math.cube on complex, real part

Percentage Accurate: 81.8% → 96.1%

Time: 6.3s

Alternatives: 6

Speedup: 1.3×

• 43.9% 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: 81.8% 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, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -6.8 \cdot 10^{+141} \lor \neg \left(x.im \leq 2.7 \cdot 10^{+135}\right):\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.im \cdot -3\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re + x.im \cdot \left(x.im \cdot -3\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -6.8e+141) (not (<= x.im 2.7e+135)))
   (* (* x.re x.im) (* x.im -3.0))
   (* x.re (+ (* x.re x.re) (* x.im (* x.im -3.0))))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -6.8e+141) || !(x_46_im <= 2.7e+135)) {
		tmp = (x_46_re * x_46_im) * (x_46_im * -3.0);
	} else {
		tmp = x_46_re * ((x_46_re * x_46_re) + (x_46_im * (x_46_im * -3.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46im <= (-6.8d+141)) .or. (.not. (x_46im <= 2.7d+135))) then
        tmp = (x_46re * x_46im) * (x_46im * (-3.0d0))
    else
        tmp = x_46re * ((x_46re * x_46re) + (x_46im * (x_46im * (-3.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -6.8e+141) || !(x_46_im <= 2.7e+135)) {
		tmp = (x_46_re * x_46_im) * (x_46_im * -3.0);
	} else {
		tmp = x_46_re * ((x_46_re * x_46_re) + (x_46_im * (x_46_im * -3.0)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_im <= -6.8e+141) or not (x_46_im <= 2.7e+135):
		tmp = (x_46_re * x_46_im) * (x_46_im * -3.0)
	else:
		tmp = x_46_re * ((x_46_re * x_46_re) + (x_46_im * (x_46_im * -3.0)))
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_im <= -6.8e+141) || !(x_46_im <= 2.7e+135))
		tmp = Float64(Float64(x_46_re * x_46_im) * Float64(x_46_im * -3.0));
	else
		tmp = Float64(x_46_re * Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * Float64(x_46_im * -3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_im <= -6.8e+141) || ~((x_46_im <= 2.7e+135)))
		tmp = (x_46_re * x_46_im) * (x_46_im * -3.0);
	else
		tmp = x_46_re * ((x_46_re * x_46_re) + (x_46_im * (x_46_im * -3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$im, -6.8e+141], N[Not[LessEqual[x$46$im, 2.7e+135]], $MachinePrecision]], N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision], N[(x$46$re * N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -6.7999999999999996e141 or 2.69999999999999985e135 < x.im

   1. Initial program 50.1%

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

      1. *-commutative 50.1%

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

      2. distribute-lft-out 50.1%

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

      3. associate-*l* 50.1%

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

      4. *-commutative 50.1%

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

      5. distribute-rgt-out-- 50.1%

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

      6. associate--l- 50.1%

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

      7. associate--l- 50.1%

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

      8. sub-neg 50.1%

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

      9. associate--l+ 50.1%

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

      10. fma-udef 62.0%

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

      11. neg-mul-1 62.0%

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

      12. count-2 62.0%

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

      13. associate-*l* 62.0%

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

      14. distribute-rgt-out-- 62.0%

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

      15. associate-*r* 62.0%

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

      16. metadata-eval 62.0%

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

   3. Simplified 62.0%

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

      1. fma-udef 50.0%

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

   5. Applied egg-rr 50.0%

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

   6. Taylor expanded in x.re around 0 62.0%

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

      1. *-commutative 62.0%

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

      2. unpow2 62.0%

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

      3. associate-*r* 90.9%

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

      4. associate-*r* 90.9%

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

      5. *-commutative 90.9%

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

      6. *-commutative 90.9%

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

   8. Simplified 90.9%

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

   if -6.7999999999999996e141 < x.im < 2.69999999999999985e135

   1. Initial program 90.8%

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

      1. *-commutative 90.8%

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

      2. distribute-lft-out 90.8%

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

      3. associate-*l* 90.7%

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

      4. *-commutative 90.7%

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

      5. distribute-rgt-out-- 99.8%

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

      6. associate--l- 99.8%

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

      7. associate--l- 99.8%

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

      8. sub-neg 99.8%

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

      9. associate--l+ 99.8%

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

      10. fma-udef 99.8%

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

      11. neg-mul-1 99.8%

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

      12. count-2 99.8%

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

      13. associate-*l* 99.8%

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

      14. distribute-rgt-out-- 99.8%

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

      15. associate-*r* 99.7%

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

      16. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. fma-udef 99.7%

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

   5. Applied egg-rr 99.7%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -6.8 \cdot 10^{+141} \lor \neg \left(x.im \leq 2.7 \cdot 10^{+135}\right):\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.im \cdot -3\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re + x.im \cdot \left(x.im \cdot -3\right)\right)\\ \end{array} \]

