math.cube on complex, real part

Percentage Accurate: 83.1% → 93.4%

Time: 9.9s

Alternatives: 8

Speedup: 0.9×

• 44.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 83.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 93.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ \begin{array}{l} \mathbf{if}\;x.im\_m \leq 7.2 \cdot 10^{+139}:\\ \;\;\;\;\mathsf{fma}\left(x.im\_m \cdot x.re, x.im\_m \cdot -3, {x.re}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im\_m, x.re \cdot \left(x.im\_m + x.re\right), \left(x.im\_m \cdot x.re\right) \cdot \left(x.im\_m \cdot -2\right)\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
(FPCore (x.re x.im_m)
 :precision binary64
 (if (<= x.im_m 7.2e+139)
   (fma (* x.im_m x.re) (* x.im_m -3.0) (pow x.re 3.0))
   (fma
    (- x.re x.im_m)
    (* x.re (+ x.im_m x.re))
    (* (* x.im_m x.re) (* x.im_m -2.0)))))

C

x.im_m = fabs(x_46_im);
double code(double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_im_m <= 7.2e+139) {
		tmp = fma((x_46_im_m * x_46_re), (x_46_im_m * -3.0), pow(x_46_re, 3.0));
	} else {
		tmp = fma((x_46_re - x_46_im_m), (x_46_re * (x_46_im_m + x_46_re)), ((x_46_im_m * x_46_re) * (x_46_im_m * -2.0)));
	}
	return tmp;
}

Julia

x.im_m = abs(x_46_im)
function code(x_46_re, x_46_im_m)
	tmp = 0.0
	if (x_46_im_m <= 7.2e+139)
		tmp = fma(Float64(x_46_im_m * x_46_re), Float64(x_46_im_m * -3.0), (x_46_re ^ 3.0));
	else
		tmp = fma(Float64(x_46_re - x_46_im_m), Float64(x_46_re * Float64(x_46_im_m + x_46_re)), Float64(Float64(x_46_im_m * x_46_re) * Float64(x_46_im_m * -2.0)));
	end
	return tmp
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
code[x$46$re_, x$46$im$95$m_] := If[LessEqual[x$46$im$95$m, 7.2e+139], N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] * N[(x$46$im$95$m * -3.0), $MachinePrecision] + N[Power[x$46$re, 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re - x$46$im$95$m), $MachinePrecision] * N[(x$46$re * N[(x$46$im$95$m + x$46$re), $MachinePrecision]), $MachinePrecision] + N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] * N[(x$46$im$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 7.19999999999999971e139

   1. Initial program 86.3%

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

   3. Simplified 85.0%

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

      1. +-commutative 85.0%

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

      2. associate-*r* 88.7%

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

      3. fma-define 91.8%

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

   6. Applied egg-rr 91.8%

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

   if 7.19999999999999971e139 < x.im

   1. Initial program 63.9%

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

      1. difference-of-squares 70.1%

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

      2. *-commutative 70.1%

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

   4. Applied egg-rr 70.1%

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

      1. associate-*l* 93.5%

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

      2. +-commutative 93.5%

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

      3. fmm-def 93.4%

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

      4. +-commutative 93.4%

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

      5. add-log-exp 67.1%

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

      6. neg-log 67.1%

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

      7. exp-prod 69.6%

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

      8. *-commutative 69.6%

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

      9. exp-sum 69.6%

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

      10. exp-lft-sqr 69.6%

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

      11. exp-prod 67.1%

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

      12. neg-log 67.1%

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

      13. add-log-exp 93.4%

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

      14. associate-*l* 93.4%

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

   6. Applied egg-rr 93.4%

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

      1. *-commutative 93.4%

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

      2. +-commutative 93.4%

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

      3. distribute-rgt-neg-in 93.4%

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

      4. *-commutative 93.4%

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

      5. distribute-lft-neg-in 93.4%

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

      6. metadata-eval 93.4%

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

   8. Simplified 93.4%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 7.2 \cdot 10^{+139}:\\ \;\;\;\;\mathsf{fma}\left(x.im \cdot x.re, x.im \cdot -3, {x.re}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im, x.re \cdot \left(x.im + x.re\right), \left(x.im \cdot x.re\right) \cdot \left(x.im \cdot -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ \begin{array}{l} \mathbf{if}\;x.re \leq 1.25 \cdot 10^{+203}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im\_m, x.re \cdot \left(x.im\_m + x.re\right), \left(x.im\_m \cdot x.re\right) \cdot \left(x.im\_m \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im\_m + x.re\right) \cdot \left(x.re \cdot \left(x.re + -27\right)\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
(FPCore (x.re x.im_m)
 :precision binary64
 (if (<= x.re 1.25e+203)
   (fma
    (- x.re x.im_m)
    (* x.re (+ x.im_m x.re))
    (* (* x.im_m x.re) (* x.im_m -2.0)))
   (* (+ x.im_m x.re) (* x.re (+ x.re -27.0)))))

C

x.im_m = fabs(x_46_im);
double code(double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_re <= 1.25e+203) {
		tmp = fma((x_46_re - x_46_im_m), (x_46_re * (x_46_im_m + x_46_re)), ((x_46_im_m * x_46_re) * (x_46_im_m * -2.0)));
	} else {
		tmp = (x_46_im_m + x_46_re) * (x_46_re * (x_46_re + -27.0));
	}
	return tmp;
}

Julia

x.im_m = abs(x_46_im)
function code(x_46_re, x_46_im_m)
	tmp = 0.0
	if (x_46_re <= 1.25e+203)
		tmp = fma(Float64(x_46_re - x_46_im_m), Float64(x_46_re * Float64(x_46_im_m + x_46_re)), Float64(Float64(x_46_im_m * x_46_re) * Float64(x_46_im_m * -2.0)));
	else
		tmp = Float64(Float64(x_46_im_m + x_46_re) * Float64(x_46_re * Float64(x_46_re + -27.0)));
	end
	return tmp
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
code[x$46$re_, x$46$im$95$m_] := If[LessEqual[x$46$re, 1.25e+203], N[(N[(x$46$re - x$46$im$95$m), $MachinePrecision] * N[(x$46$re * N[(x$46$im$95$m + x$46$re), $MachinePrecision]), $MachinePrecision] + N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] * N[(x$46$im$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im$95$m + x$46$re), $MachinePrecision] * N[(x$46$re * N[(x$46$re + -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.24999999999999999e203

