math.cube on complex, imaginary part

Percentage Accurate: 82.3% → 99.7%

Time: 7.0s

Alternatives: 16

Speedup: 1.3×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 82.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -3.4 \cdot 10^{+103} \lor \neg \left(x.im \leq 2 \cdot 10^{+87}\right):\\ \;\;\;\;\left(x.im + x.im\right) + \left(x.im + x.re\right) \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re \cdot \left(x.im \cdot x.re\right), 3, -{x.im}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -3.4e+103) (not (<= x.im 2e+87)))
   (+ (+ x.im x.im) (* (+ x.im x.re) (* x.im (- x.re x.im))))
   (fma (* x.re (* x.im x.re)) 3.0 (- (pow x.im 3.0)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -3.4e+103) || !(x_46_im <= 2e+87)) {
		tmp = (x_46_im + x_46_im) + ((x_46_im + x_46_re) * (x_46_im * (x_46_re - x_46_im)));
	} else {
		tmp = fma((x_46_re * (x_46_im * x_46_re)), 3.0, -pow(x_46_im, 3.0));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_im <= -3.4e+103) || !(x_46_im <= 2e+87))
		tmp = Float64(Float64(x_46_im + x_46_im) + Float64(Float64(x_46_im + x_46_re) * Float64(x_46_im * Float64(x_46_re - x_46_im))));
	else
		tmp = fma(Float64(x_46_re * Float64(x_46_im * x_46_re)), 3.0, Float64(-(x_46_im ^ 3.0)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$im, -3.4e+103], N[Not[LessEqual[x$46$im, 2e+87]], $MachinePrecision]], N[(N[(x$46$im + x$46$im), $MachinePrecision] + N[(N[(x$46$im + x$46$re), $MachinePrecision] * N[(x$46$im * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * 3.0 + (-N[Power[x$46$im, 3.0], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -3.3999999999999998e103 or 1.9999999999999999e87 < x.im

   1. Initial program 71.8%

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

      1. +-commutative 71.8%

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

      2. *-commutative 71.8%

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

      3. fma-def 76.5%

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

      4. *-commutative 76.5%

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

      5. distribute-rgt-out 76.5%

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

      6. *-commutative 76.5%

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

   3. Simplified 76.5%

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

      1. *-commutative 76.5%

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

      2. fma-def 71.8%

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

      3. distribute-lft-in 71.8%

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

      4. flip-+ 0.0%

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

      5. +-inverses 0.0%

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

      6. metadata-eval 0.0%

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

      7. +-inverses 0.0%

         \[\leadsto x.re \cdot \frac{\log 1}{\color{blue}{0}} + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im \]

      8. metadata-eval 0.0%

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

      9. associate-*r/ 0.0%

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

      10. metadata-eval 0.0%

          \[\leadsto \frac{x.re \cdot \color{blue}{0}}{\log 1} + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im \]

      11. +-inverses 0.0%

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

      12. distribute-lft-out-- 0.0%

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

      13. +-inverses 0.0%

          \[\leadsto \frac{\color{blue}{0}}{\log 1} + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im \]

      14. +-inverses 0.0%

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

      15. metadata-eval 0.0%

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

      16. +-inverses 0.0%

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

      17. flip-+ 83.5%

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

      18. difference-of-squares 100.0%

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

      19. associate-*l* 100.0%

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

   5. Applied egg-rr 100.0%

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

   if -3.3999999999999998e103 < x.im < 1.9999999999999999e87

   1. Initial program 88.9%

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

      1. +-commutative 88.9%

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

      2. *-commutative 88.9%

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

      3. sub-neg 88.9%

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

      4. distribute-lft-in 88.9%

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

      5. associate-+r+ 88.9%

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

      6. distribute-rgt-neg-out 88.9%

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

      7. unsub-neg 88.9%

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

      8. associate-*r* 99.7%

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

      9. distribute-rgt-out 99.7%

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

      10. *-commutative 99.7%

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

      11. count-2 99.7%

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

      12. distribute-lft1-in 99.7%

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

      13. metadata-eval 99.7%

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

      14. *-commutative 99.7%

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

      15. *-commutative 99.7%

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

      16. associate-*r* 99.7%

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

      17. cube-unmult 99.8%

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

   3. Simplified 99.8%

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

      1. associate-*r* 99.8%

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

      2. associate-*l* 99.8%

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

      3. fma-neg 99.8%

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

   5. Applied egg-rr 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -3.4 \cdot 10^{+103} \lor \neg \left(x.im \leq 2 \cdot 10^{+87}\right):\\ \;\;\;\;\left(x.im + x.im\right) + \left(x.im + x.re\right) \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re \cdot \left(x.im \cdot x.re\right), 3, -{x.im}^{3}\right)\\ \end{array} \]

Alternative 2: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -3.4 \cdot 10^{+103} \lor \neg \left(x.im \leq 2 \cdot 10^{+87}\right):\\ \;\;\;\;\left(x.im + x.im\right) + \left(x.im + x.re\right) \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.im \cdot x.re\right) \cdot 3\right) - {x.im}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -3.4e+103) (not (<= x.im 2e+87)))
   (+ (+ x.im x.im) (* (+ x.im x.re) (* x.im (- x.re x.im))))
   (- (* x.re (* (* x.im x.re) 3.0)) (pow x.im 3.0))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -3.4e+103) || !(x_46_im <= 2e+87)) {
		tmp = (x_46_im + x_46_im) + ((x_46_im + x_46_re) * (x_46_im * (x_46_re - x_46_im)));
	} else {
		tmp = (x_46_re * ((x_46_im * x_46_re) * 3.0)) - pow(x_46_im, 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 <= (-3.4d+103)) .or. (.not. (x_46im <= 2d+87))) then
        tmp = (x_46im + x_46im) + ((x_46im + x_46re) * (x_46im * (x_46re - x_46im)))
    else
        tmp = (x_46re * ((x_46im * x_46re) * 3.0d0)) - (x_46im ** 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 <= -3.4e+103) || !(x_46_im <= 2e+87)) {
		tmp = (x_46_im + x_46_im) + ((x_46_im + x_46_re) * (x_46_im * (x_46_re - x_46_im)));
	} else {
		tmp = (x_46_re * ((x_46_im * x_46_re) * 3.0)) - Math.pow(x_46_im, 3.0);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_im <= -3.4e+103) or not (x_46_im <= 2e+87):
		tmp = (x_46_im + x_46_im) + ((x_46_im + x_46_re) * (x_46_im * (x_46_re - x_46_im)))
	else:
		tmp = (x_46_re * ((x_46_im * x_46_re) * 3.0)) - math.pow(x_46_im, 3.0)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_im <= -3.4e+103) || !(x_46_im <= 2e+87))
		tmp = Float64(Float64(x_46_im + x_46_im) + Float64(Float64(x_46_im + x_46_re) * Float64(x_46_im * Float64(x_46_re - x_46_im))));
	else
		tmp = Float64(Float64(x_46_re * Float64(Float64(x_46_im * x_46_re) * 3.0)) - (x_46_im ^ 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_im <= -3.4e+103) || ~((x_46_im <= 2e+87)))
		tmp = (x_46_im + x_46_im) + ((x_46_im + x_46_re) * (x_46_im * (x_46_re - x_46_im)));
	else
		tmp = (x_46_re * ((x_46_im * x_46_re) * 3.0)) - (x_46_im ^ 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$im, -3.4e+103], N[Not[LessEqual[x$46$im, 2e+87]], $MachinePrecision]], N[(N[(x$46$im + x$46$im), $MachinePrecision] + N[(N[(x$46$im + x$46$re), $MachinePrecision] * N[(x$46$im * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * N[(N[(x$46$im * x$46$re), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] - N[Power[x$46$im, 3.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -3.3999999999999998e103 or 1.9999999999999999e87 < x.im

   1. Initial program 71.8%

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

      1. +-commutative 71.8%

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

      2. *-commutative 71.8%

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

      3. fma-def 76.5%

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

      4. *-commutative 76.5%

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

      5. distribute-rgt-out 76.5%

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

      6. *-commutative 76.5%

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

   3. Simplified 76.5%

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

      1. *-commutative 76.5%

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

      2. fma-def 71.8%

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

      3. distribute-lft-in 71.8%

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

      4. flip-+ 0.0%

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

      5. +-inverses 0.0%

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

      6. metadata-eval 0.0%

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

      7. +-inverses 0.0%

         \[\leadsto x.re \cdot \frac{\log 1}{\color{blue}{0}} + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im \]

      8. metadata-eval 0.0%

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

      9. associate-*r/ 0.0%

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

      10. metadata-eval 0.0%

          \[\leadsto \frac{x.re \cdot \color{blue}{0}}{\log 1} + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im \]

      11. +-inverses 0.0%

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

      12. distribute-lft-out-- 0.0%

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

      13. +-inverses 0.0%

          \[\leadsto \frac{\color{blue}{0}}{\log 1} + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im \]

