math.cube on complex, imaginary part

Percentage Accurate: 82.6% → 99.8%

Time: 6.0s

Alternatives: 9

Speedup: 1.4×

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

• 44.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

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_im) + (((x_46_re * x_46_im) + (x_46_im * 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_46im * x_46im)) * x_46im) + (((x_46re * x_46im) + (x_46im * 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_im * x_46_im)) * x_46_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
}

Python

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

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_im) + Float64(Float64(Float64(x_46_re * x_46_im) + Float64(x_46_im * 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_im * x_46_im)) * x_46_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
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$46$im), $MachinePrecision] + N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]

Initial Program: 82.6% accurate, 1.0× speedup

Math

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

FPCore

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

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_im) + (((x_46_re * x_46_im) + (x_46_im * 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_46im * x_46im)) * x_46im) + (((x_46re * x_46im) + (x_46im * 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_im * x_46_im)) * x_46_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
}

Python

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

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_im) + Float64(Float64(Float64(x_46_re * x_46_im) + Float64(x_46_im * 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_im * x_46_im)) * x_46_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
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$46$im), $MachinePrecision] + N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -6.4 \cdot 10^{+103} \lor \neg \left(x.im \leq 5 \cdot 10^{+86}\right):\\ \;\;\;\;x.im \cdot \left(\left(x.im + x.re\right) \cdot \left(x.re - x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(3 \cdot \left(x.im \cdot x.re\right)\right) - {x.im}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -6.4e+103) (not (<= x.im 5e+86)))
   (* x.im (* (+ x.im x.re) (- x.re x.im)))
   (- (* x.re (* 3.0 (* x.im x.re))) (pow x.im 3.0))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -6.4e+103) || !(x_46_im <= 5e+86)) {
		tmp = x_46_im * ((x_46_im + x_46_re) * (x_46_re - x_46_im));
	} else {
		tmp = (x_46_re * (3.0 * (x_46_im * x_46_re))) - pow(x_46_im, 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46im <= (-6.4d+103)) .or. (.not. (x_46im <= 5d+86))) then
        tmp = x_46im * ((x_46im + x_46re) * (x_46re - x_46im))
    else
        tmp = (x_46re * (3.0d0 * (x_46im * x_46re))) - (x_46im ** 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -6.4e+103) || !(x_46_im <= 5e+86)) {
		tmp = x_46_im * ((x_46_im + x_46_re) * (x_46_re - x_46_im));
	} else {
		tmp = (x_46_re * (3.0 * (x_46_im * x_46_re))) - Math.pow(x_46_im, 3.0);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_im <= -6.4e+103) or not (x_46_im <= 5e+86):
		tmp = x_46_im * ((x_46_im + x_46_re) * (x_46_re - x_46_im))
	else:
		tmp = (x_46_re * (3.0 * (x_46_im * x_46_re))) - math.pow(x_46_im, 3.0)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_im <= -6.4e+103) || !(x_46_im <= 5e+86))
		tmp = Float64(x_46_im * Float64(Float64(x_46_im + x_46_re) * Float64(x_46_re - x_46_im)));
	else
		tmp = Float64(Float64(x_46_re * Float64(3.0 * Float64(x_46_im * x_46_re))) - (x_46_im ^ 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_im <= -6.4e+103) || ~((x_46_im <= 5e+86)))
		tmp = x_46_im * ((x_46_im + x_46_re) * (x_46_re - x_46_im));
	else
		tmp = (x_46_re * (3.0 * (x_46_im * x_46_re))) - (x_46_im ^ 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$im, -6.4e+103], N[Not[LessEqual[x$46$im, 5e+86]], $MachinePrecision]], N[(x$46$im * N[(N[(x$46$im + x$46$re), $MachinePrecision] * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * N[(3.0 * N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[x$46$im, 3.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -6.39999999999999985e103 or 4.9999999999999998e86 < x.im

   1. Initial program 72.8%

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

      1. *-commutative 72.8%

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

      2. *-commutative 72.8%

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

      3. flip-+ 0.0%

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

      4. +-inverses 0.0%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.re \cdot \frac{\color{blue}{0}}{x.re \cdot x.im - x.re \cdot x.im} \]

      5. +-inverses 0.0%

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

      6. +-inverses 0.0%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.re \cdot \frac{x.im \cdot x.im - x.im \cdot x.im}{\color{blue}{0}} \]

      7. +-inverses 0.0%

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

      8. flip-+ 78.3%

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

      9. distribute-lft-in 78.3%

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

   3. Applied egg-rr 78.3%

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

      1. +-commutative 78.3%

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

      2. add-cube-cbrt 78.3%

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

      3. fma-def 78.3%

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

   5. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0}{0}, \frac{0}{0}, x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{0 + x.im \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)} \]

   if -6.39999999999999985e103 < x.im < 4.9999999999999998e86

   1. Initial program 89.4%

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

      1. +-commutative 89.4%

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

      2. *-commutative 89.4%

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

      3. sub-neg 89.4%

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

      4. distribute-lft-in 89.4%

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

      5. associate-+r+ 89.4%

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

      6. distribute-rgt-neg-out 89.4%

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

      7. unsub-neg 89.4%

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

      8. associate-*r* 99.6%

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

      9. distribute-rgt-out 99.7%

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

      10. *-commutative 99.7%

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

      11. count-2 99.7%

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

      12. distribute-lft1-in 99.7%

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

      13. metadata-eval 99.7%

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

      14. *-commutative 99.7%

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

      15. *-commutative 99.7%

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

      16. associate-*r* 99.6%

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

      17. cube-unmult 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x.re around 0 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -6.4 \cdot 10^{+103} \lor \neg \left(x.im \leq 5 \cdot 10^{+86}\right):\\ \;\;\;\;x.im \cdot \left(\left(x.im + x.re\right) \cdot \left(x.re - x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(3 \cdot \left(x.im \cdot x.re\right)\right) - {x.im}^{3}\\ \end{array} \]

