math.cube on complex, real part

Percentage Accurate: 82.5% → 95.9%

Time: 5.0s

Alternatives: 5

Speedup: 1.3×

• 44.7% 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: 82.5% 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: 95.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -2.05 \cdot 10^{+133}:\\ \;\;\;\;\left(x.im \cdot \left(x.im \cdot x.re\right)\right) \cdot -3\\ \mathbf{elif}\;x.im \leq 7.6 \cdot 10^{+153}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re + x.im \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(x.im \cdot x.re\right) \cdot -3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im -2.05e+133)
   (* (* x.im (* x.im x.re)) -3.0)
   (if (<= x.im 7.6e+153)
     (* x.re (+ (* x.re x.re) (* x.im (* x.im -3.0))))
     (* x.im (* (* x.im x.re) -3.0)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= -2.05e+133) {
		tmp = (x_46_im * (x_46_im * x_46_re)) * -3.0;
	} else if (x_46_im <= 7.6e+153) {
		tmp = x_46_re * ((x_46_re * x_46_re) + (x_46_im * (x_46_im * -3.0)));
	} else {
		tmp = x_46_im * ((x_46_im * x_46_re) * -3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46im <= (-2.05d+133)) then
        tmp = (x_46im * (x_46im * x_46re)) * (-3.0d0)
    else if (x_46im <= 7.6d+153) then
        tmp = x_46re * ((x_46re * x_46re) + (x_46im * (x_46im * (-3.0d0))))
    else
        tmp = x_46im * ((x_46im * x_46re) * (-3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= -2.05e+133) {
		tmp = (x_46_im * (x_46_im * x_46_re)) * -3.0;
	} else if (x_46_im <= 7.6e+153) {
		tmp = x_46_re * ((x_46_re * x_46_re) + (x_46_im * (x_46_im * -3.0)));
	} else {
		tmp = x_46_im * ((x_46_im * x_46_re) * -3.0);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= -2.05e+133:
		tmp = (x_46_im * (x_46_im * x_46_re)) * -3.0
	elif x_46_im <= 7.6e+153:
		tmp = x_46_re * ((x_46_re * x_46_re) + (x_46_im * (x_46_im * -3.0)))
	else:
		tmp = x_46_im * ((x_46_im * x_46_re) * -3.0)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= -2.05e+133)
		tmp = Float64(Float64(x_46_im * Float64(x_46_im * x_46_re)) * -3.0);
	elseif (x_46_im <= 7.6e+153)
		tmp = Float64(x_46_re * Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * Float64(x_46_im * -3.0))));
	else
		tmp = Float64(x_46_im * Float64(Float64(x_46_im * x_46_re) * -3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_im <= -2.05e+133)
		tmp = (x_46_im * (x_46_im * x_46_re)) * -3.0;
	elseif (x_46_im <= 7.6e+153)
		tmp = x_46_re * ((x_46_re * x_46_re) + (x_46_im * (x_46_im * -3.0)));
	else
		tmp = x_46_im * ((x_46_im * x_46_re) * -3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, -2.05e+133], N[(N[(x$46$im * N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * -3.0), $MachinePrecision], If[LessEqual[x$46$im, 7.6e+153], N[(x$46$re * N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(N[(x$46$im * x$46$re), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -2.05000000000000002e133