Alternative 2: 96.6% accurate, 0.1× speedup

Math

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

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.re (+ x.im x.im))))
   (* x.re (fma 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_re * (x_46_im + x_46_im)));
	} else {
		tmp = x_46_re * fma(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_re * Float64(x_46_im + x_46_im))));
	else
		tmp = Float64(x_46_re * fma(x_46_re, x_46_re, Float64(x_46_im * Float64(x_46_im * -3.0))));
	end
	return tmp
end

Wolfram

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

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 91.1%

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

      1. *-commutative 91.1%

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

      2. *-commutative 91.1%

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

      3. *-commutative 91.1%

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

      4. distribute-lft-out 91.1%

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

   3. Simplified 91.1%

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

      1. add-log-exp 51.0%

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

      2. exp-prod 52.0%

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

   5. Applied egg-rr 52.0%

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

      1. log-pow 47.1%

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

      2. difference-of-squares 47.1%

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

      3. add-log-exp 91.1%

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

      4. associate-*l* 99.7%

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

   7. Applied egg-rr 99.7%

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

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

   1. Initial program 0.0%

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

      1. *-commutative 0.0%

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

      2. distribute-lft-out 0.0%

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

      3. associate-*l* 0.0%

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

      4. *-commutative 0.0%

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

      5. distribute-rgt-out-- 54.8%

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

      6. associate--l- 54.8%

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

      7. associate--l- 54.8%

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

      8. sub-neg 54.8%

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

      9. associate--l+ 54.8%

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

      10. fma-udef 80.6%

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

      11. neg-mul-1 80.6%

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

      12. count-2 80.6%

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

      13. associate-*l* 80.6%

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

      14. distribute-rgt-out-- 80.6%

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

      15. associate-*r* 80.6%

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

      16. metadata-eval 80.6%

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

   3. Simplified 80.6%

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

4. Final simplification 97.4%

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

Alternative 3: 96.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (((x_46_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_re * (x_46_im + x_46_im)));
	} else {
		tmp = x_46_re * (x_46_re * x_46_re);
	}
	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_re * (x_46_im + x_46_im)));
	} else {
		tmp = x_46_re * (x_46_re * x_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if ((x_46_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_re * (x_46_im + x_46_im)))
	else:
		tmp = x_46_re * (x_46_re * x_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (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_re * Float64(x_46_im + x_46_im))));
	else
		tmp = Float64(x_46_re * Float64(x_46_re * x_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (((x_46_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_re * (x_46_im + x_46_im)));
	else
		tmp = x_46_re * (x_46_re * x_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 91.1%