Alternative 2: 98.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.im \cdot \left(\left(x.im + x.re\right) \cdot \left(x.re - x.im\right)\right)\\ t_1 := x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right)\\ \mathbf{if}\;x.im \leq -4 \cdot 10^{+163}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.im \leq -4.4 \cdot 10^{-95}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x.im \leq 1.32 \cdot 10^{-106}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right)\\ \mathbf{elif}\;x.im \leq 5 \cdot 10^{+86}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (* x.im (* (+ x.im x.re) (- x.re x.im))))
        (t_1
         (+
          (* x.im (- (* x.re x.re) (* x.im x.im)))
          (* x.re (+ (* x.im x.re) (* x.im x.re))))))
   (if (<= x.im -4e+163)
     t_0
     (if (<= x.im -4.4e-95)
       t_1
       (if (<= x.im 1.32e-106)
         (* x.re (* x.im (* x.re 3.0)))
         (if (<= x.im 5e+86) t_1 t_0))))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_im * ((x_46_im + x_46_re) * (x_46_re - x_46_im));
	double t_1 = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_im * x_46_re) + (x_46_im * x_46_re)));
	double tmp;
	if (x_46_im <= -4e+163) {
		tmp = t_0;
	} else if (x_46_im <= -4.4e-95) {
		tmp = t_1;
	} else if (x_46_im <= 1.32e-106) {
		tmp = x_46_re * (x_46_im * (x_46_re * 3.0));
	} else if (x_46_im <= 5e+86) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x_46im * ((x_46im + x_46re) * (x_46re - x_46im))
    t_1 = (x_46im * ((x_46re * x_46re) - (x_46im * x_46im))) + (x_46re * ((x_46im * x_46re) + (x_46im * x_46re)))
    if (x_46im <= (-4d+163)) then
        tmp = t_0
    else if (x_46im <= (-4.4d-95)) then
        tmp = t_1
    else if (x_46im <= 1.32d-106) then
        tmp = x_46re * (x_46im * (x_46re * 3.0d0))
    else if (x_46im <= 5d+86) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_im * ((x_46_im + x_46_re) * (x_46_re - x_46_im));
	double t_1 = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_im * x_46_re) + (x_46_im * x_46_re)));
	double tmp;
	if (x_46_im <= -4e+163) {
		tmp = t_0;
	} else if (x_46_im <= -4.4e-95) {
		tmp = t_1;
	} else if (x_46_im <= 1.32e-106) {
		tmp = x_46_re * (x_46_im * (x_46_re * 3.0));
	} else if (x_46_im <= 5e+86) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	t_0 = x_46_im * ((x_46_im + x_46_re) * (x_46_re - x_46_im))
	t_1 = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_im * x_46_re) + (x_46_im * x_46_re)))
	tmp = 0
	if x_46_im <= -4e+163:
		tmp = t_0
	elif x_46_im <= -4.4e-95:
		tmp = t_1
	elif x_46_im <= 1.32e-106:
		tmp = x_46_re * (x_46_im * (x_46_re * 3.0))
	elif x_46_im <= 5e+86:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(x_46_im * Float64(Float64(x_46_im + x_46_re) * Float64(x_46_re - x_46_im)))
	t_1 = Float64(Float64(x_46_im * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im))) + Float64(x_46_re * Float64(Float64(x_46_im * x_46_re) + Float64(x_46_im * x_46_re))))
	tmp = 0.0
	if (x_46_im <= -4e+163)
		tmp = t_0;
	elseif (x_46_im <= -4.4e-95)
		tmp = t_1;
	elseif (x_46_im <= 1.32e-106)
		tmp = Float64(x_46_re * Float64(x_46_im * Float64(x_46_re * 3.0)));
	elseif (x_46_im <= 5e+86)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	t_0 = x_46_im * ((x_46_im + x_46_re) * (x_46_re - x_46_im));
	t_1 = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_im * x_46_re) + (x_46_im * x_46_re)));
	tmp = 0.0;
	if (x_46_im <= -4e+163)
		tmp = t_0;
	elseif (x_46_im <= -4.4e-95)
		tmp = t_1;
	elseif (x_46_im <= 1.32e-106)
		tmp = x_46_re * (x_46_im * (x_46_re * 3.0));
	elseif (x_46_im <= 5e+86)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(x$46$im * N[(N[(x$46$im + x$46$re), $MachinePrecision] * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(N[(x$46$im * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$im, -4e+163], t$95$0, If[LessEqual[x$46$im, -4.4e-95], t$95$1, If[LessEqual[x$46$im, 1.32e-106], N[(x$46$re * N[(x$46$im * N[(x$46$re * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 5e+86], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -3.9999999999999998e163 or 4.9999999999999998e86 < x.im

   1. Initial program 67.9%

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

      1. *-commutative 67.9%

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

      2. *-commutative 67.9%

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

      3. flip-+ 0.0%

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

      4. +-inverses 0.0%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.re \cdot \frac{\color{blue}{0}}{x.re \cdot x.im - x.re \cdot x.im} \]

      5. +-inverses 0.0%

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

      6. +-inverses 0.0%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.re \cdot \frac{x.im \cdot x.im - x.im \cdot x.im}{\color{blue}{0}} \]

