

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

      10. fma-udef 81.2%

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

      11. neg-mul-1 81.2%

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

      12. count-2 81.2%

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

      13. associate-*l* 81.2%

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

      14. distribute-rgt-out-- 81.2%

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

      15. associate-*r* 81.3%

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

      16. metadata-eval 81.3%

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

   3. Simplified 81.3%

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

      1. fma-udef 76.8%

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

   5. Applied egg-rr 76.8%

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

   6. Taylor expanded in x.re around 0 60.2%

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

      1. unpow2 60.2%

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

   8. Simplified 60.2%

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

      1. add-sqr-sqrt 35.5%

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

      2. pow2 35.5%

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

      3. associate-*r* 35.5%

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

      4. sqrt-prod 35.5%

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

      5. sqrt-prod 27.7%

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

      6. add-sqr-sqrt 41.0%

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

   10. Applied egg-rr 41.0%

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

       1. unpow2 41.0%

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

       2. *-commutative 41.0%

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

       3. associate-*l* 41.1%

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

       4. associate-*l* 41.1%

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

       5. add-sqr-sqrt 74.2%

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

       6. associate-*r* 74.2%

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

       7. *-commutative 74.2%

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

   12. Applied egg-rr 74.2%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -3.3 \cdot 10^{+78}:\\ \;\;\;\;\left(x.im \cdot -3\right) \cdot \left(x.re \cdot x.im\right)\\ \mathbf{elif}\;x.im \leq -33500000000 \lor \neg \left(x.im \leq -9.8 \cdot 10^{-48}\right) \land x.im \leq 3800000:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \end{array} \]

Alternative 6: 75.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.re \cdot \left(-3 \cdot \left(x.im \cdot x.im\right)\right)\\ t_1 := x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{if}\;x.im \leq -2.8 \cdot 10^{+78}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.im \leq -52000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x.im \leq -9.8 \cdot 10^{-48}:\\ \;\;\;\;-3 \cdot \left(x.re \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{elif}\;x.im \leq 5000000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (* x.re (* -3.0 (* x.im x.im)))) (t_1 (* x.re (* x.re x.re))))
   (if (<= x.im -2.8e+78)
     t_0
     (if (<= x.im -52000000000.0)
       t_1
       (if (<= x.im -9.8e-48)
         (* -3.0 (* x.re (* x.im x.im)))
         (if (<= x.im 5000000.0) t_1 t_0))))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_re * (-3.0 * (x_46_im * x_46_im));
	double t_1 = x_46_re * (x_46_re * x_46_re);
	double tmp;
	if (x_46_im <= -2.8e+78) {
		tmp = t_0;
	} else if (x_46_im <= -52000000000.0) {
		tmp = t_1;
	} else if (x_46_im <= -9.8e-48) {
		tmp = -3.0 * (x_46_re * (x_46_im * x_46_im));
	} else if (x_46_im <= 5000000.0) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x_46re * ((-3.0d0) * (x_46im * x_46im))
    t_1 = x_46re * (x_46re * x_46re)
    if (x_46im <= (-2.8d+78)) then
        tmp = t_0
    else if (x_46im <= (-52000000000.0d0)) then
        tmp = t_1
    else if (x_46im <= (-9.8d-48)) then
        tmp = (-3.0d0) * (x_46re * (x_46im * x_46im))
    else if (x_46im <= 5000000.0d0) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_re * (-3.0 * (x_46_im * x_46_im));
	double t_1 = x_46_re * (x_46_re * x_46_re);
	double tmp;
	if (x_46_im <= -2.8e+78) {
		tmp = t_0;
	} else if (x_46_im <= -52000000000.0) {
		tmp = t_1;
	} else if (x_46_im <= -9.8e-48) {
		tmp = -3.0 * (x_46_re * (x_46_im * x_46_im));
	} else if (x_46_im <= 5000000.0) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	t_0 = x_46_re * (-3.0 * (x_46_im * x_46_im))
	t_1 = x_46_re * (x_46_re * x_46_re)
	tmp = 0
	if x_46_im <= -2.8e+78:
		tmp = t_0
	elif x_46_im <= -52000000000.0:
		tmp = t_1
	elif x_46_im <= -9.8e-48:
		tmp = -3.0 * (x_46_re * (x_46_im * x_46_im))
	elif x_46_im <= 5000000.0:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(x_46_re * Float64(-3.0 * Float64(x_46_im * x_46_im)))
	t_1 = Float64(x_46_re * Float64(x_46_re * x_46_re))
	tmp = 0.0
	if (x_46_im <= -2.8e+78)
		tmp = t_0;
	elseif (x_46_im <= -52000000000.0)
		tmp = t_1;
	elseif (x_46_im <= -9.8e-48)
		tmp = Float64(-3.0 * Float64(x_46_re * Float64(x_46_im * x_46_im)));
	elseif (x_46_im <= 5000000.0)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	t_0 = x_46_re * (-3.0 * (x_46_im * x_46_im));
	t_1 = x_46_re * (x_46_re * x_46_re);
	tmp = 0.0;
	if (x_46_im <= -2.8e+78)
		tmp = t_0;
	elseif (x_46_im <= -52000000000.0)
		tmp = t_1;
	elseif (x_46_im <= -9.8e-48)
		tmp = -3.0 * (x_46_re * (x_46_im * x_46_im));
	elseif (x_46_im <= 5000000.0)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(x$46$re * N[(-3.0 * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$im, -2.8e+78], t$95$0, If[LessEqual[x$46$im, -52000000000.0], t$95$1, If[LessEqual[x$46$im, -9.8e-48], N[(-3.0 * N[(x$46$re * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 5000000.0], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -2.8000000000000001e78 or 5e6 < x.im