      14. +-inverses 0.0%

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

      15. metadata-eval 0.0%

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

      16. +-inverses 0.0%

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

      17. flip-+ 83.5%

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

      18. difference-of-squares 100.0%

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

      19. associate-*l* 100.0%

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

   5. Applied egg-rr 100.0%

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

   if -3.3999999999999998e103 < x.im < 1.9999999999999999e87

   1. Initial program 88.9%

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

      1. +-commutative 88.9%

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

      2. *-commutative 88.9%

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

      3. sub-neg 88.9%

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

      4. distribute-lft-in 88.9%

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

      5. associate-+r+ 88.9%

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

      6. distribute-rgt-neg-out 88.9%

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

      7. unsub-neg 88.9%

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

      8. associate-*r* 99.7%

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

      9. distribute-rgt-out 99.7%

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

      10. *-commutative 99.7%

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

      11. count-2 99.7%

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

      12. distribute-lft1-in 99.7%

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

      13. metadata-eval 99.7%

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

      14. *-commutative 99.7%

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

      15. *-commutative 99.7%

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

      16. associate-*r* 99.7%

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

      17. cube-unmult 99.8%

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

   3. Simplified 99.8%

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

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

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -3.4 \cdot 10^{+103} \lor \neg \left(x.im \leq 2 \cdot 10^{+87}\right):\\ \;\;\;\;\left(x.im + x.im\right) + \left(x.im + x.re\right) \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.im \cdot x.re\right) \cdot 3\right) - {x.im}^{3}\\ \end{array} \]

Alternative 3: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -3.4 \cdot 10^{+103} \lor \neg \left(x.im \leq 2 \cdot 10^{+87}\right):\\ \;\;\;\;\left(x.im + x.im\right) + \left(x.im + x.re\right) \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right) - {x.im}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -3.4e+103) (not (<= x.im 2e+87)))
   (+ (+ x.im x.im) (* (+ x.im x.re) (* x.im (- x.re x.im))))
   (- (* x.re (* x.re (* x.im 3.0))) (pow x.im 3.0))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -3.4e+103) || !(x_46_im <= 2e+87)) {
		tmp = (x_46_im + x_46_im) + ((x_46_im + x_46_re) * (x_46_im * (x_46_re - x_46_im)));
	} else {
		tmp = (x_46_re * (x_46_re * (x_46_im * 3.0))) - pow(x_46_im, 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 <= (-3.4d+103)) .or. (.not. (x_46im <= 2d+87))) then
        tmp = (x_46im + x_46im) + ((x_46im + x_46re) * (x_46im * (x_46re - x_46im)))
    else
        tmp = (x_46re * (x_46re * (x_46im * 3.0d0))) - (x_46im ** 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 <= -3.4e+103) || !(x_46_im <= 2e+87)) {
		tmp = (x_46_im + x_46_im) + ((x_46_im + x_46_re) * (x_46_im * (x_46_re - x_46_im)));
	} else {
		tmp = (x_46_re * (x_46_re * (x_46_im * 3.0))) - Math.pow(x_46_im, 3.0);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_im <= -3.4e+103) or not (x_46_im <= 2e+87):
		tmp = (x_46_im + x_46_im) + ((x_46_im + x_46_re) * (x_46_im * (x_46_re - x_46_im)))
	else:
		tmp = (x_46_re * (x_46_re * (x_46_im * 3.0))) - math.pow(x_46_im, 3.0)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_im <= -3.4e+103) || !(x_46_im <= 2e+87))
		tmp = Float64(Float64(x_46_im + x_46_im) + Float64(Float64(x_46_im + x_46_re) * Float64(x_46_im * Float64(x_46_re - x_46_im))));
	else
		tmp = Float64(Float64(x_46_re * Float64(x_46_re * Float64(x_46_im * 3.0))) - (x_46_im ^ 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_im <= -3.4e+103) || ~((x_46_im <= 2e+87)))
		tmp = (x_46_im + x_46_im) + ((x_46_im + x_46_re) * (x_46_im * (x_46_re - x_46_im)));
	else
		tmp = (x_46_re * (x_46_re * (x_46_im * 3.0))) - (x_46_im ^ 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$im, -3.4e+103], N[Not[LessEqual[x$46$im, 2e+87]], $MachinePrecision]], N[(N[(x$46$im + x$46$im), $MachinePrecision] + N[(N[(x$46$im + x$46$re), $MachinePrecision] * N[(x$46$im * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * N[(x$46$re * N[(x$46$im * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[x$46$im, 3.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -3.3999999999999998e103 or 1.9999999999999999e87 < x.im

   1. Initial program 71.8%

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

      1. +-commutative 71.8%

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

      2. *-commutative 71.8%

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

      3. fma-def 76.5%

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

      4. *-commutative 76.5%

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

      5. distribute-rgt-out 76.5%

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

      6. *-commutative 76.5%

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

   3. Simplified 76.5%

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

      1. *-commutative 76.5%

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

      2. fma-def 71.8%

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

      3. distribute-lft-in 71.8%

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

      4. flip-+ 0.0%

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

      5. +-inverses 0.0%

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

      6. metadata-eval 0.0%

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

      7. +-inverses 0.0%

         \[\leadsto x.re \cdot \frac{\log 1}{\color{blue}{0}} + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im \]

      8. metadata-eval 0.0%

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

      9. associate-*r/ 0.0%

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

      10. metadata-eval 0.0%

          \[\leadsto \frac{x.re \cdot \color{blue}{0}}{\log 1} + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im \]

      11. +-inverses 0.0%

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

      12. distribute-lft-out-- 0.0%

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

      13. +-inverses 0.0%

          \[\leadsto \frac{\color{blue}{0}}{\log 1} + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im \]

      14. +-inverses 0.0%

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

      15. metadata-eval 0.0%

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

      16. +-inverses 0.0%

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

      17. flip-+ 83.5%

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

      18. difference-of-squares 100.0%

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

      19. associate-*l* 100.0%

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

   5. Applied egg-rr 100.0%

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

   if -3.3999999999999998e103 < x.im < 1.9999999999999999e87

   1. Initial program 88.9%

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

      1. +-commutative 88.9%

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

      2. *-commutative 88.9%

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

      3. sub-neg 88.9%

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

      4. distribute-lft-in 88.9%

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

      5. associate-+r+ 88.9%

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

      6. distribute-rgt-neg-out 88.9%

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

      7. unsub-neg 88.9%

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

      8. associate-*r* 99.7%

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

      9. distribute-rgt-out 99.7%

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

      10. *-commutative 99.7%

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

      11. count-2 99.7%

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

      12. distribute-lft1-in 99.7%

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

      13. metadata-eval 99.7%

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

      14. *-commutative 99.7%

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

      15. *-commutative 99.7%

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

      16. associate-*r* 99.7%

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

      17. cube-unmult 99.8%

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

   3. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -3.4 \cdot 10^{+103} \lor \neg \left(x.im \leq 2 \cdot 10^{+87}\right):\\ \;\;\;\;\left(x.im + x.im\right) + \left(x.im + x.re\right) \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right) - {x.im}^{3}\\ \end{array} \]

