math.cube on complex, real part

Percentage Accurate: 83.1% → 96.0%

Time: 11.5s

Alternatives: 10

Speedup: 1.3×

• Unsound rule application detected in e-graph. Results may not be sound.

• 45.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 83.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -5 \cdot 10^{+234}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{elif}\;x.re \leq -1.72 \cdot 10^{-107} \lor \neg \left(x.re \leq 6.5 \cdot 10^{-54}\right):\\ \;\;\;\;x.re \cdot \mathsf{fma}\left(x.re, x.re, x.im \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.re -5e+234)
   (* x.re (* x.re x.re))
   (if (or (<= x.re -1.72e-107) (not (<= x.re 6.5e-54)))
     (* x.re (fma x.re x.re (* x.im (* x.im -3.0))))
     (* x.im (* x.im (* x.re -3.0))))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= -5e+234) {
		tmp = x_46_re * (x_46_re * x_46_re);
	} else if ((x_46_re <= -1.72e-107) || !(x_46_re <= 6.5e-54)) {
		tmp = x_46_re * fma(x_46_re, x_46_re, (x_46_im * (x_46_im * -3.0)));
	} else {
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_re <= -5e+234)
		tmp = Float64(x_46_re * Float64(x_46_re * x_46_re));
	elseif ((x_46_re <= -1.72e-107) || !(x_46_re <= 6.5e-54))
		tmp = Float64(x_46_re * fma(x_46_re, x_46_re, Float64(x_46_im * Float64(x_46_im * -3.0))));
	else
		tmp = Float64(x_46_im * Float64(x_46_im * Float64(x_46_re * -3.0)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$re, -5e+234], N[(x$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x$46$re, -1.72e-107], N[Not[LessEqual[x$46$re, 6.5e-54]], $MachinePrecision]], N[(x$46$re * N[(x$46$re * x$46$re + N[(x$46$im * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(x$46$im * N[(x$46$re * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -5.0000000000000003e234

   1. Initial program 35.3%

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

      1. *-commutative 35.3%

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

      2. distribute-lft-out 35.3%

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

      3. associate-*l* 35.3%

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

      4. *-commutative 35.3%

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

      5. distribute-rgt-out-- 47.1%

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

      6. associate--l- 47.1%

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

      7. associate--l- 47.1%

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

      8. sub-neg 47.1%

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

      9. associate--l+ 47.1%

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

      10. fma-udef 58.8%

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

      11. neg-mul-1 58.8%

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

      12. count-2 58.8%

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

      13. associate-*l* 58.8%

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

      14. distribute-rgt-out-- 58.8%

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

      15. associate-*r* 58.8%

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

      16. metadata-eval 58.8%

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

   3. Simplified 58.8%

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

   4. Taylor expanded in x.re around inf 88.2%

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

      1. unpow2 88.2%

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

   6. Simplified 88.2%

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

   if -5.0000000000000003e234 < x.re < -1.72e-107 or 6.49999999999999991e-54 < x.re

   1. Initial program 88.9%

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

      1. *-commutative 88.9%

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

      2. distribute-lft-out 88.9%

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

      3. associate-*l* 88.9%

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

      4. *-commutative 88.9%

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

      5. distribute-rgt-out-- 92.2%

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

      6. associate--l- 92.2%

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

      7. associate--l- 92.2%

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

      8. sub-neg 92.2%

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

      9. associate--l+ 92.2%

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

      10. fma-udef 95.4%

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

      11. neg-mul-1 95.4%

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

      12. count-2 95.4%

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

      13. associate-*l* 95.4%

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

      14. distribute-rgt-out-- 95.4%

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

      15. associate-*r* 95.4%

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

      16. metadata-eval 95.4%

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

   3. Simplified 95.4%

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

   if -1.72e-107 < x.re < 6.49999999999999991e-54

   1. Initial program 87.7%

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

      1. *-commutative 87.7%

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

      2. distribute-lft-out 87.7%

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

      3. associate-*l* 87.7%

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

      4. *-commutative 87.7%

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

      5. distribute-rgt-out-- 87.7%

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

      6. associate--l- 87.7%

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

      7. associate--l- 87.7%

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

      8. sub-neg 87.7%

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

      9. associate--l+ 87.7%

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

      10. fma-udef 87.7%

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

      11. neg-mul-1 87.7%

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

      12. count-2 87.7%

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

      13. associate-*l* 87.7%

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

      14. distribute-rgt-out-- 87.7%

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

      15. associate-*r* 87.6%

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

      16. metadata-eval 87.6%

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

   3. Simplified 87.6%

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

   4. Taylor expanded in x.re around 0 87.7%

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

      1. associate-*r* 87.6%

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

      2. *-commutative 87.6%

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

      3. metadata-eval 87.6%

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

      4. distribute-rgt-out 87.6%

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

      5. metadata-eval 87.6%

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

      6. cancel-sign-sub-inv 87.6%

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

      7. *-commutative 87.6%

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

      8. unpow2 87.6%

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

      9. distribute-rgt-out-- 87.6%

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

      10. metadata-eval 87.6%

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

      11. *-commutative 87.6%

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

   6. Simplified 87.6%

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

      1. pow1 87.6%

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

      2. associate-*l* 99.8%

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

      3. *-commutative 99.8%

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

   8. Applied egg-rr 99.8%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -5 \cdot 10^{+234}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{elif}\;x.re \leq -1.72 \cdot 10^{-107} \lor \neg \left(x.re \leq 6.5 \cdot 10^{-54}\right):\\ \;\;\;\;x.re \cdot \mathsf{fma}\left(x.re, x.re, x.im \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \end{array} \]

