math.cube on complex, real part

Percentage Accurate: 83.6% → 96.6%

Time: 6.5s

Alternatives: 9

Speedup: 1.5×

• 43.6% 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: 83.6% 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: 96.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.im \leq 3.3 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(x.re, x.im \cdot \left(x.im \cdot -3\right), {x.re}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(-3 \cdot \left(x.im \cdot x.re\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im 3.3e+153)
   (fma x.re (* x.im (* x.im -3.0)) (pow x.re 3.0))
   (* x.im (* -3.0 (* x.im x.re)))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 3.3e+153) {
		tmp = fma(x_46_re, (x_46_im * (x_46_im * -3.0)), pow(x_46_re, 3.0));
	} else {
		tmp = x_46_im * (-3.0 * (x_46_im * x_46_re));
	}
	return tmp;
}

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= 3.3e+153)
		tmp = fma(x_46_re, Float64(x_46_im * Float64(x_46_im * -3.0)), (x_46_re ^ 3.0));
	else
		tmp = Float64(x_46_im * Float64(-3.0 * Float64(x_46_im * x_46_re)));
	end
	return tmp
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, 3.3e+153], N[(x$46$re * N[(x$46$im * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision] + N[Power[x$46$re, 3.0], $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(-3.0 * N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 3.29999999999999994e153

   1. Initial program 84.9%

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

      1. *-commutative 84.9%

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

      2. sub-neg 84.9%

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

      3. distribute-rgt-in 84.5%

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

      4. associate--l+ 84.5%

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

      5. associate-*r* 84.5%

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

      6. +-commutative 84.5%

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

      7. *-commutative 84.5%

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

      8. *-commutative 84.5%

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

      9. distribute-lft-out 84.5%

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

      10. associate-*l* 84.5%

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

      11. distribute-lft-out-- 84.5%

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

      12. fma-def 94.2%

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

   3. Simplified 94.3%

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

   if 3.29999999999999994e153 < x.im

   1. Initial program 49.8%

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

      1. *-commutative 49.8%

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

      2. distribute-lft-out 49.8%

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

      3. associate-*l* 49.8%

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

      4. *-commutative 49.8%

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

      5. distribute-rgt-out-- 49.8%

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

      6. associate--l- 49.8%

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

      7. associate--l- 49.8%

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

      8. sub-neg 49.8%

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

      9. associate--l+ 49.8%

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

      10. fma-udef 53.2%

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

      11. neg-mul-1 53.2%

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

      12. count-2 53.2%

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

      13. associate-*l* 53.2%

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

      14. distribute-rgt-out-- 53.2%

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

      15. associate-*r* 53.2%

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

      16. metadata-eval 53.2%

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

   3. Simplified 53.2%

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

   4. Taylor expanded in x.re around 0 53.2%

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

      1. associate-*r* 53.2%

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

      2. unpow2 53.2%

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

   6. Simplified 53.2%

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

      1. add-sqr-sqrt 28.2%

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

      2. pow2 28.2%

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

      3. *-commutative 28.2%

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

      4. sqrt-prod 28.2%

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

      5. sqrt-prod 34.3%

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

      6. add-sqr-sqrt 34.3%

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

      7. *-commutative 34.3%

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

   8. Applied egg-rr 34.3%

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

      1. unpow2 34.3%

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

      2. swap-sqr 28.2%

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

      3. add-sqr-sqrt 53.2%

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

      4. *-commutative 53.2%

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

      5. *-commutative 53.2%

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

      6. associate-*r* 53.2%

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

      7. associate-*r* 72.3%

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

      8. associate-*r* 72.4%

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

   10. Applied egg-rr 72.4%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 3.3 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(x.re, x.im \cdot \left(x.im \cdot -3\right), {x.re}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(-3 \cdot \left(x.im \cdot x.re\right)\right)\\ \end{array} \]

Alternative 2: 96.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.im \leq 7.5 \cdot 10^{+153}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.im \cdot -3\right) + x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(-3 \cdot \left(x.im \cdot x.re\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im 7.5e+153)
   (* x.re (+ (* x.im (* x.im -3.0)) (* x.re x.re)))
   (* x.im (* -3.0 (* x.im x.re)))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 7.5e+153) {
		tmp = x_46_re * ((x_46_im * (x_46_im * -3.0)) + (x_46_re * x_46_re));
	} else {
		tmp = x_46_im * (-3.0 * (x_46_im * x_46_re));
	}
	return tmp;
}

Fortran

NOTE: x.im should be positive before calling this function
real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46im <= 7.5d+153) then
        tmp = x_46re * ((x_46im * (x_46im * (-3.0d0))) + (x_46re * x_46re))
    else
        tmp = x_46im * ((-3.0d0) * (x_46im * x_46re))
    end if
    code = tmp
end function

