math.cube on complex, real part

Percentage Accurate: 83.1% → 96.1%

Time: 5.8s

Alternatives: 8

Speedup: 1.3×

• 45.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 96.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.im \cdot \left(x.im \cdot -3\right)\\ \mathbf{if}\;x.re \leq -2 \cdot 10^{+152}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - t_0\right)\\ \mathbf{elif}\;x.re \leq 7.5 \cdot 10^{-59}:\\ \;\;\;\;\left(x.re + x.im\right) \cdot \left(x.re \cdot \left(x.re - x.im\right)\right) - x.im \cdot \left(x.im \cdot \left(x.re + x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \mathsf{fma}\left(x.re, x.re, t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (* x.im (* x.im -3.0))))
   (if (<= x.re -2e+152)
     (* x.re (- (* x.re x.re) t_0))
     (if (<= x.re 7.5e-59)
       (-
        (* (+ x.re x.im) (* x.re (- x.re x.im)))
        (* x.im (* x.im (+ x.re x.re))))
       (* x.re (fma x.re x.re t_0))))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_im * (x_46_im * -3.0);
	double tmp;
	if (x_46_re <= -2e+152) {
		tmp = x_46_re * ((x_46_re * x_46_re) - t_0);
	} else if (x_46_re <= 7.5e-59) {
		tmp = ((x_46_re + x_46_im) * (x_46_re * (x_46_re - x_46_im))) - (x_46_im * (x_46_im * (x_46_re + x_46_re)));
	} else {
		tmp = x_46_re * fma(x_46_re, x_46_re, t_0);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(x_46_im * Float64(x_46_im * -3.0))
	tmp = 0.0
	if (x_46_re <= -2e+152)
		tmp = Float64(x_46_re * Float64(Float64(x_46_re * x_46_re) - t_0));
	elseif (x_46_re <= 7.5e-59)
		tmp = Float64(Float64(Float64(x_46_re + x_46_im) * Float64(x_46_re * Float64(x_46_re - x_46_im))) - Float64(x_46_im * Float64(x_46_im * Float64(x_46_re + x_46_re))));
	else
		tmp = Float64(x_46_re * fma(x_46_re, x_46_re, t_0));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(x$46$im * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$re, -2e+152], N[(x$46$re * N[(N[(x$46$re * x$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 7.5e-59], N[(N[(N[(x$46$re + x$46$im), $MachinePrecision] * N[(x$46$re * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(x$46$im * N[(x$46$re + x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re * N[(x$46$re * x$46$re + t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -2.0000000000000001e152

   1. Initial program 61.3%

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

      1. *-commutative 61.3%

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

      2. distribute-lft-out 61.3%

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

      3. associate-*l* 61.3%

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

      4. *-commutative 61.3%

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

      5. distribute-rgt-out-- 80.6%

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

      6. associate--l- 80.6%

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

      7. associate--l- 80.6%

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

      8. sub-neg 80.6%

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

      9. associate--l+ 80.6%

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

      10. fma-udef 83.9%

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

      11. neg-mul-1 83.9%

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

      12. count-2 83.9%

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

      13. associate-*l* 83.9%

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

      14. distribute-rgt-out-- 83.9%

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

      15. associate-*r* 83.9%

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

      16. metadata-eval 83.9%

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

   3. Simplified 83.9%

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

      1. fma-udef 80.6%

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

   5. Applied egg-rr 80.6%

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

      1. add-sqr-sqrt 25.8%

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

      2. sqrt-prod 87.1%

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

      3. sqr-neg 87.1%

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

      4. sqrt-unprod 61.3%

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

      5. add-sqr-sqrt 96.8%

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

      6. cancel-sign-sub-inv 96.8%

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

   7. Applied egg-rr 96.8%

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

   if -2.0000000000000001e152 < x.re < 7.50000000000000019e-59

   1. Initial program 86.7%

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

      1. *-commutative 86.7%

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

      2. *-commutative 86.7%

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

      3. *-commutative 86.7%

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

      4. distribute-lft-out 86.7%

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

   3. Simplified 86.7%

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

      1. sub-neg 86.7%

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

      2. *-commutative 86.7%

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

      3. difference-of-squares 86.7%

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

      4. associate-*l* 99.8%

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

      5. *-commutative 99.8%

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

      6. distribute-rgt-neg-in 99.8%

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

      7. distribute-rgt-in 99.8%

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

      8. distribute-lft-out 99.8%

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

   5. Applied egg-rr 99.8%

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

   if 7.50000000000000019e-59 < x.re

   1. Initial program 80.0%

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

      1. *-commutative 80.0%

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

      2. distribute-lft-out 80.0%

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

      3. associate-*l* 80.0%

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

      4. *-commutative 80.0%

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

      5. distribute-rgt-out-- 89.2%

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

      6. associate--l- 89.2%

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

      7. associate--l- 89.2%

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

      8. sub-neg 89.2%

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

      9. associate--l+ 89.2%

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

      10. fma-udef 97.1%

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

      11. neg-mul-1 97.1%

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

      12. count-2 97.1%

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

      13. associate-*l* 97.1%

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

      14. distribute-rgt-out-- 97.1%

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

      15. associate-*r* 97.1%

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

      16. metadata-eval 97.1%

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

   3. Simplified 97.1%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -2 \cdot 10^{+152}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{elif}\;x.re \leq 7.5 \cdot 10^{-59}:\\ \;\;\;\;\left(x.re + x.im\right) \cdot \left(x.re \cdot \left(x.re - x.im\right)\right) - x.im \cdot \left(x.im \cdot \left(x.re + x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \mathsf{fma}\left(x.re, x.re, x.im \cdot \left(x.im \cdot -3\right)\right)\\ \end{array} \]

Alternative 2: 96.4% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<=
      (-
       (* x.re (- (* x.re x.re) (* x.im x.im)))
       (* x.im (+ (* x.re x.im) (* x.re x.im))))
      INFINITY)
   (- (* (+ x.re x.im) (* x.re (- x.re x.im))) (* x.im (* x.im (+ 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 * ((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 = ((x_46_re + x_46_im) * (x_46_re * (x_46_re - x_46_im))) - (x_46_im * (x_46_im * (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;
}