Alternative 2: 94.2% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<=
      (-
       (* x.re (- (* x.re x.re) (* x.im x.im)))
       (* x.im (+ (* x.re x.im) (* x.re x.im))))
      INFINITY)
   (fma x.im (* x.im (* x.re -2.0)) (- (pow x.re 3.0) (* x.im (* x.re x.im))))
   (* x.re (* x.re x.re))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (((x_46_re * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re * x_46_im) + (x_46_re * x_46_im)))) <= ((double) INFINITY)) {
		tmp = fma(x_46_im, (x_46_im * (x_46_re * -2.0)), (pow(x_46_re, 3.0) - (x_46_im * (x_46_re * x_46_im))));
	} else {
		tmp = x_46_re * (x_46_re * x_46_re);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (Float64(Float64(x_46_re * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im))) - Float64(x_46_im * Float64(Float64(x_46_re * x_46_im) + Float64(x_46_re * x_46_im)))) <= Inf)
		tmp = fma(x_46_im, Float64(x_46_im * Float64(x_46_re * -2.0)), Float64((x_46_re ^ 3.0) - Float64(x_46_im * Float64(x_46_re * x_46_im))));
	else
		tmp = Float64(x_46_re * Float64(x_46_re * x_46_re));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[N[(N[(x$46$re * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(x$46$im * N[(x$46$im * N[(x$46$re * -2.0), $MachinePrecision]), $MachinePrecision] + N[(N[Power[x$46$re, 3.0], $MachinePrecision] - N[(x$46$im * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < +inf.0

   1. Initial program 95.0%

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

      1. *-commutative 95.0%

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

      2. fma-neg 95.0%

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

      3. distribute-lft-neg-in 95.0%

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

      4. *-commutative 95.0%

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

      5. *-commutative 95.0%

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

      6. count-2 95.0%

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

      7. distribute-lft-neg-in 95.0%

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

      8. metadata-eval 95.0%

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

      9. *-commutative 95.0%

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

   3. Simplified 95.0%

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

      1. fma-udef 95.0%

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

      2. fma-neg 95.0%

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

      3. distribute-rgt-neg-in 95.0%

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

      4. associate-*r* 95.0%

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

   5. Applied egg-rr 95.0%

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

      1. +-commutative 95.0%

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

      2. fma-def 95.0%

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

      3. *-commutative 95.0%

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

      4. *-commutative 95.0%

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

      5. fma-def 95.0%

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

      6. distribute-rgt-in 91.9%

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

      7. unpow3 92.0%

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

      8. *-commutative 92.0%

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

      9. associate-*r* 96.8%

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

      10. distribute-rgt-neg-in 96.8%

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

      11. associate-*r* 92.0%

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

      12. unsub-neg 92.0%

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

      13. associate-*r* 96.8%

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

      14. *-commutative 96.8%

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

      15. *-commutative 96.8%

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

   7. Simplified 96.8%

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

   if +inf.0 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im))

   1. Initial program 0.0%

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

      1. *-commutative 0.0%

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

      2. distribute-lft-out 0.0%

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

      3. associate-*l* 0.0%

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

      4. *-commutative 0.0%

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

      5. distribute-rgt-out-- 25.9%

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

      6. associate--l- 25.9%

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

      7. associate--l- 25.9%

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

      8. sub-neg 25.9%

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

      9. associate--l+ 25.9%

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

      10. fma-udef 51.9%

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

      11. neg-mul-1 51.9%

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

      12. count-2 51.9%

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

      13. associate-*l* 51.9%

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

      14. distribute-rgt-out-- 51.9%

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

      15. associate-*r* 51.9%

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

      16. metadata-eval 51.9%

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

   3. Simplified 51.9%

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

   4. Taylor expanded in x.re around inf 74.1%

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

      1. unpow2 74.1%

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

   6. Simplified 74.1%

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

4. Final simplification 94.4%

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

Alternative 3: 97.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -5.5 \cdot 10^{+155} \lor \neg \left(x.im \leq 7.6 \cdot 10^{+153}\right):\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(-x.im\right)\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re + x.im \cdot \left(x.im \cdot -3\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -5.5e+155) (not (<= x.im 7.6e+153)))
   (- (* x.im (* x.re (- x.im))) (* x.im (* x.re (+ x.im x.im))))
   (* x.re (+ (* x.re x.re) (* x.im (* x.im -3.0))))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -5.5e+155) || !(x_46_im <= 7.6e+153)) {
		tmp = (x_46_im * (x_46_re * -x_46_im)) - (x_46_im * (x_46_re * (x_46_im + x_46_im)));
	} else {
		tmp = x_46_re * ((x_46_re * x_46_re) + (x_46_im * (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 <= (-5.5d+155)) .or. (.not. (x_46im <= 7.6d+153))) then
        tmp = (x_46im * (x_46re * -x_46im)) - (x_46im * (x_46re * (x_46im + x_46im)))
    else
        tmp = x_46re * ((x_46re * x_46re) + (x_46im * (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 <= -5.5e+155) || !(x_46_im <= 7.6e+153)) {
		tmp = (x_46_im * (x_46_re * -x_46_im)) - (x_46_im * (x_46_re * (x_46_im + x_46_im)));
	} else {
		tmp = x_46_re * ((x_46_re * x_46_re) + (x_46_im * (x_46_im * -3.0)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_im <= -5.5e+155) or not (x_46_im <= 7.6e+153):
		tmp = (x_46_im * (x_46_re * -x_46_im)) - (x_46_im * (x_46_re * (x_46_im + x_46_im)))
	else:
		tmp = x_46_re * ((x_46_re * x_46_re) + (x_46_im * (x_46_im * -3.0)))
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_im <= -5.5e+155) || !(x_46_im <= 7.6e+153))
		tmp = Float64(Float64(x_46_im * Float64(x_46_re * Float64(-x_46_im))) - Float64(x_46_im * Float64(x_46_re * Float64(x_46_im + x_46_im))));
	else
		tmp = Float64(x_46_re * Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * Float64(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 <= -5.5e+155) || ~((x_46_im <= 7.6e+153)))
		tmp = (x_46_im * (x_46_re * -x_46_im)) - (x_46_im * (x_46_re * (x_46_im + x_46_im)));
	else
		tmp = x_46_re * ((x_46_re * x_46_re) + (x_46_im * (x_46_im * -3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$im, -5.5e+155], N[Not[LessEqual[x$46$im, 7.6e+153]], $MachinePrecision]], N[(N[(x$46$im * N[(x$46$re * (-x$46$im)), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(x$46$re * N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re * N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -5.5000000000000001e155 or 7.59999999999999933e153 < x.im