Java

x.im = Math.abs(x.im);
public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 7.5e+153) {
		tmp = x_46_re * ((x_46_im * (x_46_im * -3.0)) + (x_46_re * x_46_re));
	} else {
		tmp = x_46_im * (-3.0 * (x_46_im * x_46_re));
	}
	return tmp;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= 7.5e+153:
		tmp = x_46_re * ((x_46_im * (x_46_im * -3.0)) + (x_46_re * x_46_re))
	else:
		tmp = x_46_im * (-3.0 * (x_46_im * x_46_re))
	return tmp

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= 7.5e+153)
		tmp = Float64(x_46_re * Float64(Float64(x_46_im * Float64(x_46_im * -3.0)) + Float64(x_46_re * x_46_re)));
	else
		tmp = Float64(x_46_im * Float64(-3.0 * Float64(x_46_im * x_46_re)));
	end
	return tmp
end

MATLAB

x.im = abs(x.im)
function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_im <= 7.5e+153)
		tmp = x_46_re * ((x_46_im * (x_46_im * -3.0)) + (x_46_re * x_46_re));
	else
		tmp = x_46_im * (-3.0 * (x_46_im * x_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, 7.5e+153], N[(x$46$re * N[(N[(x$46$im * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(-3.0 * N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 7.50000000000000065e153

   1. Initial program 84.9%

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

      1. *-commutative 84.9%

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

      2. distribute-lft-out 84.9%

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

      3. associate-*l* 84.9%

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

      4. *-commutative 84.9%

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

      5. distribute-rgt-out-- 94.6%

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

      6. associate--l- 94.7%

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

      7. associate--l- 94.6%

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

      8. sub-neg 94.6%

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

      9. associate--l+ 94.7%

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

      10. fma-udef 95.5%

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

      11. neg-mul-1 95.5%

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

      12. count-2 95.5%

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

      13. associate-*l* 95.5%

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

      14. distribute-rgt-out-- 95.5%

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

      15. associate-*r* 95.5%

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

      16. metadata-eval 95.5%

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

   3. Simplified 95.5%

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

      1. fma-udef 94.6%

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

   5. Applied egg-rr 94.6%

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

   if 7.50000000000000065e153 < x.im

   1. Initial program 49.8%

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

      1. *-commutative 49.8%

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

      2. distribute-lft-out 49.8%

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

      3. associate-*l* 49.8%

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

      4. *-commutative 49.8%

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

      5. distribute-rgt-out-- 49.8%

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

      6. associate--l- 49.8%

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

      7. associate--l- 49.8%

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

      8. sub-neg 49.8%

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

      9. associate--l+ 49.8%

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

      10. fma-udef 53.2%

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

      11. neg-mul-1 53.2%

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

      12. count-2 53.2%

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

      13. associate-*l* 53.2%

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

      14. distribute-rgt-out-- 53.2%

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

      15. associate-*r* 53.2%

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

      16. metadata-eval 53.2%

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

   3. Simplified 53.2%

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

   4. Taylor expanded in x.re around 0 53.2%

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

      1. associate-*r* 53.2%

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

      2. unpow2 53.2%

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

   6. Simplified 53.2%

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

      1. add-sqr-sqrt 28.2%

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

      2. pow2 28.2%

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

      3. *-commutative 28.2%

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

      4. sqrt-prod 28.2%

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

      5. sqrt-prod 34.3%

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

      6. add-sqr-sqrt 34.3%

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

      7. *-commutative 34.3%

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

   8. Applied egg-rr 34.3%

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

      1. unpow2 34.3%

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

      2. swap-sqr 28.2%

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

      3. add-sqr-sqrt 53.2%

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

      4. *-commutative 53.2%

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

      5. *-commutative 53.2%

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

      6. associate-*r* 53.2%

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

      7. associate-*r* 72.3%

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

      8. associate-*r* 72.4%

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

   10. Applied egg-rr 72.4%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 7.5 \cdot 10^{+153}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.im \cdot -3\right) + x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(-3 \cdot \left(x.im \cdot x.re\right)\right)\\ \end{array} \]

Alternative 3: 74.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.re \leq -5.3 \cdot 10^{-44} \lor \neg \left(x.re \leq 1.4 \cdot 10^{+18}\right):\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \left(x.re \cdot \left(x.im \cdot x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.re -5.3e-44) (not (<= x.re 1.4e+18)))
   (* x.re (* x.re x.re))
   (* -3.0 (* x.re (* x.im x.im)))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -5.3e-44) || !(x_46_re <= 1.4e+18)) {
		tmp = x_46_re * (x_46_re * x_46_re);
	} else {
		tmp = -3.0 * (x_46_re * (x_46_im * x_46_im));
	}
	return tmp;
}