Alternative 4: 98.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right)\\ t_1 := \left(x.im + x.im\right) + \left(x.im + x.re\right) \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \mathbf{if}\;x.im \leq -1.5 \cdot 10^{+137}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x.im \leq -1.12 \cdot 10^{-107}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.im \leq 2.7 \cdot 10^{-73}:\\ \;\;\;\;\left(x.re \cdot \left(x.im \cdot x.re\right)\right) \cdot 3\\ \mathbf{elif}\;x.im \leq 10^{+58}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0
         (+
          (* x.im (- (* x.re x.re) (* x.im x.im)))
          (* x.re (+ (* x.im x.re) (* x.im x.re)))))
        (t_1 (+ (+ x.im x.im) (* (+ x.im x.re) (* x.im (- x.re x.im))))))
   (if (<= x.im -1.5e+137)
     t_1
     (if (<= x.im -1.12e-107)
       t_0
       (if (<= x.im 2.7e-73)
         (* (* x.re (* x.im x.re)) 3.0)
         (if (<= x.im 1e+58) t_0 t_1))))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_im * x_46_re) + (x_46_im * x_46_re)));
	double t_1 = (x_46_im + x_46_im) + ((x_46_im + x_46_re) * (x_46_im * (x_46_re - x_46_im)));
	double tmp;
	if (x_46_im <= -1.5e+137) {
		tmp = t_1;
	} else if (x_46_im <= -1.12e-107) {
		tmp = t_0;
	} else if (x_46_im <= 2.7e-73) {
		tmp = (x_46_re * (x_46_im * x_46_re)) * 3.0;
	} else if (x_46_im <= 1e+58) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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 * ((x_46re * x_46re) - (x_46im * x_46im))) + (x_46re * ((x_46im * x_46re) + (x_46im * x_46re)))
    t_1 = (x_46im + x_46im) + ((x_46im + x_46re) * (x_46im * (x_46re - x_46im)))
    if (x_46im <= (-1.5d+137)) then
        tmp = t_1
    else if (x_46im <= (-1.12d-107)) then
        tmp = t_0
    else if (x_46im <= 2.7d-73) then
        tmp = (x_46re * (x_46im * x_46re)) * 3.0d0
    else if (x_46im <= 1d+58) then
        tmp = t_0
    else
        tmp = t_1
    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 * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_im * x_46_re) + (x_46_im * x_46_re)));
	double t_1 = (x_46_im + x_46_im) + ((x_46_im + x_46_re) * (x_46_im * (x_46_re - x_46_im)));
	double tmp;
	if (x_46_im <= -1.5e+137) {
		tmp = t_1;
	} else if (x_46_im <= -1.12e-107) {
		tmp = t_0;
	} else if (x_46_im <= 2.7e-73) {
		tmp = (x_46_re * (x_46_im * x_46_re)) * 3.0;
	} else if (x_46_im <= 1e+58) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	t_0 = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_im * x_46_re) + (x_46_im * x_46_re)))
	t_1 = (x_46_im + x_46_im) + ((x_46_im + x_46_re) * (x_46_im * (x_46_re - x_46_im)))
	tmp = 0
	if x_46_im <= -1.5e+137:
		tmp = t_1
	elif x_46_im <= -1.12e-107:
		tmp = t_0
	elif x_46_im <= 2.7e-73:
		tmp = (x_46_re * (x_46_im * x_46_re)) * 3.0
	elif x_46_im <= 1e+58:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(Float64(x_46_im * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im))) + Float64(x_46_re * Float64(Float64(x_46_im * x_46_re) + Float64(x_46_im * x_46_re))))
	t_1 = Float64(Float64(x_46_im + x_46_im) + Float64(Float64(x_46_im + x_46_re) * Float64(x_46_im * Float64(x_46_re - x_46_im))))
	tmp = 0.0
	if (x_46_im <= -1.5e+137)
		tmp = t_1;
	elseif (x_46_im <= -1.12e-107)
		tmp = t_0;
	elseif (x_46_im <= 2.7e-73)
		tmp = Float64(Float64(x_46_re * Float64(x_46_im * x_46_re)) * 3.0);
	elseif (x_46_im <= 1e+58)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	t_0 = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_im * x_46_re) + (x_46_im * x_46_re)));
	t_1 = (x_46_im + x_46_im) + ((x_46_im + x_46_re) * (x_46_im * (x_46_re - x_46_im)));
	tmp = 0.0;
	if (x_46_im <= -1.5e+137)
		tmp = t_1;
	elseif (x_46_im <= -1.12e-107)
		tmp = t_0;
	elseif (x_46_im <= 2.7e-73)
		tmp = (x_46_re * (x_46_im * x_46_re)) * 3.0;
	elseif (x_46_im <= 1e+58)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(N[(x$46$im * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(N[(x$46$im * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im + x$46$im), $MachinePrecision] + N[(N[(x$46$im + x$46$re), $MachinePrecision] * N[(x$46$im * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$im, -1.5e+137], t$95$1, If[LessEqual[x$46$im, -1.12e-107], t$95$0, If[LessEqual[x$46$im, 2.7e-73], N[(N[(x$46$re * N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision], If[LessEqual[x$46$im, 1e+58], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -1.5e137 or 9.99999999999999944e57 < x.im

   1. Initial program 70.4%

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

      1. +-commutative 70.4%

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

      2. *-commutative 70.4%

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

      3. fma-def 75.3%

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

      4. *-commutative 75.3%

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

      5. distribute-rgt-out 75.3%

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

      6. *-commutative 75.3%

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

   3. Simplified 75.3%

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

      1. *-commutative 75.3%

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

      2. fma-def 70.4%

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

      3. distribute-lft-in 70.4%

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

      4. flip-+ 0.0%

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

      5. +-inverses 0.0%

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

      6. metadata-eval 0.0%

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

      7. +-inverses 0.0%

         \[\leadsto x.re \cdot \frac{\log 1}{\color{blue}{0}} + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im \]

      8. metadata-eval 0.0%

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

      9. associate-*r/ 0.0%

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

      10. metadata-eval 0.0%

          \[\leadsto \frac{x.re \cdot \color{blue}{0}}{\log 1} + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im \]

      11. +-inverses 0.0%

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

      12. distribute-lft-out-- 0.0%

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

      13. +-inverses 0.0%

          \[\leadsto \frac{\color{blue}{0}}{\log 1} + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im \]

      14. +-inverses 0.0%

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

      15. metadata-eval 0.0%

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

      16. +-inverses 0.0%

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

      17. flip-+ 82.7%

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

      18. difference-of-squares 100.0%

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

      19. associate-*l* 100.0%

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

   5. Applied egg-rr 100.0%

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

   if -1.5e137 < x.im < -1.12e-107 or 2.69999999999999994e-73 < x.im < 9.99999999999999944e57

   1. Initial program 97.3%

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

   if -1.12e-107 < x.im < 2.69999999999999994e-73

   1. Initial program 83.5%

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

   2. Taylor expanded in x.re around inf 83.5%

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

   4. Simplified 83.5%

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

      1. *-commutative 83.5%

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

      2. *-commutative 83.5%

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

      3. distribute-lft-out 83.5%

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

      4. add-log-exp 62.7%

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

      5. add-log-exp 62.1%

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

      6. sum-log 62.1%

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

      7. exp-lft-sqr 62.1%

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

      8. *-commutative 62.1%

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

      9. add-log-exp 83.5%

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

      10. associate-*l* 83.5%

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

      11. distribute-lft-in 83.4%

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

      12. *-un-lft-identity 83.4%

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

      13. distribute-rgt-out 83.4%

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

      14. metadata-eval 83.4%

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

      15. associate-*l* 83.5%

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

      16. associate-*r* 99.8%

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

   6. Applied egg-rr 99.8%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -1.5 \cdot 10^{+137}:\\ \;\;\;\;\left(x.im + x.im\right) + \left(x.im + x.re\right) \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \mathbf{elif}\;x.im \leq -1.12 \cdot 10^{-107}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right)\\ \mathbf{elif}\;x.im \leq 2.7 \cdot 10^{-73}:\\ \;\;\;\;\left(x.re \cdot \left(x.im \cdot x.re\right)\right) \cdot 3\\ \mathbf{elif}\;x.im \leq 10^{+58}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im + x.im\right) + \left(x.im + x.re\right) \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \end{array} \]

Alternative 5: 91.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x.im + x.im\right) + \left(x.im + x.re\right) \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \mathbf{if}\;x.im \leq -1 \cdot 10^{+134}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.im \leq -1.38 \cdot 10^{-83}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) + \left(x.re + x.re\right)\right)\\ \mathbf{elif}\;x.im \leq 10000000:\\ \;\;\;\;\left(x.re \cdot \left(x.im \cdot x.re\right)\right) \cdot 3\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (+ (+ x.im x.im) (* (+ x.im x.re) (* x.im (- x.re x.im))))))
   (if (<= x.im -1e+134)
     t_0
     (if (<= x.im -1.38e-83)
       (* x.im (+ (- (* x.re x.re) (* x.im x.im)) (+ x.re x.re)))
       (if (<= x.im 10000000.0) (* (* x.re (* x.im x.re)) 3.0) t_0)))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = (x_46_im + x_46_im) + ((x_46_im + x_46_re) * (x_46_im * (x_46_re - x_46_im)));
	double tmp;
	if (x_46_im <= -1e+134) {
		tmp = t_0;
	} else if (x_46_im <= -1.38e-83) {
		tmp = x_46_im * (((x_46_re * x_46_re) - (x_46_im * x_46_im)) + (x_46_re + x_46_re));
	} else if (x_46_im <= 10000000.0) {
		tmp = (x_46_re * (x_46_im * x_46_re)) * 3.0;
	} 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) :: tmp
    t_0 = (x_46im + x_46im) + ((x_46im + x_46re) * (x_46im * (x_46re - x_46im)))
    if (x_46im <= (-1d+134)) then
        tmp = t_0
    else if (x_46im <= (-1.38d-83)) then
        tmp = x_46im * (((x_46re * x_46re) - (x_46im * x_46im)) + (x_46re + x_46re))
    else if (x_46im <= 10000000.0d0) then
        tmp = (x_46re * (x_46im * x_46re)) * 3.0d0
    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 + x_46_im) + ((x_46_im + x_46_re) * (x_46_im * (x_46_re - x_46_im)));
	double tmp;
	if (x_46_im <= -1e+134) {
		tmp = t_0;
	} else if (x_46_im <= -1.38e-83) {
		tmp = x_46_im * (((x_46_re * x_46_re) - (x_46_im * x_46_im)) + (x_46_re + x_46_re));
	} else if (x_46_im <= 10000000.0) {
		tmp = (x_46_re * (x_46_im * x_46_re)) * 3.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	t_0 = (x_46_im + x_46_im) + ((x_46_im + x_46_re) * (x_46_im * (x_46_re - x_46_im)))
	tmp = 0
	if x_46_im <= -1e+134:
		tmp = t_0
	elif x_46_im <= -1.38e-83:
		tmp = x_46_im * (((x_46_re * x_46_re) - (x_46_im * x_46_im)) + (x_46_re + x_46_re))
	elif x_46_im <= 10000000.0:
		tmp = (x_46_re * (x_46_im * x_46_re)) * 3.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(Float64(x_46_im + x_46_im) + Float64(Float64(x_46_im + x_46_re) * Float64(x_46_im * Float64(x_46_re - x_46_im))))
	tmp = 0.0
	if (x_46_im <= -1e+134)
		tmp = t_0;
	elseif (x_46_im <= -1.38e-83)
		tmp = Float64(x_46_im * Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im)) + Float64(x_46_re + x_46_re)));
	elseif (x_46_im <= 10000000.0)
		tmp = Float64(Float64(x_46_re * Float64(x_46_im * x_46_re)) * 3.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	t_0 = (x_46_im + x_46_im) + ((x_46_im + x_46_re) * (x_46_im * (x_46_re - x_46_im)));
	tmp = 0.0;
	if (x_46_im <= -1e+134)
		tmp = t_0;
	elseif (x_46_im <= -1.38e-83)
		tmp = x_46_im * (((x_46_re * x_46_re) - (x_46_im * x_46_im)) + (x_46_re + x_46_re));
	elseif (x_46_im <= 10000000.0)
		tmp = (x_46_re * (x_46_im * x_46_re)) * 3.0;
	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 + x$46$im), $MachinePrecision] + N[(N[(x$46$im + x$46$re), $MachinePrecision] * N[(x$46$im * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$im, -1e+134], t$95$0, If[LessEqual[x$46$im, -1.38e-83], N[(x$46$im * N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] + N[(x$46$re + x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 10000000.0], N[(N[(x$46$re * N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -9.99999999999999921e133 or 1e7 < x.im