   1. Initial program 59.2%

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

      1. *-commutative 59.2%

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

      2. distribute-lft-out 59.2%

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

      3. associate-*l* 59.2%

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

      4. *-commutative 59.2%

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

      5. distribute-rgt-out-- 66.4%

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

      6. associate--l- 66.4%

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

      7. associate--l- 66.4%

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

      8. sub-neg 66.4%

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

      9. associate--l+ 66.4%

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

      10. fma-udef 73.5%

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

      11. neg-mul-1 73.5%

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

      12. count-2 73.5%

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

      13. associate-*l* 73.5%

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

      14. distribute-rgt-out-- 73.5%

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

      15. associate-*r* 73.5%

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

      16. metadata-eval 73.5%

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

   3. Simplified 73.5%

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

   4. Taylor expanded in x.re around 0 61.1%

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

      1. associate-*r* 61.1%

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

      2. *-commutative 61.1%

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

      3. *-commutative 61.1%

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

      4. metadata-eval 61.1%

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

      5. distribute-rgt-out 61.1%

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

      6. +-commutative 61.1%

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

      7. distribute-rgt-in 61.1%

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

      8. *-commutative 61.1%

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

      9. metadata-eval 61.1%

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

      10. distribute-lft-neg-in 61.1%

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

      11. count-2 61.1%

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

      12. distribute-rgt-neg-in 61.1%

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

      13. distribute-rgt-out 61.1%

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

      14. distribute-lft-out 61.1%

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

      15. distribute-rgt-neg-in 61.1%

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

      16. count-2 61.1%

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

      17. distribute-lft-neg-in 61.1%

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

      18. metadata-eval 61.1%

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

      19. mul-1-neg 61.1%

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

      20. distribute-lft-neg-in 61.1%

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

      21. distribute-rgt-neg-in 61.1%

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

      22. unpow2 61.1%

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

      23. distribute-rgt-neg-out 61.1%

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

   6. Simplified 61.2%

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

   if -2.8000000000000001e78 < x.im < -5.2e10 or -9.8000000000000005e-48 < x.im < 5e6