Fortran

NOTE: x.im should be positive before calling this function
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 <= (-5.3d-44)) .or. (.not. (x_46re <= 1.4d+18))) then
        tmp = x_46re * (x_46re * x_46re)
    else
        tmp = (-3.0d0) * (x_46re * (x_46im * x_46im))
    end if
    code = tmp
end function

Java

x.im = Math.abs(x.im);
public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -5.3e-44) || !(x_46_re <= 1.4e+18)) {
		tmp = x_46_re * (x_46_re * x_46_re);
	} else {
		tmp = -3.0 * (x_46_re * (x_46_im * x_46_im));
	}
	return tmp;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_re <= -5.3e-44) or not (x_46_re <= 1.4e+18):
		tmp = x_46_re * (x_46_re * x_46_re)
	else:
		tmp = -3.0 * (x_46_re * (x_46_im * x_46_im))
	return tmp

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_re <= -5.3e-44) || !(x_46_re <= 1.4e+18))
		tmp = Float64(x_46_re * Float64(x_46_re * x_46_re));
	else
		tmp = Float64(-3.0 * Float64(x_46_re * Float64(x_46_im * x_46_im)));
	end
	return tmp
end

MATLAB

x.im = abs(x.im)
function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_re <= -5.3e-44) || ~((x_46_re <= 1.4e+18)))
		tmp = x_46_re * (x_46_re * x_46_re);
	else
		tmp = -3.0 * (x_46_re * (x_46_im * x_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$re, -5.3e-44], N[Not[LessEqual[x$46$re, 1.4e+18]], $MachinePrecision]], N[(x$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision], N[(-3.0 * N[(x$46$re * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < -5.29999999999999971e-44 or 1.4e18 < x.re

   1. Initial program 72.5%

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

      1. *-commutative 72.5%

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

      2. distribute-lft-out 72.5%

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

      3. associate-*l* 72.5%

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

      4. *-commutative 72.5%

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

      5. distribute-rgt-out-- 89.7%

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

      6. associate--l- 89.7%

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

      7. associate--l- 89.7%

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

      8. sub-neg 89.7%

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

      9. associate--l+ 89.7%

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

      10. fma-udef 92.0%

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

      11. neg-mul-1 92.0%

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

      12. count-2 92.0%

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

      13. associate-*l* 92.0%

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

      14. distribute-rgt-out-- 92.0%

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

      15. associate-*r* 92.0%

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

      16. metadata-eval 92.0%

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

   3. Simplified 92.0%

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

   4. Taylor expanded in x.re around inf 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 -5.29999999999999971e-44 < x.re < 1.4e18

   1. Initial program 89.4%

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

      1. *-commutative 89.4%

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

      2. distribute-lft-out 89.4%

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

      3. associate-*l* 89.4%

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

      4. *-commutative 89.4%

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

      5. distribute-rgt-out-- 89.4%

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

      6. associate--l- 89.5%

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

      7. associate--l- 89.4%

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

      8. sub-neg 89.4%

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

      9. associate--l+ 89.5%

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

      10. fma-udef 89.5%

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

      11. neg-mul-1 89.5%

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

      12. count-2 89.5%

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

      13. associate-*l* 89.5%

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

      14. distribute-rgt-out-- 89.5%

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

      15. associate-*r* 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 89.4%

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

   5. Applied egg-rr 89.4%

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

   6. Taylor expanded in x.re around 0 75.4%

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

      1. unpow2 75.4%

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

   8. Simplified 75.4%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -5.3 \cdot 10^{-44} \lor \neg \left(x.re \leq 1.4 \cdot 10^{+18}\right):\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \left(x.re \cdot \left(x.im \cdot x.im\right)\right)\\ \end{array} \]

Alternative 4: 74.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.re \leq -5 \cdot 10^{-44} \lor \neg \left(x.re \leq 3.6 \cdot 10^{+17}\right):\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(-3 \cdot \left(x.im \cdot x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.re -5e-44) (not (<= x.re 3.6e+17)))
   (* x.re (* x.re x.re))
   (* x.re (* -3.0 (* x.im x.im)))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -5e-44) || !(x_46_re <= 3.6e+17)) {
		tmp = x_46_re * (x_46_re * x_46_re);
	} else {
		tmp = x_46_re * (-3.0 * (x_46_im * x_46_im));
	}
	return tmp;
}

Fortran

NOTE: x.im should be positive before calling this function
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 <= (-5d-44)) .or. (.not. (x_46re <= 3.6d+17))) then
        tmp = x_46re * (x_46re * x_46re)
    else
        tmp = x_46re * ((-3.0d0) * (x_46im * x_46im))
    end if
    code = tmp
end function

Java

x.im = Math.abs(x.im);
public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -5e-44) || !(x_46_re <= 3.6e+17)) {
		tmp = x_46_re * (x_46_re * x_46_re);
	} else {
		tmp = x_46_re * (-3.0 * (x_46_im * x_46_im));
	}
	return tmp;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_re <= -5e-44) or not (x_46_re <= 3.6e+17):
		tmp = x_46_re * (x_46_re * x_46_re)
	else:
		tmp = x_46_re * (-3.0 * (x_46_im * x_46_im))
	return tmp