   1. Initial program 73.0%

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

      1. +-commutative 73.0%

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

      2. *-commutative 73.0%

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

      3. fma-def 77.5%

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

      4. *-commutative 77.5%

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

      5. distribute-rgt-out 77.5%

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

      6. *-commutative 77.5%

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

   3. Simplified 77.5%

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

      1. *-commutative 77.5%

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

      2. fma-def 73.0%

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

      3. distribute-lft-in 73.0%

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

      4. flip-+ 0.0%

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

      5. +-inverses 0.0%

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

      6. metadata-eval 0.0%

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

      7. +-inverses 0.0%

         \[\leadsto x.re \cdot \frac{\log 1}{\color{blue}{0}} + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im \]

      8. metadata-eval 0.0%

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

      9. associate-*r/ 0.0%

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

      10. metadata-eval 0.0%

          \[\leadsto \frac{x.re \cdot \color{blue}{0}}{\log 1} + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im \]

      11. +-inverses 0.0%

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

      12. distribute-lft-out-- 0.0%

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

      13. +-inverses 0.0%

          \[\leadsto \frac{\color{blue}{0}}{\log 1} + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im \]

      14. +-inverses 0.0%

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

      15. metadata-eval 0.0%

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

      16. +-inverses 0.0%

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

      17. flip-+ 83.3%

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

      18. difference-of-squares 99.0%

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

      19. associate-*l* 99.0%

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

   5. Applied egg-rr 99.0%

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

   if -9.99999999999999921e133 < x.im < -1.37999999999999997e-83

   1. Initial program 97.9%

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

      1. *-commutative 97.9%

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

      2. *-commutative 97.9%

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

      3. flip-+ 0.0%

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

      4. +-inverses 0.0%

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

      5. metadata-eval 0.0%

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

      6. +-inverses 0.0%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.re \cdot \frac{\log 1}{\color{blue}{0}} \]

      7. metadata-eval 0.0%

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

      8. associate-*r/ 0.0%

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

      9. metadata-eval 0.0%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \frac{x.re \cdot \color{blue}{0}}{\log 1} \]

      10. +-inverses 0.0%

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

      11. distribute-lft-out-- 0.0%

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

      12. +-inverses 0.0%

          \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \frac{\color{blue}{0}}{\log 1} \]

      13. +-inverses 0.0%

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

      14. metadata-eval 0.0%

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

      15. +-inverses 0.0%

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

      16. flip-+ 90.4%

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

   3. Applied egg-rr 90.4%

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

      1. *-commutative 90.4%

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

      2. distribute-rgt-out 90.4%

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

      3. distribute-lft-out 92.7%

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

   5. Applied egg-rr 92.7%

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

   if -1.37999999999999997e-83 < x.im < 1e7

   1. Initial program 85.4%

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

   2. Taylor expanded in x.re around inf 80.9%

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

   4. Simplified 80.9%

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

      1. *-commutative 80.9%

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

      2. *-commutative 80.9%

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

      3. distribute-lft-out 80.9%

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

      4. add-log-exp 58.2%

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

      5. add-log-exp 57.6%

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

      6. sum-log 57.6%

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

      7. exp-lft-sqr 57.6%

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

      8. *-commutative 57.6%

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

      9. add-log-exp 80.9%

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

      10. associate-*l* 80.8%

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

      11. distribute-lft-in 80.8%

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

      12. *-un-lft-identity 80.8%

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

      13. distribute-rgt-out 80.8%

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

      14. metadata-eval 80.8%

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

      15. associate-*l* 80.8%

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

      16. associate-*r* 95.2%

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

   6. Applied egg-rr 95.2%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -1 \cdot 10^{+134}:\\ \;\;\;\;\left(x.im + x.im\right) + \left(x.im + x.re\right) \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \mathbf{elif}\;x.im \leq -1.38 \cdot 10^{-83}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) + \left(x.re + x.re\right)\right)\\ \mathbf{elif}\;x.im \leq 10000000:\\ \;\;\;\;\left(x.re \cdot \left(x.im \cdot x.re\right)\right) \cdot 3\\ \mathbf{else}:\\ \;\;\;\;\left(x.im + x.im\right) + \left(x.im + x.re\right) \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \end{array} \]

Alternative 6: 85.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -1.3 \cdot 10^{-82} \lor \neg \left(x.im \leq 1.25 \cdot 10^{-40}\right):\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) + \left(x.re + x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot \left(x.im \cdot x.re\right)\right) \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -1.3e-82) (not (<= x.im 1.25e-40)))
   (* x.im (+ (- (* x.re x.re) (* x.im x.im)) (+ x.re x.re)))
   (* (* x.re (* x.im x.re)) 3.0)))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -1.3e-82) || !(x_46_im <= 1.25e-40)) {
		tmp = x_46_im * (((x_46_re * x_46_re) - (x_46_im * x_46_im)) + (x_46_re + x_46_re));
	} else {
		tmp = (x_46_re * (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 <= (-1.3d-82)) .or. (.not. (x_46im <= 1.25d-40))) then
        tmp = x_46im * (((x_46re * x_46re) - (x_46im * x_46im)) + (x_46re + x_46re))
    else
        tmp = (x_46re * (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 <= -1.3e-82) || !(x_46_im <= 1.25e-40)) {
		tmp = x_46_im * (((x_46_re * x_46_re) - (x_46_im * x_46_im)) + (x_46_re + x_46_re));
	} else {
		tmp = (x_46_re * (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 <= -1.3e-82) or not (x_46_im <= 1.25e-40):
		tmp = x_46_im * (((x_46_re * x_46_re) - (x_46_im * x_46_im)) + (x_46_re + x_46_re))
	else:
		tmp = (x_46_re * (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 <= -1.3e-82) || !(x_46_im <= 1.25e-40))
		tmp = Float64(x_46_im * Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im)) + Float64(x_46_re + x_46_re)));
	else
		tmp = Float64(Float64(x_46_re * Float64(x_46_im * 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 <= -1.3e-82) || ~((x_46_im <= 1.25e-40)))
		tmp = x_46_im * (((x_46_re * x_46_re) - (x_46_im * x_46_im)) + (x_46_re + x_46_re));
	else
		tmp = (x_46_re * (x_46_im * x_46_re)) * 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$im, -1.3e-82], N[Not[LessEqual[x$46$im, 1.25e-40]], $MachinePrecision]], N[(x$46$im * N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] + N[(x$46$re + x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -1.3e-82 or 1.24999999999999991e-40 < x.im

   1. Initial program 82.2%

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

      1. *-commutative 82.2%

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

      2. *-commutative 82.2%

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

      3. flip-+ 0.0%

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

      4. +-inverses 0.0%

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

      5. metadata-eval 0.0%

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

      6. +-inverses 0.0%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.re \cdot \frac{\log 1}{\color{blue}{0}} \]

