math.cube on complex, imaginary part

Percentage Accurate: 82.7% → 99.4%

Time: 7.5s

Alternatives: 16

Speedup: 1.4×

• 44.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 82.7% 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: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -6.2 \cdot 10^{+110} \lor \neg \left(x.im \leq 7.8 \cdot 10^{+69}\right):\\ \;\;\;\;x.im \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right) - {x.im}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -6.2e+110) (not (<= x.im 7.8e+69)))
   (* x.im (* (+ x.re x.im) (- x.re x.im)))
   (- (* x.re (* x.re (* x.im 3.0))) (pow x.im 3.0))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -6.2e+110) || !(x_46_im <= 7.8e+69)) {
		tmp = x_46_im * ((x_46_re + x_46_im) * (x_46_re - x_46_im));
	} else {
		tmp = (x_46_re * (x_46_re * (x_46_im * 3.0))) - 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 <= (-6.2d+110)) .or. (.not. (x_46im <= 7.8d+69))) then
        tmp = x_46im * ((x_46re + x_46im) * (x_46re - x_46im))
    else
        tmp = (x_46re * (x_46re * (x_46im * 3.0d0))) - (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 <= -6.2e+110) || !(x_46_im <= 7.8e+69)) {
		tmp = x_46_im * ((x_46_re + x_46_im) * (x_46_re - x_46_im));
	} else {
		tmp = (x_46_re * (x_46_re * (x_46_im * 3.0))) - Math.pow(x_46_im, 3.0);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_im <= -6.2e+110) or not (x_46_im <= 7.8e+69):
		tmp = x_46_im * ((x_46_re + x_46_im) * (x_46_re - x_46_im))
	else:
		tmp = (x_46_re * (x_46_re * (x_46_im * 3.0))) - 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 <= -6.2e+110) || !(x_46_im <= 7.8e+69))
		tmp = Float64(x_46_im * Float64(Float64(x_46_re + x_46_im) * Float64(x_46_re - x_46_im)));
	else
		tmp = Float64(Float64(x_46_re * Float64(x_46_re * Float64(x_46_im * 3.0))) - (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 <= -6.2e+110) || ~((x_46_im <= 7.8e+69)))
		tmp = x_46_im * ((x_46_re + x_46_im) * (x_46_re - x_46_im));
	else
		tmp = (x_46_re * (x_46_re * (x_46_im * 3.0))) - (x_46_im ^ 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$im, -6.2e+110], N[Not[LessEqual[x$46$im, 7.8e+69]], $MachinePrecision]], N[(x$46$im * N[(N[(x$46$re + x$46$im), $MachinePrecision] * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * N[(x$46$re * N[(x$46$im * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[x$46$im, 3.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -6.20000000000000035e110 or 7.7999999999999998e69 < x.im

   1. Initial program 63.2%

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

      1. *-commutative 63.2%

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

      2. *-commutative 63.2%

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

      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%

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

      8. associate-*r/ 0.0%

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

      9. +-inverses 0.0%

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

      10. +-inverses 0.0%

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

      11. distribute-lft-out-- 0.0%

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

      12. +-inverses 0.0%

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

      13. +-inverses 0.0%

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

      14. +-inverses 0.0%

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

      15. +-inverses 0.0%

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

      16. flip-+ 67.4%

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

   3. Applied egg-rr 67.4%

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

      1. +-commutative 67.4%

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

      2. add-cube-cbrt 67.4%

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

      3. fma-def 67.4%

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

   5. Applied egg-rr 0.0%

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

   7. Simplified 100.0%

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

   if -6.20000000000000035e110 < x.im < 7.7999999999999998e69

   1. Initial program 88.8%

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

      1. +-commutative 88.8%

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

      2. *-commutative 88.8%

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

      3. distribute-lft-out 88.8%

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

      4. associate-*l* 88.7%

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

      5. *-commutative 88.7%

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

      6. distribute-lft-out 88.7%

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

      7. associate-+r- 88.7%

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

      8. distribute-lft-out-- 88.7%

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

   3. Simplified 88.8%

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

      1. sub-neg 88.8%

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

      2. associate-*l* 88.9%

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

      3. associate-*l* 99.8%

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

   5. Applied egg-rr 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -6.2 \cdot 10^{+110} \lor \neg \left(x.im \leq 7.8 \cdot 10^{+69}\right):\\ \;\;\;\;x.im \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right) - {x.im}^{3}\\ \end{array} \]

Alternative 2: 99.8% accurate, 0.1× speedup

Math

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

FPCore

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

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (((x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im)))) <= ((double) INFINITY)) {
		tmp = fma((x_46_re + x_46_im), (x_46_im * (x_46_re - x_46_im)), (x_46_re * (x_46_re * (x_46_im + x_46_im))));
	} else {
		tmp = x_46_im * ((x_46_re + x_46_im) * (x_46_re - x_46_im));
	}
	return tmp;
}