Java

public static 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.POSITIVE_INFINITY) {
		tmp = ((x_46_re + x_46_im) * (x_46_re * (x_46_re - x_46_im))) - (x_46_im * (x_46_im * (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 * ((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)))) <= math.inf:
		tmp = ((x_46_re + x_46_im) * (x_46_re * (x_46_re - x_46_im))) - (x_46_im * (x_46_im * (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 (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 = Float64(Float64(Float64(x_46_re + x_46_im) * Float64(x_46_re * Float64(x_46_re - x_46_im))) - Float64(x_46_im * Float64(x_46_im * 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 * ((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)))) <= Inf)
		tmp = ((x_46_re + x_46_im) * (x_46_re * (x_46_re - x_46_im))) - (x_46_im * (x_46_im * (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[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[(N[(N[(x$46$re + x$46$im), $MachinePrecision] * N[(x$46$re * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(x$46$im * N[(x$46$re + x$46$re), $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 (-.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 91.2%

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

      1. *-commutative 91.2%

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

      2. *-commutative 91.2%

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

      3. *-commutative 91.2%

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

      4. distribute-lft-out 91.2%

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

   3. Simplified 91.2%

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

      1. sub-neg 91.2%

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

      2. *-commutative 91.2%

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

      3. difference-of-squares 91.2%

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

      4. associate-*l* 99.8%

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

      5. *-commutative 99.8%

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

      6. distribute-rgt-neg-in 99.8%

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

      7. distribute-rgt-in 99.8%

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

      8. distribute-lft-out 99.8%

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

   5. Applied egg-rr 99.8%

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

   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-- 48.1%

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

      6. associate--l- 48.1%

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

      7. associate--l- 48.1%

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

      8. sub-neg 48.1%

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

      9. associate--l+ 48.1%

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

      10. fma-udef 74.1%

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

      11. neg-mul-1 74.1%

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

      12. count-2 74.1%

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

      13. associate-*l* 74.1%

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

      14. distribute-rgt-out-- 74.1%

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

      15. associate-*r* 74.1%

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

      16. metadata-eval 74.1%

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

   3. Simplified 74.1%

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

      1. fma-udef 48.1%

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

   5. Applied egg-rr 48.1%

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

      1. add-sqr-sqrt 18.5%

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

      2. sqrt-prod 55.6%

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

      3. sqr-neg 55.6%

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

      4. sqrt-unprod 37.0%

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

      5. add-sqr-sqrt 74.1%

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

      6. cancel-sign-sub-inv 74.1%

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

   7. Applied egg-rr 74.1%

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

4. Final simplification 97.1%

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

Alternative 3: 96.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -1.5 \cdot 10^{+147}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{elif}\;x.im \leq 7.8 \cdot 10^{+153}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re + x.im \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im -1.5e+147)
   (* x.im (* x.re (* x.im -3.0)))
   (if (<= x.im 7.8e+153)
     (* x.re (+ (* 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_im <= -1.5e+147) {
		tmp = x_46_im * (x_46_re * (x_46_im * -3.0));
	} else if (x_46_im <= 7.8e+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_im * (x_46_re * -3.0));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46im <= (-1.5d+147)) then
        tmp = x_46im * (x_46re * (x_46im * (-3.0d0)))
    else if (x_46im <= 7.8d+153) then
        tmp = x_46re * ((x_46re * x_46re) + (x_46im * (x_46im * (-3.0d0))))
    else
        tmp = x_46im * (x_46im * (x_46re * (-3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= -1.5e+147) {
		tmp = x_46_im * (x_46_re * (x_46_im * -3.0));
	} else if (x_46_im <= 7.8e+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_im * (x_46_re * -3.0));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= -1.5e+147:
		tmp = x_46_im * (x_46_re * (x_46_im * -3.0))
	elif x_46_im <= 7.8e+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_im * (x_46_re * -3.0))
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= -1.5e+147)
		tmp = Float64(x_46_im * Float64(x_46_re * Float64(x_46_im * -3.0)));
	elseif (x_46_im <= 7.8e+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(x_46_im * Float64(x_46_im * Float64(x_46_re * -3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_im <= -1.5e+147)
		tmp = x_46_im * (x_46_re * (x_46_im * -3.0));
	elseif (x_46_im <= 7.8e+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_im * (x_46_re * -3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, -1.5e+147], N[(x$46$im * N[(x$46$re * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 7.8e+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[(x$46$im * N[(x$46$im * N[(x$46$re * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -1.49999999999999997e147

   1. Initial program 43.8%

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

      1. *-commutative 43.8%

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

      2. *-commutative 43.8%

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

      3. *-commutative 43.8%

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

      4. distribute-lft-out 43.8%

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

   3. Simplified 43.8%

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

      1. sub-neg 43.8%

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

      2. *-commutative 43.8%

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

      3. difference-of-squares 57.6%

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

      4. associate-*l* 92.8%

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

      5. *-commutative 92.8%

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

      6. distribute-rgt-neg-in 92.8%

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

      7. distribute-rgt-in 92.8%

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

      8. distribute-lft-out 92.8%

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

   5. Applied egg-rr 92.8%

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

   6. Taylor expanded in x.re around 0 57.6%

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

      1. distribute-rgt-out 57.6%

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

      2. unpow2 57.6%

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

      3. metadata-eval 57.6%

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

      4. associate-*r* 57.6%

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

      5. *-commutative 57.6%

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

      6. associate-*r* 92.8%

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

      7. associate-*r* 92.8%

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

      8. *-commutative 92.8%

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

      9. associate-*l* 92.9%

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

      10. *-commutative 92.9%

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

   8. Simplified 92.9%

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

   9. Taylor expanded in x.im around 0 92.9%

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

       1. associate-*r* 92.8%

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

       2. *-commutative 92.8%

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

       3. associate-*l* 92.9%

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

       4. *-commutative 92.9%

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

   11. Simplified 92.9%

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

   if -1.49999999999999997e147 < x.im < 7.79999999999999966e153

   1. Initial program 93.2%

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

      1. *-commutative 93.2%

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

      2. distribute-lft-out 93.2%

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

      3. associate-*l* 93.1%

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

      4. *-commutative 93.1%

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

      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 7.79999999999999966e153 < x.im