      7. metadata-eval 0.0%

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

      8. associate-*r/ 0.0%

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

      9. metadata-eval 0.0%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \frac{x.re \cdot \color{blue}{0}}{\log 1} \]

      10. +-inverses 0.0%

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

      11. distribute-lft-out-- 0.0%

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

      12. +-inverses 0.0%

          \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \frac{\color{blue}{0}}{\log 1} \]

      13. +-inverses 0.0%

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

      14. metadata-eval 0.0%

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

      15. +-inverses 0.0%

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

      16. flip-+ 81.0%

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

   3. Applied egg-rr 81.0%

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

      1. *-commutative 81.0%

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

      2. distribute-rgt-out 81.0%

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

      3. distribute-lft-out 84.6%

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

   5. Applied egg-rr 84.6%

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

   if -1.3e-82 < x.im < 1.24999999999999991e-40

   1. Initial program 84.5%

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

   2. Taylor expanded in x.re around inf 82.9%

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

   4. Simplified 82.9%

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

      1. *-commutative 82.9%

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

      2. *-commutative 82.9%

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

      3. distribute-lft-out 82.9%

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

      4. add-log-exp 59.4%

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

      5. add-log-exp 58.7%

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

      6. sum-log 58.7%

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

      7. exp-lft-sqr 58.7%

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

      8. *-commutative 58.7%

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

      9. add-log-exp 82.9%

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

      10. associate-*l* 82.8%

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

      11. distribute-lft-in 82.8%

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

      12. *-un-lft-identity 82.8%

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

      13. distribute-rgt-out 82.8%

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

      14. metadata-eval 82.8%

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

      15. associate-*l* 82.8%

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

      16. associate-*r* 98.2%

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

   6. Applied egg-rr 98.2%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -1.3 \cdot 10^{-82} \lor \neg \left(x.im \leq 1.25 \cdot 10^{-40}\right):\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) + \left(x.re + x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot \left(x.im \cdot x.re\right)\right) \cdot 3\\ \end{array} \]

Alternative 7: 84.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -31000000 \lor \neg \left(x.im \leq 8000\right):\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + -3\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot \left(x.im \cdot x.re\right)\right) \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -31000000.0) (not (<= x.im 8000.0)))
   (+ (* x.im (- (* x.re x.re) (* x.im x.im))) -3.0)
   (* (* x.re (* x.im x.re)) 3.0)))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -31000000.0) || !(x_46_im <= 8000.0)) {
		tmp = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + -3.0;
	} else {
		tmp = (x_46_re * (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 <= (-31000000.0d0)) .or. (.not. (x_46im <= 8000.0d0))) then
        tmp = (x_46im * ((x_46re * x_46re) - (x_46im * x_46im))) + (-3.0d0)
    else
        tmp = (x_46re * (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 <= -31000000.0) || !(x_46_im <= 8000.0)) {
		tmp = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + -3.0;
	} else {
		tmp = (x_46_re * (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 <= -31000000.0) or not (x_46_im <= 8000.0):
		tmp = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + -3.0
	else:
		tmp = (x_46_re * (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 <= -31000000.0) || !(x_46_im <= 8000.0))
		tmp = Float64(Float64(x_46_im * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im))) + -3.0);
	else
		tmp = Float64(Float64(x_46_re * Float64(x_46_im * 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 <= -31000000.0) || ~((x_46_im <= 8000.0)))
		tmp = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + -3.0;
	else
		tmp = (x_46_re * (x_46_im * x_46_re)) * 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$im, -31000000.0], N[Not[LessEqual[x$46$im, 8000.0]], $MachinePrecision]], N[(N[(x$46$im * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -3.0), $MachinePrecision], N[(N[(x$46$re * N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -3.1e7 or 8e3 < x.im

   1. Initial program 78.9%

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

      1. *-commutative 78.9%

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

      2. *-commutative 78.9%

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

      3. flip-+ 0.0%

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

      4. +-inverses 0.0%

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

      5. metadata-eval 0.0%

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

      6. +-inverses 0.0%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.re \cdot \frac{\log 1}{\color{blue}{0}} \]

      7. metadata-eval 0.0%

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

      8. associate-*r/ 0.0%

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

      9. metadata-eval 0.0%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \frac{x.re \cdot \color{blue}{0}}{\log 1} \]

      10. +-inverses 0.0%

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

      11. distribute-lft-out-- 0.0%

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

      12. +-inverses 0.0%

          \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \frac{\color{blue}{0}}{\log 1} \]

      13. +-inverses 0.0%

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

      14. metadata-eval 0.0%

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

      15. +-inverses 0.0%

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

      16. flip-+ 81.7%

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

   3. Applied egg-rr 81.7%

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

      1. expm1-log1p-u 51.8%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x.re \cdot x.im + x.re \cdot x.im\right)\right)} \]

      2. expm1-udef 51.8%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{\left(e^{\mathsf{log1p}\left(x.re \cdot x.im + x.re \cdot x.im\right)} - 1\right)} \]

      3. flip-+ 0.0%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)}{x.re \cdot x.im - x.re \cdot x.im}}\right)} - 1\right) \]

      4. +-inverses 0.0%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{0}}{x.re \cdot x.im - x.re \cdot x.im}\right)} - 1\right) \]

      5. +-inverses 0.0%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(\frac{0}{\color{blue}{0}}\right)} - 1\right) \]

   5. Applied egg-rr 0.0%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{0}{0}\right)} - 1\right)} \]

   7. Simplified 86.9%

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

   if -3.1e7 < x.im < 8e3

   1. Initial program 86.8%

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

   2. Taylor expanded in x.re around inf 76.7%

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

   4. Simplified 76.7%

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

      1. *-commutative 76.7%

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

      2. *-commutative 76.7%

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

      3. distribute-lft-out 76.7%

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

      4. add-log-exp 56.4%

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

      5. add-log-exp 55.8%

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

      6. sum-log 55.8%

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

      7. exp-lft-sqr 55.8%

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

      8. *-commutative 55.8%

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

      9. add-log-exp 76.7%

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

      10. associate-*l* 76.7%

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

      11. distribute-lft-in 76.7%

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

      12. *-un-lft-identity 76.7%

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

      13. distribute-rgt-out 76.7%

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

      14. metadata-eval 76.7%

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

      15. associate-*l* 76.7%

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

      16. associate-*r* 89.8%

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

   6. Applied egg-rr 89.8%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -31000000 \lor \neg \left(x.im \leq 8000\right):\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + -3\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot \left(x.im \cdot x.re\right)\right) \cdot 3\\ \end{array} \]

Alternative 8: 59.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -3.4 \cdot 10^{+192} \lor \neg \left(x.im \leq 1.6 \cdot 10^{+248}\right):\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(-x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.im \cdot x.re\right) \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -3.4e+192) (not (<= x.im 1.6e+248)))
   (* x.re (* x.im (- x.re)))
   (* x.re (* (* x.im x.re) 3.0))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -3.4e+192) || !(x_46_im <= 1.6e+248)) {
		tmp = x_46_re * (x_46_im * -x_46_re);
	} else {
		tmp = x_46_re * ((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 <= (-3.4d+192)) .or. (.not. (x_46im <= 1.6d+248))) then
        tmp = x_46re * (x_46im * -x_46re)
    else
        tmp = x_46re * ((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 <= -3.4e+192) || !(x_46_im <= 1.6e+248)) {
		tmp = x_46_re * (x_46_im * -x_46_re);
	} else {
		tmp = x_46_re * ((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 <= -3.4e+192) or not (x_46_im <= 1.6e+248):
		tmp = x_46_re * (x_46_im * -x_46_re)
	else:
		tmp = x_46_re * ((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 <= -3.4e+192) || !(x_46_im <= 1.6e+248))
		tmp = Float64(x_46_re * Float64(x_46_im * Float64(-x_46_re)));
	else
		tmp = Float64(x_46_re * Float64(Float64(x_46_im * 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 <= -3.4e+192) || ~((x_46_im <= 1.6e+248)))
		tmp = x_46_re * (x_46_im * -x_46_re);
	else
		tmp = x_46_re * ((x_46_im * x_46_re) * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$im, -3.4e+192], N[Not[LessEqual[x$46$im, 1.6e+248]], $MachinePrecision]], N[(x$46$re * N[(x$46$im * (-x$46$re)), $MachinePrecision]), $MachinePrecision], N[(x$46$re * N[(N[(x$46$im * x$46$re), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -3.39999999999999996e192 or 1.59999999999999992e248 < x.im