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 91.8%

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

      1. *-commutative 91.8%

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

      2. *-commutative 91.8%

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

      3. difference-of-squares 91.8%

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

      4. associate-*l* 99.8%

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

      5. fma-def 99.8%

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

      6. *-commutative 99.8%

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

      7. *-commutative 99.8%

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

      8. *-commutative 99.8%

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

      9. distribute-lft-out 99.8%

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

   3. Simplified 99.8%

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

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

   1. Initial program 0.0%

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

      1. *-commutative 0.0%

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

      2. *-commutative 0.0%

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

      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%

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

      8. associate-*r/ 0.0%

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

      9. +-inverses 0.0%

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

      10. +-inverses 0.0%

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

      11. distribute-lft-out-- 0.0%

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

      12. +-inverses 0.0%

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

      13. +-inverses 0.0%

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

      14. +-inverses 0.0%

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

      15. +-inverses 0.0%

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

      16. flip-+ 20.0%

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

   3. Applied egg-rr 20.0%

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

      1. +-commutative 20.0%

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

      2. add-cube-cbrt 20.0%

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

      3. fma-def 20.0%

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

   5. Applied egg-rr 0.0%

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

   7. Simplified 100.0%

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

4. Final simplification 99.8%

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

Alternative 3: 95.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.im \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)\\ \mathbf{if}\;x.im \leq -2.5 \cdot 10^{-52}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.im \leq 2.12 \cdot 10^{-100}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right)\\ \mathbf{elif}\;x.im \leq 10^{+60}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot \left(x.im \cdot 3\right) - {x.im}^{3}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (* x.im (* (+ x.re x.im) (- x.re x.im)))))
   (if (<= x.im -2.5e-52)
     t_0
     (if (<= x.im 2.12e-100)
       (* x.re (* x.im (* x.re 3.0)))
       (if (<= x.im 1e+60)
         (- (* (* x.re x.re) (* x.im 3.0)) (pow x.im 3.0))
         t_0)))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_im * ((x_46_re + x_46_im) * (x_46_re - x_46_im));
	double tmp;
	if (x_46_im <= -2.5e-52) {
		tmp = t_0;
	} else if (x_46_im <= 2.12e-100) {
		tmp = x_46_re * (x_46_im * (x_46_re * 3.0));
	} else if (x_46_im <= 1e+60) {
		tmp = ((x_46_re * x_46_re) * (x_46_im * 3.0)) - pow(x_46_im, 3.0);
	} 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) :: tmp
    t_0 = x_46im * ((x_46re + x_46im) * (x_46re - x_46im))
    if (x_46im <= (-2.5d-52)) then
        tmp = t_0
    else if (x_46im <= 2.12d-100) then
        tmp = x_46re * (x_46im * (x_46re * 3.0d0))
    else if (x_46im <= 1d+60) then
        tmp = ((x_46re * x_46re) * (x_46im * 3.0d0)) - (x_46im ** 3.0d0)
    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_re + x_46_im) * (x_46_re - x_46_im));
	double tmp;
	if (x_46_im <= -2.5e-52) {
		tmp = t_0;
	} else if (x_46_im <= 2.12e-100) {
		tmp = x_46_re * (x_46_im * (x_46_re * 3.0));
	} else if (x_46_im <= 1e+60) {
		tmp = ((x_46_re * x_46_re) * (x_46_im * 3.0)) - Math.pow(x_46_im, 3.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	t_0 = x_46_im * ((x_46_re + x_46_im) * (x_46_re - x_46_im))
	tmp = 0
	if x_46_im <= -2.5e-52:
		tmp = t_0
	elif x_46_im <= 2.12e-100:
		tmp = x_46_re * (x_46_im * (x_46_re * 3.0))
	elif x_46_im <= 1e+60:
		tmp = ((x_46_re * x_46_re) * (x_46_im * 3.0)) - math.pow(x_46_im, 3.0)
	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_re + x_46_im) * Float64(x_46_re - x_46_im)))
	tmp = 0.0
	if (x_46_im <= -2.5e-52)
		tmp = t_0;
	elseif (x_46_im <= 2.12e-100)
		tmp = Float64(x_46_re * Float64(x_46_im * Float64(x_46_re * 3.0)));
	elseif (x_46_im <= 1e+60)
		tmp = Float64(Float64(Float64(x_46_re * x_46_re) * Float64(x_46_im * 3.0)) - (x_46_im ^ 3.0));
	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_re + x_46_im) * (x_46_re - x_46_im));
	tmp = 0.0;
	if (x_46_im <= -2.5e-52)
		tmp = t_0;
	elseif (x_46_im <= 2.12e-100)
		tmp = x_46_re * (x_46_im * (x_46_re * 3.0));
	elseif (x_46_im <= 1e+60)
		tmp = ((x_46_re * x_46_re) * (x_46_im * 3.0)) - (x_46_im ^ 3.0);
	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$46$re + x$46$im), $MachinePrecision] * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$im, -2.5e-52], t$95$0, If[LessEqual[x$46$im, 2.12e-100], N[(x$46$re * N[(x$46$im * N[(x$46$re * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 1e+60], N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] * N[(x$46$im * 3.0), $MachinePrecision]), $MachinePrecision] - N[Power[x$46$im, 3.0], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -2.5e-52 or 9.9999999999999995e59 < x.im

   1. Initial program 71.3%

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

      1. *-commutative 71.3%

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

      2. *-commutative 71.3%

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

      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%

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

      8. associate-*r/ 0.0%

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

      9. +-inverses 0.0%

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

      10. +-inverses 0.0%

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

      11. distribute-lft-out-- 0.0%

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

      12. +-inverses 0.0%

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

      13. +-inverses 0.0%

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

      14. +-inverses 0.0%

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

      15. +-inverses 0.0%

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

      16. flip-+ 72.5%

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

   3. Applied egg-rr 72.5%

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

      1. +-commutative 72.5%

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

      2. add-cube-cbrt 72.5%

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

      3. fma-def 72.5%

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

   5. Applied egg-rr 0.0%

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

   7. Simplified 98.4%

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

   if -2.5e-52 < x.im < 2.1200000000000001e-100

   1. Initial program 82.8%

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

      1. +-commutative 82.8%

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

      2. *-commutative 82.8%

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

      3. distribute-lft-out 82.8%

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

      4. associate-*l* 82.8%

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

      5. *-commutative 82.8%

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

      6. distribute-lft-out 82.8%

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

      7. associate-+r- 82.8%

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

      8. distribute-lft-out-- 82.8%

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

   3. Simplified 82.7%

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

      1. sub-neg 82.7%

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

      2. associate-*l* 82.8%

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

      3. associate-*l* 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. unsub-neg 99.7%

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

      2. associate-*r* 82.8%

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

   7. Applied egg-rr 82.8%

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

   8. Taylor expanded in x.re around inf 80.1%

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

   10. Simplified 80.1%

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

   11. Taylor expanded in x.im around 0 80.1%

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

       1. *-commutative 80.1%

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

       2. unpow2 80.1%

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

       3. associate-*l* 97.0%

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

       4. *-commutative 97.0%

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

       5. associate-*r* 97.1%

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

       6. associate-*l* 97.1%

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

   13. Simplified 97.1%

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

   if 2.1200000000000001e-100 < x.im < 9.9999999999999995e59

   1. Initial program 97.1%

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

      1. +-commutative 97.1%

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

      2. *-commutative 97.1%

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

      3. distribute-lft-out 97.1%

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

      4. associate-*l* 97.1%

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

      5. *-commutative 97.1%

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

      6. distribute-lft-out 97.0%

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

      7. associate-+r- 97.0%

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

      8. distribute-lft-out-- 97.1%

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

   3. Simplified 97.3%

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

      1. sub-neg 97.3%

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

      2. associate-*l* 97.3%

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

      3. associate-*l* 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. unsub-neg 100.0%

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

      2. associate-*r* 97.3%

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

   7. Applied egg-rr 97.3%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -2.5 \cdot 10^{-52}:\\ \;\;\;\;x.im \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)\\ \mathbf{elif}\;x.im \leq 2.12 \cdot 10^{-100}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right)\\ \mathbf{elif}\;x.im \leq 10^{+60}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot \left(x.im \cdot 3\right) - {x.im}^{3}\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)\\ \end{array} \]