      7. +-inverses 0.0%

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

      8. flip-+ 75.6%

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

      9. distribute-lft-in 75.6%

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

   3. Applied egg-rr 75.6%

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

      1. +-commutative 75.6%

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

      2. add-cube-cbrt 75.6%

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

      3. fma-def 75.6%

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

   5. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0}{0}, \frac{0}{0}, x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{0 + x.im \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)} \]

   if -3.9999999999999998e163 < x.im < -4.3999999999999998e-95 or 1.32000000000000001e-106 < x.im < 4.9999999999999998e86

   1. Initial program 99.7%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

   if -4.3999999999999998e-95 < x.im < 1.32000000000000001e-106

   1. Initial program 80.4%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

   2. Taylor expanded in x.re around inf 80.4%

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

   4. Simplified 80.4%

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

   5. Taylor expanded in x.re around 0 80.3%

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

   7. Simplified 80.3%

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

      1. add-log-exp 47.3%

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

      2. *-un-lft-identity 47.3%

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

      3. log-prod 47.3%

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

      4. metadata-eval 47.3%

         \[\leadsto \color{blue}{0} + \log \left(e^{\left(x.re \cdot x.re\right) \cdot \left(3 \cdot x.im\right)}\right) \]

      5. add-log-exp 80.3%

         \[\leadsto 0 + \color{blue}{\left(x.re \cdot x.re\right) \cdot \left(3 \cdot x.im\right)} \]

      6. associate-*l* 99.6%

         \[\leadsto 0 + \color{blue}{x.re \cdot \left(x.re \cdot \left(3 \cdot x.im\right)\right)} \]

      7. associate-*r* 99.7%

         \[\leadsto 0 + x.re \cdot \color{blue}{\left(\left(x.re \cdot 3\right) \cdot x.im\right)} \]

   9. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{0 + x.re \cdot \left(\left(x.re \cdot 3\right) \cdot x.im\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -4 \cdot 10^{+163}:\\ \;\;\;\;x.im \cdot \left(\left(x.im + x.re\right) \cdot \left(x.re - x.im\right)\right)\\ \mathbf{elif}\;x.im \leq -4.4 \cdot 10^{-95}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right)\\ \mathbf{elif}\;x.im \leq 1.32 \cdot 10^{-106}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right)\\ \mathbf{elif}\;x.im \leq 5 \cdot 10^{+86}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(x.im + x.re\right) \cdot \left(x.re - x.im\right)\right)\\ \end{array} \]

Alternative 3: 93.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -3.4 \cdot 10^{-51} \lor \neg \left(x.im \leq 2.05 \cdot 10^{-104}\right):\\ \;\;\;\;x.im \cdot \left(\left(x.im + x.re\right) \cdot \left(x.re - x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -3.4e-51) (not (<= x.im 2.05e-104)))
   (* x.im (* (+ x.im x.re) (- x.re x.im)))
   (* x.re (* x.im (* x.re 3.0)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -3.4e-51) || !(x_46_im <= 2.05e-104)) {
		tmp = x_46_im * ((x_46_im + x_46_re) * (x_46_re - x_46_im));
	} else {
		tmp = x_46_re * (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 <= (-3.4d-51)) .or. (.not. (x_46im <= 2.05d-104))) then
        tmp = x_46im * ((x_46im + x_46re) * (x_46re - x_46im))
    else
        tmp = x_46re * (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 <= -3.4e-51) || !(x_46_im <= 2.05e-104)) {
		tmp = x_46_im * ((x_46_im + x_46_re) * (x_46_re - x_46_im));
	} else {
		tmp = x_46_re * (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 <= -3.4e-51) or not (x_46_im <= 2.05e-104):
		tmp = x_46_im * ((x_46_im + x_46_re) * (x_46_re - x_46_im))
	else:
		tmp = x_46_re * (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 <= -3.4e-51) || !(x_46_im <= 2.05e-104))
		tmp = Float64(x_46_im * Float64(Float64(x_46_im + x_46_re) * Float64(x_46_re - x_46_im)));
	else
		tmp = Float64(x_46_re * Float64(x_46_im * Float64(x_46_re * 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_im <= -3.4e-51) || ~((x_46_im <= 2.05e-104)))
		tmp = x_46_im * ((x_46_im + x_46_re) * (x_46_re - x_46_im));
	else
		tmp = x_46_re * (x_46_im * (x_46_re * 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$im, -3.4e-51], N[Not[LessEqual[x$46$im, 2.05e-104]], $MachinePrecision]], N[(x$46$im * N[(N[(x$46$im + x$46$re), $MachinePrecision] * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re * N[(x$46$im * N[(x$46$re * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -3.40000000000000003e-51 or 2.04999999999999992e-104 < x.im

   1. Initial program 84.0%

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

      1. *-commutative 84.0%

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

      2. *-commutative 84.0%

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

      3. flip-+ 0.0%

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

      4. +-inverses 0.0%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.re \cdot \frac{\color{blue}{0}}{x.re \cdot x.im - x.re \cdot x.im} \]

      5. +-inverses 0.0%

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

      6. +-inverses 0.0%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.re \cdot \frac{x.im \cdot x.im - x.im \cdot x.im}{\color{blue}{0}} \]

      7. +-inverses 0.0%

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

      8. flip-+ 77.4%

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

      9. distribute-lft-in 77.4%

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

   3. Applied egg-rr 77.4%

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

      1. +-commutative 77.4%

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

      2. add-cube-cbrt 77.4%

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

      3. fma-def 77.4%

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

   5. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0}{0}, \frac{0}{0}, x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)} \]