   1. Initial program 58.1%

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

      1. +-commutative 58.1%

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

      2. *-commutative 58.1%

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

      3. sub-neg 58.1%

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

      4. distribute-lft-in 58.1%

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

      5. associate-+r+ 58.1%

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

      6. distribute-rgt-neg-out 58.1%

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

      7. unsub-neg 58.1%

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

      8. associate-*r* 58.1%

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

      9. distribute-rgt-out 58.1%

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

      10. *-commutative 58.1%

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

      11. count-2 58.1%

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

      12. distribute-lft1-in 58.1%

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

      13. metadata-eval 58.1%

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

      14. *-commutative 58.1%

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

      15. *-commutative 58.1%

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

      16. associate-*r* 58.1%

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

      17. cube-unmult 58.1%

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

   3. Simplified 58.1%

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

      1. associate-*r* 58.1%

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

      2. associate-*l* 58.1%

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

      3. fma-neg 58.1%

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

   5. Applied egg-rr 58.1%

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

   6. Taylor expanded in x.re around inf 0.6%

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

   8. Simplified 43.6%

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

      1. distribute-rgt-neg-out 43.6%

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

      2. *-commutative 43.6%

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

      3. associate-*r* 43.8%

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

   10. Applied egg-rr 43.8%

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

   if -3.39999999999999996e192 < x.im < 1.59999999999999992e248

   1. Initial program 86.7%

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

   2. Taylor expanded in x.re around inf 58.9%

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

   4. Simplified 58.9%

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

      1. *-commutative 58.9%

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

      2. *-commutative 58.9%

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

      3. distribute-lft-out 58.9%

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

      4. add-log-exp 45.4%

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

      5. add-log-exp 45.0%

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

      6. sum-log 45.1%

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

      7. exp-lft-sqr 45.1%

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

      8. *-commutative 45.1%

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

      9. add-log-exp 58.9%

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

      10. associate-*l* 58.9%

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

      11. distribute-lft-in 58.9%

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

      12. *-un-lft-identity 58.9%

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

      13. distribute-rgt-out 58.9%

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

      14. metadata-eval 58.9%

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

      15. associate-*l* 58.9%

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

      16. associate-*r* 67.1%

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

      17. *-commutative 67.1%

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

      18. associate-*l* 67.1%

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

      19. *-commutative 67.1%

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

   6. Applied egg-rr 67.1%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -3.4 \cdot 10^{+192} \lor \neg \left(x.im \leq 1.6 \cdot 10^{+248}\right):\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(-x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.im \cdot x.re\right) \cdot 3\right)\\ \end{array} \]

Alternative 9: 59.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -6.9 \cdot 10^{+187} \lor \neg \left(x.im \leq 1.6 \cdot 10^{+248}\right):\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(-x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im \cdot x.re\right) \cdot \left(x.re \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -6.9e+187) (not (<= x.im 1.6e+248)))
   (* x.re (* x.im (- x.re)))
   (* (* x.im x.re) (* x.re 3.0))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -6.9e+187) || !(x_46_im <= 1.6e+248)) {
		tmp = x_46_re * (x_46_im * -x_46_re);
	} else {
		tmp = (x_46_im * x_46_re) * (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 <= (-6.9d+187)) .or. (.not. (x_46im <= 1.6d+248))) then
        tmp = x_46re * (x_46im * -x_46re)
    else
        tmp = (x_46im * x_46re) * (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 <= -6.9e+187) || !(x_46_im <= 1.6e+248)) {
		tmp = x_46_re * (x_46_im * -x_46_re);
	} else {
		tmp = (x_46_im * x_46_re) * (x_46_re * 3.0);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_im <= -6.9e+187) or not (x_46_im <= 1.6e+248):
		tmp = x_46_re * (x_46_im * -x_46_re)
	else:
		tmp = (x_46_im * x_46_re) * (x_46_re * 3.0)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_im <= -6.9e+187) || !(x_46_im <= 1.6e+248))
		tmp = Float64(x_46_re * Float64(x_46_im * Float64(-x_46_re)));
	else
		tmp = Float64(Float64(x_46_im * x_46_re) * 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 <= -6.9e+187) || ~((x_46_im <= 1.6e+248)))
		tmp = x_46_re * (x_46_im * -x_46_re);
	else
		tmp = (x_46_im * x_46_re) * (x_46_re * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$im, -6.9e+187], N[Not[LessEqual[x$46$im, 1.6e+248]], $MachinePrecision]], N[(x$46$re * N[(x$46$im * (-x$46$re)), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im * x$46$re), $MachinePrecision] * N[(x$46$re * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -6.8999999999999997e187 or 1.59999999999999992e248 < x.im

   1. Initial program 58.1%

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

      1. +-commutative 58.1%

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

      2. *-commutative 58.1%

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

      3. sub-neg 58.1%

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

      4. distribute-lft-in 58.1%

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

      5. associate-+r+ 58.1%

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

      6. distribute-rgt-neg-out 58.1%

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

      7. unsub-neg 58.1%

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

      8. associate-*r* 58.1%

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

      9. distribute-rgt-out 58.1%

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

      10. *-commutative 58.1%

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

      11. count-2 58.1%

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

      12. distribute-lft1-in 58.1%

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

      13. metadata-eval 58.1%

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

      14. *-commutative 58.1%

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

      15. *-commutative 58.1%

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

      16. associate-*r* 58.1%

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

      17. cube-unmult 58.1%

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

   3. Simplified 58.1%

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

      1. associate-*r* 58.1%

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

      2. associate-*l* 58.1%

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

      3. fma-neg 58.1%

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

   5. Applied egg-rr 58.1%

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

   6. Taylor expanded in x.re around inf 0.6%

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

   8. Simplified 43.6%

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

      1. distribute-rgt-neg-out 43.6%

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

      2. *-commutative 43.6%

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

      3. associate-*r* 43.8%

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

   10. Applied egg-rr 43.8%

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

   if -6.8999999999999997e187 < x.im < 1.59999999999999992e248

   1. Initial program 86.7%

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

   2. Taylor expanded in x.re around inf 58.9%

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

   4. Simplified 58.9%

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

      1. *-commutative 58.9%

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

      2. *-commutative 58.9%

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

      3. distribute-lft-out 58.9%

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

      4. add-log-exp 45.4%

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

      5. add-log-exp 45.0%

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

      6. sum-log 45.1%

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

      7. exp-lft-sqr 45.1%

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

      8. *-commutative 45.1%

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

      9. add-log-exp 58.9%

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

      10. associate-*l* 58.9%

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

      11. distribute-lft-in 58.9%

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

      12. *-un-lft-identity 58.9%

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

      13. distribute-rgt-out 58.9%

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

      14. metadata-eval 58.9%

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

      15. associate-*l* 58.9%

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

      16. associate-*r* 67.1%

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

      17. *-commutative 67.1%

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

      18. *-commutative 67.1%

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

      19. associate-*l* 67.1%

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

      20. *-commutative 67.1%

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

   6. Applied egg-rr 67.1%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -6.9 \cdot 10^{+187} \lor \neg \left(x.im \leq 1.6 \cdot 10^{+248}\right):\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(-x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im \cdot x.re\right) \cdot \left(x.re \cdot 3\right)\\ \end{array} \]

Alternative 10: 59.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -1.8 \cdot 10^{+190} \lor \neg \left(x.im \leq 1.65 \cdot 10^{+248}\right):\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(-x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot \left(x.im \cdot x.re\right)\right) \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -1.8e+190) (not (<= x.im 1.65e+248)))
   (* x.re (* x.im (- x.re)))
   (* (* x.re (* x.im x.re)) 3.0)))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -1.8e+190) || !(x_46_im <= 1.65e+248)) {
		tmp = x_46_re * (x_46_im * -x_46_re);
	} else {
		tmp = (x_46_re * (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 <= (-1.8d+190)) .or. (.not. (x_46im <= 1.65d+248))) then
        tmp = x_46re * (x_46im * -x_46re)
    else
        tmp = (x_46re * (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 <= -1.8e+190) || !(x_46_im <= 1.65e+248)) {
		tmp = x_46_re * (x_46_im * -x_46_re);
	} else {
		tmp = (x_46_re * (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 <= -1.8e+190) or not (x_46_im <= 1.65e+248):
		tmp = x_46_re * (x_46_im * -x_46_re)
	else:
		tmp = (x_46_re * (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 <= -1.8e+190) || !(x_46_im <= 1.65e+248))
		tmp = Float64(x_46_re * Float64(x_46_im * Float64(-x_46_re)));
	else
		tmp = Float64(Float64(x_46_re * Float64(x_46_im * 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 <= -1.8e+190) || ~((x_46_im <= 1.65e+248)))
		tmp = x_46_re * (x_46_im * -x_46_re);
	else
		tmp = (x_46_re * (x_46_im * x_46_re)) * 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$im, -1.8e+190], N[Not[LessEqual[x$46$im, 1.65e+248]], $MachinePrecision]], N[(x$46$re * N[(x$46$im * (-x$46$re)), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -1.79999999999999989e190 or 1.6500000000000001e248 < x.im