   1. Initial program 84.6%

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

      1. difference-of-squares 88.4%

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

      2. *-commutative 88.4%

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

   4. Applied egg-rr 88.4%

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

      1. associate-*l* 95.1%

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

      2. +-commutative 95.1%

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

      3. fmm-def 95.1%

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

      4. +-commutative 95.1%

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

      5. add-log-exp 63.4%

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

      6. neg-log 63.4%

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

      7. exp-prod 62.0%

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

      8. *-commutative 62.0%

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

      9. exp-sum 62.0%

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

      10. exp-lft-sqr 62.0%

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

      11. exp-prod 63.4%

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

      12. neg-log 63.4%

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

      13. add-log-exp 95.1%

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

      14. associate-*l* 95.1%

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

   6. Applied egg-rr 95.1%

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

      1. *-commutative 95.1%

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

      2. +-commutative 95.1%

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

      3. distribute-rgt-neg-in 95.1%

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

      4. *-commutative 95.1%

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

      5. distribute-lft-neg-in 95.1%

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

      6. metadata-eval 95.1%

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

   8. Simplified 95.1%

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

   if 1.24999999999999999e203 < x.re

   1. Initial program 70.0%

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

      1. difference-of-squares 80.0%

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

   4. Applied egg-rr 80.0%

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

   6. Simplified 80.0%

      \[\leadsto \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   7. Step-by-step derivation

      1. *-commutative 80.0%

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

      2. *-un-lft-identity 80.0%

         \[\leadsto \left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right) \cdot x.re - \color{blue}{\left(1 \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\right)} \cdot x.im \]

      3. distribute-lft-in 80.0%

         \[\leadsto \left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right) \cdot x.re - \color{blue}{\left(1 \cdot \left(x.re \cdot x.im\right) + 1 \cdot \left(x.re \cdot x.im\right)\right)} \cdot x.im \]

      4. distribute-rgt-out 80.0%

         \[\leadsto \left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right) \cdot x.re - \color{blue}{\left(\left(x.re \cdot x.im\right) \cdot \left(1 + 1\right)\right)} \cdot x.im \]

      5. metadata-eval 80.0%

         \[\leadsto \left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right) \cdot x.re - \left(\left(x.re \cdot x.im\right) \cdot \color{blue}{2}\right) \cdot x.im \]

   8. Applied egg-rr 80.0%

      \[\leadsto \left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right) \cdot x.re - \color{blue}{\left(\left(x.re \cdot x.im\right) \cdot 2\right)} \cdot x.im \]

   9. Taylor expanded in x.im around 0 35.0%

      \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right) + {x.re}^{2} \cdot \left(x.re - 27\right)} \]
   10. Step-by-step derivation

       1. sub-neg 35.0%

          \[\leadsto x.im \cdot \left(x.re \cdot \color{blue}{\left(x.re + \left(-27\right)\right)}\right) + {x.re}^{2} \cdot \left(x.re - 27\right) \]

       2. metadata-eval 35.0%

          \[\leadsto x.im \cdot \left(x.re \cdot \left(x.re + \color{blue}{-27}\right)\right) + {x.re}^{2} \cdot \left(x.re - 27\right) \]

       3. associate-*r* 50.0%

          \[\leadsto \color{blue}{\left(x.im \cdot x.re\right) \cdot \left(x.re + -27\right)} + {x.re}^{2} \cdot \left(x.re - 27\right) \]

       4. *-commutative 50.0%

          \[\leadsto \color{blue}{\left(x.re \cdot x.im\right)} \cdot \left(x.re + -27\right) + {x.re}^{2} \cdot \left(x.re - 27\right) \]

       5. sub-neg 50.0%

          \[\leadsto \left(x.re \cdot x.im\right) \cdot \left(x.re + -27\right) + {x.re}^{2} \cdot \color{blue}{\left(x.re + \left(-27\right)\right)} \]

       6. metadata-eval 50.0%

          \[\leadsto \left(x.re \cdot x.im\right) \cdot \left(x.re + -27\right) + {x.re}^{2} \cdot \left(x.re + \color{blue}{-27}\right) \]

       7. distribute-rgt-out 80.0%

          \[\leadsto \color{blue}{\left(x.re + -27\right) \cdot \left(x.re \cdot x.im + {x.re}^{2}\right)} \]

       8. unpow2 80.0%

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

       9. distribute-lft-in 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in x.im around 0 35.0%

       \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right) + {x.re}^{2} \cdot \left(x.re - 27\right)} \]
   13. Step-by-step derivation

       1. +-commutative 35.0%

          \[\leadsto \color{blue}{{x.re}^{2} \cdot \left(x.re - 27\right) + x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right)} \]

       2. sub-neg 35.0%

          \[\leadsto {x.re}^{2} \cdot \color{blue}{\left(x.re + \left(-27\right)\right)} + x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right) \]

       3. metadata-eval 35.0%

          \[\leadsto {x.re}^{2} \cdot \left(x.re + \color{blue}{-27}\right) + x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right) \]

       4. sub-neg 35.0%

          \[\leadsto {x.re}^{2} \cdot \left(x.re + -27\right) + x.im \cdot \left(x.re \cdot \color{blue}{\left(x.re + \left(-27\right)\right)}\right) \]

       5. metadata-eval 35.0%

          \[\leadsto {x.re}^{2} \cdot \left(x.re + -27\right) + x.im \cdot \left(x.re \cdot \left(x.re + \color{blue}{-27}\right)\right) \]

       6. associate-*r* 50.0%

          \[\leadsto {x.re}^{2} \cdot \left(x.re + -27\right) + \color{blue}{\left(x.im \cdot x.re\right) \cdot \left(x.re + -27\right)} \]

       7. *-commutative 50.0%

          \[\leadsto {x.re}^{2} \cdot \left(x.re + -27\right) + \color{blue}{\left(x.re \cdot x.im\right)} \cdot \left(x.re + -27\right) \]

       8. distribute-rgt-in 80.0%

          \[\leadsto \color{blue}{\left(x.re + -27\right) \cdot \left({x.re}^{2} + x.re \cdot x.im\right)} \]