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_re <= -5e-44) || !(x_46_re <= 3.6e+17))
		tmp = Float64(x_46_re * Float64(x_46_re * x_46_re));
	else
		tmp = Float64(x_46_re * Float64(-3.0 * Float64(x_46_im * x_46_im)));
	end
	return tmp
end

MATLAB

x.im = abs(x.im)
function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_re <= -5e-44) || ~((x_46_re <= 3.6e+17)))
		tmp = x_46_re * (x_46_re * x_46_re);
	else
		tmp = x_46_re * (-3.0 * (x_46_im * x_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$re, -5e-44], N[Not[LessEqual[x$46$re, 3.6e+17]], $MachinePrecision]], N[(x$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision], N[(x$46$re * N[(-3.0 * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < -5.00000000000000039e-44 or 3.6e17 < x.re

   1. Initial program 72.5%

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

      1. *-commutative 72.5%

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

      2. distribute-lft-out 72.5%

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

      3. associate-*l* 72.5%

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

      4. *-commutative 72.5%

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

      5. distribute-rgt-out-- 89.7%

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

      6. associate--l- 89.7%

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

      7. associate--l- 89.7%

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

      8. sub-neg 89.7%

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

      9. associate--l+ 89.7%

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

      10. fma-udef 92.0%

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

      11. neg-mul-1 92.0%

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

      12. count-2 92.0%

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

      13. associate-*l* 92.0%

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

      14. distribute-rgt-out-- 92.0%

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

      15. associate-*r* 92.0%

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

      16. metadata-eval 92.0%

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

   3. Simplified 92.0%

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

   4. Taylor expanded in x.re around inf 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 -5.00000000000000039e-44 < x.re < 3.6e17

   1. Initial program 89.4%

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

      1. *-commutative 89.4%

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

      2. distribute-lft-out 89.4%

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

      3. associate-*l* 89.4%

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

      4. *-commutative 89.4%

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

      5. distribute-rgt-out-- 89.4%

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

      6. associate--l- 89.5%

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

      7. associate--l- 89.4%

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

      8. sub-neg 89.4%

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

      9. associate--l+ 89.5%

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

      10. fma-udef 89.5%

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

      11. neg-mul-1 89.5%

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

      12. count-2 89.5%

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

      13. associate-*l* 89.5%

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

      14. distribute-rgt-out-- 89.5%

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

      15. associate-*r* 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. Taylor expanded in x.re around 0 75.4%

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

      1. associate-*r* 75.5%

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

      2. *-commutative 75.5%

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

      3. metadata-eval 75.5%

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

      4. distribute-rgt-out-- 75.5%

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

      5. *-commutative 75.5%

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

      6. cancel-sign-sub-inv 75.5%

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

      7. metadata-eval 75.5%

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

      8. +-commutative 75.5%

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

      9. distribute-rgt-in 75.4%

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

   6. Simplified 75.4%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -5 \cdot 10^{-44} \lor \neg \left(x.re \leq 3.6 \cdot 10^{+17}\right):\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(-3 \cdot \left(x.im \cdot x.im\right)\right)\\ \end{array} \]

Alternative 5: 74.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.re \leq -4.9 \cdot 10^{-44} \lor \neg \left(x.re \leq 1.12 \cdot 10^{+17}\right):\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot -3\right) \cdot \left(x.im \cdot x.im\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.re -4.9e-44) (not (<= x.re 1.12e+17)))
   (* x.re (* x.re x.re))
   (* (* x.re -3.0) (* x.im x.im))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -4.9e-44) || !(x_46_re <= 1.12e+17)) {
		tmp = x_46_re * (x_46_re * x_46_re);
	} else {
		tmp = (x_46_re * -3.0) * (x_46_im * x_46_im);
	}
	return tmp;
}

Fortran

NOTE: x.im should be positive before calling this function
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 <= (-4.9d-44)) .or. (.not. (x_46re <= 1.12d+17))) then
        tmp = x_46re * (x_46re * x_46re)
    else
        tmp = (x_46re * (-3.0d0)) * (x_46im * x_46im)
    end if
    code = tmp
end function

Java

x.im = Math.abs(x.im);
public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -4.9e-44) || !(x_46_re <= 1.12e+17)) {
		tmp = x_46_re * (x_46_re * x_46_re);
	} else {
		tmp = (x_46_re * -3.0) * (x_46_im * x_46_im);
	}
	return tmp;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_re <= -4.9e-44) or not (x_46_re <= 1.12e+17):
		tmp = x_46_re * (x_46_re * x_46_re)
	else:
		tmp = (x_46_re * -3.0) * (x_46_im * x_46_im)
	return tmp