   1. Initial program 94.2%

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

      1. *-commutative 94.2%

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

      2. distribute-lft-out 94.2%

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

      3. associate-*l* 94.2%

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

      4. *-commutative 94.2%

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

      5. distribute-rgt-out-- 99.9%

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

      6. associate--l- 99.9%

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

      7. associate--l- 99.9%

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

      8. sub-neg 99.9%

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

      9. associate--l+ 99.9%

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

      10. fma-udef 99.9%

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

      11. neg-mul-1 99.9%

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

      12. count-2 99.9%

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

      13. associate-*l* 99.9%

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

      14. distribute-rgt-out-- 99.9%

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

      15. associate-*r* 99.9%

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

      16. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x.re around inf 92.6%

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

      1. unpow2 92.6%

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

   6. Simplified 92.6%

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

   if -5.2e10 < x.im < -9.8000000000000005e-48

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. distribute-lft-out 99.8%

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

      3. associate-*l* 99.5%

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

      4. *-commutative 99.5%

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

      5. distribute-rgt-out-- 99.3%

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

      6. associate--l- 99.3%

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

      7. associate--l- 99.3%

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

      8. sub-neg 99.3%

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

      9. associate--l+ 99.3%

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

      10. fma-udef 99.3%

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

      11. neg-mul-1 99.3%

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

      12. count-2 99.3%

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

      13. associate-*l* 99.3%

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

      14. distribute-rgt-out-- 99.3%

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

      15. associate-*r* 99.5%

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

      16. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

      1. fma-udef 99.5%

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

   5. Applied egg-rr 99.5%

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

   6. Taylor expanded in x.re around 0 74.7%

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

      1. unpow2 74.7%

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

   8. Simplified 74.7%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -2.8 \cdot 10^{+78}:\\ \;\;\;\;x.re \cdot \left(-3 \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{elif}\;x.im \leq -52000000000:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{elif}\;x.im \leq -9.8 \cdot 10^{-48}:\\ \;\;\;\;-3 \cdot \left(x.re \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{elif}\;x.im \leq 5000000:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(-3 \cdot \left(x.im \cdot x.im\right)\right)\\ \end{array} \]

Alternative 7: 81.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x.im \cdot -3\right) \cdot \left(x.re \cdot x.im\right)\\ t_1 := x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{if}\;x.im \leq -2.8 \cdot 10^{+78}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.im \leq -36000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x.im \leq -2.2 \cdot 10^{-48}:\\ \;\;\;\;-3 \cdot \left(x.re \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{elif}\;x.im \leq 2550000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (* (* x.im -3.0) (* x.re x.im))) (t_1 (* x.re (* x.re x.re))))
   (if (<= x.im -2.8e+78)
     t_0
     (if (<= x.im -36000000000.0)
       t_1
       (if (<= x.im -2.2e-48)
         (* -3.0 (* x.re (* x.im x.im)))
         (if (<= x.im 2550000.0) t_1 t_0))))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = (x_46_im * -3.0) * (x_46_re * x_46_im);
	double t_1 = x_46_re * (x_46_re * x_46_re);
	double tmp;
	if (x_46_im <= -2.8e+78) {
		tmp = t_0;
	} else if (x_46_im <= -36000000000.0) {
		tmp = t_1;
	} else if (x_46_im <= -2.2e-48) {
		tmp = -3.0 * (x_46_re * (x_46_im * x_46_im));
	} else if (x_46_im <= 2550000.0) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x_46im * (-3.0d0)) * (x_46re * x_46im)
    t_1 = x_46re * (x_46re * x_46re)
    if (x_46im <= (-2.8d+78)) then
        tmp = t_0
    else if (x_46im <= (-36000000000.0d0)) then
        tmp = t_1
    else if (x_46im <= (-2.2d-48)) then
        tmp = (-3.0d0) * (x_46re * (x_46im * x_46im))
    else if (x_46im <= 2550000.0d0) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double t_0 = (x_46_im * -3.0) * (x_46_re * x_46_im);
	double t_1 = x_46_re * (x_46_re * x_46_re);
	double tmp;
	if (x_46_im <= -2.8e+78) {
		tmp = t_0;
	} else if (x_46_im <= -36000000000.0) {
		tmp = t_1;
	} else if (x_46_im <= -2.2e-48) {
		tmp = -3.0 * (x_46_re * (x_46_im * x_46_im));
	} else if (x_46_im <= 2550000.0) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	t_0 = (x_46_im * -3.0) * (x_46_re * x_46_im)
	t_1 = x_46_re * (x_46_re * x_46_re)
	tmp = 0
	if x_46_im <= -2.8e+78:
		tmp = t_0
	elif x_46_im <= -36000000000.0:
		tmp = t_1
	elif x_46_im <= -2.2e-48:
		tmp = -3.0 * (x_46_re * (x_46_im * x_46_im))
	elif x_46_im <= 2550000.0:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(Float64(x_46_im * -3.0) * Float64(x_46_re * x_46_im))
	t_1 = Float64(x_46_re * Float64(x_46_re * x_46_re))
	tmp = 0.0
	if (x_46_im <= -2.8e+78)
		tmp = t_0;
	elseif (x_46_im <= -36000000000.0)
		tmp = t_1;
	elseif (x_46_im <= -2.2e-48)
		tmp = Float64(-3.0 * Float64(x_46_re * Float64(x_46_im * x_46_im)));
	elseif (x_46_im <= 2550000.0)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	t_0 = (x_46_im * -3.0) * (x_46_re * x_46_im);
	t_1 = x_46_re * (x_46_re * x_46_re);
	tmp = 0.0;
	if (x_46_im <= -2.8e+78)
		tmp = t_0;
	elseif (x_46_im <= -36000000000.0)
		tmp = t_1;
	elseif (x_46_im <= -2.2e-48)
		tmp = -3.0 * (x_46_re * (x_46_im * x_46_im));
	elseif (x_46_im <= 2550000.0)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(N[(x$46$im * -3.0), $MachinePrecision] * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$im, -2.8e+78], t$95$0, If[LessEqual[x$46$im, -36000000000.0], t$95$1, If[LessEqual[x$46$im, -2.2e-48], N[(-3.0 * N[(x$46$re * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 2550000.0], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -2.8000000000000001e78 or 2.55e6 < x.im