   7. Simplified 92.6%

      \[\leadsto \color{blue}{0 + x.im \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)} \]

   if -3.40000000000000003e-51 < x.im < 2.04999999999999992e-104

   1. Initial program 82.5%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

   2. Taylor expanded in x.re around inf 81.7%

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

   4. Simplified 81.7%

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

   5. Taylor expanded in x.re around 0 81.7%

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

   7. Simplified 81.7%

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

      1. add-log-exp 46.4%

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

      2. *-un-lft-identity 46.4%

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

      3. log-prod 46.4%

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

      4. metadata-eval 46.4%

         \[\leadsto \color{blue}{0} + \log \left(e^{\left(x.re \cdot x.re\right) \cdot \left(3 \cdot x.im\right)}\right) \]

      5. add-log-exp 81.7%

         \[\leadsto 0 + \color{blue}{\left(x.re \cdot x.re\right) \cdot \left(3 \cdot x.im\right)} \]

      6. associate-*l* 98.8%

         \[\leadsto 0 + \color{blue}{x.re \cdot \left(x.re \cdot \left(3 \cdot x.im\right)\right)} \]

      7. associate-*r* 98.9%

         \[\leadsto 0 + x.re \cdot \color{blue}{\left(\left(x.re \cdot 3\right) \cdot x.im\right)} \]

   9. Applied egg-rr 98.9%

      \[\leadsto \color{blue}{0 + x.re \cdot \left(\left(x.re \cdot 3\right) \cdot x.im\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -3.4 \cdot 10^{-51} \lor \neg \left(x.im \leq 2.05 \cdot 10^{-104}\right):\\ \;\;\;\;x.im \cdot \left(\left(x.im + x.re\right) \cdot \left(x.re - x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right)\\ \end{array} \]

Alternative 4: 77.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -1.95 \cdot 10^{+109} \lor \neg \left(x.im \leq 1.4 \cdot 10^{-57}\right):\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.im \cdot x.re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -1.95e+109) (not (<= x.im 1.4e-57)))
   (* x.im (* x.im (- x.im)))
   (* 3.0 (* x.re (* x.im x.re)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -1.95e+109) || !(x_46_im <= 1.4e-57)) {
		tmp = x_46_im * (x_46_im * -x_46_im);
	} else {
		tmp = 3.0 * (x_46_re * (x_46_im * 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 <= (-1.95d+109)) .or. (.not. (x_46im <= 1.4d-57))) then
        tmp = x_46im * (x_46im * -x_46im)
    else
        tmp = 3.0d0 * (x_46re * (x_46im * 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 <= -1.95e+109) || !(x_46_im <= 1.4e-57)) {
		tmp = x_46_im * (x_46_im * -x_46_im);
	} else {
		tmp = 3.0 * (x_46_re * (x_46_im * x_46_re));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_im <= -1.95e+109) or not (x_46_im <= 1.4e-57):
		tmp = x_46_im * (x_46_im * -x_46_im)
	else:
		tmp = 3.0 * (x_46_re * (x_46_im * x_46_re))
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_im <= -1.95e+109) || !(x_46_im <= 1.4e-57))
		tmp = Float64(x_46_im * Float64(x_46_im * Float64(-x_46_im)));
	else
		tmp = Float64(3.0 * Float64(x_46_re * Float64(x_46_im * 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 <= -1.95e+109) || ~((x_46_im <= 1.4e-57)))
		tmp = x_46_im * (x_46_im * -x_46_im);
	else
		tmp = 3.0 * (x_46_re * (x_46_im * x_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$im, -1.95e+109], N[Not[LessEqual[x$46$im, 1.4e-57]], $MachinePrecision]], N[(x$46$im * N[(x$46$im * (-x$46$im)), $MachinePrecision]), $MachinePrecision], N[(3.0 * N[(x$46$re * N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -1.95000000000000008e109 or 1.4e-57 < x.im

   1. Initial program 78.4%

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

      1. *-commutative 78.4%

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

      2. *-commutative 78.4%

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

      3. flip-+ 0.0%

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

      4. +-inverses 0.0%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.re \cdot \frac{\color{blue}{0}}{x.re \cdot x.im - x.re \cdot x.im} \]

      5. +-inverses 0.0%

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

      6. +-inverses 0.0%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.re \cdot \frac{x.im \cdot x.im - x.im \cdot x.im}{\color{blue}{0}} \]

      7. +-inverses 0.0%

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

      8. flip-+ 78.9%

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

      9. distribute-lft-in 78.9%

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

   3. Applied egg-rr 78.9%

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

      1. +-commutative 78.9%

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

      2. add-cube-cbrt 78.9%

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

      3. fma-def 78.9%

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

   5. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0}{0}, \frac{0}{0}, x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)} \]

   7. Simplified 97.1%

      \[\leadsto \color{blue}{0 + x.im \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)} \]

   8. Taylor expanded in x.re around 0 84.5%

      \[\leadsto 0 + x.im \cdot \color{blue}{\left(-1 \cdot {x.im}^{2}\right)} \]
   9. Step-by-step derivation

      1. unpow2 84.5%

         \[\leadsto 0 + x.im \cdot \left(-1 \cdot \color{blue}{\left(x.im \cdot x.im\right)}\right) \]

      2. neg-mul-1 84.5%

         \[\leadsto 0 + x.im \cdot \color{blue}{\left(-x.im \cdot x.im\right)} \]

      3. distribute-rgt-neg-in 84.5%

         \[\leadsto 0 + x.im \cdot \color{blue}{\left(x.im \cdot \left(-x.im\right)\right)} \]