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_re <= -4.9e-44) || !(x_46_re <= 1.12e+17))
		tmp = Float64(x_46_re * Float64(x_46_re * x_46_re));
	else
		tmp = Float64(Float64(x_46_re * -3.0) * Float64(x_46_im * x_46_im));
	end
	return tmp
end

MATLAB

x.im = abs(x.im)
function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_re <= -4.9e-44) || ~((x_46_re <= 1.12e+17)))
		tmp = x_46_re * (x_46_re * x_46_re);
	else
		tmp = (x_46_re * -3.0) * (x_46_im * x_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$re, -4.9e-44], N[Not[LessEqual[x$46$re, 1.12e+17]], $MachinePrecision]], N[(x$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * -3.0), $MachinePrecision] * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < -4.9000000000000003e-44 or 1.12e17 < x.re

   1. Initial program 72.5%

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

      1. *-commutative 72.5%

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

      2. distribute-lft-out 72.5%

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

      3. associate-*l* 72.5%

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

      4. *-commutative 72.5%

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

      5. distribute-rgt-out-- 89.7%

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

      6. associate--l- 89.7%

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

      7. associate--l- 89.7%

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

      8. sub-neg 89.7%

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

      9. associate--l+ 89.7%

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

      10. fma-udef 92.0%

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

      11. neg-mul-1 92.0%

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

      12. count-2 92.0%

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

      13. associate-*l* 92.0%

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

      14. distribute-rgt-out-- 92.0%

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

      15. associate-*r* 92.0%

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

      16. metadata-eval 92.0%

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

   3. Simplified 92.0%

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

   4. Taylor expanded in x.re around inf 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 -4.9000000000000003e-44 < x.re < 1.12e17

   1. Initial program 89.4%

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

      1. *-commutative 89.4%

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

      2. distribute-lft-out 89.4%

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

      3. associate-*l* 89.4%

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

      4. *-commutative 89.4%

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

      5. distribute-rgt-out-- 89.4%

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

      6. associate--l- 89.5%

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

      7. associate--l- 89.4%

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

      8. sub-neg 89.4%

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

      9. associate--l+ 89.5%

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

      10. fma-udef 89.5%

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

      11. neg-mul-1 89.5%

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

      12. count-2 89.5%

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

      13. associate-*l* 89.5%

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

      14. distribute-rgt-out-- 89.5%

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

      15. associate-*r* 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. Taylor expanded in x.re around 0 75.4%

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

      1. associate-*r* 75.5%

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

      2. unpow2 75.5%

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

   6. Simplified 75.5%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -4.9 \cdot 10^{-44} \lor \neg \left(x.re \leq 1.12 \cdot 10^{+17}\right):\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot -3\right) \cdot \left(x.im \cdot x.im\right)\\ \end{array} \]

Alternative 6: 79.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.re \leq -3.65 \cdot 10^{-44} \lor \neg \left(x.re \leq 3.6 \cdot 10^{+17}\right):\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(-3 \cdot \left(x.im \cdot x.re\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.re -3.65e-44) (not (<= x.re 3.6e+17)))
   (* x.re (* x.re x.re))
   (* x.im (* -3.0 (* x.im x.re)))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -3.65e-44) || !(x_46_re <= 3.6e+17)) {
		tmp = x_46_re * (x_46_re * x_46_re);
	} else {
		tmp = x_46_im * (-3.0 * (x_46_im * x_46_re));
	}
	return tmp;
}

Fortran

NOTE: x.im should be positive before calling this function
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 <= (-3.65d-44)) .or. (.not. (x_46re <= 3.6d+17))) then
        tmp = x_46re * (x_46re * x_46re)
    else
        tmp = x_46im * ((-3.0d0) * (x_46im * x_46re))
    end if
    code = tmp
end function

Java

x.im = Math.abs(x.im);
public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -3.65e-44) || !(x_46_re <= 3.6e+17)) {
		tmp = x_46_re * (x_46_re * x_46_re);
	} else {
		tmp = x_46_im * (-3.0 * (x_46_im * x_46_re));
	}
	return tmp;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_re <= -3.65e-44) or not (x_46_re <= 3.6e+17):
		tmp = x_46_re * (x_46_re * x_46_re)
	else:
		tmp = x_46_im * (-3.0 * (x_46_im * x_46_re))
	return tmp

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_re <= -3.65e-44) || !(x_46_re <= 3.6e+17))
		tmp = Float64(x_46_re * Float64(x_46_re * x_46_re));
	else
		tmp = Float64(x_46_im * Float64(-3.0 * Float64(x_46_im * x_46_re)));
	end
	return tmp
end