   1. Initial program 45.5%

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

      1. *-commutative 45.5%

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

      2. *-commutative 45.5%

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

      3. *-commutative 45.5%

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

      4. distribute-lft-out 45.5%

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

   3. Simplified 45.5%

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

      1. sub-neg 45.5%

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

      2. *-commutative 45.5%

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

      3. difference-of-squares 51.7%

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

      4. associate-*l* 81.0%

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

      5. *-commutative 81.0%

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

      6. distribute-rgt-neg-in 81.0%

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

      7. distribute-rgt-in 81.0%

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

      8. distribute-lft-out 81.0%

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

   5. Applied egg-rr 81.0%

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

   6. Taylor expanded in x.re around 0 54.9%

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

      1. distribute-rgt-out 54.9%

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

      2. unpow2 54.9%

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

      3. metadata-eval 54.9%

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

      4. associate-*r* 54.9%

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

      5. *-commutative 54.9%

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

      6. associate-*r* 84.3%

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

      7. associate-*r* 84.1%

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

      8. *-commutative 84.1%

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

      9. associate-*l* 84.2%

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

      10. *-commutative 84.2%

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

   8. Simplified 84.2%

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

   9. Taylor expanded in x.im around 0 84.2%

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

       1. associate-*r* 84.3%

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

       2. *-commutative 84.3%

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

       3. *-commutative 84.3%

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

   11. Simplified 84.3%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -1.5 \cdot 10^{+147}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{elif}\;x.im \leq 7.8 \cdot 10^{+153}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re + x.im \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \end{array} \]

Alternative 4: 60.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -4.6 \cdot 10^{+147} \lor \neg \left(x.re \leq 5.7 \cdot 10^{+145}\right) \land x.re \leq 5.8 \cdot 10^{+281}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.re -4.6e+147) (and (not (<= x.re 5.7e+145)) (<= x.re 5.8e+281)))
   (* 3.0 (* x.re (* x.im x.im)))
   (* 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 <= -4.6e+147) || (!(x_46_re <= 5.7e+145) && (x_46_re <= 5.8e+281))) {
		tmp = 3.0 * (x_46_re * (x_46_im * x_46_im));
	} else {
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46re <= (-4.6d+147)) .or. (.not. (x_46re <= 5.7d+145)) .and. (x_46re <= 5.8d+281)) then
        tmp = 3.0d0 * (x_46re * (x_46im * x_46im))
    else
        tmp = x_46im * (x_46im * (x_46re * (-3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -4.6e+147) || (!(x_46_re <= 5.7e+145) && (x_46_re <= 5.8e+281))) {
		tmp = 3.0 * (x_46_re * (x_46_im * x_46_im));
	} else {
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_re <= -4.6e+147) or (not (x_46_re <= 5.7e+145) and (x_46_re <= 5.8e+281)):
		tmp = 3.0 * (x_46_re * (x_46_im * x_46_im))
	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 <= -4.6e+147) || (!(x_46_re <= 5.7e+145) && (x_46_re <= 5.8e+281)))
		tmp = Float64(3.0 * Float64(x_46_re * Float64(x_46_im * x_46_im)));
	else
		tmp = Float64(x_46_im * Float64(x_46_im * Float64(x_46_re * -3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_re <= -4.6e+147) || (~((x_46_re <= 5.7e+145)) && (x_46_re <= 5.8e+281)))
		tmp = 3.0 * (x_46_re * (x_46_im * x_46_im));
	else
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$re, -4.6e+147], And[N[Not[LessEqual[x$46$re, 5.7e+145]], $MachinePrecision], LessEqual[x$46$re, 5.8e+281]]], N[(3.0 * N[(x$46$re * N[(x$46$im * x$46$im), $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 2 regimes

2. if x.re < -4.5999999999999998e147 or 5.6999999999999999e145 < x.re < 5.80000000000000019e281

   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-- 81.2%

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

      6. associate--l- 81.2%

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

      7. associate--l- 81.2%

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

      8. sub-neg 81.2%

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

      9. associate--l+ 81.2%

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

      10. fma-udef 89.9%

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

      11. neg-mul-1 89.9%

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

      12. count-2 89.9%

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

      13. associate-*l* 89.9%

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

      14. distribute-rgt-out-- 89.9%

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

      15. associate-*r* 89.9%

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

      16. metadata-eval 89.9%

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

   3. Simplified 89.9%

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

      1. fma-udef 81.2%

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

   5. Applied egg-rr 81.2%

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

      1. add-sqr-sqrt 26.1%

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

      2. sqrt-prod 84.1%

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

      3. sqr-neg 84.1%

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

      4. sqrt-unprod 58.0%

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

      5. add-sqr-sqrt 91.3%

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

      6. cancel-sign-sub-inv 91.3%

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

   7. Applied egg-rr 91.3%

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

   8. Taylor expanded in x.re around 0 31.0%

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

      1. unpow2 31.0%

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

   10. Simplified 31.0%

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

   if -4.5999999999999998e147 < x.re < 5.6999999999999999e145 or 5.80000000000000019e281 < x.re

   1. Initial program 88.7%

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

      1. *-commutative 88.7%

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

      2. *-commutative 88.7%

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

      3. *-commutative 88.7%

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

      4. distribute-lft-out 88.7%

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

   3. Simplified 88.7%

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

      1. sub-neg 88.7%

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

      2. *-commutative 88.7%

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

      3. difference-of-squares 89.3%

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

      4. associate-*l* 99.7%

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

      5. *-commutative 99.7%

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

      6. distribute-rgt-neg-in 99.7%

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

      7. distribute-rgt-in 99.7%

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

      8. distribute-lft-out 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in x.re around 0 65.9%

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

      1. distribute-rgt-out 65.9%

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

      2. unpow2 65.9%

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

      3. metadata-eval 65.9%

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

      4. associate-*r* 65.9%

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

      5. *-commutative 65.9%

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

      6. associate-*r* 76.4%

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

      7. associate-*r* 76.4%

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

      8. *-commutative 76.4%

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

      9. associate-*l* 76.4%

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

      10. *-commutative 76.4%

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

   8. Simplified 76.4%

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

   9. Taylor expanded in x.im around 0 76.4%

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

       1. associate-*r* 76.4%

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

       2. *-commutative 76.4%

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

       3. *-commutative 76.4%

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

   11. Simplified 76.4%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -4.6 \cdot 10^{+147} \lor \neg \left(x.re \leq 5.7 \cdot 10^{+145}\right) \land x.re \leq 5.8 \cdot 10^{+281}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \end{array} \]