   10. Simplified 84.5%

       \[\leadsto 0 + x.im \cdot \color{blue}{\left(x.im \cdot \left(-x.im\right)\right)} \]

   if -1.95000000000000008e109 < x.im < 1.4e-57

   1. Initial program 87.7%

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

      1. +-commutative 87.7%

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

      2. *-commutative 87.7%

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

      3. sub-neg 87.7%

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

      4. distribute-lft-in 86.2%

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

      5. associate-+r+ 86.2%

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

      6. distribute-rgt-neg-out 86.2%

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

      7. unsub-neg 86.2%

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

      8. associate-*r* 98.2%

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

      9. distribute-rgt-out 98.3%

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

      10. *-commutative 98.3%

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

      11. count-2 98.3%

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

      12. distribute-lft1-in 98.3%

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

      13. metadata-eval 98.3%

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

      14. *-commutative 98.3%

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

      15. *-commutative 98.3%

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

      16. associate-*r* 98.2%

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

      17. cube-unmult 98.3%

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

   3. Simplified 98.3%

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

   4. Taylor expanded in x.re around 0 98.4%

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

      1. associate-*r* 98.3%

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

      2. fma-neg 98.3%

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

   6. Applied egg-rr 98.3%

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

   7. Taylor expanded in x.re around inf 74.9%

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

      1. unpow2 74.9%

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

      2. *-commutative 74.9%

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

   9. Simplified 74.9%

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

   10. Taylor expanded in x.im around 0 74.9%

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

       1. unpow2 74.9%

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

       2. associate-*l* 86.8%

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

   12. Simplified 86.8%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -1.95 \cdot 10^{+109} \lor \neg \left(x.im \leq 1.4 \cdot 10^{-57}\right):\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.im \cdot x.re\right)\right)\\ \end{array} \]

Alternative 5: 77.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -1.95 \cdot 10^{+109} \lor \neg \left(x.im \leq 1.4 \cdot 10^{-57}\right):\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -1.95e+109) (not (<= x.im 1.4e-57)))
   (* x.im (* x.im (- x.im)))
   (* x.re (* x.im (* x.re 3.0)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -1.95e+109) || !(x_46_im <= 1.4e-57)) {
		tmp = x_46_im * (x_46_im * -x_46_im);
	} else {
		tmp = x_46_re * (x_46_im * (x_46_re * 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46im <= (-1.95d+109)) .or. (.not. (x_46im <= 1.4d-57))) then
        tmp = x_46im * (x_46im * -x_46im)
    else
        tmp = x_46re * (x_46im * (x_46re * 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -1.95e+109) || !(x_46_im <= 1.4e-57)) {
		tmp = x_46_im * (x_46_im * -x_46_im);
	} else {
		tmp = x_46_re * (x_46_im * (x_46_re * 3.0));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_im <= -1.95e+109) or not (x_46_im <= 1.4e-57):
		tmp = x_46_im * (x_46_im * -x_46_im)
	else:
		tmp = x_46_re * (x_46_im * (x_46_re * 3.0))
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_im <= -1.95e+109) || !(x_46_im <= 1.4e-57))
		tmp = Float64(x_46_im * Float64(x_46_im * Float64(-x_46_im)));
	else
		tmp = Float64(x_46_re * Float64(x_46_im * Float64(x_46_re * 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_im <= -1.95e+109) || ~((x_46_im <= 1.4e-57)))
		tmp = x_46_im * (x_46_im * -x_46_im);
	else
		tmp = x_46_re * (x_46_im * (x_46_re * 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$im, -1.95e+109], N[Not[LessEqual[x$46$im, 1.4e-57]], $MachinePrecision]], N[(x$46$im * N[(x$46$im * (-x$46$im)), $MachinePrecision]), $MachinePrecision], N[(x$46$re * N[(x$46$im * N[(x$46$re * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -1.95000000000000008e109 or 1.4e-57 < x.im

   1. Initial program 78.4%

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

      1. *-commutative 78.4%

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

      2. *-commutative 78.4%

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

      3. flip-+ 0.0%

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

      4. +-inverses 0.0%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.re \cdot \frac{\color{blue}{0}}{x.re \cdot x.im - x.re \cdot x.im} \]

      5. +-inverses 0.0%

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

      6. +-inverses 0.0%

         \[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + x.re \cdot \frac{x.im \cdot x.im - x.im \cdot x.im}{\color{blue}{0}} \]

      7. +-inverses 0.0%

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

      8. flip-+ 78.9%

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

      9. distribute-lft-in 78.9%

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

   3. Applied egg-rr 78.9%

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

      1. +-commutative 78.9%

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

      2. add-cube-cbrt 78.9%

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

      3. fma-def 78.9%

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

   5. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0}{0}, \frac{0}{0}, x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)} \]

   7. Simplified 97.1%

      \[\leadsto \color{blue}{0 + x.im \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)} \]

   8. Taylor expanded in x.re around 0 84.5%

      \[\leadsto 0 + x.im \cdot \color{blue}{\left(-1 \cdot {x.im}^{2}\right)} \]
   9. Step-by-step derivation

      1. unpow2 84.5%

         \[\leadsto 0 + x.im \cdot \left(-1 \cdot \color{blue}{\left(x.im \cdot x.im\right)}\right) \]

      2. neg-mul-1 84.5%

         \[\leadsto 0 + x.im \cdot \color{blue}{\left(-x.im \cdot x.im\right)} \]

      3. distribute-rgt-neg-in 84.5%

         \[\leadsto 0 + x.im \cdot \color{blue}{\left(x.im \cdot \left(-x.im\right)\right)} \]

   10. Simplified 84.5%

       \[\leadsto 0 + x.im \cdot \color{blue}{\left(x.im \cdot \left(-x.im\right)\right)} \]

   if -1.95000000000000008e109 < x.im < 1.4e-57

   1. Initial program 87.7%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

   2. Taylor expanded in x.re around inf 74.8%

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

   4. Simplified 74.8%

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

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

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

   7. Simplified 74.8%

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

      1. add-log-exp 43.4%

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

      2. *-un-lft-identity 43.4%

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

      3. log-prod 43.4%

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

      4. metadata-eval 43.4%

         \[\leadsto \color{blue}{0} + \log \left(e^{\left(x.re \cdot x.re\right) \cdot \left(3 \cdot x.im\right)}\right) \]