       9. unpow2 80.0%

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

       10. distribute-lft-in 100.0%

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

       11. associate-*r* 100.0%

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

       12. *-commutative 100.0%

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

   14. Simplified 100.0%

       \[\leadsto \color{blue}{\left(x.re \cdot \left(x.re + -27\right)\right) \cdot \left(x.re + x.im\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 1.25 \cdot 10^{+203}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im, x.re \cdot \left(x.im + x.re\right), \left(x.im \cdot x.re\right) \cdot \left(x.im \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im + x.re\right) \cdot \left(x.re \cdot \left(x.re + -27\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ \begin{array}{l} \mathbf{if}\;x.re \leq 2 \cdot 10^{+90}:\\ \;\;\;\;{x.re}^{3} + \left(x.im\_m \cdot x.re\right) \cdot \left(x.im\_m \cdot -3\right)\\ \mathbf{elif}\;x.re \leq 1.4 \cdot 10^{+200}:\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\_m\right) \cdot \left(x.im\_m + x.re\right)\right) - x.im\_m \cdot \left(\left(x.im\_m \cdot x.re\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im\_m + x.re\right) \cdot \left(x.re \cdot \left(x.re + -27\right)\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
(FPCore (x.re x.im_m)
 :precision binary64
 (if (<= x.re 2e+90)
   (+ (pow x.re 3.0) (* (* x.im_m x.re) (* x.im_m -3.0)))
   (if (<= x.re 1.4e+200)
     (-
      (* x.re (* (- x.re x.im_m) (+ x.im_m x.re)))
      (* x.im_m (* (* x.im_m x.re) 2.0)))
     (* (+ x.im_m x.re) (* x.re (+ x.re -27.0))))))

C

x.im_m = fabs(x_46_im);
double code(double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_re <= 2e+90) {
		tmp = pow(x_46_re, 3.0) + ((x_46_im_m * x_46_re) * (x_46_im_m * -3.0));
	} else if (x_46_re <= 1.4e+200) {
		tmp = (x_46_re * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))) - (x_46_im_m * ((x_46_im_m * x_46_re) * 2.0));
	} else {
		tmp = (x_46_im_m + x_46_re) * (x_46_re * (x_46_re + -27.0));
	}
	return tmp;
}

Fortran

x.im_m = abs(x_46im)
real(8) function code(x_46re, x_46im_m)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46re <= 2d+90) then
        tmp = (x_46re ** 3.0d0) + ((x_46im_m * x_46re) * (x_46im_m * (-3.0d0)))
    else if (x_46re <= 1.4d+200) then
        tmp = (x_46re * ((x_46re - x_46im_m) * (x_46im_m + x_46re))) - (x_46im_m * ((x_46im_m * x_46re) * 2.0d0))
    else
        tmp = (x_46im_m + x_46re) * (x_46re * (x_46re + (-27.0d0)))
    end if
    code = tmp
end function

Java

x.im_m = Math.abs(x_46_im);
public static double code(double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_re <= 2e+90) {
		tmp = Math.pow(x_46_re, 3.0) + ((x_46_im_m * x_46_re) * (x_46_im_m * -3.0));
	} else if (x_46_re <= 1.4e+200) {
		tmp = (x_46_re * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))) - (x_46_im_m * ((x_46_im_m * x_46_re) * 2.0));
	} else {
		tmp = (x_46_im_m + x_46_re) * (x_46_re * (x_46_re + -27.0));
	}
	return tmp;
}

Python

x.im_m = math.fabs(x_46_im)
def code(x_46_re, x_46_im_m):
	tmp = 0
	if x_46_re <= 2e+90:
		tmp = math.pow(x_46_re, 3.0) + ((x_46_im_m * x_46_re) * (x_46_im_m * -3.0))
	elif x_46_re <= 1.4e+200:
		tmp = (x_46_re * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))) - (x_46_im_m * ((x_46_im_m * x_46_re) * 2.0))
	else:
		tmp = (x_46_im_m + x_46_re) * (x_46_re * (x_46_re + -27.0))
	return tmp

Julia

x.im_m = abs(x_46_im)
function code(x_46_re, x_46_im_m)
	tmp = 0.0
	if (x_46_re <= 2e+90)
		tmp = Float64((x_46_re ^ 3.0) + Float64(Float64(x_46_im_m * x_46_re) * Float64(x_46_im_m * -3.0)));
	elseif (x_46_re <= 1.4e+200)
		tmp = Float64(Float64(x_46_re * Float64(Float64(x_46_re - x_46_im_m) * Float64(x_46_im_m + x_46_re))) - Float64(x_46_im_m * Float64(Float64(x_46_im_m * x_46_re) * 2.0)));
	else
		tmp = Float64(Float64(x_46_im_m + x_46_re) * Float64(x_46_re * Float64(x_46_re + -27.0)));
	end
	return tmp
end

MATLAB

x.im_m = abs(x_46_im);
function tmp_2 = code(x_46_re, x_46_im_m)
	tmp = 0.0;
	if (x_46_re <= 2e+90)
		tmp = (x_46_re ^ 3.0) + ((x_46_im_m * x_46_re) * (x_46_im_m * -3.0));
	elseif (x_46_re <= 1.4e+200)
		tmp = (x_46_re * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))) - (x_46_im_m * ((x_46_im_m * x_46_re) * 2.0));
	else
		tmp = (x_46_im_m + x_46_re) * (x_46_re * (x_46_re + -27.0));
	end
	tmp_2 = tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
code[x$46$re_, x$46$im$95$m_] := If[LessEqual[x$46$re, 2e+90], N[(N[Power[x$46$re, 3.0], $MachinePrecision] + N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] * N[(x$46$im$95$m * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 1.4e+200], N[(N[(x$46$re * N[(N[(x$46$re - x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m + x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im$95$m * N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im$95$m + x$46$re), $MachinePrecision] * N[(x$46$re * N[(x$46$re + -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.re < 1.99999999999999993e90

   1. Initial program 84.7%

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

   3. Simplified 83.4%

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

      1. +-commutative 83.4%

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

      2. associate-*r* 90.8%

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

      3. fma-define 93.6%

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

   6. Applied egg-rr 93.6%

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

      1. fma-undefine 90.8%

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

      2. +-commutative 90.8%

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

      3. *-commutative 90.8%

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

      4. *-commutative 90.8%

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

      5. *-commutative 90.8%

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

   8. Applied egg-rr 90.8%

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

   if 1.99999999999999993e90 < x.re < 1.39999999999999992e200

   1. Initial program 83.3%

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

      1. difference-of-squares 100.0%

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

      2. *-commutative 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. *-commutative 83.3%

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

      2. *-un-lft-identity 83.3%

         \[\leadsto \left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right) \cdot x.re - \color{blue}{\left(1 \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\right)} \cdot x.im \]

      3. distribute-lft-in 83.3%

         \[\leadsto \left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right) \cdot x.re - \color{blue}{\left(1 \cdot \left(x.re \cdot x.im\right) + 1 \cdot \left(x.re \cdot x.im\right)\right)} \cdot x.im \]

      4. distribute-rgt-out 83.3%

         \[\leadsto \left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right) \cdot x.re - \color{blue}{\left(\left(x.re \cdot x.im\right) \cdot \left(1 + 1\right)\right)} \cdot x.im \]