Alternative 5: 60.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -2.4 \cdot 10^{+151} \lor \neg \left(x.re \leq 10^{+144}\right) \land x.re \leq 5.8 \cdot 10^{+281}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.re -2.4e+151) (and (not (<= x.re 1e+144)) (<= x.re 5.8e+281)))
   (* x.im (* x.im (* x.re 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 <= -2.4e+151) || (!(x_46_re <= 1e+144) && (x_46_re <= 5.8e+281))) {
		tmp = x_46_im * (x_46_im * (x_46_re * 3.0));
	} else {
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46re <= (-2.4d+151)) .or. (.not. (x_46re <= 1d+144)) .and. (x_46re <= 5.8d+281)) then
        tmp = x_46im * (x_46im * (x_46re * 3.0d0))
    else
        tmp = x_46im * (x_46im * (x_46re * (-3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -2.4e+151) || (!(x_46_re <= 1e+144) && (x_46_re <= 5.8e+281))) {
		tmp = x_46_im * (x_46_im * (x_46_re * 3.0));
	} else {
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_re <= -2.4e+151) or (not (x_46_re <= 1e+144) and (x_46_re <= 5.8e+281)):
		tmp = x_46_im * (x_46_im * (x_46_re * 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 <= -2.4e+151) || (!(x_46_re <= 1e+144) && (x_46_re <= 5.8e+281)))
		tmp = Float64(x_46_im * Float64(x_46_im * Float64(x_46_re * 3.0)));
	else
		tmp = Float64(x_46_im * Float64(x_46_im * Float64(x_46_re * -3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_re <= -2.4e+151) || (~((x_46_re <= 1e+144)) && (x_46_re <= 5.8e+281)))
		tmp = x_46_im * (x_46_im * (x_46_re * 3.0));
	else
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$re, -2.4e+151], And[N[Not[LessEqual[x$46$re, 1e+144]], $MachinePrecision], LessEqual[x$46$re, 5.8e+281]]], N[(x$46$im * N[(x$46$im * N[(x$46$re * 3.0), $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 2 regimes

2. if x.re < -2.4000000000000001e151 or 1.00000000000000002e144 < x.re < 5.80000000000000019e281

   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%

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

      2. *-commutative 62.3%

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

      3. *-commutative 62.3%

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

      4. distribute-lft-out 62.3%

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

   3. Simplified 62.3%

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

      1. sub-neg 62.3%

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

      2. *-commutative 62.3%

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

      3. difference-of-squares 69.6%

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

      4. associate-*l* 69.6%

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

      5. *-commutative 69.6%

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

      6. distribute-rgt-neg-in 69.6%

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

      7. distribute-rgt-in 69.6%

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

      8. distribute-lft-out 69.6%

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

   5. Applied egg-rr 69.6%

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

   6. Taylor expanded in x.re around 0 9.3%

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

      1. distribute-rgt-out 9.3%

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

      2. unpow2 9.3%

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

      3. metadata-eval 9.3%

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

      4. associate-*r* 9.3%

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

      5. *-commutative 9.3%

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

      6. associate-*r* 9.2%

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

      7. associate-*r* 9.2%

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

      8. *-commutative 9.2%

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

      9. associate-*l* 9.2%

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

      10. *-commutative 9.2%

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

   8. Simplified 9.2%

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

   9. Taylor expanded in x.im around 0 9.2%

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

       1. associate-*r* 9.2%

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

       2. *-commutative 9.2%

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

       3. *-commutative 9.2%

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

   11. Simplified 9.2%

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

       1. expm1-log1p-u 6.1%

          \[\leadsto x.im \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x.im \cdot \left(x.re \cdot -3\right)\right)\right)} \]

       2. expm1-udef 6.1%

          \[\leadsto x.im \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(x.im \cdot \left(x.re \cdot -3\right)\right)} - 1\right)} \]

       3. add-sqr-sqrt 1.6%

          \[\leadsto x.im \cdot \left(e^{\mathsf{log1p}\left(x.im \cdot \color{blue}{\left(\sqrt{x.re \cdot -3} \cdot \sqrt{x.re \cdot -3}\right)}\right)} - 1\right) \]

       4. sqrt-unprod 17.5%

          \[\leadsto x.im \cdot \left(e^{\mathsf{log1p}\left(x.im \cdot \color{blue}{\sqrt{\left(x.re \cdot -3\right) \cdot \left(x.re \cdot -3\right)}}\right)} - 1\right) \]

       5. swap-sqr 17.5%

          \[\leadsto x.im \cdot \left(e^{\mathsf{log1p}\left(x.im \cdot \sqrt{\color{blue}{\left(x.re \cdot x.re\right) \cdot \left(-3 \cdot -3\right)}}\right)} - 1\right) \]

       6. metadata-eval 17.5%

          \[\leadsto x.im \cdot \left(e^{\mathsf{log1p}\left(x.im \cdot \sqrt{\left(x.re \cdot x.re\right) \cdot \color{blue}{9}}\right)} - 1\right) \]

       7. metadata-eval 17.5%

          \[\leadsto x.im \cdot \left(e^{\mathsf{log1p}\left(x.im \cdot \sqrt{\left(x.re \cdot x.re\right) \cdot \color{blue}{\left(3 \cdot 3\right)}}\right)} - 1\right) \]

       8. swap-sqr 17.5%

          \[\leadsto x.im \cdot \left(e^{\mathsf{log1p}\left(x.im \cdot \sqrt{\color{blue}{\left(x.re \cdot 3\right) \cdot \left(x.re \cdot 3\right)}}\right)} - 1\right) \]

       9. sqrt-unprod 7.7%

          \[\leadsto x.im \cdot \left(e^{\mathsf{log1p}\left(x.im \cdot \color{blue}{\left(\sqrt{x.re \cdot 3} \cdot \sqrt{x.re \cdot 3}\right)}\right)} - 1\right) \]

       10. add-sqr-sqrt 17.0%

           \[\leadsto x.im \cdot \left(e^{\mathsf{log1p}\left(x.im \cdot \color{blue}{\left(x.re \cdot 3\right)}\right)} - 1\right) \]