Alternative 4: 95.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.im \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)\\ \mathbf{if}\;x.im \leq -1.35 \cdot 10^{-50}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.im \leq 2.15 \cdot 10^{-100}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right)\\ \mathbf{elif}\;x.im \leq 4 \cdot 10^{+67}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (* x.im (* (+ x.re x.im) (- x.re x.im)))))
   (if (<= x.im -1.35e-50)
     t_0
     (if (<= x.im 2.15e-100)
       (* x.re (* x.im (* x.re 3.0)))
       (if (<= x.im 4e+67)
         (+
          (* x.im (- (* x.re x.re) (* x.im x.im)))
          (* x.re (+ (* x.re x.im) (* x.re x.im))))
         t_0)))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_im * ((x_46_re + x_46_im) * (x_46_re - x_46_im));
	double tmp;
	if (x_46_im <= -1.35e-50) {
		tmp = t_0;
	} else if (x_46_im <= 2.15e-100) {
		tmp = x_46_re * (x_46_im * (x_46_re * 3.0));
	} else if (x_46_im <= 4e+67) {
		tmp = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im)));
	} 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) :: tmp
    t_0 = x_46im * ((x_46re + x_46im) * (x_46re - x_46im))
    if (x_46im <= (-1.35d-50)) then
        tmp = t_0
    else if (x_46im <= 2.15d-100) then
        tmp = x_46re * (x_46im * (x_46re * 3.0d0))
    else if (x_46im <= 4d+67) then
        tmp = (x_46im * ((x_46re * x_46re) - (x_46im * x_46im))) + (x_46re * ((x_46re * x_46im) + (x_46re * x_46im)))
    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_re + x_46_im) * (x_46_re - x_46_im));
	double tmp;
	if (x_46_im <= -1.35e-50) {
		tmp = t_0;
	} else if (x_46_im <= 2.15e-100) {
		tmp = x_46_re * (x_46_im * (x_46_re * 3.0));
	} else if (x_46_im <= 4e+67) {
		tmp = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	t_0 = x_46_im * ((x_46_re + x_46_im) * (x_46_re - x_46_im))
	tmp = 0
	if x_46_im <= -1.35e-50:
		tmp = t_0
	elif x_46_im <= 2.15e-100:
		tmp = x_46_re * (x_46_im * (x_46_re * 3.0))
	elif x_46_im <= 4e+67:
		tmp = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im)))
	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_re + x_46_im) * Float64(x_46_re - x_46_im)))
	tmp = 0.0
	if (x_46_im <= -1.35e-50)
		tmp = t_0;
	elseif (x_46_im <= 2.15e-100)
		tmp = Float64(x_46_re * Float64(x_46_im * Float64(x_46_re * 3.0)));
	elseif (x_46_im <= 4e+67)
		tmp = 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_re * x_46_im) + Float64(x_46_re * x_46_im))));
	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_re + x_46_im) * (x_46_re - x_46_im));
	tmp = 0.0;
	if (x_46_im <= -1.35e-50)
		tmp = t_0;
	elseif (x_46_im <= 2.15e-100)
		tmp = x_46_re * (x_46_im * (x_46_re * 3.0));
	elseif (x_46_im <= 4e+67)
		tmp = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im)));
	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$46$re + x$46$im), $MachinePrecision] * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$im, -1.35e-50], t$95$0, If[LessEqual[x$46$im, 2.15e-100], N[(x$46$re * N[(x$46$im * N[(x$46$re * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 4e+67], 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$46$re * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -1.35e-50 or 3.99999999999999993e67 < x.im

   1. Initial program 71.3%

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

      1. *-commutative 71.3%

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

      2. *-commutative 71.3%

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

      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%

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

      8. associate-*r/ 0.0%

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

      9. +-inverses 0.0%

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

      10. +-inverses 0.0%

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

      11. distribute-lft-out-- 0.0%

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

      12. +-inverses 0.0%

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

      13. +-inverses 0.0%

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

      14. +-inverses 0.0%

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

      15. +-inverses 0.0%

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

      16. flip-+ 72.5%

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

   3. Applied egg-rr 72.5%

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

      1. +-commutative 72.5%

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

      2. add-cube-cbrt 72.5%

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

      3. fma-def 72.5%

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

   5. Applied egg-rr 0.0%

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

   7. Simplified 98.4%

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

   if -1.35e-50 < x.im < 2.14999999999999999e-100

   1. Initial program 82.8%

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

      1. +-commutative 82.8%

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

      2. *-commutative 82.8%

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

      3. distribute-lft-out 82.8%

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

      4. associate-*l* 82.8%

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

      5. *-commutative 82.8%

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

      6. distribute-lft-out 82.8%

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

      7. associate-+r- 82.8%

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

      8. distribute-lft-out-- 82.8%

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

   3. Simplified 82.7%

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

      1. sub-neg 82.7%

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

      2. associate-*l* 82.8%

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

      3. associate-*l* 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. unsub-neg 99.7%

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

      2. associate-*r* 82.8%

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

   7. Applied egg-rr 82.8%

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

   8. Taylor expanded in x.re around inf 80.1%

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

   10. Simplified 80.1%

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

   11. Taylor expanded in x.im around 0 80.1%

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

       1. *-commutative 80.1%

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

       2. unpow2 80.1%

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

       3. associate-*l* 97.0%

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

       4. *-commutative 97.0%

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

       5. associate-*r* 97.1%

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

       6. associate-*l* 97.1%

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

   13. Simplified 97.1%

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

   if 2.14999999999999999e-100 < x.im < 3.99999999999999993e67

   1. Initial program 97.1%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -1.35 \cdot 10^{-50}:\\ \;\;\;\;x.im \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)\\ \mathbf{elif}\;x.im \leq 2.15 \cdot 10^{-100}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right)\\ \mathbf{elif}\;x.im \leq 4 \cdot 10^{+67}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)\\ \end{array} \]

Alternative 5: 92.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -1.55 \cdot 10^{-50}:\\ \;\;\;\;x.im \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)\\ \mathbf{elif}\;x.im \leq 9.6 \cdot 10^{-96}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\frac{\frac{1}{x.re - x.im}}{x.re + x.im}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im -1.55e-50)
   (* x.im (* (+ x.re x.im) (- x.re x.im)))
   (if (<= x.im 9.6e-96)
     (* x.re (* x.im (* x.re 3.0)))
     (/ x.im (/ (/ 1.0 (- x.re x.im)) (+ x.re x.im))))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= -1.55e-50) {
		tmp = x_46_im * ((x_46_re + x_46_im) * (x_46_re - x_46_im));
	} else if (x_46_im <= 9.6e-96) {
		tmp = x_46_re * (x_46_im * (x_46_re * 3.0));
	} else {
		tmp = x_46_im / ((1.0 / (x_46_re - x_46_im)) / (x_46_re + x_46_im));
	}
	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 <= (-1.55d-50)) then
        tmp = x_46im * ((x_46re + x_46im) * (x_46re - x_46im))
    else if (x_46im <= 9.6d-96) then
        tmp = x_46re * (x_46im * (x_46re * 3.0d0))
    else
        tmp = x_46im / ((1.0d0 / (x_46re - x_46im)) / (x_46re + x_46im))
    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 <= -1.55e-50) {
		tmp = x_46_im * ((x_46_re + x_46_im) * (x_46_re - x_46_im));
	} else if (x_46_im <= 9.6e-96) {
		tmp = x_46_re * (x_46_im * (x_46_re * 3.0));
	} else {
		tmp = x_46_im / ((1.0 / (x_46_re - x_46_im)) / (x_46_re + x_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= -1.55e-50:
		tmp = x_46_im * ((x_46_re + x_46_im) * (x_46_re - x_46_im))
	elif x_46_im <= 9.6e-96:
		tmp = x_46_re * (x_46_im * (x_46_re * 3.0))
	else:
		tmp = x_46_im / ((1.0 / (x_46_re - x_46_im)) / (x_46_re + x_46_im))
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= -1.55e-50)
		tmp = Float64(x_46_im * Float64(Float64(x_46_re + x_46_im) * Float64(x_46_re - x_46_im)));
	elseif (x_46_im <= 9.6e-96)
		tmp = Float64(x_46_re * Float64(x_46_im * Float64(x_46_re * 3.0)));
	else
		tmp = Float64(x_46_im / Float64(Float64(1.0 / Float64(x_46_re - x_46_im)) / Float64(x_46_re + x_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_im <= -1.55e-50)
		tmp = x_46_im * ((x_46_re + x_46_im) * (x_46_re - x_46_im));
	elseif (x_46_im <= 9.6e-96)
		tmp = x_46_re * (x_46_im * (x_46_re * 3.0));
	else
		tmp = x_46_im / ((1.0 / (x_46_re - x_46_im)) / (x_46_re + x_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, -1.55e-50], N[(x$46$im * N[(N[(x$46$re + x$46$im), $MachinePrecision] * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 9.6e-96], N[(x$46$re * N[(x$46$im * N[(x$46$re * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im / N[(N[(1.0 / N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision] / N[(x$46$re + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -1.5500000000000001e-50