      5. metadata-eval 83.3%

         \[\leadsto \left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right) \cdot x.re - \left(\left(x.re \cdot x.im\right) \cdot \color{blue}{2}\right) \cdot x.im \]

   6. Applied egg-rr 100.0%

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

   if 1.39999999999999992e200 < x.re

   1. Initial program 70.0%

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

      1. difference-of-squares 80.0%

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

   4. Applied egg-rr 80.0%

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

   6. Simplified 80.0%

      \[\leadsto \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   7. Step-by-step derivation

      1. *-commutative 80.0%

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

      2. *-un-lft-identity 80.0%

         \[\leadsto \left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right) \cdot x.re - \color{blue}{\left(1 \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\right)} \cdot x.im \]

      3. distribute-lft-in 80.0%

         \[\leadsto \left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right) \cdot x.re - \color{blue}{\left(1 \cdot \left(x.re \cdot x.im\right) + 1 \cdot \left(x.re \cdot x.im\right)\right)} \cdot x.im \]

      4. distribute-rgt-out 80.0%

         \[\leadsto \left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right) \cdot x.re - \color{blue}{\left(\left(x.re \cdot x.im\right) \cdot \left(1 + 1\right)\right)} \cdot x.im \]

      5. metadata-eval 80.0%

         \[\leadsto \left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right) \cdot x.re - \left(\left(x.re \cdot x.im\right) \cdot \color{blue}{2}\right) \cdot x.im \]

   8. Applied egg-rr 80.0%

      \[\leadsto \left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right) \cdot x.re - \color{blue}{\left(\left(x.re \cdot x.im\right) \cdot 2\right)} \cdot x.im \]

   9. Taylor expanded in x.im around 0 35.0%

      \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right) + {x.re}^{2} \cdot \left(x.re - 27\right)} \]
   10. Step-by-step derivation

       1. sub-neg 35.0%

          \[\leadsto x.im \cdot \left(x.re \cdot \color{blue}{\left(x.re + \left(-27\right)\right)}\right) + {x.re}^{2} \cdot \left(x.re - 27\right) \]

       2. metadata-eval 35.0%

          \[\leadsto x.im \cdot \left(x.re \cdot \left(x.re + \color{blue}{-27}\right)\right) + {x.re}^{2} \cdot \left(x.re - 27\right) \]

       3. associate-*r* 50.0%

          \[\leadsto \color{blue}{\left(x.im \cdot x.re\right) \cdot \left(x.re + -27\right)} + {x.re}^{2} \cdot \left(x.re - 27\right) \]

       4. *-commutative 50.0%

          \[\leadsto \color{blue}{\left(x.re \cdot x.im\right)} \cdot \left(x.re + -27\right) + {x.re}^{2} \cdot \left(x.re - 27\right) \]

       5. sub-neg 50.0%

          \[\leadsto \left(x.re \cdot x.im\right) \cdot \left(x.re + -27\right) + {x.re}^{2} \cdot \color{blue}{\left(x.re + \left(-27\right)\right)} \]

       6. metadata-eval 50.0%

          \[\leadsto \left(x.re \cdot x.im\right) \cdot \left(x.re + -27\right) + {x.re}^{2} \cdot \left(x.re + \color{blue}{-27}\right) \]

       7. distribute-rgt-out 80.0%

          \[\leadsto \color{blue}{\left(x.re + -27\right) \cdot \left(x.re \cdot x.im + {x.re}^{2}\right)} \]

       8. unpow2 80.0%

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

       9. distribute-lft-in 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in x.im around 0 35.0%

       \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right) + {x.re}^{2} \cdot \left(x.re - 27\right)} \]
   13. Step-by-step derivation

       1. +-commutative 35.0%

          \[\leadsto \color{blue}{{x.re}^{2} \cdot \left(x.re - 27\right) + x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right)} \]

       2. sub-neg 35.0%

          \[\leadsto {x.re}^{2} \cdot \color{blue}{\left(x.re + \left(-27\right)\right)} + x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right) \]

       3. metadata-eval 35.0%

          \[\leadsto {x.re}^{2} \cdot \left(x.re + \color{blue}{-27}\right) + x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right) \]

       4. sub-neg 35.0%

          \[\leadsto {x.re}^{2} \cdot \left(x.re + -27\right) + x.im \cdot \left(x.re \cdot \color{blue}{\left(x.re + \left(-27\right)\right)}\right) \]

       5. metadata-eval 35.0%

          \[\leadsto {x.re}^{2} \cdot \left(x.re + -27\right) + x.im \cdot \left(x.re \cdot \left(x.re + \color{blue}{-27}\right)\right) \]

       6. associate-*r* 50.0%

          \[\leadsto {x.re}^{2} \cdot \left(x.re + -27\right) + \color{blue}{\left(x.im \cdot x.re\right) \cdot \left(x.re + -27\right)} \]

       7. *-commutative 50.0%

          \[\leadsto {x.re}^{2} \cdot \left(x.re + -27\right) + \color{blue}{\left(x.re \cdot x.im\right)} \cdot \left(x.re + -27\right) \]

       8. distribute-rgt-in 80.0%

          \[\leadsto \color{blue}{\left(x.re + -27\right) \cdot \left({x.re}^{2} + x.re \cdot x.im\right)} \]

       9. unpow2 80.0%

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

       10. distribute-lft-in 100.0%

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

       11. associate-*r* 100.0%

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

       12. *-commutative 100.0%

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

   14. Simplified 100.0%

       \[\leadsto \color{blue}{\left(x.re \cdot \left(x.re + -27\right)\right) \cdot \left(x.re + x.im\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 2 \cdot 10^{+90}:\\ \;\;\;\;{x.re}^{3} + \left(x.im \cdot x.re\right) \cdot \left(x.im \cdot -3\right)\\ \mathbf{elif}\;x.re \leq 1.4 \cdot 10^{+200}:\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right) - x.im \cdot \left(\left(x.im \cdot x.re\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im + x.re\right) \cdot \left(x.re \cdot \left(x.re + -27\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ \begin{array}{l} \mathbf{if}\;x.re \leq 1.16 \cdot 10^{+198}:\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\_m\right) \cdot \left(x.im\_m + x.re\right)\right) - x.im\_m \cdot \left(\left(x.im\_m \cdot x.re\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im\_m + x.re\right) \cdot \left(x.re \cdot \left(x.re + -27\right)\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
(FPCore (x.re x.im_m)
 :precision binary64
 (if (<= x.re 1.16e+198)
   (-
    (* x.re (* (- x.re x.im_m) (+ x.im_m x.re)))
    (* x.im_m (* (* x.im_m x.re) 2.0)))
   (* (+ x.im_m x.re) (* x.re (+ x.re -27.0)))))