   1. Initial program 64.3%

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

      1. *-commutative 64.3%

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

      2. distribute-lft-out 64.3%

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

      3. associate-*l* 64.3%

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

      4. *-commutative 64.3%

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

      5. distribute-rgt-out-- 64.2%

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

      6. associate--l- 64.2%

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

      7. associate--l- 64.2%

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

      8. sub-neg 64.2%

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

      9. associate--l+ 64.2%

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

      10. fma-udef 73.5%

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

      11. neg-mul-1 73.5%

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

      12. count-2 73.5%

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

      13. associate-*l* 73.5%

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

      14. distribute-rgt-out-- 73.5%

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

      15. associate-*r* 73.5%

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

      16. metadata-eval 73.5%

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

   3. Simplified 73.5%

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

   4. Taylor expanded in x.re around 0 73.6%

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

      1. associate-*r* 73.5%

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

      2. *-commutative 73.5%

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

      3. *-commutative 73.5%

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

      4. metadata-eval 73.5%

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

      5. distribute-rgt-out 73.5%

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

      6. distribute-lft-in 73.6%

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

      7. metadata-eval 73.6%

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

      8. distribute-lft-neg-in 73.6%

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

      9. count-2 73.6%

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

      10. distribute-rgt-neg-in 73.6%

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

      11. distribute-rgt-out 73.6%

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

      12. distribute-lft-out 73.6%

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

      13. distribute-rgt-neg-in 73.6%

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

      14. count-2 73.6%

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

      15. distribute-lft-neg-in 73.6%

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

      16. metadata-eval 73.6%

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

      17. mul-1-neg 73.6%

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

      18. distribute-rgt-neg-in 73.6%

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

      19. distribute-lft-neg-in 73.6%

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

      20. unpow2 73.6%

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

      21. distribute-rgt-neg-out 73.6%

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

      22. *-commutative 73.6%

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

   6. Simplified 73.5%

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

      1. add-sqr-sqrt 35.8%

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

      2. pow2 35.8%

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

      3. associate-*r* 35.8%

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

      4. sqrt-prod 35.8%

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

      5. sqrt-prod 0.0%

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

      6. add-sqr-sqrt 46.4%

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

   8. Applied egg-rr 46.4%

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

      1. unpow2 46.4%

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

      2. swap-sqr 35.7%

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

      3. add-sqr-sqrt 73.5%

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

      4. *-commutative 73.5%

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

      5. associate-*l* 73.6%

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

      6. *-commutative 73.6%

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

      7. *-commutative 73.6%

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

      8. associate-*l* 93.0%

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

   10. Applied egg-rr 93.0%

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

   if -2.05000000000000002e133 < x.im < 7.59999999999999933e153

   1. Initial program 91.9%

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

      1. *-commutative 91.9%

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

      2. distribute-lft-out 91.9%

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

      3. associate-*l* 91.9%

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

      4. *-commutative 91.9%

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

      5. distribute-rgt-out-- 99.8%

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

      6. associate--l- 99.8%

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

      7. associate--l- 99.8%

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

      8. sub-neg 99.8%

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

      9. associate--l+ 99.8%

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

      10. fma-udef 99.8%

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

      11. neg-mul-1 99.8%

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

      12. count-2 99.8%

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

      13. associate-*l* 99.8%

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

      14. distribute-rgt-out-- 99.8%

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

      15. associate-*r* 99.8%

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

      16. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. fma-udef 99.8%

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

   5. Applied egg-rr 99.8%

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

   if 7.59999999999999933e153 < x.im

   1. Initial program 53.7%

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

      1. *-commutative 53.7%

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