   1. Initial program 59.2%

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

      1. *-commutative 59.2%

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

      2. distribute-lft-out 59.2%

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

      3. associate-*l* 59.2%

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

      4. *-commutative 59.2%

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

      5. distribute-rgt-out-- 66.4%

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

      6. associate--l- 66.4%

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

      7. associate--l- 66.4%

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

      8. sub-neg 66.4%

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

      9. associate--l+ 66.4%

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

      10. fma-udef 73.5%

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

      11. neg-mul-1 73.5%

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

      12. count-2 73.5%

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

      13. associate-*l* 73.5%

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

      14. distribute-rgt-out-- 73.5%

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

      15. associate-*r* 73.5%

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

      16. metadata-eval 73.5%

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

   3. Simplified 73.5%

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

      1. fma-udef 66.4%

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

   5. Applied egg-rr 66.4%

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

   6. Taylor expanded in x.re around 0 61.1%

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

      1. unpow2 61.1%

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

   8. Simplified 61.1%

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

      1. add-sqr-sqrt 33.9%

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

      2. pow2 33.9%

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

      3. associate-*r* 33.9%

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

      4. sqrt-prod 33.8%

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

      5. sqrt-prod 18.9%

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

      6. add-sqr-sqrt 43.3%

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

   10. Applied egg-rr 43.3%

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

       1. unpow2 43.3%

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

       2. swap-sqr 33.8%

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

       3. add-sqr-sqrt 61.1%

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

       4. associate-*l* 79.3%

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

       5. associate-*r* 79.3%

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

       6. *-commutative 79.3%

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

       7. associate-*l* 79.3%

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

       8. *-commutative 79.3%

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

   12. Applied egg-rr 79.3%

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

   if -2.8000000000000001e78 < x.im < -3.6e10 or -2.20000000000000013e-48 < x.im < 2.55e6

   1. Initial program 94.2%

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

      1. *-commutative 94.2%

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

      2. distribute-lft-out 94.2%

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

      3. associate-*l* 94.2%

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

      4. *-commutative 94.2%

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

      5. distribute-rgt-out-- 99.9%

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

      6. associate--l- 99.9%

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

      7. associate--l- 99.9%

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

      8. sub-neg 99.9%

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

      9. associate--l+ 99.9%

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

      10. fma-udef 99.9%

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

      11. neg-mul-1 99.9%

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

      12. count-2 99.9%

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

      13. associate-*l* 99.9%

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

      14. distribute-rgt-out-- 99.9%

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

      15. associate-*r* 99.9%

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

      16. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x.re around inf 92.6%

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

      1. unpow2 92.6%

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

   6. Simplified 92.6%

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

   if -3.6e10 < x.im < -2.20000000000000013e-48

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. distribute-lft-out 99.8%

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

      3. associate-*l* 99.5%

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

      4. *-commutative 99.5%

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

      5. distribute-rgt-out-- 99.3%

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

      6. associate--l- 99.3%

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

      7. associate--l- 99.3%

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

      8. sub-neg 99.3%

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

      9. associate--l+ 99.3%

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

      10. fma-udef 99.3%

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

      11. neg-mul-1 99.3%

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

      12. count-2 99.3%

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

      13. associate-*l* 99.3%

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

      14. distribute-rgt-out-- 99.3%

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

      15. associate-*r* 99.5%

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

      16. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

      1. fma-udef 99.5%

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

   5. Applied egg-rr 99.5%

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

   6. Taylor expanded in x.re around 0 74.7%

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

      1. unpow2 74.7%

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

   8. Simplified 74.7%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -2.8 \cdot 10^{+78}:\\ \;\;\;\;\left(x.im \cdot -3\right) \cdot \left(x.re \cdot x.im\right)\\ \mathbf{elif}\;x.im \leq -36000000000:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{elif}\;x.im \leq -2.2 \cdot 10^{-48}:\\ \;\;\;\;-3 \cdot \left(x.re \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{elif}\;x.im \leq 2550000:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im \cdot -3\right) \cdot \left(x.re \cdot x.im\right)\\ \end{array} \]