   1. Initial program 79.4%

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

      1. *-commutative 79.4%

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

      2. *-commutative 79.4%

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

      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%

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

      8. associate-*r/ 0.0%

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

      9. +-inverses 0.0%

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

      10. +-inverses 0.0%

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

      11. distribute-lft-out-- 0.0%

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

      12. +-inverses 0.0%

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

      13. +-inverses 0.0%

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

      14. +-inverses 0.0%

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

      15. +-inverses 0.0%

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

      16. flip-+ 77.3%

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

   3. Applied egg-rr 77.3%

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

      1. +-commutative 77.3%

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

      2. add-cube-cbrt 77.3%

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

      3. fma-def 77.3%

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

   5. Applied egg-rr 0.0%

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

   7. Simplified 97.4%

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

   if -1.5500000000000001e-50 < x.im < 9.60000000000000076e-96

   1. Initial program 82.8%

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

      1. +-commutative 82.8%

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

      2. *-commutative 82.8%

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

      3. distribute-lft-out 82.8%

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

      4. associate-*l* 82.8%

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

      5. *-commutative 82.8%

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

      6. distribute-lft-out 82.8%

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

      7. associate-+r- 82.8%

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

      8. distribute-lft-out-- 82.8%

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

   3. Simplified 82.7%

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

      1. sub-neg 82.7%

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

      2. associate-*l* 82.8%

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

      3. associate-*l* 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. unsub-neg 99.7%

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

      2. associate-*r* 82.8%

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

   7. Applied egg-rr 82.8%

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

   8. Taylor expanded in x.re around inf 80.1%

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

   10. Simplified 80.1%

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

   11. Taylor expanded in x.im around 0 80.1%

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

       1. *-commutative 80.1%

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

       2. unpow2 80.1%

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

       3. associate-*l* 97.0%

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

       4. *-commutative 97.0%

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

       5. associate-*r* 97.1%

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

       6. associate-*l* 97.1%

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

   13. Simplified 97.1%

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

   if 9.60000000000000076e-96 < x.im

   1. Initial program 75.0%

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

      1. *-commutative 75.0%

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

      2. *-commutative 75.0%

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

      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%

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

      8. associate-*r/ 0.0%

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

      9. +-inverses 0.0%

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

      10. +-inverses 0.0%

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

      11. distribute-lft-out-- 0.0%

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

      12. +-inverses 0.0%

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

      13. +-inverses 0.0%

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

      14. +-inverses 0.0%

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

      15. +-inverses 0.0%

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

      16. flip-+ 67.5%

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

   3. Applied egg-rr 67.5%

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

      1. +-commutative 67.5%

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

      2. add-cube-cbrt 67.5%

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

      3. fma-def 67.5%

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

   5. Applied egg-rr 0.0%

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

   7. Simplified 91.0%

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

      1. associate-*r* 91.1%

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

      2. distribute-rgt-in 81.6%

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

      3. flip-- 73.1%

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

      4. +-commutative 73.1%

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

      5. associate-*r/ 67.5%

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

      6. distribute-rgt-in 69.9%

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

   9. Applied egg-rr 69.9%

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

       1. associate-/l* 75.4%

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

       2. associate-/l* 75.5%

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

       3. difference-of-squares 90.9%

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

       4. +-commutative 90.9%

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

       5. associate-/r* 91.1%

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

       6. *-inverses 91.1%

          \[\leadsto 0 + \frac{x.im}{\frac{\frac{\color{blue}{1}}{x.re - x.im}}{x.im + x.re}} \]

       7. +-commutative 91.1%

          \[\leadsto 0 + \frac{x.im}{\frac{\frac{1}{x.re - x.im}}{\color{blue}{x.re + x.im}}} \]

   11. Simplified 91.1%

       \[\leadsto 0 + \color{blue}{\frac{x.im}{\frac{\frac{1}{x.re - x.im}}{x.re + x.im}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -1.55 \cdot 10^{-50}:\\ \;\;\;\;x.im \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)\\ \mathbf{elif}\;x.im \leq 9.6 \cdot 10^{-96}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\frac{\frac{1}{x.re - x.im}}{x.re + x.im}}\\ \end{array} \]

Alternative 6: 92.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -1.7 \cdot 10^{-49} \lor \neg \left(x.im \leq 5.5 \cdot 10^{-98}\right):\\ \;\;\;\;x.im \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -1.7e-49) (not (<= x.im 5.5e-98)))
   (* x.im (* (+ x.re x.im) (- 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 <= -1.7e-49) || !(x_46_im <= 5.5e-98)) {
		tmp = x_46_im * ((x_46_re + x_46_im) * (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 <= (-1.7d-49)) .or. (.not. (x_46im <= 5.5d-98))) then
        tmp = x_46im * ((x_46re + x_46im) * (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 <= -1.7e-49) || !(x_46_im <= 5.5e-98)) {
		tmp = x_46_im * ((x_46_re + x_46_im) * (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 <= -1.7e-49) or not (x_46_im <= 5.5e-98):
		tmp = x_46_im * ((x_46_re + x_46_im) * (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 <= -1.7e-49) || !(x_46_im <= 5.5e-98))
		tmp = Float64(x_46_im * Float64(Float64(x_46_re + x_46_im) * 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 <= -1.7e-49) || ~((x_46_im <= 5.5e-98)))
		tmp = x_46_im * ((x_46_re + x_46_im) * (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, -1.7e-49], N[Not[LessEqual[x$46$im, 5.5e-98]], $MachinePrecision]], N[(x$46$im * N[(N[(x$46$re + x$46$im), $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 < -1.70000000000000002e-49 or 5.4999999999999997e-98 < x.im