      2. distribute-lft-out 53.7%

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

      3. associate-*l* 53.7%

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

      4. *-commutative 53.7%

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

      5. distribute-rgt-out-- 53.7%

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

      6. associate--l- 53.7%

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

      7. associate--l- 53.7%

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

      8. sub-neg 53.7%

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

      9. associate--l+ 53.7%

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

      10. fma-udef 66.8%

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

      11. neg-mul-1 66.8%

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

      12. count-2 66.8%

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

      13. associate-*l* 66.8%

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

      14. distribute-rgt-out-- 66.8%

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

      15. associate-*r* 66.8%

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

      16. metadata-eval 66.8%

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

   3. Simplified 66.8%

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

   4. Taylor expanded in x.re around 0 66.8%

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

      1. associate-*r* 66.8%

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

      2. *-commutative 66.8%

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

      3. *-commutative 66.8%

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

      4. metadata-eval 66.8%

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

      5. distribute-rgt-out 66.8%

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

      6. distribute-lft-in 66.8%

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

      7. metadata-eval 66.8%

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

      8. distribute-lft-neg-in 66.8%

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

      9. count-2 66.8%

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

      10. distribute-rgt-neg-in 66.8%

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

      11. distribute-rgt-out 66.8%

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

      12. distribute-lft-out 66.8%

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

      13. distribute-rgt-neg-in 66.8%

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

      14. count-2 66.8%

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

      15. distribute-lft-neg-in 66.8%

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

      16. metadata-eval 66.8%

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

      17. mul-1-neg 66.8%

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

      18. distribute-rgt-neg-in 66.8%

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

      19. distribute-lft-neg-in 66.8%

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

      20. unpow2 66.8%

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

      21. distribute-rgt-neg-out 66.8%

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

      22. *-commutative 66.8%

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

   6. Simplified 66.8%

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

      1. add-sqr-sqrt 22.5%

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

      2. pow2 22.5%

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

      3. associate-*r* 22.5%

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

      4. sqrt-prod 22.5%

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

      5. sqrt-prod 34.7%

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

      6. add-sqr-sqrt 34.7%

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

   8. Applied egg-rr 34.7%

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

      1. unpow2 34.7%

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

      2. swap-sqr 22.5%

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

      3. add-sqr-sqrt 66.8%

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

      4. associate-*r* 86.8%

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

      5. *-commutative 86.8%

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

      6. associate-*r* 87.0%

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

      7. *-commutative 87.0%

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

   10. Applied egg-rr 87.0%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -2.05 \cdot 10^{+133}:\\ \;\;\;\;\left(x.im \cdot \left(x.im \cdot x.re\right)\right) \cdot -3\\ \mathbf{elif}\;x.im \leq 7.6 \cdot 10^{+153}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re + x.im \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(x.im \cdot x.re\right) \cdot -3\right)\\ \end{array} \]

Alternative 2: 74.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -2.4 \cdot 10^{+109} \lor \neg \left(x.im \leq 1.4 \cdot 10^{-57}\right):\\ \;\;\;\;-3 \cdot \left(x.re \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -2.4e+109) (not (<= x.im 1.4e-57)))
   (* -3.0 (* x.re (* x.im x.im)))
   (* x.re (* x.re x.re))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -2.4e+109) || !(x_46_im <= 1.4e-57)) {
		tmp = -3.0 * (x_46_re * (x_46_im * x_46_im));
	} else {
		tmp = x_46_re * (x_46_re * x_46_re);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46im <= (-2.4d+109)) .or. (.not. (x_46im <= 1.4d-57))) then
        tmp = (-3.0d0) * (x_46re * (x_46im * x_46im))
    else
        tmp = x_46re * (x_46re * x_46re)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -2.4e+109) || !(x_46_im <= 1.4e-57)) {
		tmp = -3.0 * (x_46_re * (x_46_im * x_46_im));
	} else {
		tmp = x_46_re * (x_46_re * x_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_im <= -2.4e+109) or not (x_46_im <= 1.4e-57):
		tmp = -3.0 * (x_46_re * (x_46_im * x_46_im))
	else:
		tmp = x_46_re * (x_46_re * x_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_im <= -2.4e+109) || !(x_46_im <= 1.4e-57))
		tmp = Float64(-3.0 * Float64(x_46_re * Float64(x_46_im * x_46_im)));
	else
		tmp = Float64(x_46_re * Float64(x_46_re * x_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_im <= -2.4e+109) || ~((x_46_im <= 1.4e-57)))
		tmp = -3.0 * (x_46_re * (x_46_im * x_46_im));
	else
		tmp = x_46_re * (x_46_re * x_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$im, -2.4e+109], N[Not[LessEqual[x$46$im, 1.4e-57]], $MachinePrecision]], N[(-3.0 * N[(x$46$re * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -2.39999999999999987e109 or 1.4e-57 < x.im