   1. Initial program 51.2%

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

      1. *-commutative 51.2%

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

      2. *-commutative 51.2%

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

      3. *-commutative 51.2%

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

      4. distribute-lft-out 51.2%

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

   3. Simplified 51.2%

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

   4. Taylor expanded in x.re around 0 62.1%

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

      1. mul-1-neg 62.1%

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

      2. unpow2 62.1%

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

      3. associate-*r* 79.6%

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

      4. *-commutative 79.6%

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

      5. distribute-rgt-neg-in 79.6%

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

      6. distribute-rgt-neg-in 79.6%

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

   6. Simplified 79.6%

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

   if -5.5000000000000001e155 < x.im < 7.59999999999999933e153

   1. Initial program 96.2%

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

      1. *-commutative 96.2%

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

      2. distribute-lft-out 96.2%

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

      3. associate-*l* 96.1%

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

      4. *-commutative 96.1%

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

      5. distribute-rgt-out-- 99.8%

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

      6. associate--l- 99.8%

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

      7. associate--l- 99.8%

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

      8. sub-neg 99.8%

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

      9. associate--l+ 99.8%

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

      10. fma-udef 99.8%

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

      11. neg-mul-1 99.8%

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

      12. count-2 99.8%

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

      13. associate-*l* 99.8%

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

      14. distribute-rgt-out-- 99.8%

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

      15. associate-*r* 99.8%

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

      16. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. fma-udef 99.8%

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

   5. Applied egg-rr 99.8%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -5.5 \cdot 10^{+155} \lor \neg \left(x.im \leq 7.6 \cdot 10^{+153}\right):\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(-x.im\right)\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re + x.im \cdot \left(x.im \cdot -3\right)\right)\\ \end{array} \]

Alternative 4: 97.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -5.5 \cdot 10^{+155}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \mathbf{elif}\;x.im \leq 7.6 \cdot 10^{+153}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re + x.im \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(-x.im\right)\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im -5.5e+155)
   (* x.im (* x.im (* x.re -3.0)))
   (if (<= x.im 7.6e+153)
     (* x.re (+ (* x.re x.re) (* x.im (* x.im -3.0))))
     (- (* x.im (* x.re (- x.im))) (* x.im (* x.re (+ x.im x.im)))))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= -5.5e+155) {
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0));
	} else if (x_46_im <= 7.6e+153) {
		tmp = x_46_re * ((x_46_re * x_46_re) + (x_46_im * (x_46_im * -3.0)));
	} else {
		tmp = (x_46_im * (x_46_re * -x_46_im)) - (x_46_im * (x_46_re * (x_46_im + x_46_im)));
	}
	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 <= (-5.5d+155)) then
        tmp = x_46im * (x_46im * (x_46re * (-3.0d0)))
    else if (x_46im <= 7.6d+153) then
        tmp = x_46re * ((x_46re * x_46re) + (x_46im * (x_46im * (-3.0d0))))
    else
        tmp = (x_46im * (x_46re * -x_46im)) - (x_46im * (x_46re * (x_46im + x_46im)))
    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 <= -5.5e+155) {
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0));
	} else if (x_46_im <= 7.6e+153) {
		tmp = x_46_re * ((x_46_re * x_46_re) + (x_46_im * (x_46_im * -3.0)));
	} else {
		tmp = (x_46_im * (x_46_re * -x_46_im)) - (x_46_im * (x_46_re * (x_46_im + x_46_im)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= -5.5e+155:
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0))
	elif x_46_im <= 7.6e+153:
		tmp = x_46_re * ((x_46_re * x_46_re) + (x_46_im * (x_46_im * -3.0)))
	else:
		tmp = (x_46_im * (x_46_re * -x_46_im)) - (x_46_im * (x_46_re * (x_46_im + x_46_im)))
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= -5.5e+155)
		tmp = Float64(x_46_im * Float64(x_46_im * Float64(x_46_re * -3.0)));
	elseif (x_46_im <= 7.6e+153)
		tmp = Float64(x_46_re * Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * Float64(x_46_im * -3.0))));
	else
		tmp = Float64(Float64(x_46_im * Float64(x_46_re * Float64(-x_46_im))) - Float64(x_46_im * Float64(x_46_re * Float64(x_46_im + x_46_im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_im <= -5.5e+155)
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0));
	elseif (x_46_im <= 7.6e+153)
		tmp = x_46_re * ((x_46_re * x_46_re) + (x_46_im * (x_46_im * -3.0)));
	else
		tmp = (x_46_im * (x_46_re * -x_46_im)) - (x_46_im * (x_46_re * (x_46_im + x_46_im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, -5.5e+155], N[(x$46$im * N[(x$46$im * N[(x$46$re * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 7.6e+153], N[(x$46$re * N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im * N[(x$46$re * (-x$46$im)), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(x$46$re * N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -5.5000000000000001e155