Alternative 8: 96.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -7.8 \cdot 10^{+153}:\\ \;\;\;\;\left(x.im \cdot -3\right) \cdot \left(x.re \cdot x.im\right)\\ \mathbf{elif}\;x.im \leq 1.05 \cdot 10^{+136}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.im \cdot -3\right) + x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im -7.8e+153)
   (* (* x.im -3.0) (* x.re x.im))
   (if (<= x.im 1.05e+136)
     (* x.re (+ (* x.im (* x.im -3.0)) (* x.re x.re)))
     (* x.im (* x.im (* x.re -3.0))))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= -7.8e+153) {
		tmp = (x_46_im * -3.0) * (x_46_re * x_46_im);
	} else if (x_46_im <= 1.05e+136) {
		tmp = x_46_re * ((x_46_im * (x_46_im * -3.0)) + (x_46_re * x_46_re));
	} else {
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46im <= (-7.8d+153)) then
        tmp = (x_46im * (-3.0d0)) * (x_46re * x_46im)
    else if (x_46im <= 1.05d+136) then
        tmp = x_46re * ((x_46im * (x_46im * (-3.0d0))) + (x_46re * x_46re))
    else
        tmp = x_46im * (x_46im * (x_46re * (-3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= -7.8e+153) {
		tmp = (x_46_im * -3.0) * (x_46_re * x_46_im);
	} else if (x_46_im <= 1.05e+136) {
		tmp = x_46_re * ((x_46_im * (x_46_im * -3.0)) + (x_46_re * x_46_re));
	} else {
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= -7.8e+153:
		tmp = (x_46_im * -3.0) * (x_46_re * x_46_im)
	elif x_46_im <= 1.05e+136:
		tmp = x_46_re * ((x_46_im * (x_46_im * -3.0)) + (x_46_re * x_46_re))
	else:
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0))
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= -7.8e+153)
		tmp = Float64(Float64(x_46_im * -3.0) * Float64(x_46_re * x_46_im));
	elseif (x_46_im <= 1.05e+136)
		tmp = Float64(x_46_re * Float64(Float64(x_46_im * Float64(x_46_im * -3.0)) + Float64(x_46_re * x_46_re)));
	else
		tmp = Float64(x_46_im * Float64(x_46_im * Float64(x_46_re * -3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_im <= -7.8e+153)
		tmp = (x_46_im * -3.0) * (x_46_re * x_46_im);
	elseif (x_46_im <= 1.05e+136)
		tmp = x_46_re * ((x_46_im * (x_46_im * -3.0)) + (x_46_re * x_46_re));
	else
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, -7.8e+153], N[(N[(x$46$im * -3.0), $MachinePrecision] * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 1.05e+136], N[(x$46$re * N[(N[(x$46$im * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(x$46$im * N[(x$46$re * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -7.79999999999999966e153