   13. Applied egg-rr 17.0%

       \[\leadsto x.im \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(x.im \cdot \left(x.re \cdot 3\right)\right)} - 1\right)} \]
   14. Step-by-step derivation

       1. expm1-def 17.0%

          \[\leadsto x.im \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x.im \cdot \left(x.re \cdot 3\right)\right)\right)} \]

       2. expm1-log1p 31.1%

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

   15. Simplified 31.1%

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

   if -2.4000000000000001e151 < x.re < 1.00000000000000002e144 or 5.80000000000000019e281 < x.re

   1. Initial program 88.7%

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

      1. *-commutative 88.7%

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

      2. *-commutative 88.7%

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

      3. *-commutative 88.7%

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

      4. distribute-lft-out 88.7%

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

   3. Simplified 88.7%

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

      1. sub-neg 88.7%

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

      2. *-commutative 88.7%

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

      3. difference-of-squares 89.3%

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

      4. associate-*l* 99.7%

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

      5. *-commutative 99.7%

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

      6. distribute-rgt-neg-in 99.7%

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

      7. distribute-rgt-in 99.7%

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

      8. distribute-lft-out 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in x.re around 0 65.9%

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

      1. distribute-rgt-out 65.9%

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

      2. unpow2 65.9%

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

      3. metadata-eval 65.9%

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

      4. associate-*r* 65.9%

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

      5. *-commutative 65.9%

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

      6. associate-*r* 76.4%

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

      7. associate-*r* 76.4%

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

      8. *-commutative 76.4%

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

      9. associate-*l* 76.4%

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

      10. *-commutative 76.4%

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

   8. Simplified 76.4%

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

   9. Taylor expanded in x.im around 0 76.4%

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

       1. associate-*r* 76.4%

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

       2. *-commutative 76.4%

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

       3. *-commutative 76.4%

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

   11. Simplified 76.4%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -2.4 \cdot 10^{+151} \lor \neg \left(x.re \leq 10^{+144}\right) \land x.re \leq 5.8 \cdot 10^{+281}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \end{array} \]

Alternative 6: 60.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.im \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right)\\ \mathbf{if}\;x.re \leq -1.9 \cdot 10^{+151}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.re \leq 4.6 \cdot 10^{+142}:\\ \;\;\;\;x.im \cdot \left(-3 \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{elif}\;x.re \leq 5.8 \cdot 10^{+281}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (* x.im (* x.im (* x.re 3.0)))))
   (if (<= x.re -1.9e+151)
     t_0
     (if (<= x.re 4.6e+142)
       (* x.im (* -3.0 (* x.re x.im)))
       (if (<= x.re 5.8e+281) t_0 (* x.im (* x.im (* x.re -3.0))))))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_im * (x_46_im * (x_46_re * 3.0));
	double tmp;
	if (x_46_re <= -1.9e+151) {
		tmp = t_0;
	} else if (x_46_re <= 4.6e+142) {
		tmp = x_46_im * (-3.0 * (x_46_re * x_46_im));
	} else if (x_46_re <= 5.8e+281) {
		tmp = t_0;
	} else {
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x_46im * (x_46im * (x_46re * 3.0d0))
    if (x_46re <= (-1.9d+151)) then
        tmp = t_0
    else if (x_46re <= 4.6d+142) then
        tmp = x_46im * ((-3.0d0) * (x_46re * x_46im))
    else if (x_46re <= 5.8d+281) then
        tmp = t_0
    else
        tmp = x_46im * (x_46im * (x_46re * (-3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_im * (x_46_im * (x_46_re * 3.0));
	double tmp;
	if (x_46_re <= -1.9e+151) {
		tmp = t_0;
	} else if (x_46_re <= 4.6e+142) {
		tmp = x_46_im * (-3.0 * (x_46_re * x_46_im));
	} else if (x_46_re <= 5.8e+281) {
		tmp = t_0;
	} else {
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	t_0 = x_46_im * (x_46_im * (x_46_re * 3.0))
	tmp = 0
	if x_46_re <= -1.9e+151:
		tmp = t_0
	elif x_46_re <= 4.6e+142:
		tmp = x_46_im * (-3.0 * (x_46_re * x_46_im))
	elif x_46_re <= 5.8e+281:
		tmp = t_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)
	t_0 = Float64(x_46_im * Float64(x_46_im * Float64(x_46_re * 3.0)))
	tmp = 0.0
	if (x_46_re <= -1.9e+151)
		tmp = t_0;
	elseif (x_46_re <= 4.6e+142)
		tmp = Float64(x_46_im * Float64(-3.0 * Float64(x_46_re * x_46_im)));
	elseif (x_46_re <= 5.8e+281)
		tmp = t_0;
	else
		tmp = Float64(x_46_im * Float64(x_46_im * Float64(x_46_re * -3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	t_0 = x_46_im * (x_46_im * (x_46_re * 3.0));
	tmp = 0.0;
	if (x_46_re <= -1.9e+151)
		tmp = t_0;
	elseif (x_46_re <= 4.6e+142)
		tmp = x_46_im * (-3.0 * (x_46_re * x_46_im));
	elseif (x_46_re <= 5.8e+281)
		tmp = t_0;
	else
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(x$46$im * N[(x$46$im * N[(x$46$re * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$re, -1.9e+151], t$95$0, If[LessEqual[x$46$re, 4.6e+142], N[(x$46$im * N[(-3.0 * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 5.8e+281], t$95$0, 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 < -1.9e151 or 4.60000000000000004e142 < x.re < 5.80000000000000019e281