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

      1. *-commutative 91.1%

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

      2. *-commutative 91.1%

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

      3. *-commutative 91.1%

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

      4. distribute-lft-out 91.1%

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

   3. Simplified 91.1%

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

      1. add-log-exp 51.0%

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

      2. exp-prod 52.0%

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

   5. Applied egg-rr 52.0%

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

      1. log-pow 47.1%

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

      2. difference-of-squares 47.1%

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

      3. add-log-exp 91.1%

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

      4. associate-*l* 99.7%

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

   7. Applied egg-rr 99.7%

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

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

   1. Initial program 0.0%

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

      1. *-commutative 0.0%

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

      2. distribute-lft-out 0.0%

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

      3. associate-*l* 0.0%

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

      4. *-commutative 0.0%

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

      5. distribute-rgt-out-- 54.8%

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

      6. associate--l- 54.8%

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

      7. associate--l- 54.8%

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

      8. sub-neg 54.8%

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

      9. associate--l+ 54.8%

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

      10. fma-udef 80.6%

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

      11. neg-mul-1 80.6%

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

      12. count-2 80.6%

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

      13. associate-*l* 80.6%

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

      14. distribute-rgt-out-- 80.6%

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

      15. associate-*r* 80.6%

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

      16. metadata-eval 80.6%

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

   3. Simplified 80.6%

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

   4. Taylor expanded in x.re around inf 74.2%

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

      1. unpow2 74.2%

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

   6. Simplified 74.2%

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

4. Final simplification 96.6%

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

Alternative 4: 72.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -6.6 \lor \neg \left(x.re \leq 1.02 \cdot 10^{+26}\right) \land \left(x.re \leq 2.2 \cdot 10^{+78} \lor \neg \left(x.re \leq 4.9 \cdot 10^{+113}\right)\right):\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.im \cdot x.im\right) \cdot -3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.re -6.6)
         (and (not (<= x.re 1.02e+26))
              (or (<= x.re 2.2e+78) (not (<= x.re 4.9e+113)))))
   (* 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 <= -6.6) || (!(x_46_re <= 1.02e+26) && ((x_46_re <= 2.2e+78) || !(x_46_re <= 4.9e+113)))) {
		tmp = x_46_re * (x_46_re * x_46_re);
	} 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) :: tmp
    if ((x_46re <= (-6.6d0)) .or. (.not. (x_46re <= 1.02d+26)) .and. (x_46re <= 2.2d+78) .or. (.not. (x_46re <= 4.9d+113))) then
        tmp = x_46re * (x_46re * x_46re)
    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 tmp;
	if ((x_46_re <= -6.6) || (!(x_46_re <= 1.02e+26) && ((x_46_re <= 2.2e+78) || !(x_46_re <= 4.9e+113)))) {
		tmp = x_46_re * (x_46_re * x_46_re);
	} else {
		tmp = 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 <= -6.6) or (not (x_46_re <= 1.02e+26) and ((x_46_re <= 2.2e+78) or not (x_46_re <= 4.9e+113))):
		tmp = x_46_re * (x_46_re * x_46_re)
	else:
		tmp = x_46_re * ((x_46_im * x_46_im) * -3.0)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_re <= -6.6) || (!(x_46_re <= 1.02e+26) && ((x_46_re <= 2.2e+78) || !(x_46_re <= 4.9e+113))))
		tmp = Float64(x_46_re * Float64(x_46_re * x_46_re));
	else
		tmp = Float64(x_46_re * Float64(Float64(x_46_im * 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 <= -6.6) || (~((x_46_re <= 1.02e+26)) && ((x_46_re <= 2.2e+78) || ~((x_46_re <= 4.9e+113)))))
		tmp = x_46_re * (x_46_re * x_46_re);
	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_] := If[Or[LessEqual[x$46$re, -6.6], And[N[Not[LessEqual[x$46$re, 1.02e+26]], $MachinePrecision], Or[LessEqual[x$46$re, 2.2e+78], N[Not[LessEqual[x$46$re, 4.9e+113]], $MachinePrecision]]]], N[(x$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision], N[(x$46$re * N[(N[(x$46$im * x$46$im), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < -6.5999999999999996 or 1.0200000000000001e26 < x.re < 2.20000000000000014e78 or 4.90000000000000021e113 < x.re

   1. Initial program 73.6%

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

      1. *-commutative 73.6%

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

      2. distribute-lft-out 73.6%

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

      3. associate-*l* 73.6%

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

      4. *-commutative 73.6%

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

      5. distribute-rgt-out-- 88.0%

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

      6. associate--l- 88.0%

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

      7. associate--l- 88.0%

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

      8. sub-neg 88.0%

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

      9. associate--l+ 88.0%

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

      10. fma-udef 94.8%

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

      11. neg-mul-1 94.8%

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

      12. count-2 94.8%

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

      13. associate-*l* 94.8%

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

      14. distribute-rgt-out-- 94.8%

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

      15. associate-*r* 94.8%

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

      16. metadata-eval 94.8%

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

   3. Simplified 94.8%

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

   4. Taylor expanded in x.re around inf 83.0%

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

      1. unpow2 83.0%

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

   6. Simplified 83.0%

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

   if -6.5999999999999996 < x.re < 1.0200000000000001e26 or 2.20000000000000014e78 < x.re < 4.90000000000000021e113