      5. add-log-exp 74.8%

         \[\leadsto 0 + \color{blue}{\left(x.re \cdot x.re\right) \cdot \left(3 \cdot x.im\right)} \]

      6. associate-*l* 86.8%

         \[\leadsto 0 + \color{blue}{x.re \cdot \left(x.re \cdot \left(3 \cdot x.im\right)\right)} \]

      7. associate-*r* 86.9%

         \[\leadsto 0 + x.re \cdot \color{blue}{\left(\left(x.re \cdot 3\right) \cdot x.im\right)} \]

   9. Applied egg-rr 86.9%

      \[\leadsto \color{blue}{0 + x.re \cdot \left(\left(x.re \cdot 3\right) \cdot x.im\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -1.95 \cdot 10^{+109} \lor \neg \left(x.im \leq 1.4 \cdot 10^{-57}\right):\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right)\\ \end{array} \]

Alternative 6: 51.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -2.5 \cdot 10^{+134}:\\ \;\;\;\;x.im \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x.im \cdot \left(x.re \cdot x.re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im -2.5e+134) (* x.im x.re) (* 3.0 (* x.im (* x.re x.re)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= -2.5e+134) {
		tmp = x_46_im * x_46_re;
	} else {
		tmp = 3.0 * (x_46_im * (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 <= (-2.5d+134)) then
        tmp = x_46im * x_46re
    else
        tmp = 3.0d0 * (x_46im * (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 <= -2.5e+134) {
		tmp = x_46_im * x_46_re;
	} else {
		tmp = 3.0 * (x_46_im * (x_46_re * x_46_re));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= -2.5e+134:
		tmp = x_46_im * x_46_re
	else:
		tmp = 3.0 * (x_46_im * (x_46_re * x_46_re))
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= -2.5e+134)
		tmp = Float64(x_46_im * x_46_re);
	else
		tmp = Float64(3.0 * Float64(x_46_im * 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 <= -2.5e+134)
		tmp = x_46_im * x_46_re;
	else
		tmp = 3.0 * (x_46_im * (x_46_re * x_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, -2.5e+134], N[(x$46$im * x$46$re), $MachinePrecision], N[(3.0 * N[(x$46$im * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -2.4999999999999999e134

   1. Initial program 64.3%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

   2. Taylor expanded in x.re around 0 64.3%

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

   4. Simplified 7.5%

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

      1. *-commutative 7.5%

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

      2. *-commutative 7.5%

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

      3. flip-+ 0.0%

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

      4. +-inverses 0.0%

         \[\leadsto -3 + x.re \cdot \frac{\color{blue}{0}}{x.re \cdot x.im - x.re \cdot x.im} \]

      5. +-inverses 0.0%

         \[\leadsto -3 + x.re \cdot \frac{\color{blue}{x.im \cdot x.im - x.im \cdot x.im}}{x.re \cdot x.im - x.re \cdot x.im} \]

      6. +-inverses 0.0%

         \[\leadsto -3 + x.re \cdot \frac{x.im \cdot x.im - x.im \cdot x.im}{\color{blue}{0}} \]

      7. +-inverses 0.0%

         \[\leadsto -3 + x.re \cdot \frac{x.im \cdot x.im - x.im \cdot x.im}{\color{blue}{x.im - x.im}} \]

      8. flip-+ 20.7%

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

      9. distribute-lft-in 20.7%

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

      10. *-un-lft-identity 20.7%

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

      11. fma-def 20.7%

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

   6. Applied egg-rr 20.7%

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

   8. Simplified 20.7%

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

   9. Taylor expanded in x.im around inf 20.7%

      \[\leadsto \color{blue}{x.re \cdot x.im} \]

   if -2.4999999999999999e134 < x.im

   1. Initial program 87.2%

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

      1. +-commutative 87.2%

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

      2. *-commutative 87.2%

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

      3. sub-neg 87.2%

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

      4. distribute-lft-in 84.4%

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

      5. associate-+r+ 84.4%

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

      6. distribute-rgt-neg-out 84.4%

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

      7. unsub-neg 84.4%

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

      8. associate-*r* 92.2%

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

      9. distribute-rgt-out 92.3%

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

      10. *-commutative 92.3%

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

      11. count-2 92.3%

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

      12. distribute-lft1-in 92.3%

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

      13. metadata-eval 92.3%

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

      14. *-commutative 92.3%

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

      15. *-commutative 92.3%

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

      16. associate-*r* 92.2%

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

      17. cube-unmult 92.3%

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

   3. Simplified 92.3%

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

   4. Taylor expanded in x.re around 0 92.4%

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

      1. associate-*r* 92.4%

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

      2. fma-neg 93.3%

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

   6. Applied egg-rr 93.3%

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

   7. Taylor expanded in x.re around inf 56.3%

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

      1. unpow2 56.3%

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

      2. *-commutative 56.3%

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

   9. Simplified 56.3%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -2.5 \cdot 10^{+134}:\\ \;\;\;\;x.im \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x.im \cdot \left(x.re \cdot x.re\right)\right)\\ \end{array} \]