C

x.im_m = fabs(x_46_im);
double code(double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_re <= 1.16e+198) {
		tmp = (x_46_re * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))) - (x_46_im_m * ((x_46_im_m * x_46_re) * 2.0));
	} else {
		tmp = (x_46_im_m + x_46_re) * (x_46_re * (x_46_re + -27.0));
	}
	return tmp;
}

Fortran

x.im_m = abs(x_46im)
real(8) function code(x_46re, x_46im_m)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46re <= 1.16d+198) then
        tmp = (x_46re * ((x_46re - x_46im_m) * (x_46im_m + x_46re))) - (x_46im_m * ((x_46im_m * x_46re) * 2.0d0))
    else
        tmp = (x_46im_m + x_46re) * (x_46re * (x_46re + (-27.0d0)))
    end if
    code = tmp
end function

Java

x.im_m = Math.abs(x_46_im);
public static double code(double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_re <= 1.16e+198) {
		tmp = (x_46_re * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))) - (x_46_im_m * ((x_46_im_m * x_46_re) * 2.0));
	} else {
		tmp = (x_46_im_m + x_46_re) * (x_46_re * (x_46_re + -27.0));
	}
	return tmp;
}

Python

x.im_m = math.fabs(x_46_im)
def code(x_46_re, x_46_im_m):
	tmp = 0
	if x_46_re <= 1.16e+198:
		tmp = (x_46_re * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))) - (x_46_im_m * ((x_46_im_m * x_46_re) * 2.0))
	else:
		tmp = (x_46_im_m + x_46_re) * (x_46_re * (x_46_re + -27.0))
	return tmp

Julia

x.im_m = abs(x_46_im)
function code(x_46_re, x_46_im_m)
	tmp = 0.0
	if (x_46_re <= 1.16e+198)
		tmp = Float64(Float64(x_46_re * Float64(Float64(x_46_re - x_46_im_m) * Float64(x_46_im_m + x_46_re))) - Float64(x_46_im_m * Float64(Float64(x_46_im_m * x_46_re) * 2.0)));
	else
		tmp = Float64(Float64(x_46_im_m + x_46_re) * Float64(x_46_re * Float64(x_46_re + -27.0)));
	end
	return tmp
end

MATLAB

x.im_m = abs(x_46_im);
function tmp_2 = code(x_46_re, x_46_im_m)
	tmp = 0.0;
	if (x_46_re <= 1.16e+198)
		tmp = (x_46_re * ((x_46_re - x_46_im_m) * (x_46_im_m + x_46_re))) - (x_46_im_m * ((x_46_im_m * x_46_re) * 2.0));
	else
		tmp = (x_46_im_m + x_46_re) * (x_46_re * (x_46_re + -27.0));
	end
	tmp_2 = tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
code[x$46$re_, x$46$im$95$m_] := If[LessEqual[x$46$re, 1.16e+198], N[(N[(x$46$re * N[(N[(x$46$re - x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m + x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im$95$m * N[(N[(x$46$im$95$m * x$46$re), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im$95$m + x$46$re), $MachinePrecision] * N[(x$46$re * N[(x$46$re + -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.16000000000000001e198

   1. Initial program 84.6%

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

      1. difference-of-squares 88.4%

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

      2. *-commutative 88.4%

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

   4. Applied egg-rr 88.4%

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

      1. *-commutative 55.7%

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

      2. *-un-lft-identity 55.7%

         \[\leadsto \left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right) \cdot x.re - \color{blue}{\left(1 \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\right)} \cdot x.im \]

      3. distribute-lft-in 55.7%

         \[\leadsto \left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right) \cdot x.re - \color{blue}{\left(1 \cdot \left(x.re \cdot x.im\right) + 1 \cdot \left(x.re \cdot x.im\right)\right)} \cdot x.im \]

      4. distribute-rgt-out 55.7%

         \[\leadsto \left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right) \cdot x.re - \color{blue}{\left(\left(x.re \cdot x.im\right) \cdot \left(1 + 1\right)\right)} \cdot x.im \]

      5. metadata-eval 55.7%

         \[\leadsto \left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right) \cdot x.re - \left(\left(x.re \cdot x.im\right) \cdot \color{blue}{2}\right) \cdot x.im \]

   6. Applied egg-rr 88.4%

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

   if 1.16000000000000001e198 < x.re

   1. Initial program 70.0%

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

      1. difference-of-squares 80.0%

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

   4. Applied egg-rr 80.0%

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

   6. Simplified 80.0%

      \[\leadsto \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   7. Step-by-step derivation

      1. *-commutative 80.0%

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

      2. *-un-lft-identity 80.0%

         \[\leadsto \left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right) \cdot x.re - \color{blue}{\left(1 \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\right)} \cdot x.im \]

      3. distribute-lft-in 80.0%

         \[\leadsto \left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right) \cdot x.re - \color{blue}{\left(1 \cdot \left(x.re \cdot x.im\right) + 1 \cdot \left(x.re \cdot x.im\right)\right)} \cdot x.im \]

      4. distribute-rgt-out 80.0%

         \[\leadsto \left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right) \cdot x.re - \color{blue}{\left(\left(x.re \cdot x.im\right) \cdot \left(1 + 1\right)\right)} \cdot x.im \]

      5. metadata-eval 80.0%

         \[\leadsto \left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right) \cdot x.re - \left(\left(x.re \cdot x.im\right) \cdot \color{blue}{2}\right) \cdot x.im \]

   8. Applied egg-rr 80.0%

      \[\leadsto \left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right) \cdot x.re - \color{blue}{\left(\left(x.re \cdot x.im\right) \cdot 2\right)} \cdot x.im \]

   9. Taylor expanded in x.im around 0 35.0%

      \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right) + {x.re}^{2} \cdot \left(x.re - 27\right)} \]
   10. Step-by-step derivation

       1. sub-neg 35.0%

          \[\leadsto x.im \cdot \left(x.re \cdot \color{blue}{\left(x.re + \left(-27\right)\right)}\right) + {x.re}^{2} \cdot \left(x.re - 27\right) \]

       2. metadata-eval 35.0%

          \[\leadsto x.im \cdot \left(x.re \cdot \left(x.re + \color{blue}{-27}\right)\right) + {x.re}^{2} \cdot \left(x.re - 27\right) \]

       3. associate-*r* 50.0%

          \[\leadsto \color{blue}{\left(x.im \cdot x.re\right) \cdot \left(x.re + -27\right)} + {x.re}^{2} \cdot \left(x.re - 27\right) \]