   1. Initial program 77.0%

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

      1. *-commutative 77.0%

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

      2. *-commutative 77.0%

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

      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%

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

      8. associate-*r/ 0.0%

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

      9. +-inverses 0.0%

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

      10. +-inverses 0.0%

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

      11. distribute-lft-out-- 0.0%

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

      12. +-inverses 0.0%

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

      13. +-inverses 0.0%

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

      14. +-inverses 0.0%

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

      15. +-inverses 0.0%

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

      16. flip-+ 72.0%

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

   3. Applied egg-rr 72.0%

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

      1. +-commutative 72.0%

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

      2. add-cube-cbrt 72.0%

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

      3. fma-def 72.0%

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

   5. Applied egg-rr 0.0%

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

   7. Simplified 94.0%

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

   if -1.70000000000000002e-49 < x.im < 5.4999999999999997e-98

   1. Initial program 82.8%

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

      1. +-commutative 82.8%

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

      2. *-commutative 82.8%

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

      3. distribute-lft-out 82.8%

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

      4. associate-*l* 82.8%

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

      5. *-commutative 82.8%

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

      6. distribute-lft-out 82.8%

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

      7. associate-+r- 82.8%

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

      8. distribute-lft-out-- 82.8%

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

   3. Simplified 82.7%

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

      1. sub-neg 82.7%

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

      2. associate-*l* 82.8%

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

      3. associate-*l* 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. unsub-neg 99.7%

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

      2. associate-*r* 82.8%

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

   7. Applied egg-rr 82.8%

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

   8. Taylor expanded in x.re around inf 80.1%

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

   10. Simplified 80.1%

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

   11. Taylor expanded in x.im around 0 80.1%

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

       1. *-commutative 80.1%

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

       2. unpow2 80.1%

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

       3. associate-*l* 97.0%

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

       4. *-commutative 97.0%

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

       5. associate-*r* 97.1%

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

       6. associate-*l* 97.1%

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

   13. Simplified 97.1%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -1.7 \cdot 10^{-49} \lor \neg \left(x.im \leq 5.5 \cdot 10^{-98}\right):\\ \;\;\;\;x.im \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right)\\ \end{array} \]

Alternative 7: 54.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -2.2 \cdot 10^{+145}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\left(x.re \cdot x.re\right) \cdot x.im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im -2.2e+145)
   (* x.re (* x.im x.im))
   (* 3.0 (* (* x.re x.re) x.im))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= -2.2e+145) {
		tmp = x_46_re * (x_46_im * x_46_im);
	} else {
		tmp = 3.0 * ((x_46_re * x_46_re) * x_46_im);
	}
	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 <= (-2.2d+145)) then
        tmp = x_46re * (x_46im * x_46im)
    else
        tmp = 3.0d0 * ((x_46re * x_46re) * x_46im)
    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 <= -2.2e+145) {
		tmp = x_46_re * (x_46_im * x_46_im);
	} else {
		tmp = 3.0 * ((x_46_re * x_46_re) * x_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= -2.2e+145:
		tmp = x_46_re * (x_46_im * x_46_im)
	else:
		tmp = 3.0 * ((x_46_re * x_46_re) * x_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= -2.2e+145)
		tmp = Float64(x_46_re * Float64(x_46_im * x_46_im));
	else
		tmp = Float64(3.0 * Float64(Float64(x_46_re * x_46_re) * x_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_im <= -2.2e+145)
		tmp = x_46_re * (x_46_im * x_46_im);
	else
		tmp = 3.0 * ((x_46_re * x_46_re) * x_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, -2.2e+145], N[(x$46$re * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision], N[(3.0 * N[(N[(x$46$re * x$46$re), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -2.20000000000000009e145

   1. Initial program 61.5%

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

      1. *-commutative 61.5%

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

      2. *-commutative 61.5%

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

      3. difference-of-squares 74.4%

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

      4. associate-*l* 74.4%

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

      5. fma-def 74.4%

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

      6. *-commutative 74.4%

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

      7. *-commutative 74.4%

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

      8. *-commutative 74.4%

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

      9. distribute-lft-out 74.4%

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

   3. Simplified 74.4%

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

   4. Taylor expanded in x.im around 0 29.0%

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

   5. Taylor expanded in x.re around 0 46.2%

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

      1. unpow2 46.2%

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

   7. Simplified 46.2%

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

   if -2.20000000000000009e145 < x.im

   1. Initial program 82.4%

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

      1. +-commutative 82.4%

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

      2. *-commutative 82.4%

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

      3. distribute-lft-out 82.4%

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

      4. associate-*l* 82.4%

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

      5. *-commutative 82.4%

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

      6. distribute-lft-out 85.6%

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

      7. associate-+r- 85.6%

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

      8. distribute-lft-out-- 78.3%

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

   3. Simplified 78.3%

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

      1. sub-neg 78.3%

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

      2. associate-*l* 78.4%

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

      3. associate-*l* 86.5%

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

   5. Applied egg-rr 86.5%

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

      1. unsub-neg 86.5%

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

      2. associate-*r* 78.4%

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

   7. Applied egg-rr 78.4%

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

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

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

   10. 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 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -2.2 \cdot 10^{+145}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\left(x.re \cdot x.re\right) \cdot x.im\right)\\ \end{array} \]

Alternative 8: 54.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -1.35 \cdot 10^{+148}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.re\right) \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im -1.35e+148)
   (* x.re (* x.im x.im))
   (* x.im (* (* x.re x.re) 3.0))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= -1.35e+148) {
		tmp = x_46_re * (x_46_im * x_46_im);
	} else {
		tmp = x_46_im * ((x_46_re * 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 <= (-1.35d+148)) then
        tmp = x_46re * (x_46im * x_46im)
    else
        tmp = x_46im * ((x_46re * 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 <= -1.35e+148) {
		tmp = x_46_re * (x_46_im * x_46_im);
	} else {
		tmp = x_46_im * ((x_46_re * x_46_re) * 3.0);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= -1.35e+148:
		tmp = x_46_re * (x_46_im * x_46_im)
	else:
		tmp = x_46_im * ((x_46_re * x_46_re) * 3.0)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= -1.35e+148)
		tmp = Float64(x_46_re * Float64(x_46_im * x_46_im));
	else
		tmp = Float64(x_46_im * Float64(Float64(x_46_re * 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 <= -1.35e+148)
		tmp = x_46_re * (x_46_im * x_46_im);
	else
		tmp = x_46_im * ((x_46_re * x_46_re) * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, -1.35e+148], N[(x$46$re * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(N[(x$46$re * x$46$re), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -1.35000000000000009e148