Alternative 7: 56.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -2.5 \cdot 10^{+134}:\\ \;\;\;\;x.im \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.im \cdot x.re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im -2.5e+134) (* x.im x.re) (* 3.0 (* x.re (* x.im x.re)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= -2.5e+134) {
		tmp = x_46_im * x_46_re;
	} else {
		tmp = 3.0 * (x_46_re * (x_46_im * 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 <= (-2.5d+134)) then
        tmp = x_46im * x_46re
    else
        tmp = 3.0d0 * (x_46re * (x_46im * 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 <= -2.5e+134) {
		tmp = x_46_im * x_46_re;
	} else {
		tmp = 3.0 * (x_46_re * (x_46_im * x_46_re));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= -2.5e+134:
		tmp = x_46_im * x_46_re
	else:
		tmp = 3.0 * (x_46_re * (x_46_im * x_46_re))
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= -2.5e+134)
		tmp = Float64(x_46_im * x_46_re);
	else
		tmp = Float64(3.0 * Float64(x_46_re * Float64(x_46_im * 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 <= -2.5e+134)
		tmp = x_46_im * x_46_re;
	else
		tmp = 3.0 * (x_46_re * (x_46_im * x_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, -2.5e+134], N[(x$46$im * x$46$re), $MachinePrecision], N[(3.0 * N[(x$46$re * N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -2.4999999999999999e134

   1. Initial program 64.3%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

   2. Taylor expanded in x.re around 0 64.3%

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

   4. Simplified 7.5%

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

      1. *-commutative 7.5%

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

      2. *-commutative 7.5%

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

      3. flip-+ 0.0%

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

      4. +-inverses 0.0%

         \[\leadsto -3 + x.re \cdot \frac{\color{blue}{0}}{x.re \cdot x.im - x.re \cdot x.im} \]

      5. +-inverses 0.0%

         \[\leadsto -3 + x.re \cdot \frac{\color{blue}{x.im \cdot x.im - x.im \cdot x.im}}{x.re \cdot x.im - x.re \cdot x.im} \]

      6. +-inverses 0.0%

         \[\leadsto -3 + x.re \cdot \frac{x.im \cdot x.im - x.im \cdot x.im}{\color{blue}{0}} \]

      7. +-inverses 0.0%

         \[\leadsto -3 + x.re \cdot \frac{x.im \cdot x.im - x.im \cdot x.im}{\color{blue}{x.im - x.im}} \]

      8. flip-+ 20.7%

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

      9. distribute-lft-in 20.7%

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

      10. *-un-lft-identity 20.7%

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

      11. fma-def 20.7%

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

   6. Applied egg-rr 20.7%

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

   8. Simplified 20.7%

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

   9. Taylor expanded in x.im around inf 20.7%

      \[\leadsto \color{blue}{x.re \cdot x.im} \]

   if -2.4999999999999999e134 < x.im

   1. Initial program 87.2%

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

      1. +-commutative 87.2%

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

      2. *-commutative 87.2%

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

      3. sub-neg 87.2%

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

      4. distribute-lft-in 84.4%

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

      5. associate-+r+ 84.4%

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

      6. distribute-rgt-neg-out 84.4%

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

      7. unsub-neg 84.4%

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

      8. associate-*r* 92.2%

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

      9. distribute-rgt-out 92.3%

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

      10. *-commutative 92.3%

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

      11. count-2 92.3%

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

      12. distribute-lft1-in 92.3%

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

      13. metadata-eval 92.3%

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

      14. *-commutative 92.3%

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

      15. *-commutative 92.3%

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

      16. associate-*r* 92.2%

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

      17. cube-unmult 92.3%

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

   3. Simplified 92.3%

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

   4. Taylor expanded in x.re around 0 92.4%

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

      1. associate-*r* 92.4%

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

      2. fma-neg 93.3%

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

   6. Applied egg-rr 93.3%

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

   7. Taylor expanded in x.re around inf 56.3%

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

      1. unpow2 56.3%

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

      2. *-commutative 56.3%

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

   9. Simplified 56.3%

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

   10. Taylor expanded in x.im around 0 56.3%

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

       1. unpow2 56.3%

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

       2. associate-*l* 64.2%

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

   12. Simplified 64.2%

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

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -2.5 \cdot 10^{+134}:\\ \;\;\;\;x.im \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.im \cdot x.re\right)\right)\\ \end{array} \]

Alternative 8: 19.7% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ x.im \cdot x.re \end{array} \]

FPCore

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

C

double code(double x_46_re, double x_46_im) {
	return x_46_im * 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_46im * x_46re
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.4%

   \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

2. Taylor expanded in x.re around 0 65.5%

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

4. Simplified 21.2%

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

   1. *-commutative 21.2%

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

   2. *-commutative 21.2%

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

   3. flip-+ 0.0%

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

   4. +-inverses 0.0%

      \[\leadsto -3 + x.re \cdot \frac{\color{blue}{0}}{x.re \cdot x.im - x.re \cdot x.im} \]

   5. +-inverses 0.0%

      \[\leadsto -3 + x.re \cdot \frac{\color{blue}{x.im \cdot x.im - x.im \cdot x.im}}{x.re \cdot x.im - x.re \cdot x.im} \]

   6. +-inverses 0.0%

      \[\leadsto -3 + x.re \cdot \frac{x.im \cdot x.im - x.im \cdot x.im}{\color{blue}{0}} \]

   7. +-inverses 0.0%

      \[\leadsto -3 + x.re \cdot \frac{x.im \cdot x.im - x.im \cdot x.im}{\color{blue}{x.im - x.im}} \]

   8. flip-+ 10.1%

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

   9. distribute-lft-in 10.1%

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

   10. *-un-lft-identity 10.1%

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

   11. fma-def 10.1%

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

6. Applied egg-rr 10.1%

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

8. Simplified 10.1%

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

9. Taylor expanded in x.im around inf 19.4%

   \[\leadsto \color{blue}{x.re \cdot x.im} \]

10. Final simplification 19.4%

    \[\leadsto x.im \cdot x.re \]