   1. Initial program 50.0%

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

      1. *-commutative 50.0%

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

      2. distribute-lft-out 50.0%

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

      3. associate-*l* 50.0%

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

      4. *-commutative 50.0%

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

      5. distribute-rgt-out-- 50.0%

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

      6. associate--l- 50.0%

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

      7. associate--l- 50.0%

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

      8. sub-neg 50.0%

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

      9. associate--l+ 50.0%

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

      10. fma-udef 61.5%

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

      11. neg-mul-1 61.5%

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

      12. count-2 61.5%

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

      13. associate-*l* 61.5%

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

      14. distribute-rgt-out-- 61.5%

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

      15. associate-*r* 61.5%

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

      16. metadata-eval 61.5%

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

   3. Simplified 61.5%

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

      1. fma-udef 50.0%

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

   5. Applied egg-rr 50.0%

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

   6. Taylor expanded in x.re around 0 61.5%

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

      1. unpow2 61.5%

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

   8. Simplified 61.5%

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

      1. add-sqr-sqrt 29.6%

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

      2. pow2 29.6%

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

      3. associate-*r* 29.6%

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

      4. sqrt-prod 29.6%

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

      5. sqrt-prod 0.0%

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

      6. add-sqr-sqrt 45.5%

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

   10. Applied egg-rr 45.5%

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

       1. unpow2 45.5%

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

       2. swap-sqr 29.6%

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

       3. add-sqr-sqrt 61.5%

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

       4. associate-*l* 85.7%

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

       5. associate-*r* 85.6%

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

       6. *-commutative 85.6%

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

       7. associate-*l* 85.7%

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

       8. *-commutative 85.7%

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

   12. Applied egg-rr 85.7%

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

   if -7.79999999999999966e153 < x.im < 1.05e136

   1. Initial program 92.0%

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

      1. *-commutative 92.0%

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

      2. distribute-lft-out 92.0%

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

      3. associate-*l* 91.9%

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

      4. *-commutative 91.9%

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

      5. distribute-rgt-out-- 99.8%

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

      6. associate--l- 99.8%

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

      7. associate--l- 99.8%

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

      8. sub-neg 99.8%

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

      9. associate--l+ 99.8%

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

      10. fma-udef 99.8%

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

      11. neg-mul-1 99.8%

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

      12. count-2 99.8%

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

      13. associate-*l* 99.8%

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

      14. distribute-rgt-out-- 99.8%

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

      15. associate-*r* 99.8%

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

      16. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. fma-udef 99.8%

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

   5. Applied egg-rr 99.8%

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

   if 1.05e136 < x.im

   1. Initial program 48.7%

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

      1. *-commutative 48.7%

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

      2. distribute-lft-out 48.7%

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

      3. associate-*l* 48.8%

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

      4. *-commutative 48.8%

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

      5. distribute-rgt-out-- 48.8%

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

      6. associate--l- 48.8%

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

      7. associate--l- 48.8%

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

      8. sub-neg 48.8%

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

      9. associate--l+ 48.8%

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

      10. fma-udef 58.8%

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

      11. neg-mul-1 58.8%

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

      12. count-2 58.8%

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

      13. associate-*l* 58.8%

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

      14. distribute-rgt-out-- 58.8%

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

      15. associate-*r* 58.8%

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

      16. metadata-eval 58.8%

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

   3. Simplified 58.8%

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

      1. fma-udef 48.8%

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

   5. Applied egg-rr 48.8%

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

   6. Taylor expanded in x.re around 0 58.8%

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

      1. unpow2 58.8%

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

   8. Simplified 58.8%

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

      1. add-sqr-sqrt 30.8%

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

      2. pow2 30.8%

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

      3. associate-*r* 30.8%

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

      4. sqrt-prod 30.8%

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

      5. sqrt-prod 43.2%

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

      6. add-sqr-sqrt 43.2%

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

   10. Applied egg-rr 43.2%

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

       1. unpow2 43.2%

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

       2. *-commutative 43.2%

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

       3. associate-*l* 43.3%

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

       4. associate-*l* 43.3%

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

       5. add-sqr-sqrt 89.9%

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

       6. associate-*r* 89.9%

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

       7. *-commutative 89.9%

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

   12. Applied egg-rr 89.9%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -7.8 \cdot 10^{+153}:\\ \;\;\;\;\left(x.im \cdot -3\right) \cdot \left(x.re \cdot x.im\right)\\ \mathbf{elif}\;x.im \leq 1.05 \cdot 10^{+136}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.im \cdot -3\right) + x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \end{array} \]

Alternative 9: 70.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -2.4 \cdot 10^{+115} \lor \neg \left(x.im \leq 6.5 \cdot 10^{+135}\right):\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -2.4e+115) (not (<= x.im 6.5e+135)))
   (* x.re (* x.im (- x.im)))
   (* x.re (* x.re x.re))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -2.4e+115) || !(x_46_im <= 6.5e+135)) {
		tmp = x_46_re * (x_46_im * -x_46_im);
	} else {
		tmp = x_46_re * (x_46_re * x_46_re);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46im <= (-2.4d+115)) .or. (.not. (x_46im <= 6.5d+135))) then
        tmp = x_46re * (x_46im * -x_46im)
    else
        tmp = x_46re * (x_46re * x_46re)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -2.4e+115) || !(x_46_im <= 6.5e+135)) {
		tmp = x_46_re * (x_46_im * -x_46_im);
	} else {
		tmp = x_46_re * (x_46_re * x_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_im <= -2.4e+115) or not (x_46_im <= 6.5e+135):
		tmp = x_46_re * (x_46_im * -x_46_im)
	else:
		tmp = x_46_re * (x_46_re * x_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_im <= -2.4e+115) || !(x_46_im <= 6.5e+135))
		tmp = Float64(x_46_re * Float64(x_46_im * Float64(-x_46_im)));
	else
		tmp = Float64(x_46_re * Float64(x_46_re * x_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_im <= -2.4e+115) || ~((x_46_im <= 6.5e+135)))
		tmp = x_46_re * (x_46_im * -x_46_im);
	else
		tmp = x_46_re * (x_46_re * x_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$im, -2.4e+115], N[Not[LessEqual[x$46$im, 6.5e+135]], $MachinePrecision]], N[(x$46$re * N[(x$46$im * (-x$46$im)), $MachinePrecision]), $MachinePrecision], N[(x$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -2.4e115 or 6.5000000000000003e135 < x.im