   1. Initial program 58.1%

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

      1. +-commutative 58.1%

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

      2. *-commutative 58.1%

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

      3. sub-neg 58.1%

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

      4. distribute-lft-in 58.1%

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

      5. associate-+r+ 58.1%

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

      6. distribute-rgt-neg-out 58.1%

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

      7. unsub-neg 58.1%

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

      8. associate-*r* 58.1%

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

      9. distribute-rgt-out 58.1%

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

      10. *-commutative 58.1%

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

      11. count-2 58.1%

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

      12. distribute-lft1-in 58.1%

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

      13. metadata-eval 58.1%

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

      14. *-commutative 58.1%

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

      15. *-commutative 58.1%

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

      16. associate-*r* 58.1%

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

      17. cube-unmult 58.1%

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

   3. Simplified 58.1%

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

      1. associate-*r* 58.1%

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

      2. associate-*l* 58.1%

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

      3. fma-neg 58.1%

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

   5. Applied egg-rr 58.1%

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

   6. Taylor expanded in x.re around inf 0.6%

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

   8. Simplified 43.6%

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

      1. distribute-rgt-neg-out 43.6%

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

      2. *-commutative 43.6%

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

      3. associate-*r* 43.8%

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

   10. Applied egg-rr 43.8%

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

   if -1.79999999999999989e190 < x.im < 1.6500000000000001e248

   1. Initial program 86.7%

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

   2. Taylor expanded in x.re around inf 58.9%

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

   4. Simplified 58.9%

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

      1. *-commutative 58.9%

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

      2. *-commutative 58.9%

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

      3. distribute-lft-out 58.9%

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

      4. add-log-exp 45.4%

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

      5. add-log-exp 45.0%

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

      6. sum-log 45.1%

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

      7. exp-lft-sqr 45.1%

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

      8. *-commutative 45.1%

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

      9. add-log-exp 58.9%

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

      10. associate-*l* 58.9%

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

      11. distribute-lft-in 58.9%

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

      12. *-un-lft-identity 58.9%

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

      13. distribute-rgt-out 58.9%

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

      14. metadata-eval 58.9%

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

      15. associate-*l* 58.9%

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

      16. associate-*r* 67.1%

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

   6. Applied egg-rr 67.1%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -1.8 \cdot 10^{+190} \lor \neg \left(x.im \leq 1.65 \cdot 10^{+248}\right):\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(-x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot \left(x.im \cdot x.re\right)\right) \cdot 3\\ \end{array} \]

Alternative 11: 39.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -6 \cdot 10^{+187} \lor \neg \left(x.im \leq 1.6 \cdot 10^{+248}\right):\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(-x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -6e+187) (not (<= x.im 1.6e+248)))
   (* x.re (* x.im (- x.re)))
   (* x.im (* x.re x.re))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -6e+187) || !(x_46_im <= 1.6e+248)) {
		tmp = x_46_re * (x_46_im * -x_46_re);
	} else {
		tmp = x_46_im * (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 <= (-6d+187)) .or. (.not. (x_46im <= 1.6d+248))) then
        tmp = x_46re * (x_46im * -x_46re)
    else
        tmp = x_46im * (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 <= -6e+187) || !(x_46_im <= 1.6e+248)) {
		tmp = x_46_re * (x_46_im * -x_46_re);
	} else {
		tmp = x_46_im * (x_46_re * x_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_im <= -6e+187) or not (x_46_im <= 1.6e+248):
		tmp = x_46_re * (x_46_im * -x_46_re)
	else:
		tmp = x_46_im * (x_46_re * x_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_im <= -6e+187) || !(x_46_im <= 1.6e+248))
		tmp = Float64(x_46_re * Float64(x_46_im * Float64(-x_46_re)));
	else
		tmp = Float64(x_46_im * 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 <= -6e+187) || ~((x_46_im <= 1.6e+248)))
		tmp = x_46_re * (x_46_im * -x_46_re);
	else
		tmp = x_46_im * (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, -6e+187], N[Not[LessEqual[x$46$im, 1.6e+248]], $MachinePrecision]], N[(x$46$re * N[(x$46$im * (-x$46$re)), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -5.9999999999999998e187 or 1.59999999999999992e248 < x.im

   1. Initial program 58.1%

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

      1. +-commutative 58.1%

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

      2. *-commutative 58.1%

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

      3. sub-neg 58.1%

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

      4. distribute-lft-in 58.1%

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

      5. associate-+r+ 58.1%

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

      6. distribute-rgt-neg-out 58.1%

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

      7. unsub-neg 58.1%

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

      8. associate-*r* 58.1%

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

      9. distribute-rgt-out 58.1%

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

      10. *-commutative 58.1%

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

      11. count-2 58.1%

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

      12. distribute-lft1-in 58.1%

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

      13. metadata-eval 58.1%

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

      14. *-commutative 58.1%

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

      15. *-commutative 58.1%

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

      16. associate-*r* 58.1%

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

      17. cube-unmult 58.1%

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

   3. Simplified 58.1%

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

      1. associate-*r* 58.1%

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

      2. associate-*l* 58.1%

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

      3. fma-neg 58.1%

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

   5. Applied egg-rr 58.1%

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

   6. Taylor expanded in x.re around inf 0.6%

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

   8. Simplified 43.6%

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

      1. distribute-rgt-neg-out 43.6%

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

      2. *-commutative 43.6%

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

      3. associate-*r* 43.8%

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

   10. Applied egg-rr 43.8%

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

   if -5.9999999999999998e187 < x.im < 1.59999999999999992e248

   1. Initial program 86.7%

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

      1. *-commutative 86.7%

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

      2. *-commutative 86.7%

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

      3. flip-+ 0.0%

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

      4. +-inverses 0.0%

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

      5. metadata-eval 0.0%

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

      6. +-inverses 0.0%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.re \cdot \frac{\log 1}{\color{blue}{0}} \]

      7. metadata-eval 0.0%

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

      8. associate-*r/ 0.0%

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

      9. metadata-eval 0.0%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \frac{x.re \cdot \color{blue}{0}}{\log 1} \]

      10. +-inverses 0.0%

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

      11. distribute-lft-out-- 0.0%

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

      12. +-inverses 0.0%

          \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \frac{\color{blue}{0}}{\log 1} \]

      13. +-inverses 0.0%

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

      14. metadata-eval 0.0%

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

      15. +-inverses 0.0%

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

      16. flip-+ 66.1%

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

   3. Applied egg-rr 66.1%

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

   4. Taylor expanded in x.re around inf 44.8%

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

   6. Simplified 44.8%

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

4. Final simplification 44.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -6 \cdot 10^{+187} \lor \neg \left(x.im \leq 1.6 \cdot 10^{+248}\right):\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(-x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re\right)\\ \end{array} \]

Alternative 12: 34.3% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.2%

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

   1. *-commutative 83.2%

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

   2. *-commutative 83.2%

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

   3. flip-+ 0.0%

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

   4. +-inverses 0.0%

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

   5. metadata-eval 0.0%

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

   6. +-inverses 0.0%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.re \cdot \frac{\log 1}{\color{blue}{0}} \]

   7. metadata-eval 0.0%

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

   8. associate-*r/ 0.0%

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

   9. metadata-eval 0.0%

      \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \frac{x.re \cdot \color{blue}{0}}{\log 1} \]

   10. +-inverses 0.0%

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

   11. distribute-lft-out-- 0.0%

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

   12. +-inverses 0.0%

       \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \frac{\color{blue}{0}}{\log 1} \]

   13. +-inverses 0.0%

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

   14. metadata-eval 0.0%

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

   15. +-inverses 0.0%

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

   16. flip-+ 66.7%

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

3. Applied egg-rr 66.7%

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

4. Taylor expanded in x.re around inf 39.4%

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

6. Simplified 39.4%

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

7. Final simplification 39.4%

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

Alternative 13: 3.6% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ x.re \cdot -3 \end{array} \]

FPCore

(FPCore (x.re x.im) :precision binary64 (* x.re -3.0))

C

double code(double x_46_re, double x_46_im) {
	return x_46_re * -3.0;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = x_46re * (-3.0d0)
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return x_46_re * -3.0;
}

Python

def code(x_46_re, x_46_im):
	return x_46_re * -3.0

Julia

function code(x_46_re, x_46_im)
	return Float64(x_46_re * -3.0)
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = x_46_re * -3.0;
end

Wolfram

code[x$46$re_, x$46$im_] := N[(x$46$re * -3.0), $MachinePrecision]

Derivation

1. Initial program 83.2%

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

2. Taylor expanded in x.re around inf 51.9%

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

4. Simplified 51.9%

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

   1. expm1-log1p-u 41.5%

      \[\leadsto \left(x.re \cdot x.re\right) \cdot x.im + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x.re \cdot x.im + x.im \cdot x.re\right)\right)} \cdot x.re \]

   2. expm1-udef 31.0%

      \[\leadsto \left(x.re \cdot x.re\right) \cdot x.im + \color{blue}{\left(e^{\mathsf{log1p}\left(x.re \cdot x.im + x.im \cdot x.re\right)} - 1\right)} \cdot x.re \]