MATLAB

x.im = abs(x.im)
function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_re <= -3.65e-44) || ~((x_46_re <= 3.6e+17)))
		tmp = x_46_re * (x_46_re * x_46_re);
	else
		tmp = x_46_im * (-3.0 * (x_46_im * x_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$re, -3.65e-44], N[Not[LessEqual[x$46$re, 3.6e+17]], $MachinePrecision]], N[(x$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(-3.0 * N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < -3.64999999999999993e-44 or 3.6e17 < x.re

   1. Initial program 72.5%

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

      1. *-commutative 72.5%

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

      2. distribute-lft-out 72.5%

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

      3. associate-*l* 72.5%

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

      4. *-commutative 72.5%

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

      5. distribute-rgt-out-- 89.7%

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

      6. associate--l- 89.7%

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

      7. associate--l- 89.7%

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

      8. sub-neg 89.7%

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

      9. associate--l+ 89.7%

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

      10. fma-udef 92.0%

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

      11. neg-mul-1 92.0%

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

      12. count-2 92.0%

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

      13. associate-*l* 92.0%

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

      14. distribute-rgt-out-- 92.0%

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

      15. associate-*r* 92.0%

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

      16. metadata-eval 92.0%

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

   3. Simplified 92.0%

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

   4. Taylor expanded in x.re around inf 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 -3.64999999999999993e-44 < x.re < 3.6e17

   1. Initial program 89.4%

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

      1. *-commutative 89.4%

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

      2. distribute-lft-out 89.4%

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

      3. associate-*l* 89.4%

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

      4. *-commutative 89.4%

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

      5. distribute-rgt-out-- 89.4%

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

      6. associate--l- 89.5%

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

      7. associate--l- 89.4%

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

      8. sub-neg 89.4%

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

      9. associate--l+ 89.5%

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

      10. fma-udef 89.5%

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

      11. neg-mul-1 89.5%

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

      12. count-2 89.5%

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

      13. associate-*l* 89.5%

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

      14. distribute-rgt-out-- 89.5%

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

      15. associate-*r* 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. Taylor expanded in x.re around 0 75.4%

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

      1. associate-*r* 75.5%

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

      2. unpow2 75.5%

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

   6. Simplified 75.5%

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

      1. add-sqr-sqrt 45.7%

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

      2. pow2 45.7%

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

      3. *-commutative 45.7%

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

      4. sqrt-prod 35.9%

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

      5. sqrt-prod 16.4%

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

      6. add-sqr-sqrt 38.0%

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

      7. *-commutative 38.0%

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

   8. Applied egg-rr 38.0%

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

      1. unpow2 38.0%

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

      2. swap-sqr 35.8%

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

      3. add-sqr-sqrt 75.5%

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

      4. *-commutative 75.5%

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

      5. *-commutative 75.5%

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

      6. associate-*r* 75.4%

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

      7. associate-*r* 85.6%

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

      8. associate-*r* 85.6%

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

   10. Applied egg-rr 85.6%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -3.65 \cdot 10^{-44} \lor \neg \left(x.re \leq 3.6 \cdot 10^{+17}\right):\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(-3 \cdot \left(x.im \cdot x.re\right)\right)\\ \end{array} \]

Alternative 7: 80.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.re \leq -5.6 \cdot 10^{-44} \lor \neg \left(x.re \leq 4.7 \cdot 10^{+16}\right):\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im \cdot x.re\right) \cdot \left(x.im \cdot -3\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.re -5.6e-44) (not (<= x.re 4.7e+16)))
   (* x.re (* x.re x.re))
   (* (* x.im x.re) (* x.im -3.0))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -5.6e-44) || !(x_46_re <= 4.7e+16)) {
		tmp = x_46_re * (x_46_re * x_46_re);
	} else {
		tmp = (x_46_im * x_46_re) * (x_46_im * -3.0);
	}
	return tmp;
}

Fortran

NOTE: x.im should be positive before calling this function
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 <= (-5.6d-44)) .or. (.not. (x_46re <= 4.7d+16))) then
        tmp = x_46re * (x_46re * x_46re)
    else
        tmp = (x_46im * x_46re) * (x_46im * (-3.0d0))
    end if
    code = tmp
end function

Java

x.im = Math.abs(x.im);
public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -5.6e-44) || !(x_46_re <= 4.7e+16)) {
		tmp = x_46_re * (x_46_re * x_46_re);
	} else {
		tmp = (x_46_im * x_46_re) * (x_46_im * -3.0);
	}
	return tmp;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_re <= -5.6e-44) or not (x_46_re <= 4.7e+16):
		tmp = x_46_re * (x_46_re * x_46_re)
	else:
		tmp = (x_46_im * x_46_re) * (x_46_im * -3.0)
	return tmp

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_re <= -5.6e-44) || !(x_46_re <= 4.7e+16))
		tmp = Float64(x_46_re * Float64(x_46_re * x_46_re));
	else
		tmp = Float64(Float64(x_46_im * x_46_re) * Float64(x_46_im * -3.0));
	end
	return tmp
end