   1. Initial program 53.0%

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

      1. *-commutative 53.0%

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

      2. *-commutative 53.0%

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

      3. *-commutative 53.0%

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

      4. distribute-lft-out 53.0%

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

   3. Simplified 53.0%

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

   4. Taylor expanded in x.re around 0 63.0%

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

      1. associate-*r* 63.0%

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

      2. mul-1-neg 63.0%

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

      3. unpow2 63.0%

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

   6. Simplified 63.0%

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

      1. associate-*r* 88.3%

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

      2. *-commutative 88.3%

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

      3. distribute-rgt-out-- 88.5%

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

      4. *-commutative 88.5%

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

      5. add-sqr-sqrt 45.6%

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

      6. sqrt-unprod 52.7%

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

      7. sqr-neg 52.7%

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

      8. sqrt-unprod 17.1%

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

      9. add-sqr-sqrt 38.7%

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

      10. distribute-rgt-in 38.7%

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

      11. distribute-lft-out 38.7%

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

   8. Applied egg-rr 38.7%

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

      1. distribute-lft-out-- 61.5%

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

   10. Simplified 61.5%

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

   11. Taylor expanded in x.im around 0 58.4%

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

       1. count-2 58.4%

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

       2. associate--r+ 58.4%

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

       3. +-inverses 58.4%

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

       4. sub0-neg 58.4%

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

       5. unpow2 58.4%

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

       6. distribute-lft-neg-in 58.4%

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

       7. distribute-rgt-neg-in 58.4%

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

   13. Simplified 58.4%

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

   if -2.4e115 < x.im < 6.5000000000000003e135

   1. Initial program 91.8%

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

      1. *-commutative 91.8%

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

      2. distribute-lft-out 91.8%

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

      3. associate-*l* 91.7%

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

      4. *-commutative 91.7%

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

      5. distribute-rgt-out-- 99.8%

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

      6. associate--l- 99.8%

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

      7. associate--l- 99.8%

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

      8. sub-neg 99.8%

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

      9. associate--l+ 99.8%

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

      10. fma-udef 99.8%

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

      11. neg-mul-1 99.8%

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

      12. count-2 99.8%

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

      13. associate-*l* 99.8%

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

      14. distribute-rgt-out-- 99.8%

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

      15. associate-*r* 99.8%

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

      16. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x.re around inf 78.6%

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

      1. unpow2 78.6%

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

   6. Simplified 78.6%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -2.4 \cdot 10^{+115} \lor \neg \left(x.im \leq 6.5 \cdot 10^{+135}\right):\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \end{array} \]

Alternative 10: 71.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -2.3 \cdot 10^{+115} \lor \neg \left(x.im \leq 4 \cdot 10^{+135}\right):\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -2.3e+115) (not (<= x.im 4e+135)))
   (* x.im (* x.re (- x.im)))
   (* x.re (* x.re x.re))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -2.3e+115) || !(x_46_im <= 4e+135)) {
		tmp = x_46_im * (x_46_re * -x_46_im);
	} else {
		tmp = x_46_re * (x_46_re * x_46_re);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46im <= (-2.3d+115)) .or. (.not. (x_46im <= 4d+135))) then
        tmp = x_46im * (x_46re * -x_46im)
    else
        tmp = x_46re * (x_46re * x_46re)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -2.3e+115) || !(x_46_im <= 4e+135)) {
		tmp = x_46_im * (x_46_re * -x_46_im);
	} else {
		tmp = x_46_re * (x_46_re * x_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_im <= -2.3e+115) or not (x_46_im <= 4e+135):
		tmp = x_46_im * (x_46_re * -x_46_im)
	else:
		tmp = x_46_re * (x_46_re * x_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_im <= -2.3e+115) || !(x_46_im <= 4e+135))
		tmp = Float64(x_46_im * Float64(x_46_re * Float64(-x_46_im)));
	else
		tmp = Float64(x_46_re * Float64(x_46_re * x_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_im <= -2.3e+115) || ~((x_46_im <= 4e+135)))
		tmp = x_46_im * (x_46_re * -x_46_im);
	else
		tmp = x_46_re * (x_46_re * x_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$im, -2.3e+115], N[Not[LessEqual[x$46$im, 4e+135]], $MachinePrecision]], N[(x$46$im * N[(x$46$re * (-x$46$im)), $MachinePrecision]), $MachinePrecision], N[(x$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -2.30000000000000004e115 or 3.99999999999999985e135 < x.im