       4. *-commutative 50.0%

          \[\leadsto \color{blue}{\left(x.re \cdot x.im\right)} \cdot \left(x.re + -27\right) + {x.re}^{2} \cdot \left(x.re - 27\right) \]

       5. sub-neg 50.0%

          \[\leadsto \left(x.re \cdot x.im\right) \cdot \left(x.re + -27\right) + {x.re}^{2} \cdot \color{blue}{\left(x.re + \left(-27\right)\right)} \]

       6. metadata-eval 50.0%

          \[\leadsto \left(x.re \cdot x.im\right) \cdot \left(x.re + -27\right) + {x.re}^{2} \cdot \left(x.re + \color{blue}{-27}\right) \]

       7. distribute-rgt-out 80.0%

          \[\leadsto \color{blue}{\left(x.re + -27\right) \cdot \left(x.re \cdot x.im + {x.re}^{2}\right)} \]

       8. unpow2 80.0%

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

       9. distribute-lft-in 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in x.im around 0 35.0%

       \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right) + {x.re}^{2} \cdot \left(x.re - 27\right)} \]
   13. Step-by-step derivation

       1. +-commutative 35.0%

          \[\leadsto \color{blue}{{x.re}^{2} \cdot \left(x.re - 27\right) + x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right)} \]

       2. sub-neg 35.0%

          \[\leadsto {x.re}^{2} \cdot \color{blue}{\left(x.re + \left(-27\right)\right)} + x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right) \]

       3. metadata-eval 35.0%

          \[\leadsto {x.re}^{2} \cdot \left(x.re + \color{blue}{-27}\right) + x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right) \]

       4. sub-neg 35.0%

          \[\leadsto {x.re}^{2} \cdot \left(x.re + -27\right) + x.im \cdot \left(x.re \cdot \color{blue}{\left(x.re + \left(-27\right)\right)}\right) \]

       5. metadata-eval 35.0%

          \[\leadsto {x.re}^{2} \cdot \left(x.re + -27\right) + x.im \cdot \left(x.re \cdot \left(x.re + \color{blue}{-27}\right)\right) \]

       6. associate-*r* 50.0%

          \[\leadsto {x.re}^{2} \cdot \left(x.re + -27\right) + \color{blue}{\left(x.im \cdot x.re\right) \cdot \left(x.re + -27\right)} \]

       7. *-commutative 50.0%

          \[\leadsto {x.re}^{2} \cdot \left(x.re + -27\right) + \color{blue}{\left(x.re \cdot x.im\right)} \cdot \left(x.re + -27\right) \]

       8. distribute-rgt-in 80.0%

          \[\leadsto \color{blue}{\left(x.re + -27\right) \cdot \left({x.re}^{2} + x.re \cdot x.im\right)} \]

       9. unpow2 80.0%

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

       10. distribute-lft-in 100.0%

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

       11. associate-*r* 100.0%

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

       12. *-commutative 100.0%

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

   14. Simplified 100.0%

       \[\leadsto \color{blue}{\left(x.re \cdot \left(x.re + -27\right)\right) \cdot \left(x.re + x.im\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 1.16 \cdot 10^{+198}:\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right) - x.im \cdot \left(\left(x.im \cdot x.re\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im + x.re\right) \cdot \left(x.re \cdot \left(x.re + -27\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ \begin{array}{l} \mathbf{if}\;x.re \leq 4 \cdot 10^{+130}:\\ \;\;\;\;x.re \cdot \left(\left(x.im\_m + x.re\right) \cdot \left(x.re + -27\right) - x.im\_m \cdot \left(x.im\_m \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im\_m + x.re\right) \cdot \left(x.re \cdot \left(x.re + -27\right)\right)\\ \end{array} \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
(FPCore (x.re x.im_m)
 :precision binary64
 (if (<= x.re 4e+130)
   (* x.re (- (* (+ x.im_m x.re) (+ x.re -27.0)) (* x.im_m (* x.im_m 2.0))))
   (* (+ x.im_m x.re) (* x.re (+ x.re -27.0)))))

C

x.im_m = fabs(x_46_im);
double code(double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_re <= 4e+130) {
		tmp = x_46_re * (((x_46_im_m + x_46_re) * (x_46_re + -27.0)) - (x_46_im_m * (x_46_im_m * 2.0)));
	} else {
		tmp = (x_46_im_m + x_46_re) * (x_46_re * (x_46_re + -27.0));
	}
	return tmp;
}

Fortran

x.im_m = abs(x_46im)
real(8) function code(x_46re, x_46im_m)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    real(8) :: tmp
    if (x_46re <= 4d+130) then
        tmp = x_46re * (((x_46im_m + x_46re) * (x_46re + (-27.0d0))) - (x_46im_m * (x_46im_m * 2.0d0)))
    else
        tmp = (x_46im_m + x_46re) * (x_46re * (x_46re + (-27.0d0)))
    end if
    code = tmp
end function

Java

x.im_m = Math.abs(x_46_im);
public static double code(double x_46_re, double x_46_im_m) {
	double tmp;
	if (x_46_re <= 4e+130) {
		tmp = x_46_re * (((x_46_im_m + x_46_re) * (x_46_re + -27.0)) - (x_46_im_m * (x_46_im_m * 2.0)));
	} else {
		tmp = (x_46_im_m + x_46_re) * (x_46_re * (x_46_re + -27.0));
	}
	return tmp;
}

Python

x.im_m = math.fabs(x_46_im)
def code(x_46_re, x_46_im_m):
	tmp = 0
	if x_46_re <= 4e+130:
		tmp = x_46_re * (((x_46_im_m + x_46_re) * (x_46_re + -27.0)) - (x_46_im_m * (x_46_im_m * 2.0)))
	else:
		tmp = (x_46_im_m + x_46_re) * (x_46_re * (x_46_re + -27.0))
	return tmp

Julia

x.im_m = abs(x_46_im)
function code(x_46_re, x_46_im_m)
	tmp = 0.0
	if (x_46_re <= 4e+130)
		tmp = Float64(x_46_re * Float64(Float64(Float64(x_46_im_m + x_46_re) * Float64(x_46_re + -27.0)) - Float64(x_46_im_m * Float64(x_46_im_m * 2.0))));
	else
		tmp = Float64(Float64(x_46_im_m + x_46_re) * Float64(x_46_re * Float64(x_46_re + -27.0)));
	end
	return tmp
end