   1. Initial program 70.6%

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

      1. *-commutative 70.6%

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

      2. distribute-lft-out 70.6%

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

      3. associate-*l* 70.5%

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

      4. *-commutative 70.5%

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

      5. distribute-rgt-out-- 77.4%

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

      6. associate--l- 77.4%

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

      7. associate--l- 77.4%

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

      8. sub-neg 77.4%

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

      9. associate--l+ 77.4%

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

      10. fma-udef 83.4%

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

      11. neg-mul-1 83.4%

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

      12. count-2 83.4%

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

      13. associate-*l* 83.4%

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

      14. distribute-rgt-out-- 83.4%

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

      15. associate-*r* 83.4%

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

      16. metadata-eval 83.4%

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

   3. Simplified 83.4%

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

   4. Taylor expanded in x.re around 0 73.2%

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

      1. unpow2 73.2%

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

   6. Simplified 73.2%

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

   if -2.39999999999999987e109 < x.im < 1.4e-57

   1. Initial program 94.9%

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

      1. *-commutative 94.9%

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

      2. distribute-lft-out 94.9%

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

      3. associate-*l* 94.9%

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

      4. *-commutative 94.9%

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

      5. distribute-rgt-out-- 99.8%

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

      6. associate--l- 99.8%

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

      7. associate--l- 99.8%

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

      8. sub-neg 99.8%

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

      9. associate--l+ 99.8%

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

      10. fma-udef 99.8%

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

      11. neg-mul-1 99.8%

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

      12. count-2 99.8%

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

      13. associate-*l* 99.8%

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

      14. distribute-rgt-out-- 99.8%

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

      15. associate-*r* 99.9%

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

      16. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x.re around inf 89.5%

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

      1. unpow2 89.5%

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

   6. Simplified 89.5%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -2.4 \cdot 10^{+109} \lor \neg \left(x.im \leq 1.4 \cdot 10^{-57}\right):\\ \;\;\;\;-3 \cdot \left(x.re \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \end{array} \]

Alternative 3: 80.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -4 \cdot 10^{+109} \lor \neg \left(x.im \leq 5.2 \cdot 10^{-58}\right):\\ \;\;\;\;x.im \cdot \left(\left(x.im \cdot x.re\right) \cdot -3\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -4e+109) (not (<= x.im 5.2e-58)))
   (* x.im (* (* x.im x.re) -3.0))
   (* x.re (* x.re x.re))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -4e+109) || !(x_46_im <= 5.2e-58)) {
		tmp = x_46_im * ((x_46_im * x_46_re) * -3.0);
	} else {
		tmp = x_46_re * (x_46_re * x_46_re);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46im <= (-4d+109)) .or. (.not. (x_46im <= 5.2d-58))) then
        tmp = x_46im * ((x_46im * x_46re) * (-3.0d0))
    else
        tmp = x_46re * (x_46re * x_46re)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -4e+109) || !(x_46_im <= 5.2e-58)) {
		tmp = x_46_im * ((x_46_im * x_46_re) * -3.0);
	} else {
		tmp = x_46_re * (x_46_re * x_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_im <= -4e+109) or not (x_46_im <= 5.2e-58):
		tmp = x_46_im * ((x_46_im * x_46_re) * -3.0)
	else:
		tmp = x_46_re * (x_46_re * x_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_im <= -4e+109) || !(x_46_im <= 5.2e-58))
		tmp = Float64(x_46_im * Float64(Float64(x_46_im * x_46_re) * -3.0));
	else
		tmp = Float64(x_46_re * Float64(x_46_re * x_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_im <= -4e+109) || ~((x_46_im <= 5.2e-58)))
		tmp = x_46_im * ((x_46_im * x_46_re) * -3.0);
	else
		tmp = x_46_re * (x_46_re * x_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$im, -4e+109], N[Not[LessEqual[x$46$im, 5.2e-58]], $MachinePrecision]], N[(x$46$im * N[(N[(x$46$im * x$46$re), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision], N[(x$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -3.99999999999999993e109 or 5.20000000000000013e-58 < x.im