   1. Initial program 85.6%

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

      1. *-commutative 85.6%

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

      2. distribute-lft-out 85.6%

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

      3. associate-*l* 85.6%

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

      4. *-commutative 85.6%

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

      5. distribute-rgt-out-- 85.6%

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

      6. associate--l- 85.6%

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

      7. associate--l- 85.6%

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

      8. sub-neg 85.6%

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

      9. associate--l+ 85.6%

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

      10. fma-udef 85.6%

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

      11. neg-mul-1 85.6%

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

      12. count-2 85.6%

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

      13. associate-*l* 85.6%

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

      14. distribute-rgt-out-- 85.6%

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

      15. associate-*r* 85.6%

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

      16. metadata-eval 85.6%

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

   3. Simplified 85.6%

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

   4. Taylor expanded in x.re around 0 72.9%

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

      1. associate-*r* 73.0%

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

      2. *-commutative 73.0%

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

      3. metadata-eval 73.0%

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

      4. distribute-rgt-out-- 73.0%

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

      5. *-commutative 73.0%

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

      6. cancel-sign-sub-inv 73.0%

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

      7. metadata-eval 73.0%

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

      8. +-commutative 73.0%

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

      9. distribute-rgt-in 72.9%

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

      10. *-commutative 72.9%

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

      11. metadata-eval 72.9%

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

      12. distribute-lft-neg-in 72.9%

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

      13. count-2 72.9%

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

      14. distribute-rgt-neg-in 72.9%

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

      15. distribute-rgt-out 72.9%

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

      16. distribute-lft-out 72.9%

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

      17. distribute-rgt-neg-in 72.9%

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

      18. count-2 72.9%

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

      19. distribute-lft-neg-in 72.9%

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

      20. metadata-eval 72.9%

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

      21. mul-1-neg 72.9%

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

      22. distribute-lft-neg-in 72.9%

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

   6. Simplified 73.0%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -6.6 \lor \neg \left(x.re \leq 1.02 \cdot 10^{+26}\right) \land \left(x.re \leq 2.2 \cdot 10^{+78} \lor \neg \left(x.re \leq 4.9 \cdot 10^{+113}\right)\right):\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.im \cdot x.im\right) \cdot -3\right)\\ \end{array} \]

Alternative 5: 79.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{if}\;x.re \leq -15.2:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.re \leq 8 \cdot 10^{+24}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.im \cdot -3\right)\\ \mathbf{elif}\;x.re \leq 2.2 \cdot 10^{+78} \lor \neg \left(x.re \leq 1.95 \cdot 10^{+108}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.im \cdot x.im\right) \cdot -3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (* x.re (* x.re x.re))))
   (if (<= x.re -15.2)
     t_0
     (if (<= x.re 8e+24)
       (* (* x.re x.im) (* x.im -3.0))
       (if (or (<= x.re 2.2e+78) (not (<= x.re 1.95e+108)))
         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_re * (x_46_re * x_46_re);
	double tmp;
	if (x_46_re <= -15.2) {
		tmp = t_0;
	} else if (x_46_re <= 8e+24) {
		tmp = (x_46_re * x_46_im) * (x_46_im * -3.0);
	} else if ((x_46_re <= 2.2e+78) || !(x_46_re <= 1.95e+108)) {
		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_46re * (x_46re * x_46re)
    if (x_46re <= (-15.2d0)) then
        tmp = t_0
    else if (x_46re <= 8d+24) then
        tmp = (x_46re * x_46im) * (x_46im * (-3.0d0))
    else if ((x_46re <= 2.2d+78) .or. (.not. (x_46re <= 1.95d+108))) 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_re * (x_46_re * x_46_re);
	double tmp;
	if (x_46_re <= -15.2) {
		tmp = t_0;
	} else if (x_46_re <= 8e+24) {
		tmp = (x_46_re * x_46_im) * (x_46_im * -3.0);
	} else if ((x_46_re <= 2.2e+78) || !(x_46_re <= 1.95e+108)) {
		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_re * (x_46_re * x_46_re)
	tmp = 0
	if x_46_re <= -15.2:
		tmp = t_0
	elif x_46_re <= 8e+24:
		tmp = (x_46_re * x_46_im) * (x_46_im * -3.0)
	elif (x_46_re <= 2.2e+78) or not (x_46_re <= 1.95e+108):
		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_re * Float64(x_46_re * x_46_re))
	tmp = 0.0
	if (x_46_re <= -15.2)
		tmp = t_0;
	elseif (x_46_re <= 8e+24)
		tmp = Float64(Float64(x_46_re * x_46_im) * Float64(x_46_im * -3.0));
	elseif ((x_46_re <= 2.2e+78) || !(x_46_re <= 1.95e+108))
		tmp = t_0;
	else
		tmp = Float64(x_46_re * Float64(Float64(x_46_im * x_46_im) * -3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	t_0 = x_46_re * (x_46_re * x_46_re);
	tmp = 0.0;
	if (x_46_re <= -15.2)
		tmp = t_0;
	elseif (x_46_re <= 8e+24)
		tmp = (x_46_re * x_46_im) * (x_46_im * -3.0);
	elseif ((x_46_re <= 2.2e+78) || ~((x_46_re <= 1.95e+108)))
		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$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$re, -15.2], t$95$0, If[LessEqual[x$46$re, 8e+24], N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x$46$re, 2.2e+78], N[Not[LessEqual[x$46$re, 1.95e+108]], $MachinePrecision]], t$95$0, N[(x$46$re * N[(N[(x$46$im * x$46$im), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -15.199999999999999 or 7.9999999999999999e24 < x.re < 2.20000000000000014e78 or 1.94999999999999992e108 < x.re