   1. Initial program 61.5%

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

      1. *-commutative 61.5%

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

      2. *-commutative 61.5%

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

      3. difference-of-squares 74.4%

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

      4. associate-*l* 74.4%

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

      5. fma-def 74.4%

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

      6. *-commutative 74.4%

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

      7. *-commutative 74.4%

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

      8. *-commutative 74.4%

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

      9. distribute-lft-out 74.4%

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

   3. Simplified 74.4%

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

   4. Taylor expanded in x.im around 0 29.0%

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

   5. Taylor expanded in x.re around 0 46.2%

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

      1. unpow2 46.2%

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

   7. Simplified 46.2%

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

   if -1.35000000000000009e148 < x.im

   1. Initial program 82.4%

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

      1. +-commutative 82.4%

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

      2. *-commutative 82.4%

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

      3. distribute-lft-out 82.4%

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

      4. associate-*l* 82.4%

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

      5. *-commutative 82.4%

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

      6. distribute-lft-out 85.6%

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

      7. associate-+r- 85.6%

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

      8. distribute-lft-out-- 78.3%

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

   3. Simplified 78.3%

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

      1. sub-neg 78.3%

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

      2. associate-*l* 78.4%

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

      3. associate-*l* 86.5%

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

   5. Applied egg-rr 86.5%

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

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

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

      1. *-commutative 56.3%

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

      2. *-commutative 56.3%

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

      3. associate-*l* 56.4%

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

      4. *-commutative 56.4%

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

      5. unpow2 56.3%

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

   8. Simplified 56.3%

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

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -1.35 \cdot 10^{+148}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.re\right) \cdot 3\right)\\ \end{array} \]

Alternative 9: 60.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -4.6 \cdot 10^{+149}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re \cdot x.im\right) \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im -4.6e+149)
   (* x.re (* x.im x.im))
   (* x.re (* (* x.re x.im) 3.0))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= -4.6e+149) {
		tmp = x_46_re * (x_46_im * x_46_im);
	} else {
		tmp = x_46_re * ((x_46_re * 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 <= (-4.6d+149)) then
        tmp = x_46re * (x_46im * x_46im)
    else
        tmp = x_46re * ((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 <= -4.6e+149) {
		tmp = x_46_re * (x_46_im * x_46_im);
	} else {
		tmp = x_46_re * ((x_46_re * x_46_im) * 3.0);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= -4.6e+149:
		tmp = x_46_re * (x_46_im * x_46_im)
	else:
		tmp = x_46_re * ((x_46_re * x_46_im) * 3.0)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= -4.6e+149)
		tmp = Float64(x_46_re * Float64(x_46_im * x_46_im));
	else
		tmp = Float64(x_46_re * Float64(Float64(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 <= -4.6e+149)
		tmp = x_46_re * (x_46_im * x_46_im);
	else
		tmp = x_46_re * ((x_46_re * x_46_im) * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, -4.6e+149], N[(x$46$re * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision], N[(x$46$re * N[(N[(x$46$re * x$46$im), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -4.5999999999999997e149

   1. Initial program 61.5%

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

      1. *-commutative 61.5%

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

      2. *-commutative 61.5%

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

      3. difference-of-squares 74.4%

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

      4. associate-*l* 74.4%

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

      5. fma-def 74.4%

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

      6. *-commutative 74.4%

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

      7. *-commutative 74.4%

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

      8. *-commutative 74.4%

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

      9. distribute-lft-out 74.4%

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

   3. Simplified 74.4%

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

   4. Taylor expanded in x.im around 0 29.0%

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

   5. Taylor expanded in x.re around 0 46.2%

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

      1. unpow2 46.2%

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

   7. Simplified 46.2%

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

   if -4.5999999999999997e149 < x.im

   1. Initial program 82.4%

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

      1. *-commutative 82.4%

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

      2. *-commutative 82.4%

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

      3. difference-of-squares 85.2%

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

      4. associate-*l* 93.3%

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

      5. fma-def 93.3%

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

      6. *-commutative 93.3%

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

      7. *-commutative 93.3%

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

      8. *-commutative 93.3%

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

      9. distribute-lft-out 93.3%

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

   3. Simplified 93.3%

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

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

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

   5. Taylor expanded in x.re around 0 54.0%

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

      1. +-commutative 54.0%

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

      2. unpow2 54.0%

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

      3. distribute-rgt1-in 54.0%

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

      4. metadata-eval 54.0%

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

      5. *-commutative 54.0%

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

      6. unpow2 54.0%

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

      7. associate-*l* 62.1%

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

      8. distribute-lft-out 66.8%

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

      9. *-commutative 66.8%

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

      10. associate-*l* 66.8%

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

   7. Simplified 66.8%

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

   8. Taylor expanded in x.im around 0 64.5%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -4.6 \cdot 10^{+149}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re \cdot x.im\right) \cdot 3\right)\\ \end{array} \]

Alternative 10: 60.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -1.76 \cdot 10^{+149}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im -1.76e+149)
   (* x.re (* 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 <= -1.76e+149) {
		tmp = x_46_re * (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 <= (-1.76d+149)) then
        tmp = x_46re * (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 <= -1.76e+149) {
		tmp = x_46_re * (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 <= -1.76e+149:
		tmp = x_46_re * (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 <= -1.76e+149)
		tmp = Float64(x_46_re * Float64(x_46_im * 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 <= -1.76e+149)
		tmp = x_46_re * (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[LessEqual[x$46$im, -1.76e+149], N[(x$46$re * 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 < -1.7599999999999999e149

   1. Initial program 61.5%

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

      1. *-commutative 61.5%

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

      2. *-commutative 61.5%

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

      3. difference-of-squares 74.4%

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

      4. associate-*l* 74.4%

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

      5. fma-def 74.4%

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

      6. *-commutative 74.4%

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

      7. *-commutative 74.4%

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

      8. *-commutative 74.4%

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

      9. distribute-lft-out 74.4%

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

   3. Simplified 74.4%

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

   4. Taylor expanded in x.im around 0 29.0%

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

   5. Taylor expanded in x.re around 0 46.2%

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

      1. unpow2 46.2%

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

   7. Simplified 46.2%

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

   if -1.7599999999999999e149 < x.im

   1. Initial program 82.4%

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

      1. +-commutative 82.4%

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

      2. *-commutative 82.4%

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

      3. distribute-lft-out 82.4%

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

      4. associate-*l* 82.4%

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

      5. *-commutative 82.4%

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

      6. distribute-lft-out 85.6%

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

      7. associate-+r- 85.6%

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

      8. distribute-lft-out-- 78.3%

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

   3. Simplified 78.3%

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

      1. sub-neg 78.3%

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

      2. associate-*l* 78.4%

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

      3. associate-*l* 86.5%

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

   5. Applied egg-rr 86.5%

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

      1. unsub-neg 86.5%

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

      2. associate-*r* 78.4%

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

   7. Applied egg-rr 78.4%

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

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

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

   10. Simplified 56.3%

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

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

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

       1. *-commutative 56.3%

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

       2. unpow2 56.3%

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

       3. associate-*l* 64.5%

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

       4. *-commutative 64.5%

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

       5. associate-*r* 64.5%

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

       6. associate-*l* 64.5%

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

   13. Simplified 64.5%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -1.76 \cdot 10^{+149}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right)\\ \end{array} \]