   1. Initial program 70.6%

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

      1. *-commutative 70.6%

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

      2. distribute-lft-out 70.6%

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

      3. associate-*l* 70.5%

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

      4. *-commutative 70.5%

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

      5. distribute-rgt-out-- 77.4%

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

      6. associate--l- 77.4%

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

      7. associate--l- 77.4%

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

      8. sub-neg 77.4%

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

      9. associate--l+ 77.4%

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

      10. fma-udef 83.4%

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

      11. neg-mul-1 83.4%

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

      12. count-2 83.4%

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

      13. associate-*l* 83.4%

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

      14. distribute-rgt-out-- 83.4%

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

      15. associate-*r* 83.4%

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

      16. metadata-eval 83.4%

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

   3. Simplified 83.4%

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

   4. Taylor expanded in x.re around 0 73.2%

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

      1. associate-*r* 73.1%

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

      2. *-commutative 73.1%

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

      3. *-commutative 73.1%

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

      4. metadata-eval 73.1%

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

      5. distribute-rgt-out 73.1%

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

      6. distribute-lft-in 73.2%

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

      7. metadata-eval 73.2%

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

      8. distribute-lft-neg-in 73.2%

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

      9. count-2 73.2%

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

      10. distribute-rgt-neg-in 73.2%

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

      11. distribute-rgt-out 73.2%

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

      12. distribute-lft-out 73.2%

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

      13. distribute-rgt-neg-in 73.2%

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

      14. count-2 73.2%

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

      15. distribute-lft-neg-in 73.2%

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

      16. metadata-eval 73.2%

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

      17. mul-1-neg 73.2%

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

      18. distribute-rgt-neg-in 73.2%

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

      19. distribute-lft-neg-in 73.2%

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

      20. unpow2 73.2%

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

      21. distribute-rgt-neg-out 73.2%

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

      22. *-commutative 73.2%

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

   6. Simplified 73.1%

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

      1. add-sqr-sqrt 36.6%

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

      2. pow2 36.6%

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

      3. associate-*r* 36.5%

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

      4. sqrt-prod 33.9%

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

      5. sqrt-prod 21.2%

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

      6. add-sqr-sqrt 40.3%

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

   8. Applied egg-rr 40.3%

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

      1. unpow2 40.3%

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

      2. swap-sqr 34.0%

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

      3. add-sqr-sqrt 73.1%

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

      4. associate-*r* 84.2%

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

      5. *-commutative 84.2%

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

      6. associate-*r* 84.2%

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

      7. *-commutative 84.2%

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

   10. Applied egg-rr 84.2%

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

   if -3.99999999999999993e109 < x.im < 5.20000000000000013e-58

   1. Initial program 94.9%

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

      1. *-commutative 94.9%

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

      2. distribute-lft-out 94.9%

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

      3. associate-*l* 94.9%

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

      4. *-commutative 94.9%

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

      5. distribute-rgt-out-- 99.8%

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

      6. associate--l- 99.8%

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

      7. associate--l- 99.8%

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

      8. sub-neg 99.8%

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

      9. associate--l+ 99.8%

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

      10. fma-udef 99.8%

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

      11. neg-mul-1 99.8%

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

      12. count-2 99.8%

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

      13. associate-*l* 99.8%

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

      14. distribute-rgt-out-- 99.8%

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

      15. associate-*r* 99.9%

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

      16. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x.re around inf 89.5%

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

      1. unpow2 89.5%

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

   6. Simplified 89.5%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -4 \cdot 10^{+109} \lor \neg \left(x.im \leq 5.2 \cdot 10^{-58}\right):\\ \;\;\;\;x.im \cdot \left(\left(x.im \cdot x.re\right) \cdot -3\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \end{array} \]

Alternative 4: 80.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -2 \cdot 10^{+109} \lor \neg \left(x.im \leq 1.4 \cdot 10^{-57}\right):\\ \;\;\;\;\left(x.im \cdot \left(x.im \cdot x.re\right)\right) \cdot -3\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -2e+109) (not (<= x.im 1.4e-57)))
   (* (* x.im (* x.im x.re)) -3.0)
   (* x.re (* x.re x.re))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -2e+109) || !(x_46_im <= 1.4e-57)) {
		tmp = (x_46_im * (x_46_im * x_46_re)) * -3.0;
	} else {
		tmp = x_46_re * (x_46_re * x_46_re);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46im <= (-2d+109)) .or. (.not. (x_46im <= 1.4d-57))) then
        tmp = (x_46im * (x_46im * x_46re)) * (-3.0d0)
    else
        tmp = x_46re * (x_46re * x_46re)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -2e+109) || !(x_46_im <= 1.4e-57)) {
		tmp = (x_46_im * (x_46_im * x_46_re)) * -3.0;
	} else {
		tmp = x_46_re * (x_46_re * x_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_im <= -2e+109) or not (x_46_im <= 1.4e-57):
		tmp = (x_46_im * (x_46_im * x_46_re)) * -3.0
	else:
		tmp = x_46_re * (x_46_re * x_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_im <= -2e+109) || !(x_46_im <= 1.4e-57))
		tmp = Float64(Float64(x_46_im * Float64(x_46_im * x_46_re)) * -3.0);
	else
		tmp = Float64(x_46_re * Float64(x_46_re * x_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_im <= -2e+109) || ~((x_46_im <= 1.4e-57)))
		tmp = (x_46_im * (x_46_im * x_46_re)) * -3.0;
	else
		tmp = x_46_re * (x_46_re * x_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$im, -2e+109], N[Not[LessEqual[x$46$im, 1.4e-57]], $MachinePrecision]], N[(N[(x$46$im * N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * -3.0), $MachinePrecision], N[(x$46$re * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -1.99999999999999996e109 or 1.4e-57 < x.im