   1. Initial program 73.6%

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

      1. *-commutative 73.6%

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

      2. distribute-lft-out 73.6%

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

      3. associate-*l* 73.6%

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

      4. *-commutative 73.6%

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

      5. distribute-rgt-out-- 88.0%

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

      6. associate--l- 88.0%

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

      7. associate--l- 88.0%

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

      8. sub-neg 88.0%

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

      9. associate--l+ 88.0%

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

      10. fma-udef 94.8%

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

      11. neg-mul-1 94.8%

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

      12. count-2 94.8%

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

      13. associate-*l* 94.8%

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

      14. distribute-rgt-out-- 94.8%

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

      15. associate-*r* 94.8%

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

      16. metadata-eval 94.8%

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

   3. Simplified 94.8%

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

   4. Taylor expanded in x.re around inf 83.0%

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

      1. unpow2 83.0%

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

   6. Simplified 83.0%

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

   if -15.199999999999999 < x.re < 7.9999999999999999e24

   1. Initial program 84.6%

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

      1. *-commutative 84.6%

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

      2. distribute-lft-out 84.6%

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

      3. associate-*l* 84.6%

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

      4. *-commutative 84.6%

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

      5. distribute-rgt-out-- 84.6%

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

      6. associate--l- 84.6%

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

      7. associate--l- 84.6%

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

      8. sub-neg 84.6%

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

      9. associate--l+ 84.6%

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

      10. fma-udef 84.6%

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

      11. neg-mul-1 84.6%

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

      12. count-2 84.6%

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

      13. associate-*l* 84.6%

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

      14. distribute-rgt-out-- 84.6%

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

      15. associate-*r* 84.6%

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

      16. metadata-eval 84.6%

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

   3. Simplified 84.6%

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

      1. fma-udef 84.6%

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

   5. Applied egg-rr 84.6%

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

   6. Taylor expanded in x.re around 0 72.6%

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

      1. *-commutative 72.6%

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

      2. unpow2 72.6%

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

      3. associate-*r* 87.6%

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

      4. associate-*r* 87.6%

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

      5. *-commutative 87.6%

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

      6. *-commutative 87.6%

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

   8. Simplified 87.6%

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

   if 2.20000000000000014e78 < x.re < 1.94999999999999992e108

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. distribute-lft-out 100.0%

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

      3. associate-*l* 100.0%

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

      4. *-commutative 100.0%

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

      5. distribute-rgt-out-- 100.0%

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

      6. associate--l- 100.0%

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

      7. associate--l- 100.0%

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

      8. sub-neg 100.0%

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

      9. associate--l+ 100.0%

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

      10. fma-udef 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. Taylor expanded in x.re around 0 77.9%

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

      1. associate-*r* 77.9%

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

      2. *-commutative 77.9%

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

      3. metadata-eval 77.9%

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

      4. distribute-rgt-out-- 77.9%

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

      5. *-commutative 77.9%

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

      6. cancel-sign-sub-inv 77.9%

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

      7. metadata-eval 77.9%

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

      8. +-commutative 77.9%

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

      9. distribute-rgt-in 77.9%

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

      10. *-commutative 77.9%

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

      11. metadata-eval 77.9%

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

      12. distribute-lft-neg-in 77.9%

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

      13. count-2 77.9%

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

      14. distribute-rgt-neg-in 77.9%

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

      15. distribute-rgt-out 77.9%

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

      16. distribute-lft-out 77.9%

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

      17. distribute-rgt-neg-in 77.9%

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

      18. count-2 77.9%

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

      19. distribute-lft-neg-in 77.9%

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

      20. metadata-eval 77.9%

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

      21. mul-1-neg 77.9%

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

      22. distribute-lft-neg-in 77.9%

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

   6. Simplified 77.9%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -15.2:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{elif}\;x.re \leq 8 \cdot 10^{+24}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.im \cdot -3\right)\\ \mathbf{elif}\;x.re \leq 2.2 \cdot 10^{+78} \lor \neg \left(x.re \leq 1.95 \cdot 10^{+108}\right):\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.im \cdot x.im\right) \cdot -3\right)\\ \end{array} \]

Alternative 6: 58.6% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.1%

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

   1. *-commutative 80.1%

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

   2. distribute-lft-out 80.1%

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

   3. associate-*l* 80.1%

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

   4. *-commutative 80.1%

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

   5. distribute-rgt-out-- 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 89.9%

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

   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. Taylor expanded in x.re around inf 55.3%

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

   1. unpow2 55.3%

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

6. Simplified 55.3%

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

7. Final simplification 55.3%

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

Developer target: 86.8% 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 2023195 
(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)))