   1. Initial program 62.3%

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

      1. *-commutative 62.3%

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

      2. distribute-lft-out 62.3%

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

      3. associate-*l* 62.3%

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

      4. *-commutative 62.3%

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

      5. distribute-rgt-out-- 62.3%

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

      6. associate--l- 62.3%

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

      7. associate--l- 62.3%

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

      8. sub-neg 62.3%

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

      9. associate--l+ 62.3%

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

      10. fma-udef 70.6%

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

      11. neg-mul-1 70.6%

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

      12. count-2 70.6%

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

      13. associate-*l* 70.6%

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

      14. distribute-rgt-out-- 70.6%

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

      15. associate-*r* 70.6%

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

      16. metadata-eval 70.6%

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

   3. Simplified 70.6%

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

   4. Taylor expanded in x.re around 0 70.6%

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

      1. associate-*r* 70.6%

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

      2. *-commutative 70.6%

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

      3. metadata-eval 70.6%

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

      4. distribute-rgt-out 70.6%

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

      5. metadata-eval 70.6%

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

      6. cancel-sign-sub-inv 70.6%

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

      7. *-commutative 70.6%

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

      8. unpow2 70.6%

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

      9. distribute-rgt-out-- 70.6%

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

      10. metadata-eval 70.6%

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

      11. *-commutative 70.6%

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

   6. Simplified 70.6%

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

      1. pow1 70.6%

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

      2. associate-*l* 88.8%

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

      3. *-commutative 88.8%

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

   8. Applied egg-rr 88.8%

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

   if -5.5000000000000001e155 < x.im < 7.59999999999999933e153

   1. Initial program 96.2%

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

      1. *-commutative 96.2%

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

      2. distribute-lft-out 96.2%

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

      3. associate-*l* 96.1%

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

      4. *-commutative 96.1%

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

      5. distribute-rgt-out-- 99.8%

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

      6. associate--l- 99.8%

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

      7. associate--l- 99.8%

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

      8. sub-neg 99.8%

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

      9. associate--l+ 99.8%

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

      10. fma-udef 99.8%

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

      11. neg-mul-1 99.8%

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

      12. count-2 99.8%

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

      13. associate-*l* 99.8%

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

      14. distribute-rgt-out-- 99.8%

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

      15. associate-*r* 99.8%

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

      16. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. fma-udef 99.8%

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

   5. Applied egg-rr 99.8%

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

   if 7.59999999999999933e153 < x.im

   1. Initial program 36.9%

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

      1. *-commutative 36.9%

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

      2. *-commutative 36.9%

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

      3. *-commutative 36.9%

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

      4. distribute-lft-out 36.9%

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

   3. Simplified 36.9%

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

   4. Taylor expanded in x.re around 0 51.2%

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

      1. mul-1-neg 51.2%

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

      2. unpow2 51.2%

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

      3. associate-*r* 67.8%

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

      4. *-commutative 67.8%

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

      5. distribute-rgt-neg-in 67.8%

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

      6. distribute-rgt-neg-in 67.8%

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

   6. Simplified 67.8%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -5.5 \cdot 10^{+155}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \mathbf{elif}\;x.im \leq 7.6 \cdot 10^{+153}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re + x.im \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(-x.im\right)\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\\ \end{array} \]

Alternative 5: 97.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -5.5 \cdot 10^{+155}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{elif}\;x.im \leq 7.6 \cdot 10^{+153}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re + x.im \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(-x.im\right)\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im -5.5e+155)
   (* x.im (* x.re (* x.im -3.0)))
   (if (<= x.im 7.6e+153)
     (* x.re (+ (* x.re x.re) (* x.im (* x.im -3.0))))
     (- (* x.im (* x.re (- x.im))) (* x.im (* x.re (+ x.im x.im)))))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= -5.5e+155) {
		tmp = x_46_im * (x_46_re * (x_46_im * -3.0));
	} else if (x_46_im <= 7.6e+153) {
		tmp = x_46_re * ((x_46_re * x_46_re) + (x_46_im * (x_46_im * -3.0)));
	} else {
		tmp = (x_46_im * (x_46_re * -x_46_im)) - (x_46_im * (x_46_re * (x_46_im + x_46_im)));
	}
	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 <= (-5.5d+155)) then
        tmp = x_46im * (x_46re * (x_46im * (-3.0d0)))
    else if (x_46im <= 7.6d+153) then
        tmp = x_46re * ((x_46re * x_46re) + (x_46im * (x_46im * (-3.0d0))))
    else
        tmp = (x_46im * (x_46re * -x_46im)) - (x_46im * (x_46re * (x_46im + x_46im)))
    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 <= -5.5e+155) {
		tmp = x_46_im * (x_46_re * (x_46_im * -3.0));
	} else if (x_46_im <= 7.6e+153) {
		tmp = x_46_re * ((x_46_re * x_46_re) + (x_46_im * (x_46_im * -3.0)));
	} else {
		tmp = (x_46_im * (x_46_re * -x_46_im)) - (x_46_im * (x_46_re * (x_46_im + x_46_im)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= -5.5e+155:
		tmp = x_46_im * (x_46_re * (x_46_im * -3.0))
	elif x_46_im <= 7.6e+153:
		tmp = x_46_re * ((x_46_re * x_46_re) + (x_46_im * (x_46_im * -3.0)))
	else:
		tmp = (x_46_im * (x_46_re * -x_46_im)) - (x_46_im * (x_46_re * (x_46_im + x_46_im)))
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= -5.5e+155)
		tmp = Float64(x_46_im * Float64(x_46_re * Float64(x_46_im * -3.0)));
	elseif (x_46_im <= 7.6e+153)
		tmp = Float64(x_46_re * Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * Float64(x_46_im * -3.0))));
	else
		tmp = Float64(Float64(x_46_im * Float64(x_46_re * Float64(-x_46_im))) - Float64(x_46_im * Float64(x_46_re * Float64(x_46_im + x_46_im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_im <= -5.5e+155)
		tmp = x_46_im * (x_46_re * (x_46_im * -3.0));
	elseif (x_46_im <= 7.6e+153)
		tmp = x_46_re * ((x_46_re * x_46_re) + (x_46_im * (x_46_im * -3.0)));
	else
		tmp = (x_46_im * (x_46_re * -x_46_im)) - (x_46_im * (x_46_re * (x_46_im + x_46_im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, -5.5e+155], N[(x$46$im * N[(x$46$re * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 7.6e+153], N[(x$46$re * N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im * N[(x$46$re * (-x$46$im)), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(x$46$re * N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -5.5000000000000001e155