   1. Initial program 70.6%

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

      1. *-commutative 70.6%

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

      2. distribute-lft-out 70.6%

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

      3. associate-*l* 70.5%

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

      4. *-commutative 70.5%

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

      5. distribute-rgt-out-- 77.4%

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

      6. associate--l- 77.4%

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

      7. associate--l- 77.4%

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

      8. sub-neg 77.4%

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

      9. associate--l+ 77.4%

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

      10. fma-udef 83.4%

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

      11. neg-mul-1 83.4%

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

      12. count-2 83.4%

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

      13. associate-*l* 83.4%

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

      14. distribute-rgt-out-- 83.4%

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

      15. associate-*r* 83.4%

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

      16. metadata-eval 83.4%

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

   3. Simplified 83.4%

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

   4. Taylor expanded in x.re around 0 73.2%

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

      1. associate-*r* 73.1%

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

      2. *-commutative 73.1%

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

      3. *-commutative 73.1%

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

      4. metadata-eval 73.1%

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

      5. distribute-rgt-out 73.1%

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

      6. distribute-lft-in 73.2%

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

      7. metadata-eval 73.2%

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

      8. distribute-lft-neg-in 73.2%

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

      9. count-2 73.2%

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

      10. distribute-rgt-neg-in 73.2%

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

      11. distribute-rgt-out 73.2%

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

      12. distribute-lft-out 73.2%

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

      13. distribute-rgt-neg-in 73.2%

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

      14. count-2 73.2%

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

      15. distribute-lft-neg-in 73.2%

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

      16. metadata-eval 73.2%

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

      17. mul-1-neg 73.2%

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

      18. distribute-rgt-neg-in 73.2%

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

      19. distribute-lft-neg-in 73.2%

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

      20. unpow2 73.2%

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

      21. distribute-rgt-neg-out 73.2%

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

      22. *-commutative 73.2%

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

   6. Simplified 73.1%

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

      1. add-sqr-sqrt 36.6%

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

      2. pow2 36.6%

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

      3. associate-*r* 36.5%

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

      4. sqrt-prod 33.9%

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

      5. sqrt-prod 21.2%

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

      6. add-sqr-sqrt 40.3%

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

   8. Applied egg-rr 40.3%

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

      1. unpow2 40.3%

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

      2. swap-sqr 34.0%

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

      3. add-sqr-sqrt 73.1%

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

      4. *-commutative 73.1%

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

      5. associate-*l* 73.2%

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

      6. *-commutative 73.2%

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

      7. *-commutative 73.2%

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

      8. associate-*l* 84.3%

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

   10. Applied egg-rr 84.3%

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

   if -1.99999999999999996e109 < x.im < 1.4e-57

   1. Initial program 94.9%

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

      1. *-commutative 94.9%

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

      2. distribute-lft-out 94.9%

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

      3. associate-*l* 94.9%

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

      4. *-commutative 94.9%

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

      5. distribute-rgt-out-- 99.8%

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

      6. associate--l- 99.8%

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

      7. associate--l- 99.8%

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

      8. sub-neg 99.8%

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

      9. associate--l+ 99.8%

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

      10. fma-udef 99.8%

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

      11. neg-mul-1 99.8%

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

      12. count-2 99.8%

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

      13. associate-*l* 99.8%

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

      14. distribute-rgt-out-- 99.8%

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

      15. associate-*r* 99.9%

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

      16. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x.re around inf 89.5%

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

      1. unpow2 89.5%

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

   6. Simplified 89.5%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -2 \cdot 10^{+109} \lor \neg \left(x.im \leq 1.4 \cdot 10^{-57}\right):\\ \;\;\;\;\left(x.im \cdot \left(x.im \cdot x.re\right)\right) \cdot -3\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \end{array} \]

Alternative 5: 59.1% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.9%

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

   1. *-commutative 83.9%

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

   2. distribute-lft-out 83.9%

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

   3. associate-*l* 83.8%

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

   4. *-commutative 83.8%

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

   5. distribute-rgt-out-- 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 92.4%

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

   11. neg-mul-1 92.4%

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

   12. count-2 92.4%

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

   13. associate-*l* 92.4%

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

   14. distribute-rgt-out-- 92.4%

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

   15. associate-*r* 92.4%

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

   16. metadata-eval 92.4%

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

3. Simplified 92.4%

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

4. Taylor expanded in x.re around inf 57.8%

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

   1. unpow2 57.8%

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

6. Simplified 57.8%

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

7. Final simplification 57.8%

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

Developer target: 86.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023178 
(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)))