   1. Initial program 53.0%

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

      1. *-commutative 53.0%

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

      2. *-commutative 53.0%

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

      3. *-commutative 53.0%

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

      4. distribute-lft-out 53.0%

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

   3. Simplified 53.0%

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

   4. Taylor expanded in x.re around 0 63.0%

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

      1. associate-*r* 63.0%

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

      2. mul-1-neg 63.0%

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

      3. unpow2 63.0%

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

   6. Simplified 63.0%

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

      1. associate-*r* 88.3%

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

      2. *-commutative 88.3%

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

      3. distribute-rgt-out-- 88.5%

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

      4. *-commutative 88.5%

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

      5. add-sqr-sqrt 45.6%

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

      6. sqrt-unprod 52.7%

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

      7. sqr-neg 52.7%

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

      8. sqrt-unprod 17.1%

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

      9. add-sqr-sqrt 38.7%

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

      10. distribute-rgt-in 38.7%

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

      11. distribute-lft-out 38.7%

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

   8. Applied egg-rr 38.7%

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

      1. distribute-lft-out-- 61.5%

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

   10. Simplified 61.5%

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

   11. Taylor expanded in x.im around 0 58.4%

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

       1. count-2 58.4%

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

       2. associate--r+ 58.4%

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

       3. +-inverses 58.4%

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

       4. sub0-neg 58.4%

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

       5. unpow2 58.4%

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

       6. distribute-lft-neg-in 58.4%

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

       7. associate-*r* 61.5%

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

       8. *-commutative 61.5%

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

       9. distribute-rgt-neg-in 61.5%

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

       10. *-commutative 61.5%

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

   13. Simplified 61.5%

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

   if -2.30000000000000004e115 < x.im < 3.99999999999999985e135

   1. Initial program 91.8%

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

      1. *-commutative 91.8%

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

      2. distribute-lft-out 91.8%

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

      3. associate-*l* 91.7%

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

      4. *-commutative 91.7%

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

      5. distribute-rgt-out-- 99.8%

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

      6. associate--l- 99.8%

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

      7. associate--l- 99.8%

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

      8. sub-neg 99.8%

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

      9. associate--l+ 99.8%

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

      10. fma-udef 99.8%

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

      11. neg-mul-1 99.8%

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

      12. count-2 99.8%

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

      13. associate-*l* 99.8%

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

      14. distribute-rgt-out-- 99.8%

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

      15. associate-*r* 99.8%

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

      16. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x.re around inf 78.6%

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

      1. unpow2 78.6%

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

   6. Simplified 78.6%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -2.3 \cdot 10^{+115} \lor \neg \left(x.im \leq 4 \cdot 10^{+135}\right):\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \end{array} \]

Alternative 11: 57.8% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.2%

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

   1. *-commutative 81.2%

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

   2. distribute-lft-out 81.2%

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

   3. associate-*l* 81.1%

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

   4. *-commutative 81.1%

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

   5. distribute-rgt-out-- 87.0%

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

   6. associate--l- 87.0%

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

   7. associate--l- 87.0%

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

   8. sub-neg 87.0%

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

   9. associate--l+ 87.0%

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

   10. fma-udef 89.7%

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

   11. neg-mul-1 89.7%

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

   12. count-2 89.7%

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

   13. associate-*l* 89.7%

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

   14. distribute-rgt-out-- 89.7%

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

   15. associate-*r* 89.8%

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

   16. metadata-eval 89.8%

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

3. Simplified 89.8%

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

4. Taylor expanded in x.re around inf 60.5%

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

   1. unpow2 60.5%

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

6. Simplified 60.5%

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

7. Final simplification 60.5%

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

Developer target: 87.3% 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 2023171 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64

  :herbie-target
  (+ (* (* x.re x.re) (- x.re x.im)) (* (* x.re x.im) (- x.re (* 3.0 x.im))))

  (- (* (- (* x.re x.re) (* x.im x.im)) x.re) (* (+ (* x.re x.im) (* x.im x.re)) x.im)))