MATLAB

x.im = abs(x.im)
function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_re <= -5.6e-44) || ~((x_46_re <= 4.7e+16)))
		tmp = x_46_re * (x_46_re * x_46_re);
	else
		tmp = (x_46_im * x_46_re) * (x_46_im * -3.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$re, -5.6e-44], N[Not[LessEqual[x$46$re, 4.7e+16]], $MachinePrecision]], N[(x$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im * x$46$re), $MachinePrecision] * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < -5.6e-44 or 4.7e16 < x.re

   1. Initial program 72.5%

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

      1. *-commutative 72.5%

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

      2. distribute-lft-out 72.5%

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

      3. associate-*l* 72.5%

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

      4. *-commutative 72.5%

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

      5. distribute-rgt-out-- 89.7%

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

      6. associate--l- 89.7%

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

      7. associate--l- 89.7%

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

      8. sub-neg 89.7%

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

      9. associate--l+ 89.7%

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

      10. fma-udef 92.0%

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

      11. neg-mul-1 92.0%

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

      12. count-2 92.0%

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

      13. associate-*l* 92.0%

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

      14. distribute-rgt-out-- 92.0%

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

      15. associate-*r* 92.0%

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

      16. metadata-eval 92.0%

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

   3. Simplified 92.0%

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

   4. Taylor expanded in x.re around inf 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 -5.6e-44 < x.re < 4.7e16

   1. Initial program 89.4%

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

      1. *-commutative 89.4%

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

      2. distribute-lft-out 89.4%

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

      3. associate-*l* 89.4%

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

      4. *-commutative 89.4%

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

      5. distribute-rgt-out-- 89.4%

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

      6. associate--l- 89.5%

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

      7. associate--l- 89.4%

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

      8. sub-neg 89.4%

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

      9. associate--l+ 89.5%

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

      10. fma-udef 89.5%

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

      11. neg-mul-1 89.5%

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

      12. count-2 89.5%

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

      13. associate-*l* 89.5%

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

      14. distribute-rgt-out-- 89.5%

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

      15. associate-*r* 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. Taylor expanded in x.re around 0 75.4%

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

      1. associate-*r* 75.5%

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

      2. unpow2 75.5%

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

   6. Simplified 75.5%

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

      1. add-sqr-sqrt 45.7%

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

      2. pow2 45.7%

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

      3. *-commutative 45.7%

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

      4. sqrt-prod 35.9%

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

      5. sqrt-prod 16.4%

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

      6. add-sqr-sqrt 38.0%

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

      7. *-commutative 38.0%

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

   8. Applied egg-rr 38.0%

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

      1. unpow2 38.0%

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

      2. swap-sqr 35.8%

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

      3. add-sqr-sqrt 75.5%

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

      4. associate-*l* 75.4%

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

      5. *-commutative 75.4%

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

      6. associate-*r* 85.6%

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

      7. associate-*l* 85.7%

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

      8. *-commutative 85.7%

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

   10. Applied egg-rr 85.7%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -5.6 \cdot 10^{-44} \lor \neg \left(x.re \leq 4.7 \cdot 10^{+16}\right):\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im \cdot x.re\right) \cdot \left(x.im \cdot -3\right)\\ \end{array} \]

Alternative 8: 71.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} x.im = |x.im|\\ \\ \begin{array}{l} \mathbf{if}\;x.im \leq 2.45 \cdot 10^{+178}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im 2.45e+178) (* x.re (* x.re x.re)) (* x.re (* x.im (- x.im)))))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 2.45e+178) {
		tmp = x_46_re * (x_46_re * x_46_re);
	} else {
		tmp = x_46_re * (x_46_im * -x_46_im);
	}
	return tmp;
}

Fortran

NOTE: x.im should be positive before calling this function
real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46im <= 2.45d+178) then
        tmp = x_46re * (x_46re * x_46re)
    else
        tmp = x_46re * (x_46im * -x_46im)
    end if
    code = tmp
end function

Java

x.im = Math.abs(x.im);
public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 2.45e+178) {
		tmp = x_46_re * (x_46_re * x_46_re);
	} else {
		tmp = x_46_re * (x_46_im * -x_46_im);
	}
	return tmp;
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= 2.45e+178:
		tmp = x_46_re * (x_46_re * x_46_re)
	else:
		tmp = x_46_re * (x_46_im * -x_46_im)
	return tmp

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= 2.45e+178)
		tmp = Float64(x_46_re * Float64(x_46_re * x_46_re));
	else
		tmp = Float64(x_46_re * Float64(x_46_im * Float64(-x_46_im)));
	end
	return tmp
end