   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%

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

      2. *-commutative 62.3%

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

      3. *-commutative 62.3%

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

      4. distribute-lft-out 62.3%

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

   3. Simplified 62.3%

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

      1. sub-neg 62.3%

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

      2. *-commutative 62.3%

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

      3. difference-of-squares 69.6%

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

      4. associate-*l* 69.6%

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

      5. *-commutative 69.6%

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

      6. distribute-rgt-neg-in 69.6%

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

      7. distribute-rgt-in 69.6%

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

      8. distribute-lft-out 69.6%

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

   5. Applied egg-rr 69.6%

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

   6. Taylor expanded in x.re around 0 9.3%

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

      1. distribute-rgt-out 9.3%

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

      2. unpow2 9.3%

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

      3. metadata-eval 9.3%

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

      4. associate-*r* 9.3%

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

      5. *-commutative 9.3%

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

      6. associate-*r* 9.2%

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

      7. associate-*r* 9.2%

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

      8. *-commutative 9.2%

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

      9. associate-*l* 9.2%

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

      10. *-commutative 9.2%

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

   8. Simplified 9.2%

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

   9. Taylor expanded in x.im around 0 9.2%

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

       1. associate-*r* 9.2%

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

       2. *-commutative 9.2%

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

       3. *-commutative 9.2%

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

   11. Simplified 9.2%

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

       1. expm1-log1p-u 6.1%

          \[\leadsto x.im \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x.im \cdot \left(x.re \cdot -3\right)\right)\right)} \]

       2. expm1-udef 6.1%

          \[\leadsto x.im \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(x.im \cdot \left(x.re \cdot -3\right)\right)} - 1\right)} \]

       3. add-sqr-sqrt 1.6%

          \[\leadsto x.im \cdot \left(e^{\mathsf{log1p}\left(x.im \cdot \color{blue}{\left(\sqrt{x.re \cdot -3} \cdot \sqrt{x.re \cdot -3}\right)}\right)} - 1\right) \]

       4. sqrt-unprod 17.5%

          \[\leadsto x.im \cdot \left(e^{\mathsf{log1p}\left(x.im \cdot \color{blue}{\sqrt{\left(x.re \cdot -3\right) \cdot \left(x.re \cdot -3\right)}}\right)} - 1\right) \]

       5. swap-sqr 17.5%

          \[\leadsto x.im \cdot \left(e^{\mathsf{log1p}\left(x.im \cdot \sqrt{\color{blue}{\left(x.re \cdot x.re\right) \cdot \left(-3 \cdot -3\right)}}\right)} - 1\right) \]

       6. metadata-eval 17.5%

          \[\leadsto x.im \cdot \left(e^{\mathsf{log1p}\left(x.im \cdot \sqrt{\left(x.re \cdot x.re\right) \cdot \color{blue}{9}}\right)} - 1\right) \]

       7. metadata-eval 17.5%

          \[\leadsto x.im \cdot \left(e^{\mathsf{log1p}\left(x.im \cdot \sqrt{\left(x.re \cdot x.re\right) \cdot \color{blue}{\left(3 \cdot 3\right)}}\right)} - 1\right) \]

       8. swap-sqr 17.5%

          \[\leadsto x.im \cdot \left(e^{\mathsf{log1p}\left(x.im \cdot \sqrt{\color{blue}{\left(x.re \cdot 3\right) \cdot \left(x.re \cdot 3\right)}}\right)} - 1\right) \]

       9. sqrt-unprod 7.7%

          \[\leadsto x.im \cdot \left(e^{\mathsf{log1p}\left(x.im \cdot \color{blue}{\left(\sqrt{x.re \cdot 3} \cdot \sqrt{x.re \cdot 3}\right)}\right)} - 1\right) \]

       10. add-sqr-sqrt 17.0%

           \[\leadsto x.im \cdot \left(e^{\mathsf{log1p}\left(x.im \cdot \color{blue}{\left(x.re \cdot 3\right)}\right)} - 1\right) \]

   13. Applied egg-rr 17.0%

       \[\leadsto x.im \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(x.im \cdot \left(x.re \cdot 3\right)\right)} - 1\right)} \]
   14. Step-by-step derivation

       1. expm1-def 17.0%

          \[\leadsto x.im \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x.im \cdot \left(x.re \cdot 3\right)\right)\right)} \]

       2. expm1-log1p 31.1%

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

   15. Simplified 31.1%

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

   if -1.9e151 < x.re < 4.60000000000000004e142

   1. Initial program 89.1%

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

      1. *-commutative 89.1%

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

      2. *-commutative 89.1%

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

      3. *-commutative 89.1%

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

      4. distribute-lft-out 89.1%

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

   3. Simplified 89.1%

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

      1. sub-neg 89.1%

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

      2. *-commutative 89.1%

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

      3. difference-of-squares 89.1%

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

      4. associate-*l* 99.7%

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

      5. *-commutative 99.7%

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

      6. distribute-rgt-neg-in 99.7%

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

      7. distribute-rgt-in 99.7%

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

      8. distribute-lft-out 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in x.re around 0 66.4%

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

      1. distribute-rgt-out 66.4%

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

      2. unpow2 66.4%

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

      3. metadata-eval 66.4%

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

      4. associate-*r* 66.4%

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

      5. *-commutative 66.4%

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

      6. associate-*r* 77.1%

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

      7. associate-*r* 77.1%

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

      8. *-commutative 77.1%

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

      9. associate-*l* 77.1%

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

      10. *-commutative 77.1%

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

   8. Simplified 77.1%

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

   if 5.80000000000000019e281 < x.re

   1. Initial program 66.7%

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

      1. *-commutative 66.7%

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

      2. *-commutative 66.7%

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

      3. *-commutative 66.7%

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

      4. distribute-lft-out 66.7%

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

   3. Simplified 66.7%

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

      1. sub-neg 66.7%

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

      2. *-commutative 66.7%

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

      3. difference-of-squares 100.0%

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

      4. associate-*l* 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-in 100.0%

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

      7. distribute-rgt-in 100.0%

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

      8. distribute-lft-out 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x.re around 0 34.4%

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

      1. distribute-rgt-out 34.4%

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

      2. unpow2 34.4%

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

      3. metadata-eval 34.4%

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

      4. associate-*r* 34.4%

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

      5. *-commutative 34.4%

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

      6. associate-*r* 34.1%

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

      7. associate-*r* 34.1%

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

      8. *-commutative 34.1%

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

      9. associate-*l* 34.1%

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

      10. *-commutative 34.1%

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

   8. Simplified 34.1%

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

   9. Taylor expanded in x.im around 0 34.1%

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

       1. associate-*r* 34.1%

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

       2. *-commutative 34.1%

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

       3. *-commutative 34.1%

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

   11. Simplified 34.1%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -1.9 \cdot 10^{+151}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right)\\ \mathbf{elif}\;x.re \leq 4.6 \cdot 10^{+142}:\\ \;\;\;\;x.im \cdot \left(-3 \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{elif}\;x.re \leq 5.8 \cdot 10^{+281}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \end{array} \]