MATLAB

x.im_m = abs(x_46_im);
function tmp_2 = code(x_46_re, x_46_im_m)
	tmp = 0.0;
	if (x_46_re <= 4e+130)
		tmp = x_46_re * (((x_46_im_m + x_46_re) * (x_46_re + -27.0)) - (x_46_im_m * (x_46_im_m * 2.0)));
	else
		tmp = (x_46_im_m + x_46_re) * (x_46_re * (x_46_re + -27.0));
	end
	tmp_2 = tmp;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
code[x$46$re_, x$46$im$95$m_] := If[LessEqual[x$46$re, 4e+130], N[(x$46$re * N[(N[(N[(x$46$im$95$m + x$46$re), $MachinePrecision] * N[(x$46$re + -27.0), $MachinePrecision]), $MachinePrecision] - N[(x$46$im$95$m * N[(x$46$im$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im$95$m + x$46$re), $MachinePrecision] * N[(x$46$re * N[(x$46$re + -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 4.0000000000000002e130

   1. Initial program 85.2%

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

      1. difference-of-squares 87.4%

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

   4. Applied egg-rr 87.4%

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

   6. Simplified 52.9%

      \[\leadsto \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   7. Step-by-step derivation

      1. *-commutative 52.9%

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

      2. *-un-lft-identity 52.9%

         \[\leadsto \left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right) \cdot x.re - \color{blue}{\left(1 \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\right)} \cdot x.im \]

      3. distribute-lft-in 52.9%

         \[\leadsto \left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right) \cdot x.re - \color{blue}{\left(1 \cdot \left(x.re \cdot x.im\right) + 1 \cdot \left(x.re \cdot x.im\right)\right)} \cdot x.im \]

      4. distribute-rgt-out 52.9%

         \[\leadsto \left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right) \cdot x.re - \color{blue}{\left(\left(x.re \cdot x.im\right) \cdot \left(1 + 1\right)\right)} \cdot x.im \]

      5. metadata-eval 52.9%

         \[\leadsto \left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right) \cdot x.re - \left(\left(x.re \cdot x.im\right) \cdot \color{blue}{2}\right) \cdot x.im \]

   8. Applied egg-rr 52.9%

      \[\leadsto \left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right) \cdot x.re - \color{blue}{\left(\left(x.re \cdot x.im\right) \cdot 2\right)} \cdot x.im \]
   9. Step-by-step derivation

      1. sub-neg 52.9%

         \[\leadsto \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right) \cdot x.re + \left(-\left(\left(x.re \cdot x.im\right) \cdot 2\right) \cdot x.im\right)} \]

      2. associate-*l* 52.9%

         \[\leadsto \color{blue}{\left(x.im + x.re\right) \cdot \left(\left(x.re + -27\right) \cdot x.re\right)} + \left(-\left(\left(x.re \cdot x.im\right) \cdot 2\right) \cdot x.im\right) \]

      3. +-commutative 52.9%

         \[\leadsto \color{blue}{\left(x.re + x.im\right)} \cdot \left(\left(x.re + -27\right) \cdot x.re\right) + \left(-\left(\left(x.re \cdot x.im\right) \cdot 2\right) \cdot x.im\right) \]

      4. associate-*l* 52.9%

         \[\leadsto \left(x.re + x.im\right) \cdot \left(\left(x.re + -27\right) \cdot x.re\right) + \left(-\color{blue}{\left(x.re \cdot x.im\right) \cdot \left(2 \cdot x.im\right)}\right) \]

   10. Applied egg-rr 52.9%

       \[\leadsto \color{blue}{\left(x.re + x.im\right) \cdot \left(\left(x.re + -27\right) \cdot x.re\right) + \left(-\left(x.re \cdot x.im\right) \cdot \left(2 \cdot x.im\right)\right)} \]
   11. Step-by-step derivation

       1. sub-neg 52.9%

          \[\leadsto \color{blue}{\left(x.re + x.im\right) \cdot \left(\left(x.re + -27\right) \cdot x.re\right) - \left(x.re \cdot x.im\right) \cdot \left(2 \cdot x.im\right)} \]

       2. associate-*l* 51.8%

          \[\leadsto \left(x.re + x.im\right) \cdot \left(\left(x.re + -27\right) \cdot x.re\right) - \color{blue}{x.re \cdot \left(x.im \cdot \left(2 \cdot x.im\right)\right)} \]

       3. associate-*r* 51.8%

          \[\leadsto \color{blue}{\left(\left(x.re + x.im\right) \cdot \left(x.re + -27\right)\right) \cdot x.re} - x.re \cdot \left(x.im \cdot \left(2 \cdot x.im\right)\right) \]

       4. *-commutative 51.8%

          \[\leadsto \color{blue}{x.re \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re + -27\right)\right)} - x.re \cdot \left(x.im \cdot \left(2 \cdot x.im\right)\right) \]

       5. distribute-lft-out-- 57.8%

          \[\leadsto \color{blue}{x.re \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re + -27\right) - x.im \cdot \left(2 \cdot x.im\right)\right)} \]

       6. +-commutative 57.8%

          \[\leadsto x.re \cdot \left(\color{blue}{\left(x.im + x.re\right)} \cdot \left(x.re + -27\right) - x.im \cdot \left(2 \cdot x.im\right)\right) \]

       7. *-commutative 57.8%

          \[\leadsto x.re \cdot \left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right) - x.im \cdot \color{blue}{\left(x.im \cdot 2\right)}\right) \]

   12. Simplified 57.8%

       \[\leadsto \color{blue}{x.re \cdot \left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right) - x.im \cdot \left(x.im \cdot 2\right)\right)} \]

   if 4.0000000000000002e130 < x.re

   1. Initial program 73.7%

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

      1. difference-of-squares 89.5%

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

   4. Applied egg-rr 89.5%

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

   6. Simplified 84.2%

      \[\leadsto \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
   7. Step-by-step derivation

      1. *-commutative 84.2%

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

      2. *-un-lft-identity 84.2%

         \[\leadsto \left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right) \cdot x.re - \color{blue}{\left(1 \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\right)} \cdot x.im \]

      3. distribute-lft-in 84.2%

         \[\leadsto \left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right) \cdot x.re - \color{blue}{\left(1 \cdot \left(x.re \cdot x.im\right) + 1 \cdot \left(x.re \cdot x.im\right)\right)} \cdot x.im \]

      4. distribute-rgt-out 84.2%

         \[\leadsto \left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right) \cdot x.re - \color{blue}{\left(\left(x.re \cdot x.im\right) \cdot \left(1 + 1\right)\right)} \cdot x.im \]

      5. metadata-eval 84.2%

         \[\leadsto \left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right) \cdot x.re - \left(\left(x.re \cdot x.im\right) \cdot \color{blue}{2}\right) \cdot x.im \]