   1. Initial program 62.3%

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

      1. *-commutative 62.3%

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

      2. distribute-lft-out 62.3%

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

      3. associate-*l* 62.3%

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

      4. *-commutative 62.3%

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

      5. distribute-rgt-out-- 62.3%

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

      6. associate--l- 62.3%

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

      7. associate--l- 62.3%

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

      8. sub-neg 62.3%

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

      9. associate--l+ 62.3%

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

      10. fma-udef 70.6%

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

      11. neg-mul-1 70.6%

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

      12. count-2 70.6%

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

      13. associate-*l* 70.6%

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

      14. distribute-rgt-out-- 70.6%

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

      15. associate-*r* 70.6%

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

      16. metadata-eval 70.6%

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

   3. Simplified 70.6%

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

   4. Taylor expanded in x.re around 0 70.6%

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

      1. associate-*r* 70.6%

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

      2. *-commutative 70.6%

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

      3. metadata-eval 70.6%

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

      4. distribute-rgt-out 70.6%

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

      5. metadata-eval 70.6%

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

      6. cancel-sign-sub-inv 70.6%

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

      7. *-commutative 70.6%

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

      8. unpow2 70.6%

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

      9. distribute-rgt-out-- 70.6%

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

      10. metadata-eval 70.6%

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

      11. *-commutative 70.6%

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

   6. Simplified 70.6%

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

      1. pow1 70.6%

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

      2. associate-*l* 88.8%

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

      3. *-commutative 88.8%

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

   8. Applied egg-rr 88.8%

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

   9. Taylor expanded in x.im around 0 88.8%

      \[\leadsto {\left(x.im \cdot \color{blue}{\left(-3 \cdot \left(x.re \cdot x.im\right)\right)}\right)}^{1} \]
   10. Step-by-step derivation

       1. *-commutative 88.8%

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

       2. *-commutative 88.8%

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

       3. *-commutative 88.8%

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

       4. associate-*l* 88.9%

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

   11. Simplified 88.9%

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

   if -5.5000000000000001e155 < x.im < 7.59999999999999933e153

   1. Initial program 96.2%

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

      1. *-commutative 96.2%

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

      2. distribute-lft-out 96.2%

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

      3. associate-*l* 96.1%

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

      4. *-commutative 96.1%

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

      5. distribute-rgt-out-- 99.8%

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

      6. associate--l- 99.8%

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

      7. associate--l- 99.8%

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

      8. sub-neg 99.8%

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

      9. associate--l+ 99.8%

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

      10. fma-udef 99.8%

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

      11. neg-mul-1 99.8%

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

      12. count-2 99.8%

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

      13. associate-*l* 99.8%

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

      14. distribute-rgt-out-- 99.8%

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

      15. associate-*r* 99.8%

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

      16. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. fma-udef 99.8%

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

   5. Applied egg-rr 99.8%

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

   if 7.59999999999999933e153 < x.im

   1. Initial program 36.9%

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

      1. *-commutative 36.9%

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

      2. *-commutative 36.9%

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

      3. *-commutative 36.9%

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

      4. distribute-lft-out 36.9%

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

   3. Simplified 36.9%

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

   4. Taylor expanded in x.re around 0 51.2%

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

      1. mul-1-neg 51.2%

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

      2. unpow2 51.2%

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

      3. associate-*r* 67.8%

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

      4. *-commutative 67.8%

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

      5. distribute-rgt-neg-in 67.8%

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

      6. distribute-rgt-neg-in 67.8%

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

   6. Simplified 67.8%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -5.5 \cdot 10^{+155}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{elif}\;x.im \leq 7.6 \cdot 10^{+153}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re + x.im \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(-x.im\right)\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\\ \end{array} \]

Alternative 6: 75.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -5.5 \cdot 10^{+30} \lor \neg \left(x.im \leq 6.6 \cdot 10^{-18}\right) \land \left(x.im \leq 5.5 \cdot 10^{+133} \lor \neg \left(x.im \leq 1.4 \cdot 10^{+214}\right)\right):\\ \;\;\;\;-3 \cdot \left(x.re \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -5.5e+30)
         (and (not (<= x.im 6.6e-18))
              (or (<= x.im 5.5e+133) (not (<= x.im 1.4e+214)))))
   (* -3.0 (* x.re (* x.im x.im)))
   (* x.re (* x.re x.re))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -5.5e+30) || (!(x_46_im <= 6.6e-18) && ((x_46_im <= 5.5e+133) || !(x_46_im <= 1.4e+214)))) {
		tmp = -3.0 * (x_46_re * (x_46_im * x_46_im));
	} else {
		tmp = x_46_re * (x_46_re * x_46_re);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46im <= (-5.5d+30)) .or. (.not. (x_46im <= 6.6d-18)) .and. (x_46im <= 5.5d+133) .or. (.not. (x_46im <= 1.4d+214))) then
        tmp = (-3.0d0) * (x_46re * (x_46im * x_46im))
    else
        tmp = x_46re * (x_46re * x_46re)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -5.5e+30) || (!(x_46_im <= 6.6e-18) && ((x_46_im <= 5.5e+133) || !(x_46_im <= 1.4e+214)))) {
		tmp = -3.0 * (x_46_re * (x_46_im * x_46_im));
	} else {
		tmp = x_46_re * (x_46_re * x_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_im <= -5.5e+30) or (not (x_46_im <= 6.6e-18) and ((x_46_im <= 5.5e+133) or not (x_46_im <= 1.4e+214))):
		tmp = -3.0 * (x_46_re * (x_46_im * x_46_im))
	else:
		tmp = x_46_re * (x_46_re * x_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_im <= -5.5e+30) || (!(x_46_im <= 6.6e-18) && ((x_46_im <= 5.5e+133) || !(x_46_im <= 1.4e+214))))
		tmp = Float64(-3.0 * Float64(x_46_re * Float64(x_46_im * x_46_im)));
	else
		tmp = Float64(x_46_re * Float64(x_46_re * x_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_im <= -5.5e+30) || (~((x_46_im <= 6.6e-18)) && ((x_46_im <= 5.5e+133) || ~((x_46_im <= 1.4e+214)))))
		tmp = -3.0 * (x_46_re * (x_46_im * x_46_im));
	else
		tmp = x_46_re * (x_46_re * x_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$im, -5.5e+30], And[N[Not[LessEqual[x$46$im, 6.6e-18]], $MachinePrecision], Or[LessEqual[x$46$im, 5.5e+133], N[Not[LessEqual[x$46$im, 1.4e+214]], $MachinePrecision]]]], N[(-3.0 * N[(x$46$re * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -5.50000000000000025e30 or 6.6000000000000003e-18 < x.im < 5.5e133 or 1.3999999999999999e214 < x.im