Alternative 7: 60.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.im \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right)\\ \mathbf{if}\;x.re \leq -9 \cdot 10^{+151}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.re \leq 1.25 \cdot 10^{+140}:\\ \;\;\;\;x.im \cdot \left(-3 \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{elif}\;x.re \leq 5.8 \cdot 10^{+281}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.im \cdot -3\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (* x.im (* x.im (* x.re 3.0)))))
   (if (<= x.re -9e+151)
     t_0
     (if (<= x.re 1.25e+140)
       (* x.im (* -3.0 (* x.re x.im)))
       (if (<= x.re 5.8e+281) t_0 (* x.re (* x.im (* x.im -3.0))))))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_im * (x_46_im * (x_46_re * 3.0));
	double tmp;
	if (x_46_re <= -9e+151) {
		tmp = t_0;
	} else if (x_46_re <= 1.25e+140) {
		tmp = x_46_im * (-3.0 * (x_46_re * x_46_im));
	} else if (x_46_re <= 5.8e+281) {
		tmp = t_0;
	} else {
		tmp = 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) :: t_0
    real(8) :: tmp
    t_0 = x_46im * (x_46im * (x_46re * 3.0d0))
    if (x_46re <= (-9d+151)) then
        tmp = t_0
    else if (x_46re <= 1.25d+140) then
        tmp = x_46im * ((-3.0d0) * (x_46re * x_46im))
    else if (x_46re <= 5.8d+281) then
        tmp = t_0
    else
        tmp = 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 t_0 = x_46_im * (x_46_im * (x_46_re * 3.0));
	double tmp;
	if (x_46_re <= -9e+151) {
		tmp = t_0;
	} else if (x_46_re <= 1.25e+140) {
		tmp = x_46_im * (-3.0 * (x_46_re * x_46_im));
	} else if (x_46_re <= 5.8e+281) {
		tmp = t_0;
	} else {
		tmp = x_46_re * (x_46_im * (x_46_im * -3.0));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	t_0 = x_46_im * (x_46_im * (x_46_re * 3.0))
	tmp = 0
	if x_46_re <= -9e+151:
		tmp = t_0
	elif x_46_re <= 1.25e+140:
		tmp = x_46_im * (-3.0 * (x_46_re * x_46_im))
	elif x_46_re <= 5.8e+281:
		tmp = t_0
	else:
		tmp = x_46_re * (x_46_im * (x_46_im * -3.0))
	return tmp

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(x_46_im * Float64(x_46_im * Float64(x_46_re * 3.0)))
	tmp = 0.0
	if (x_46_re <= -9e+151)
		tmp = t_0;
	elseif (x_46_re <= 1.25e+140)
		tmp = Float64(x_46_im * Float64(-3.0 * Float64(x_46_re * x_46_im)));
	elseif (x_46_re <= 5.8e+281)
		tmp = t_0;
	else
		tmp = Float64(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)
	t_0 = x_46_im * (x_46_im * (x_46_re * 3.0));
	tmp = 0.0;
	if (x_46_re <= -9e+151)
		tmp = t_0;
	elseif (x_46_re <= 1.25e+140)
		tmp = x_46_im * (-3.0 * (x_46_re * x_46_im));
	elseif (x_46_re <= 5.8e+281)
		tmp = t_0;
	else
		tmp = x_46_re * (x_46_im * (x_46_im * -3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(x$46$im * N[(x$46$im * N[(x$46$re * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$re, -9e+151], t$95$0, If[LessEqual[x$46$re, 1.25e+140], N[(x$46$im * N[(-3.0 * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 5.8e+281], t$95$0, N[(x$46$re * N[(x$46$im * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -8.9999999999999997e151 or 1.25000000000000002e140 < x.re < 5.80000000000000019e281

   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%

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

      2. *-commutative 62.3%

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

      3. *-commutative 62.3%

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

      4. distribute-lft-out 62.3%

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

   3. Simplified 62.3%

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

      1. sub-neg 62.3%

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

      2. *-commutative 62.3%

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

      3. difference-of-squares 69.6%

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

      4. associate-*l* 69.6%

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

      5. *-commutative 69.6%

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

      6. distribute-rgt-neg-in 69.6%

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

      7. distribute-rgt-in 69.6%

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

      8. distribute-lft-out 69.6%

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

   5. Applied egg-rr 69.6%

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

   6. Taylor expanded in x.re around 0 9.3%

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

      1. distribute-rgt-out 9.3%

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

      2. unpow2 9.3%

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

      3. metadata-eval 9.3%

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

      4. associate-*r* 9.3%

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

      5. *-commutative 9.3%

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

      6. associate-*r* 9.2%

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

      7. associate-*r* 9.2%

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

      8. *-commutative 9.2%

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

      9. associate-*l* 9.2%

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

      10. *-commutative 9.2%

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

   8. Simplified 9.2%

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

   9. Taylor expanded in x.im around 0 9.2%

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

       1. associate-*r* 9.2%

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

       2. *-commutative 9.2%

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

       3. *-commutative 9.2%

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

   11. Simplified 9.2%

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

       1. expm1-log1p-u 6.1%

          \[\leadsto x.im \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x.im \cdot \left(x.re \cdot -3\right)\right)\right)} \]

       2. expm1-udef 6.1%

          \[\leadsto x.im \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(x.im \cdot \left(x.re \cdot -3\right)\right)} - 1\right)} \]

       3. add-sqr-sqrt 1.6%

          \[\leadsto x.im \cdot \left(e^{\mathsf{log1p}\left(x.im \cdot \color{blue}{\left(\sqrt{x.re \cdot -3} \cdot \sqrt{x.re \cdot -3}\right)}\right)} - 1\right) \]

       4. sqrt-unprod 17.5%

          \[\leadsto x.im \cdot \left(e^{\mathsf{log1p}\left(x.im \cdot \color{blue}{\sqrt{\left(x.re \cdot -3\right) \cdot \left(x.re \cdot -3\right)}}\right)} - 1\right) \]