Alternative 9: 2.7% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ -10 \end{array} \]

FPCore

(FPCore (x.re x.im) :precision binary64 -10.0)

C

double code(double x_46_re, double x_46_im) {
	return -10.0;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = -10.0d0
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return -10.0;
}

Python

def code(x_46_re, x_46_im):
	return -10.0

Julia

function code(x_46_re, x_46_im)
	return -10.0
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = -10.0;
end

Wolfram

code[x$46$re_, x$46$im_] := -10.0

Derivation

1. Initial program 83.4%

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

   1. +-commutative 83.4%

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

   2. *-commutative 83.4%

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

   3. sub-neg 83.4%

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

   4. distribute-lft-in 81.1%

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

   5. associate-+r+ 81.1%

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

   6. distribute-rgt-neg-out 81.1%

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

   7. unsub-neg 81.1%

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

   8. associate-*r* 87.7%

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

   9. distribute-rgt-out 87.7%

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

   10. *-commutative 87.7%

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

   11. count-2 87.7%

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

   12. distribute-lft1-in 87.7%

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

   13. metadata-eval 87.7%

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

   14. *-commutative 87.7%

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

   15. *-commutative 87.7%

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

   16. associate-*r* 87.6%

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

   17. cube-unmult 87.7%

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

3. Simplified 87.7%

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

   1. associate-*r* 87.8%

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

   2. associate-*l* 87.7%

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

   3. flip-- 26.8%

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

   4. div-inv 26.4%

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

   5. swap-sqr 26.3%

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

   6. pow2 26.3%

      \[\leadsto \left(\color{blue}{{\left(x.re \cdot \left(x.re \cdot x.im\right)\right)}^{2}} \cdot \left(3 \cdot 3\right) - {x.im}^{3} \cdot {x.im}^{3}\right) \cdot \frac{1}{\left(x.re \cdot \left(x.re \cdot x.im\right)\right) \cdot 3 + {x.im}^{3}} \]

   7. metadata-eval 26.3%

      \[\leadsto \left({\left(x.re \cdot \left(x.re \cdot x.im\right)\right)}^{2} \cdot \color{blue}{9} - {x.im}^{3} \cdot {x.im}^{3}\right) \cdot \frac{1}{\left(x.re \cdot \left(x.re \cdot x.im\right)\right) \cdot 3 + {x.im}^{3}} \]

   8. pow-prod-up 26.2%

      \[\leadsto \left({\left(x.re \cdot \left(x.re \cdot x.im\right)\right)}^{2} \cdot 9 - \color{blue}{{x.im}^{\left(3 + 3\right)}}\right) \cdot \frac{1}{\left(x.re \cdot \left(x.re \cdot x.im\right)\right) \cdot 3 + {x.im}^{3}} \]

   9. metadata-eval 26.2%

      \[\leadsto \left({\left(x.re \cdot \left(x.re \cdot x.im\right)\right)}^{2} \cdot 9 - {x.im}^{\color{blue}{6}}\right) \cdot \frac{1}{\left(x.re \cdot \left(x.re \cdot x.im\right)\right) \cdot 3 + {x.im}^{3}} \]

   10. associate-*l* 26.2%

       \[\leadsto \left({\left(x.re \cdot \left(x.re \cdot x.im\right)\right)}^{2} \cdot 9 - {x.im}^{6}\right) \cdot \frac{1}{\color{blue}{x.re \cdot \left(\left(x.re \cdot x.im\right) \cdot 3\right)} + {x.im}^{3}} \]

   11. associate-*r* 26.2%

       \[\leadsto \left({\left(x.re \cdot \left(x.re \cdot x.im\right)\right)}^{2} \cdot 9 - {x.im}^{6}\right) \cdot \frac{1}{x.re \cdot \color{blue}{\left(x.re \cdot \left(x.im \cdot 3\right)\right)} + {x.im}^{3}} \]

   12. fma-def 26.2%

       \[\leadsto \left({\left(x.re \cdot \left(x.re \cdot x.im\right)\right)}^{2} \cdot 9 - {x.im}^{6}\right) \cdot \frac{1}{\color{blue}{\mathsf{fma}\left(x.re, x.re \cdot \left(x.im \cdot 3\right), {x.im}^{3}\right)}} \]

5. Applied egg-rr 26.2%

   \[\leadsto \color{blue}{\left({\left(x.re \cdot \left(x.re \cdot x.im\right)\right)}^{2} \cdot 9 - {x.im}^{6}\right) \cdot \frac{1}{\mathsf{fma}\left(x.re, x.re \cdot \left(x.im \cdot 3\right), {x.im}^{3}\right)}} \]

7. Simplified 2.9%

   \[\leadsto \color{blue}{-10} \]

8. Final simplification 2.9%

   \[\leadsto -10 \]

Developer target: 91.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(x.re \cdot x.im\right) \cdot \left(2 \cdot x.re\right) + \left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.re + x.im\right) \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (+ (* (* x.re x.im) (* 2.0 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im))))

C

double code(double x_46_re, double x_46_im) {
	return ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - 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_46im) * (2.0d0 * x_46re)) + ((x_46im * (x_46re - 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_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
}

Python

def code(x_46_re, x_46_im):
	return ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im))

Julia

function code(x_46_re, x_46_im)
	return Float64(Float64(Float64(x_46_re * x_46_im) * Float64(2.0 * x_46_re)) + Float64(Float64(x_46_im * Float64(x_46_re - x_46_im)) * Float64(x_46_re + x_46_im)))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
end

Wolfram

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

Reproduce

herbie shell --seed 2023178 
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"
  :precision binary64

  :herbie-target
  (+ (* (* x.re x.im) (* 2.0 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im)))

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