   1. Initial program 75.8%

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

      1. *-commutative 75.8%

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

      2. *-commutative 75.8%

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

      3. *-commutative 75.8%

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

      4. distribute-lft-out 75.8%

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

   3. Simplified 75.8%

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

      1. add-cbrt-cube 50.9%

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

      2. pow3 50.9%

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

   5. Applied egg-rr 50.9%

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

   6. Taylor expanded in x.re around 0 74.5%

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

      1. unpow2 74.5%

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

      2. unpow2 74.5%

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

      3. distribute-rgt-out-- 74.5%

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

      4. metadata-eval 74.5%

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

      5. associate-*r* 74.4%

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

   8. Simplified 74.4%

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

   9. Taylor expanded in x.re around 0 74.4%

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

       1. unpow2 74.4%

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

   11. Simplified 74.4%

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

   if -5.50000000000000025e30 < x.im < 6.6000000000000003e-18 or 5.5e133 < x.im < 1.3999999999999999e214

   1. Initial program 91.4%

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

      1. *-commutative 91.4%

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

      2. distribute-lft-out 91.4%

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

      3. associate-*l* 91.3%

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

      4. *-commutative 91.3%

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

      5. distribute-rgt-out-- 94.0%

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

      6. associate--l- 94.0%

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

      7. associate--l- 94.0%

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

      8. sub-neg 94.0%

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

      9. associate--l+ 94.0%

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

      10. fma-udef 94.0%

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

      11. neg-mul-1 94.0%

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

      12. count-2 94.0%

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

      13. associate-*l* 94.0%

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

      14. distribute-rgt-out-- 94.0%

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

      15. associate-*r* 94.0%

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

      16. metadata-eval 94.0%

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

   3. Simplified 94.0%

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

   4. Taylor expanded in x.re around inf 92.8%

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

      1. unpow2 92.8%

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

   6. Simplified 92.8%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -5.5 \cdot 10^{+30} \lor \neg \left(x.im \leq 6.6 \cdot 10^{-18}\right) \land \left(x.im \leq 5.5 \cdot 10^{+133} \lor \neg \left(x.im \leq 1.4 \cdot 10^{+214}\right)\right):\\ \;\;\;\;-3 \cdot \left(x.re \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \end{array} \]

Alternative 7: 75.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -3 \cdot \left(x.re \cdot \left(x.im \cdot x.im\right)\right)\\ t_1 := x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{if}\;x.im \leq -1.95 \cdot 10^{+30}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.im \leq 5.1 \cdot 10^{-18}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x.im \leq 5.5 \cdot 10^{+133}:\\ \;\;\;\;x.re \cdot \left(\left(x.im \cdot x.im\right) \cdot -3\right)\\ \mathbf{elif}\;x.im \leq 1.4 \cdot 10^{+214}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (* -3.0 (* x.re (* x.im x.im)))) (t_1 (* x.re (* x.re x.re))))
   (if (<= x.im -1.95e+30)
     t_0
     (if (<= x.im 5.1e-18)
       t_1
       (if (<= x.im 5.5e+133)
         (* x.re (* (* x.im x.im) -3.0))
         (if (<= x.im 1.4e+214) t_1 t_0))))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = -3.0 * (x_46_re * (x_46_im * x_46_im));
	double t_1 = x_46_re * (x_46_re * x_46_re);
	double tmp;
	if (x_46_im <= -1.95e+30) {
		tmp = t_0;
	} else if (x_46_im <= 5.1e-18) {
		tmp = t_1;
	} else if (x_46_im <= 5.5e+133) {
		tmp = x_46_re * ((x_46_im * x_46_im) * -3.0);
	} else if (x_46_im <= 1.4e+214) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (-3.0d0) * (x_46re * (x_46im * x_46im))
    t_1 = x_46re * (x_46re * x_46re)
    if (x_46im <= (-1.95d+30)) then
        tmp = t_0
    else if (x_46im <= 5.1d-18) then
        tmp = t_1
    else if (x_46im <= 5.5d+133) then
        tmp = x_46re * ((x_46im * x_46im) * (-3.0d0))
    else if (x_46im <= 1.4d+214) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double t_0 = -3.0 * (x_46_re * (x_46_im * x_46_im));
	double t_1 = x_46_re * (x_46_re * x_46_re);
	double tmp;
	if (x_46_im <= -1.95e+30) {
		tmp = t_0;
	} else if (x_46_im <= 5.1e-18) {
		tmp = t_1;
	} else if (x_46_im <= 5.5e+133) {
		tmp = x_46_re * ((x_46_im * x_46_im) * -3.0);
	} else if (x_46_im <= 1.4e+214) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	t_0 = -3.0 * (x_46_re * (x_46_im * x_46_im))
	t_1 = x_46_re * (x_46_re * x_46_re)
	tmp = 0
	if x_46_im <= -1.95e+30:
		tmp = t_0
	elif x_46_im <= 5.1e-18:
		tmp = t_1
	elif x_46_im <= 5.5e+133:
		tmp = x_46_re * ((x_46_im * x_46_im) * -3.0)
	elif x_46_im <= 1.4e+214:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(-3.0 * Float64(x_46_re * Float64(x_46_im * x_46_im)))
	t_1 = Float64(x_46_re * Float64(x_46_re * x_46_re))
	tmp = 0.0
	if (x_46_im <= -1.95e+30)
		tmp = t_0;
	elseif (x_46_im <= 5.1e-18)
		tmp = t_1;
	elseif (x_46_im <= 5.5e+133)
		tmp = Float64(x_46_re * Float64(Float64(x_46_im * x_46_im) * -3.0));
	elseif (x_46_im <= 1.4e+214)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	t_0 = -3.0 * (x_46_re * (x_46_im * x_46_im));
	t_1 = x_46_re * (x_46_re * x_46_re);
	tmp = 0.0;
	if (x_46_im <= -1.95e+30)
		tmp = t_0;
	elseif (x_46_im <= 5.1e-18)
		tmp = t_1;
	elseif (x_46_im <= 5.5e+133)
		tmp = x_46_re * ((x_46_im * x_46_im) * -3.0);
	elseif (x_46_im <= 1.4e+214)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(-3.0 * N[(x$46$re * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$im, -1.95e+30], t$95$0, If[LessEqual[x$46$im, 5.1e-18], t$95$1, If[LessEqual[x$46$im, 5.5e+133], N[(x$46$re * N[(N[(x$46$im * x$46$im), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 1.4e+214], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -1.95000000000000005e30 or 1.3999999999999999e214 < x.im