Alternative 11: 67.4% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

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 * (x_46im + (x_46re * 3.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

   1. *-commutative 79.3%

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

   2. *-commutative 79.3%

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

   3. difference-of-squares 83.6%

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

   4. associate-*l* 90.4%

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

   5. fma-def 90.5%

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

   6. *-commutative 90.5%

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

   7. *-commutative 90.5%

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

   8. *-commutative 90.5%

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

   9. distribute-lft-out 90.5%

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

3. Simplified 90.5%

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

4. Taylor expanded in x.im around 0 61.0%

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

5. Taylor expanded in x.re around 0 51.3%

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

   1. +-commutative 51.3%

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

   2. unpow2 51.3%

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

   3. distribute-rgt1-in 51.3%

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

   4. metadata-eval 51.3%

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

   5. *-commutative 51.3%

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

   6. unpow2 51.3%

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

   7. associate-*l* 58.1%

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

   8. distribute-lft-out 62.1%

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

   9. *-commutative 62.1%

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

   10. associate-*l* 62.1%

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

7. Simplified 62.1%

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

   1. distribute-lft-out 67.1%

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

   2. *-commutative 67.1%

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

   3. *-commutative 67.1%

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

9. Applied egg-rr 67.1%

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

10. Final simplification 67.1%

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

Alternative 12: 37.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -4 \cdot 10^{+147}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.im\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im -4e+147) (* x.im (* x.re x.im)) (* (* x.re x.re) x.im)))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= -4e+147) {
		tmp = x_46_im * (x_46_re * x_46_im);
	} else {
		tmp = (x_46_re * x_46_re) * x_46_im;
	}
	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+147)) then
        tmp = x_46im * (x_46re * x_46im)
    else
        tmp = (x_46re * x_46re) * x_46im
    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+147) {
		tmp = x_46_im * (x_46_re * x_46_im);
	} else {
		tmp = (x_46_re * x_46_re) * x_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= -4e+147:
		tmp = x_46_im * (x_46_re * x_46_im)
	else:
		tmp = (x_46_re * x_46_re) * x_46_im
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= -4e+147)
		tmp = Float64(x_46_im * Float64(x_46_re * x_46_im));
	else
		tmp = Float64(Float64(x_46_re * x_46_re) * x_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_im <= -4e+147)
		tmp = x_46_im * (x_46_re * x_46_im);
	else
		tmp = (x_46_re * x_46_re) * x_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, -4e+147], N[(x$46$im * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * x$46$re), $MachinePrecision] * x$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -3.9999999999999999e147

   1. Initial program 61.5%

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

      1. *-commutative 61.5%

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

      2. *-commutative 61.5%

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

      3. difference-of-squares 74.4%

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

      4. associate-*l* 74.4%

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

      5. fma-def 74.4%

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

      6. *-commutative 74.4%

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

      7. *-commutative 74.4%

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

      8. *-commutative 74.4%

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

      9. distribute-lft-out 74.4%

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

   3. Simplified 74.4%

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

   4. Taylor expanded in x.im around 0 29.0%

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

   5. Taylor expanded in x.re around 0 46.2%

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

      1. unpow2 46.2%

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

      2. *-commutative 46.2%

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

      3. associate-*l* 36.7%

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

   7. Simplified 36.7%

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

   if -3.9999999999999999e147 < x.im

   1. Initial program 82.4%

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

      1. *-commutative 82.4%

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

      2. *-commutative 82.4%

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

      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%

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

      8. associate-*r/ 0.0%

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

      9. +-inverses 0.0%

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

      10. +-inverses 0.0%

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

      11. distribute-lft-out-- 0.0%

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

      12. +-inverses 0.0%

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

      13. +-inverses 0.0%

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

      14. +-inverses 0.0%

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

      15. +-inverses 0.0%

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

      16. clear-num 0.0%

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

      17. +-inverses 0.0%

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

      18. +-inverses 0.0%

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

      19. +-inverses 0.0%

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

      20. +-inverses 0.0%

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

      21. flip-+ 43.2%

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

   3. Applied egg-rr 43.2%

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

   4. Taylor expanded in x.re around inf 40.5%

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

      1. *-commutative 40.5%

         \[\leadsto \color{blue}{x.im \cdot {x.re}^{2}} \]

      2. unpow2 40.5%

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

   6. Simplified 40.5%

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

4. Final simplification 39.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -4 \cdot 10^{+147}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.im\\ \end{array} \]

Alternative 13: 38.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -9.2 \cdot 10^{+140}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.im\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im -9.2e+140) (* x.re (* x.im x.im)) (* (* x.re x.re) x.im)))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= -9.2e+140) {
		tmp = x_46_re * (x_46_im * x_46_im);
	} else {
		tmp = (x_46_re * x_46_re) * x_46_im;
	}
	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 <= (-9.2d+140)) then
        tmp = x_46re * (x_46im * x_46im)
    else
        tmp = (x_46re * x_46re) * x_46im
    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 <= -9.2e+140) {
		tmp = x_46_re * (x_46_im * x_46_im);
	} else {
		tmp = (x_46_re * x_46_re) * x_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= -9.2e+140:
		tmp = x_46_re * (x_46_im * x_46_im)
	else:
		tmp = (x_46_re * x_46_re) * x_46_im
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= -9.2e+140)
		tmp = Float64(x_46_re * Float64(x_46_im * x_46_im));
	else
		tmp = Float64(Float64(x_46_re * x_46_re) * x_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_im <= -9.2e+140)
		tmp = x_46_re * (x_46_im * x_46_im);
	else
		tmp = (x_46_re * x_46_re) * x_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, -9.2e+140], N[(x$46$re * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * x$46$re), $MachinePrecision] * x$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -9.19999999999999961e140

   1. Initial program 61.5%

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

      1. *-commutative 61.5%

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

      2. *-commutative 61.5%

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

      3. difference-of-squares 74.4%

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

      4. associate-*l* 74.4%

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

      5. fma-def 74.4%

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

      6. *-commutative 74.4%

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

      7. *-commutative 74.4%

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

      8. *-commutative 74.4%

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

      9. distribute-lft-out 74.4%

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

   3. Simplified 74.4%

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

   4. Taylor expanded in x.im around 0 29.0%

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

   5. Taylor expanded in x.re around 0 46.2%

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

      1. unpow2 46.2%

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

   7. Simplified 46.2%

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

   if -9.19999999999999961e140 < x.im

   1. Initial program 82.4%

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

      1. *-commutative 82.4%

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

      2. *-commutative 82.4%

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

      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%

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

      8. associate-*r/ 0.0%

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

      9. +-inverses 0.0%

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

      10. +-inverses 0.0%

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

      11. distribute-lft-out-- 0.0%

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

      12. +-inverses 0.0%

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

      13. +-inverses 0.0%

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

      14. +-inverses 0.0%

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

      15. +-inverses 0.0%

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

      16. clear-num 0.0%

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

      17. +-inverses 0.0%

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

      18. +-inverses 0.0%

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

      19. +-inverses 0.0%

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

      20. +-inverses 0.0%

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

      21. flip-+ 43.2%

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

   3. Applied egg-rr 43.2%

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

   4. Taylor expanded in x.re around inf 40.5%

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

      1. *-commutative 40.5%

         \[\leadsto \color{blue}{x.im \cdot {x.re}^{2}} \]