       5. swap-sqr 17.5%

          \[\leadsto x.im \cdot \left(e^{\mathsf{log1p}\left(x.im \cdot \sqrt{\color{blue}{\left(x.re \cdot x.re\right) \cdot \left(-3 \cdot -3\right)}}\right)} - 1\right) \]

       6. metadata-eval 17.5%

          \[\leadsto x.im \cdot \left(e^{\mathsf{log1p}\left(x.im \cdot \sqrt{\left(x.re \cdot x.re\right) \cdot \color{blue}{9}}\right)} - 1\right) \]

       7. metadata-eval 17.5%

          \[\leadsto x.im \cdot \left(e^{\mathsf{log1p}\left(x.im \cdot \sqrt{\left(x.re \cdot x.re\right) \cdot \color{blue}{\left(3 \cdot 3\right)}}\right)} - 1\right) \]

       8. swap-sqr 17.5%

          \[\leadsto x.im \cdot \left(e^{\mathsf{log1p}\left(x.im \cdot \sqrt{\color{blue}{\left(x.re \cdot 3\right) \cdot \left(x.re \cdot 3\right)}}\right)} - 1\right) \]

       9. sqrt-unprod 7.7%

          \[\leadsto x.im \cdot \left(e^{\mathsf{log1p}\left(x.im \cdot \color{blue}{\left(\sqrt{x.re \cdot 3} \cdot \sqrt{x.re \cdot 3}\right)}\right)} - 1\right) \]

       10. add-sqr-sqrt 17.0%

           \[\leadsto x.im \cdot \left(e^{\mathsf{log1p}\left(x.im \cdot \color{blue}{\left(x.re \cdot 3\right)}\right)} - 1\right) \]

   13. Applied egg-rr 17.0%

       \[\leadsto x.im \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(x.im \cdot \left(x.re \cdot 3\right)\right)} - 1\right)} \]
   14. Step-by-step derivation

       1. expm1-def 17.0%

          \[\leadsto x.im \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x.im \cdot \left(x.re \cdot 3\right)\right)\right)} \]

       2. expm1-log1p 31.1%

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

   15. Simplified 31.1%

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

   if -8.9999999999999997e151 < x.re < 1.25000000000000002e140

   1. Initial program 89.1%

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

      1. *-commutative 89.1%

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

      2. *-commutative 89.1%

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

      3. *-commutative 89.1%

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

      4. distribute-lft-out 89.1%

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

   3. Simplified 89.1%

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

      1. sub-neg 89.1%

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

      2. *-commutative 89.1%

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

      3. difference-of-squares 89.1%

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

      4. associate-*l* 99.7%

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

      5. *-commutative 99.7%

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

      6. distribute-rgt-neg-in 99.7%

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

      7. distribute-rgt-in 99.7%

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

      8. distribute-lft-out 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in x.re around 0 66.4%

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

      1. distribute-rgt-out 66.4%

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

      2. unpow2 66.4%

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

      3. metadata-eval 66.4%

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

      4. associate-*r* 66.4%

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

      5. *-commutative 66.4%

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

      6. associate-*r* 77.1%

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

      7. associate-*r* 77.1%

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

      8. *-commutative 77.1%

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

      9. associate-*l* 77.1%

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

      10. *-commutative 77.1%

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

   8. Simplified 77.1%

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

   if 5.80000000000000019e281 < x.re

   1. Initial program 66.7%

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

      1. *-commutative 66.7%

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

      2. distribute-lft-out 66.7%

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

      3. associate-*l* 66.7%

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

      4. *-commutative 66.7%

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

      5. distribute-rgt-out-- 66.7%

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

      6. associate--l- 66.7%

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

      7. associate--l- 66.7%

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

      8. sub-neg 66.7%

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

      9. associate--l+ 66.7%

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

      10. fma-udef 100.0%

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

      11. neg-mul-1 100.0%

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

      12. count-2 100.0%

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

      13. associate-*l* 100.0%

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

      14. distribute-rgt-out-- 100.0%

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

      15. associate-*r* 100.0%

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

      16. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. fma-udef 66.7%

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

   5. Applied egg-rr 66.7%

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

   6. Taylor expanded in x.re around 0 34.4%

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

      1. *-commutative 34.4%

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

      2. unpow2 34.4%

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

      3. associate-*r* 34.4%

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

   8. Simplified 34.4%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -9 \cdot 10^{+151}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right)\\ \mathbf{elif}\;x.re \leq 1.25 \cdot 10^{+140}:\\ \;\;\;\;x.im \cdot \left(-3 \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{elif}\;x.re \leq 5.8 \cdot 10^{+281}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.im \cdot -3\right)\right)\\ \end{array} \]

Alternative 8: 23.6% accurate, 2.7× speedup

Math

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

FPCore

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

C

double code(double x_46_re, double x_46_im) {
	return 3.0 * (x_46_re * (x_46_im * 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 = 3.0d0 * (x_46re * (x_46im * x_46im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.6%

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

   1. *-commutative 81.6%

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

   2. distribute-lft-out 81.6%

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

   3. associate-*l* 81.6%

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

   4. *-commutative 81.6%

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

   5. distribute-rgt-out-- 86.7%

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

   6. associate--l- 86.7%

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

   7. associate--l- 86.7%

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

   8. sub-neg 86.7%

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

   9. associate--l+ 86.7%

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

   10. fma-udef 89.4%

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

   11. neg-mul-1 89.4%

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

   12. count-2 89.4%

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

   13. associate-*l* 89.4%

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

   14. distribute-rgt-out-- 89.4%

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

   15. associate-*r* 89.4%

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

   16. metadata-eval 89.4%

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

3. Simplified 89.4%

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

   1. fma-udef 86.7%

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

5. Applied egg-rr 86.7%

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

   1. add-sqr-sqrt 38.7%

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

   2. sqrt-prod 72.4%

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

   3. sqr-neg 72.4%

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

   4. sqrt-unprod 33.6%

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

   5. add-sqr-sqrt 60.6%

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

   6. cancel-sign-sub-inv 60.6%

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

7. Applied egg-rr 60.6%

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

8. Taylor expanded in x.re around 0 27.5%

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

   1. unpow2 27.5%

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

10. Simplified 27.5%

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

11. Final simplification 27.5%

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

Developer target: 86.7% 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 2023182 
(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)))