   1. Initial program 70.1%

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

      1. *-commutative 70.1%

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

      2. *-commutative 70.1%

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

      3. *-commutative 70.1%

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

      4. distribute-lft-out 70.1%

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

   3. Simplified 70.1%

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

      1. add-cbrt-cube 52.6%

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

      2. pow3 52.6%

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

   5. Applied egg-rr 52.6%

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

   6. Taylor expanded in x.re around 0 74.4%

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

      1. unpow2 74.4%

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

      2. unpow2 74.4%

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

      3. distribute-rgt-out-- 74.4%

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

      4. metadata-eval 74.4%

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

      5. associate-*r* 74.3%

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

   8. Simplified 74.3%

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

   9. Taylor expanded in x.re around 0 74.4%

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

       1. unpow2 74.4%

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

   11. Simplified 74.4%

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

   if -1.95000000000000005e30 < x.im < 5.09999999999999983e-18 or 5.5e133 < x.im < 1.3999999999999999e214

   1. Initial program 91.4%

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

      1. *-commutative 91.4%

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

      2. distribute-lft-out 91.4%

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

      3. associate-*l* 91.3%

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

      4. *-commutative 91.3%

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

      5. distribute-rgt-out-- 94.0%

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

      6. associate--l- 94.0%

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

      7. associate--l- 94.0%

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

      8. sub-neg 94.0%

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

      9. associate--l+ 94.0%

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

      10. fma-udef 94.0%

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

      11. neg-mul-1 94.0%

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

      12. count-2 94.0%

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

      13. associate-*l* 94.0%

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

      14. distribute-rgt-out-- 94.0%

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

      15. associate-*r* 94.0%

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

      16. metadata-eval 94.0%

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

   3. Simplified 94.0%

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

   4. Taylor expanded in x.re around inf 92.8%

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

      1. unpow2 92.8%

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

   6. Simplified 92.8%

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

   if 5.09999999999999983e-18 < x.im < 5.5e133

   1. Initial program 95.6%

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

      1. *-commutative 95.6%

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

      2. distribute-lft-out 95.6%

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

      3. associate-*l* 95.4%

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

      4. *-commutative 95.4%

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

      5. distribute-rgt-out-- 99.8%

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

      6. associate--l- 99.8%

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

      7. associate--l- 99.8%

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

      8. sub-neg 99.8%

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

      9. associate--l+ 99.8%

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

      10. fma-udef 99.8%

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

      11. neg-mul-1 99.8%

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

      12. count-2 99.8%

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

      13. associate-*l* 99.8%

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

      14. distribute-rgt-out-- 99.8%

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

      15. associate-*r* 99.8%

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

      16. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x.re around 0 74.8%

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

      1. unpow2 74.8%

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

   6. Simplified 74.8%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -1.95 \cdot 10^{+30}:\\ \;\;\;\;-3 \cdot \left(x.re \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{elif}\;x.im \leq 5.1 \cdot 10^{-18}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{elif}\;x.im \leq 5.5 \cdot 10^{+133}:\\ \;\;\;\;x.re \cdot \left(\left(x.im \cdot x.im\right) \cdot -3\right)\\ \mathbf{elif}\;x.im \leq 1.4 \cdot 10^{+214}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \left(x.re \cdot \left(x.im \cdot x.im\right)\right)\\ \end{array} \]