MATLAB

x.im = abs(x.im)
function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_im <= 2.45e+178)
		tmp = x_46_re * (x_46_re * x_46_re);
	else
		tmp = x_46_re * (x_46_im * -x_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, 2.45e+178], N[(x$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision], N[(x$46$re * N[(x$46$im * (-x$46$im)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 2.4500000000000001e178

   1. Initial program 82.8%

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

      1. *-commutative 82.8%

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

      2. distribute-lft-out 82.8%

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

      3. associate-*l* 82.8%

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

      4. *-commutative 82.8%

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

      5. distribute-rgt-out-- 92.3%

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

      6. associate--l- 92.3%

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

      7. associate--l- 92.3%

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

      8. sub-neg 92.3%

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

      9. associate--l+ 92.3%

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

      10. fma-udef 93.1%

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

      11. neg-mul-1 93.1%

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

      12. count-2 93.1%

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

      13. associate-*l* 93.1%

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

      14. distribute-rgt-out-- 93.1%

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

      15. associate-*r* 93.1%

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

      16. metadata-eval 93.1%

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

   3. Simplified 93.1%

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

   4. Taylor expanded in x.re around inf 66.6%

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

      1. unpow2 66.6%

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

   6. Simplified 66.6%

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

   if 2.4500000000000001e178 < x.im

   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. fma-neg 62.3%

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

      3. distribute-lft-neg-in 62.3%

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

      4. *-commutative 62.3%

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

      5. *-commutative 62.3%

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

      6. count-2 62.3%

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

      7. distribute-lft-neg-in 62.3%

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

      8. metadata-eval 62.3%

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

      9. *-commutative 62.3%

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

   3. Simplified 62.3%

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

   4. Taylor expanded in x.re around 0 66.6%

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

      1. neg-mul-1 66.6%

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

      2. unpow2 66.6%

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

      3. distribute-rgt-neg-in 66.6%

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

   6. Simplified 66.6%

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

      1. fma-udef 66.6%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 0.0%

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

      4. swap-sqr 0.0%

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

      5. sqr-neg 0.0%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 0.0%

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

      8. associate-*r* 0.0%

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

   8. Applied egg-rr 0.0%

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

      1. associate-*r* 3.1%

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

      2. *-commutative 3.1%

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

      3. distribute-lft-out 68.3%

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

   10. Simplified 68.3%

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

   11. Taylor expanded in x.re around 0 68.3%

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

       1. *-commutative 68.3%

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

       2. distribute-lft1-in 68.3%

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

       3. metadata-eval 68.3%

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

       4. *-commutative 68.3%

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

       5. associate-*r* 66.6%

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

       6. *-commutative 66.6%

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

       7. neg-mul-1 66.6%

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

   13. Simplified 66.6%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 2.45 \cdot 10^{+178}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \end{array} \]

Alternative 9: 59.0% accurate, 3.8× speedup

Math

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

FPCore

NOTE: x.im should be positive before calling this function
(FPCore (x.re x.im) :precision binary64 (* x.re (* x.re x.re)))

C

x.im = abs(x.im);
double code(double x_46_re, double x_46_im) {
	return x_46_re * (x_46_re * x_46_re);
}

Fortran

NOTE: x.im should be positive before calling this function
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

x.im = Math.abs(x.im);
public static double code(double x_46_re, double x_46_im) {
	return x_46_re * (x_46_re * x_46_re);
}

Python

x.im = abs(x.im)
def code(x_46_re, x_46_im):
	return x_46_re * (x_46_re * x_46_re)

Julia

x.im = abs(x.im)
function code(x_46_re, x_46_im)
	return Float64(x_46_re * Float64(x_46_re * x_46_re))
end

MATLAB

x.im = abs(x.im)
function tmp = code(x_46_re, x_46_im)
	tmp = x_46_re * (x_46_re * x_46_re);
end

Wolfram

NOTE: x.im should be positive before calling this function
code[x$46$re_, x$46$im_] := N[(x$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.9%

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

   1. *-commutative 80.9%

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

   2. distribute-lft-out 80.9%

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

   3. associate-*l* 80.9%

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

   4. *-commutative 80.9%

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

   5. distribute-rgt-out-- 89.6%

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

   6. associate--l- 89.6%

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

   7. associate--l- 89.6%

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

   8. sub-neg 89.6%

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

   9. associate--l+ 89.6%

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

   10. fma-udef 90.7%

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

   11. neg-mul-1 90.7%

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

   12. count-2 90.7%

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

   13. associate-*l* 90.7%

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

   14. distribute-rgt-out-- 90.7%

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

   15. associate-*r* 90.7%

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

   16. metadata-eval 90.7%

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

3. Simplified 90.7%

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

4. Taylor expanded in x.re around inf 62.3%

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

   1. unpow2 62.3%

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

6. Simplified 62.3%

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

7. Final simplification 62.3%

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

Developer target: 87.1% 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 2023189 
(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)))