      2. unpow2 40.5%

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

   6. Simplified 40.5%

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

4. Final simplification 41.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -9.2 \cdot 10^{+140}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.re\right) \cdot x.im\\ \end{array} \]

Alternative 14: 27.7% accurate, 3.8× speedup

Math

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

FPCore

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

C

double code(double x_46_re, double x_46_im) {
	return 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_46im * (x_46re * x_46im)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

   1. *-commutative 79.3%

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

   2. *-commutative 79.3%

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

   3. difference-of-squares 83.6%

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

   4. associate-*l* 90.4%

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

   5. fma-def 90.5%

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

   6. *-commutative 90.5%

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

   7. *-commutative 90.5%

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

   8. *-commutative 90.5%

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

   9. distribute-lft-out 90.5%

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

3. Simplified 90.5%

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

4. Taylor expanded in x.im around 0 61.0%

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

5. Taylor expanded in x.re around 0 30.7%

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

   1. unpow2 30.7%

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

   2. *-commutative 30.7%

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

   3. associate-*l* 28.5%

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

7. Simplified 28.5%

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

8. Final simplification 28.5%

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

Alternative 15: 4.5% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ x.im \cdot -3 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

2. Taylor expanded in x.re around 0 63.5%

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

4. Simplified 25.7%

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

5. Taylor expanded in x.re around 0 4.5%

   \[\leadsto \color{blue}{-3 \cdot x.im} \]
6. Step-by-step derivation

   1. *-commutative 4.5%

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

7. Simplified 4.5%

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

8. Final simplification 4.5%

   \[\leadsto x.im \cdot -3 \]

Alternative 16: 2.7% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ -2.6 \end{array} \]

FPCore

(FPCore (x.re x.im) :precision binary64 -2.6)

C

double code(double x_46_re, double x_46_im) {
	return -2.6;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = -2.6d0
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return -2.6;
}

Python

def code(x_46_re, x_46_im):
	return -2.6

Julia

function code(x_46_re, x_46_im)
	return -2.6
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = -2.6;
end

Wolfram

code[x$46$re_, x$46$im_] := -2.6

Derivation

1. Initial program 79.3%

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

   1. +-commutative 79.3%

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

   2. *-commutative 79.3%

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

   3. distribute-lft-out 79.3%

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

   4. associate-*l* 79.2%

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

   5. *-commutative 79.2%

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

   6. distribute-lft-out 83.1%

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

   7. associate-+r- 83.1%

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

   8. distribute-lft-out-- 75.7%

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

3. Simplified 75.8%

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

   1. flip3-- 10.7%

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

   2. frac-2neg 10.7%

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

   3. *-commutative 10.7%

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

   4. unpow-prod-down 10.7%

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

   5. metadata-eval 10.7%

      \[\leadsto \frac{-\left(\color{blue}{27} \cdot {\left(\left(x.re \cdot x.re\right) \cdot x.im\right)}^{3} - {\left({x.im}^{3}\right)}^{3}\right)}{-\left(\left(\left(\left(x.re \cdot x.re\right) \cdot x.im\right) \cdot 3\right) \cdot \left(\left(\left(x.re \cdot x.re\right) \cdot x.im\right) \cdot 3\right) + \left({x.im}^{3} \cdot {x.im}^{3} + \left(\left(\left(x.re \cdot x.re\right) \cdot x.im\right) \cdot 3\right) \cdot {x.im}^{3}\right)\right)} \]

   6. associate-*l* 10.7%

      \[\leadsto \frac{-\left(27 \cdot {\color{blue}{\left(x.re \cdot \left(x.re \cdot x.im\right)\right)}}^{3} - {\left({x.im}^{3}\right)}^{3}\right)}{-\left(\left(\left(\left(x.re \cdot x.re\right) \cdot x.im\right) \cdot 3\right) \cdot \left(\left(\left(x.re \cdot x.re\right) \cdot x.im\right) \cdot 3\right) + \left({x.im}^{3} \cdot {x.im}^{3} + \left(\left(\left(x.re \cdot x.re\right) \cdot x.im\right) \cdot 3\right) \cdot {x.im}^{3}\right)\right)} \]

   7. pow-pow 10.6%

      \[\leadsto \frac{-\left(27 \cdot {\left(x.re \cdot \left(x.re \cdot x.im\right)\right)}^{3} - \color{blue}{{x.im}^{\left(3 \cdot 3\right)}}\right)}{-\left(\left(\left(\left(x.re \cdot x.re\right) \cdot x.im\right) \cdot 3\right) \cdot \left(\left(\left(x.re \cdot x.re\right) \cdot x.im\right) \cdot 3\right) + \left({x.im}^{3} \cdot {x.im}^{3} + \left(\left(\left(x.re \cdot x.re\right) \cdot x.im\right) \cdot 3\right) \cdot {x.im}^{3}\right)\right)} \]

   8. metadata-eval 10.6%

      \[\leadsto \frac{-\left(27 \cdot {\left(x.re \cdot \left(x.re \cdot x.im\right)\right)}^{3} - {x.im}^{\color{blue}{9}}\right)}{-\left(\left(\left(\left(x.re \cdot x.re\right) \cdot x.im\right) \cdot 3\right) \cdot \left(\left(\left(x.re \cdot x.re\right) \cdot x.im\right) \cdot 3\right) + \left({x.im}^{3} \cdot {x.im}^{3} + \left(\left(\left(x.re \cdot x.re\right) \cdot x.im\right) \cdot 3\right) \cdot {x.im}^{3}\right)\right)} \]

5. Applied egg-rr 7.2%

   \[\leadsto \color{blue}{\frac{-\left(27 \cdot {\left(x.re \cdot \left(x.re \cdot x.im\right)\right)}^{3} - {x.im}^{9}\right)}{-\left(\left(9 \cdot \left(\left(x.im \cdot x.im\right) \cdot {x.re}^{4}\right) + {x.im}^{6}\right) + \left(x.re \cdot x.re\right) \cdot \left(\left(x.im \cdot 3\right) \cdot {x.im}^{3}\right)\right)}} \]

7. Simplified 2.7%

   \[\leadsto \color{blue}{-2.6} \]

8. Final simplification 2.7%

   \[\leadsto -2.6 \]

Developer target: 91.9% 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 2023268 
(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)))