Alternative 8: 90.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -2.7 \cdot 10^{+173} \lor \neg \left(x.re \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re + x.im \cdot \left(x.im \cdot -3\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.re -2.7e+173) (not (<= x.re 1.35e+154)))
   (* x.re (* x.re x.re))
   (* x.re (+ (* x.re x.re) (* x.im (* x.im -3.0))))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -2.7e+173) || !(x_46_re <= 1.35e+154)) {
		tmp = x_46_re * (x_46_re * x_46_re);
	} else {
		tmp = x_46_re * ((x_46_re * x_46_re) + (x_46_im * (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_46re <= (-2.7d+173)) .or. (.not. (x_46re <= 1.35d+154))) then
        tmp = x_46re * (x_46re * x_46re)
    else
        tmp = x_46re * ((x_46re * x_46re) + (x_46im * (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_re <= -2.7e+173) || !(x_46_re <= 1.35e+154)) {
		tmp = x_46_re * (x_46_re * x_46_re);
	} else {
		tmp = x_46_re * ((x_46_re * x_46_re) + (x_46_im * (x_46_im * -3.0)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_re <= -2.7e+173) or not (x_46_re <= 1.35e+154):
		tmp = x_46_re * (x_46_re * x_46_re)
	else:
		tmp = x_46_re * ((x_46_re * x_46_re) + (x_46_im * (x_46_im * -3.0)))
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_re <= -2.7e+173) || !(x_46_re <= 1.35e+154))
		tmp = Float64(x_46_re * Float64(x_46_re * x_46_re));
	else
		tmp = Float64(x_46_re * Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * Float64(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_re <= -2.7e+173) || ~((x_46_re <= 1.35e+154)))
		tmp = x_46_re * (x_46_re * x_46_re);
	else
		tmp = x_46_re * ((x_46_re * x_46_re) + (x_46_im * (x_46_im * -3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$re, -2.7e+173], N[Not[LessEqual[x$46$re, 1.35e+154]], $MachinePrecision]], N[(x$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision], N[(x$46$re * N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < -2.7000000000000001e173 or 1.35000000000000003e154 < x.re

   1. Initial program 59.7%

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

      1. *-commutative 59.7%

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

      2. distribute-lft-out 59.7%

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

      3. associate-*l* 59.7%

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

      4. *-commutative 59.7%

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

      5. distribute-rgt-out-- 67.7%

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

      6. associate--l- 67.7%

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

      7. associate--l- 67.7%

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

      8. sub-neg 67.7%

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

      9. associate--l+ 67.7%

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

      10. fma-udef 79.0%

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

      11. neg-mul-1 79.0%

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

      12. count-2 79.0%

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

      13. associate-*l* 79.0%

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

      14. distribute-rgt-out-- 79.0%

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

      15. associate-*r* 79.0%

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

      16. metadata-eval 79.0%

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

   3. Simplified 79.0%

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

   4. Taylor expanded in x.re around inf 88.7%

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

      1. unpow2 88.7%

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

   6. Simplified 88.7%

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

   if -2.7000000000000001e173 < x.re < 1.35000000000000003e154

   1. Initial program 93.0%

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

      1. *-commutative 93.0%

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

      2. distribute-lft-out 93.0%

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

      3. associate-*l* 93.0%

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

      4. *-commutative 93.0%

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

      5. distribute-rgt-out-- 94.0%

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

      6. associate--l- 94.0%

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

      7. associate--l- 94.0%

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

      8. sub-neg 94.0%

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

      9. associate--l+ 94.0%

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

      10. fma-udef 94.0%

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

      11. neg-mul-1 94.0%

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

      12. count-2 94.0%

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

      13. associate-*l* 94.0%

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

      14. distribute-rgt-out-- 94.0%

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

      15. associate-*r* 94.0%

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

      16. metadata-eval 94.0%

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

   3. Simplified 94.0%

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

      1. fma-udef 94.0%

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

   5. Applied egg-rr 94.0%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -2.7 \cdot 10^{+173} \lor \neg \left(x.re \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re + x.im \cdot \left(x.im \cdot -3\right)\right)\\ \end{array} \]

Alternative 9: 58.2% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.9%

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

   1. *-commutative 84.9%

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

   2. distribute-lft-out 84.9%

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

   3. associate-*l* 84.9%

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

   4. *-commutative 84.9%

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

   5. distribute-rgt-out-- 87.7%

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

   6. associate--l- 87.7%

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

   7. associate--l- 87.7%

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

   8. sub-neg 87.7%

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

   9. associate--l+ 87.7%

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

   10. fma-udef 90.4%

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

   11. neg-mul-1 90.4%

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

   12. count-2 90.4%

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

   13. associate-*l* 90.4%

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

   14. distribute-rgt-out-- 90.4%

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

   15. associate-*r* 90.4%

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

   16. metadata-eval 90.4%

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

3. Simplified 90.4%

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

4. Taylor expanded in x.re around inf 62.0%

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

   1. unpow2 62.0%

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

6. Simplified 62.0%

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

7. Final simplification 62.0%

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

Alternative 10: 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 84.9%

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

   1. *-commutative 84.9%

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

   2. distribute-lft-out 84.9%

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

   3. associate-*l* 84.9%

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

   4. *-commutative 84.9%

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

   5. distribute-rgt-out-- 87.7%

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

   6. associate--l- 87.7%

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

   7. associate--l- 87.7%

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

   8. sub-neg 87.7%

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

   9. associate--l+ 87.7%

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

   10. fma-udef 90.4%

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

   11. neg-mul-1 90.4%

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

   12. count-2 90.4%

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

   13. associate-*l* 90.4%

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

   14. distribute-rgt-out-- 90.4%

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

   15. associate-*r* 90.4%

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

   16. metadata-eval 90.4%

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

3. Simplified 90.4%

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

   1. fma-udef 87.7%

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

5. Applied egg-rr 87.7%

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

6. Taylor expanded in x.re around 0 50.6%

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

8. Simplified 2.6%

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

9. Final simplification 2.6%

   \[\leadsto -3 \]

Developer target: 87.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(x.re \cdot x.re\right) \cdot \left(x.re - x.im\right) + \left(x.re \cdot x.im\right) \cdot \left(x.re - 3 \cdot x.im\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023187 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64

  :herbie-target
  (+ (* (* x.re x.re) (- x.re x.im)) (* (* x.re x.im) (- x.re (* 3.0 x.im))))

  (- (* (- (* x.re x.re) (* x.im x.im)) x.re) (* (+ (* x.re x.im) (* x.im x.re)) x.im)))