   3. *-commutative 31.0%

      \[\leadsto \left(x.re \cdot x.re\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(\color{blue}{x.im \cdot x.re} + x.im \cdot x.re\right)} - 1\right) \cdot x.re \]

   4. flip-+ 0.0%

      \[\leadsto \left(x.re \cdot x.re\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\left(x.im \cdot x.re\right) \cdot \left(x.im \cdot x.re\right) - \left(x.im \cdot x.re\right) \cdot \left(x.im \cdot x.re\right)}{x.im \cdot x.re - x.im \cdot x.re}}\right)} - 1\right) \cdot x.re \]

   5. +-inverses 0.0%

      \[\leadsto \left(x.re \cdot x.re\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{0}}{x.im \cdot x.re - x.im \cdot x.re}\right)} - 1\right) \cdot x.re \]

   6. +-inverses 0.0%

      \[\leadsto \left(x.re \cdot x.re\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(\frac{0}{\color{blue}{0}}\right)} - 1\right) \cdot x.re \]

6. Applied egg-rr 0.0%

   \[\leadsto \left(x.re \cdot x.re\right) \cdot x.im + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{0}{0}\right)} - 1\right)} \cdot x.re \]

8. Simplified 19.5%

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

9. Taylor expanded in x.re around 0 3.5%

   \[\leadsto \color{blue}{-3 \cdot x.re} \]

10. Final simplification 3.5%

    \[\leadsto x.re \cdot -3 \]

Alternative 14: 3.6% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ x.re + x.re \end{array} \]

FPCore

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

C

double code(double x_46_re, double x_46_im) {
	return 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
end function

Java

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

Python

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

Julia

function code(x_46_re, x_46_im)
	return 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;
end

Wolfram

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

Derivation

1. Initial program 83.2%

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

2. Taylor expanded in x.re around inf 51.9%

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

4. Simplified 51.9%

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

   1. expm1-log1p-u 41.5%

      \[\leadsto \left(x.re \cdot x.re\right) \cdot x.im + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x.re \cdot x.im + x.im \cdot x.re\right)\right)} \cdot x.re \]

   2. expm1-udef 31.0%

      \[\leadsto \left(x.re \cdot x.re\right) \cdot x.im + \color{blue}{\left(e^{\mathsf{log1p}\left(x.re \cdot x.im + x.im \cdot x.re\right)} - 1\right)} \cdot x.re \]

   3. *-commutative 31.0%

      \[\leadsto \left(x.re \cdot x.re\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(\color{blue}{x.im \cdot x.re} + x.im \cdot x.re\right)} - 1\right) \cdot x.re \]

   4. flip-+ 0.0%

      \[\leadsto \left(x.re \cdot x.re\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\left(x.im \cdot x.re\right) \cdot \left(x.im \cdot x.re\right) - \left(x.im \cdot x.re\right) \cdot \left(x.im \cdot x.re\right)}{x.im \cdot x.re - x.im \cdot x.re}}\right)} - 1\right) \cdot x.re \]

   5. +-inverses 0.0%

      \[\leadsto \left(x.re \cdot x.re\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{0}}{x.im \cdot x.re - x.im \cdot x.re}\right)} - 1\right) \cdot x.re \]

   6. +-inverses 0.0%

      \[\leadsto \left(x.re \cdot x.re\right) \cdot x.im + \left(e^{\mathsf{log1p}\left(\frac{0}{\color{blue}{0}}\right)} - 1\right) \cdot x.re \]

6. Applied egg-rr 0.0%

   \[\leadsto \left(x.re \cdot x.re\right) \cdot x.im + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{0}{0}\right)} - 1\right)} \cdot x.re \]

8. Simplified 19.5%

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

9. Taylor expanded in x.re around 0 3.5%

   \[\leadsto \color{blue}{-3 \cdot x.re} \]

11. Simplified 3.9%

    \[\leadsto \color{blue}{x.re + x.re} \]

12. Final simplification 3.9%

    \[\leadsto x.re + x.re \]

Alternative 15: 2.7% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ -3 \end{array} \]

FPCore

(FPCore (x.re x.im) :precision binary64 -3.0)

C

double code(double x_46_re, double x_46_im) {
	return -3.0;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = -3.0d0
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return -3.0;
}

Python

def code(x_46_re, x_46_im):
	return -3.0

Julia

function code(x_46_re, x_46_im)
	return -3.0
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = -3.0;
end

Wolfram

code[x$46$re_, x$46$im_] := -3.0

Derivation

1. Initial program 83.2%

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

   1. +-commutative 83.2%

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

   2. *-commutative 83.2%

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

   3. sub-neg 83.2%

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

   4. distribute-lft-in 80.1%

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

   5. associate-+r+ 80.1%

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

   6. distribute-rgt-neg-out 80.1%

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

   7. unsub-neg 80.1%

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

   8. associate-*r* 87.3%

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

   9. distribute-rgt-out 87.3%

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

   10. *-commutative 87.3%

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

   11. count-2 87.3%

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

   12. distribute-lft1-in 87.3%

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

   13. metadata-eval 87.3%

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

   14. *-commutative 87.3%

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

   15. *-commutative 87.3%

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

   16. associate-*r* 87.3%

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

   17. cube-unmult 87.4%

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

3. Simplified 87.4%

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

4. Taylor expanded in x.re around 0 61.0%

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

6. Simplified 2.7%

   \[\leadsto \color{blue}{-3} \]

7. Final simplification 2.7%

   \[\leadsto -3 \]

Alternative 16: 2.7% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ 0.1 \end{array} \]

FPCore

(FPCore (x.re x.im) :precision binary64 0.1)

C

double code(double x_46_re, double x_46_im) {
	return 0.1;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = 0.1d0
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return 0.1;
}

Python

def code(x_46_re, x_46_im):
	return 0.1

Julia

function code(x_46_re, x_46_im)
	return 0.1
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = 0.1;
end

Wolfram

code[x$46$re_, x$46$im_] := 0.1

Derivation

1. Initial program 83.2%

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

   1. +-commutative 83.2%

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

   2. *-commutative 83.2%

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

   3. sub-neg 83.2%

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

   4. distribute-lft-in 80.1%

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

   5. associate-+r+ 80.1%

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

   6. distribute-rgt-neg-out 80.1%

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

   7. unsub-neg 80.1%

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

   8. associate-*r* 87.3%

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

   9. distribute-rgt-out 87.3%

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

   10. *-commutative 87.3%

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

   11. count-2 87.3%

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

   12. distribute-lft1-in 87.3%

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

   13. metadata-eval 87.3%

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

   14. *-commutative 87.3%

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

   15. *-commutative 87.3%

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

   16. associate-*r* 87.3%

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

   17. cube-unmult 87.4%

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

3. Simplified 87.4%

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

   1. sub-neg 87.4%

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

   2. associate-*r* 87.3%

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

   3. associate-*l* 87.4%

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

   4. flip3-+ 12.1%

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

   5. associate-*r* 11.7%

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

   6. associate-*r* 11.7%

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

   7. unpow-prod-down 7.4%

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

   8. pow2 7.4%

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

   9. pow-pow 7.4%

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

   10. metadata-eval 7.4%

       \[\leadsto \frac{{x.re}^{\color{blue}{6}} \cdot {\left(x.im \cdot 3\right)}^{3} + {\left(-{x.im}^{3}\right)}^{3}}{\left(\left(x.re \cdot \left(x.re \cdot x.im\right)\right) \cdot 3\right) \cdot \left(\left(x.re \cdot \left(x.re \cdot x.im\right)\right) \cdot 3\right) + \left(\left(-{x.im}^{3}\right) \cdot \left(-{x.im}^{3}\right) - \left(\left(x.re \cdot \left(x.re \cdot x.im\right)\right) \cdot 3\right) \cdot \left(-{x.im}^{3}\right)\right)} \]

5. Applied egg-rr 7.3%

   \[\leadsto \color{blue}{\frac{{x.re}^{6} \cdot {\left(x.im \cdot 3\right)}^{3} + {\left(-{x.im}^{3}\right)}^{3}}{{x.re}^{4} \cdot \left(\left(x.im \cdot x.im\right) \cdot 9\right) + \left(\left(-{x.im}^{3}\right) \cdot \left(-{x.im}^{3}\right) - \left(x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right)\right) \cdot \left(-{x.im}^{3}\right)\right)}} \]

7. Simplified 2.7%

   \[\leadsto \color{blue}{0.1} \]

8. Final simplification 2.7%

   \[\leadsto 0.1 \]

Developer target: 91.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023182 
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"
  :precision binary64

  :herbie-target
  (+ (* (* x.re x.im) (* 2.0 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im)))

  (+ (* (- (* x.re x.re) (* x.im x.im)) x.im) (* (+ (* x.re x.im) (* x.im x.re)) x.re)))