   8. Applied egg-rr 84.2%

      \[\leadsto \left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right) \cdot x.re - \color{blue}{\left(\left(x.re \cdot x.im\right) \cdot 2\right)} \cdot x.im \]

   9. Taylor expanded in x.im around 0 36.8%

      \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right) + {x.re}^{2} \cdot \left(x.re - 27\right)} \]
   10. Step-by-step derivation

       1. sub-neg 36.8%

          \[\leadsto x.im \cdot \left(x.re \cdot \color{blue}{\left(x.re + \left(-27\right)\right)}\right) + {x.re}^{2} \cdot \left(x.re - 27\right) \]

       2. metadata-eval 36.8%

          \[\leadsto x.im \cdot \left(x.re \cdot \left(x.re + \color{blue}{-27}\right)\right) + {x.re}^{2} \cdot \left(x.re - 27\right) \]

       3. associate-*r* 57.9%

          \[\leadsto \color{blue}{\left(x.im \cdot x.re\right) \cdot \left(x.re + -27\right)} + {x.re}^{2} \cdot \left(x.re - 27\right) \]

       4. *-commutative 57.9%

          \[\leadsto \color{blue}{\left(x.re \cdot x.im\right)} \cdot \left(x.re + -27\right) + {x.re}^{2} \cdot \left(x.re - 27\right) \]

       5. sub-neg 57.9%

          \[\leadsto \left(x.re \cdot x.im\right) \cdot \left(x.re + -27\right) + {x.re}^{2} \cdot \color{blue}{\left(x.re + \left(-27\right)\right)} \]

       6. metadata-eval 57.9%

          \[\leadsto \left(x.re \cdot x.im\right) \cdot \left(x.re + -27\right) + {x.re}^{2} \cdot \left(x.re + \color{blue}{-27}\right) \]

       7. distribute-rgt-out 78.9%

          \[\leadsto \color{blue}{\left(x.re + -27\right) \cdot \left(x.re \cdot x.im + {x.re}^{2}\right)} \]

       8. unpow2 78.9%

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

       9. distribute-lft-in 94.7%

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

   11. Simplified 94.7%

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

   12. Taylor expanded in x.im around 0 36.8%

       \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right) + {x.re}^{2} \cdot \left(x.re - 27\right)} \]
   13. Step-by-step derivation

       1. +-commutative 36.8%

          \[\leadsto \color{blue}{{x.re}^{2} \cdot \left(x.re - 27\right) + x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right)} \]

       2. sub-neg 36.8%

          \[\leadsto {x.re}^{2} \cdot \color{blue}{\left(x.re + \left(-27\right)\right)} + x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right) \]

       3. metadata-eval 36.8%

          \[\leadsto {x.re}^{2} \cdot \left(x.re + \color{blue}{-27}\right) + x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right) \]

       4. sub-neg 36.8%

          \[\leadsto {x.re}^{2} \cdot \left(x.re + -27\right) + x.im \cdot \left(x.re \cdot \color{blue}{\left(x.re + \left(-27\right)\right)}\right) \]

       5. metadata-eval 36.8%

          \[\leadsto {x.re}^{2} \cdot \left(x.re + -27\right) + x.im \cdot \left(x.re \cdot \left(x.re + \color{blue}{-27}\right)\right) \]

       6. associate-*r* 57.9%

          \[\leadsto {x.re}^{2} \cdot \left(x.re + -27\right) + \color{blue}{\left(x.im \cdot x.re\right) \cdot \left(x.re + -27\right)} \]

       7. *-commutative 57.9%

          \[\leadsto {x.re}^{2} \cdot \left(x.re + -27\right) + \color{blue}{\left(x.re \cdot x.im\right)} \cdot \left(x.re + -27\right) \]

       8. distribute-rgt-in 78.9%

          \[\leadsto \color{blue}{\left(x.re + -27\right) \cdot \left({x.re}^{2} + x.re \cdot x.im\right)} \]

       9. unpow2 78.9%

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

       10. distribute-lft-in 94.7%

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

       11. associate-*r* 94.7%

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

       12. *-commutative 94.7%

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

   14. Simplified 94.7%

       \[\leadsto \color{blue}{\left(x.re \cdot \left(x.re + -27\right)\right) \cdot \left(x.re + x.im\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 4 \cdot 10^{+130}:\\ \;\;\;\;x.re \cdot \left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right) - x.im \cdot \left(x.im \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im + x.re\right) \cdot \left(x.re \cdot \left(x.re + -27\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 56.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ \left(x.re + -27\right) \cdot \left(x.re \cdot \left(x.re \cdot \left(1 + \frac{x.im\_m}{x.re}\right)\right)\right) \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
(FPCore (x.re x.im_m)
 :precision binary64
 (* (+ x.re -27.0) (* x.re (* x.re (+ 1.0 (/ x.im_m x.re))))))

C

x.im_m = fabs(x_46_im);
double code(double x_46_re, double x_46_im_m) {
	return (x_46_re + -27.0) * (x_46_re * (x_46_re * (1.0 + (x_46_im_m / x_46_re))));
}

Fortran

x.im_m = abs(x_46im)
real(8) function code(x_46re, x_46im_m)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    code = (x_46re + (-27.0d0)) * (x_46re * (x_46re * (1.0d0 + (x_46im_m / x_46re))))
end function

Java

x.im_m = Math.abs(x_46_im);
public static double code(double x_46_re, double x_46_im_m) {
	return (x_46_re + -27.0) * (x_46_re * (x_46_re * (1.0 + (x_46_im_m / x_46_re))));
}

Python

x.im_m = math.fabs(x_46_im)
def code(x_46_re, x_46_im_m):
	return (x_46_re + -27.0) * (x_46_re * (x_46_re * (1.0 + (x_46_im_m / x_46_re))))

Julia

x.im_m = abs(x_46_im)
function code(x_46_re, x_46_im_m)
	return Float64(Float64(x_46_re + -27.0) * Float64(x_46_re * Float64(x_46_re * Float64(1.0 + Float64(x_46_im_m / x_46_re)))))
end

MATLAB

x.im_m = abs(x_46_im);
function tmp = code(x_46_re, x_46_im_m)
	tmp = (x_46_re + -27.0) * (x_46_re * (x_46_re * (1.0 + (x_46_im_m / x_46_re))));
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
code[x$46$re_, x$46$im$95$m_] := N[(N[(x$46$re + -27.0), $MachinePrecision] * N[(x$46$re * N[(x$46$re * N[(1.0 + N[(x$46$im$95$m / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.5%

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

   1. difference-of-squares 87.8%

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

4. Applied egg-rr 87.8%

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

6. Simplified 57.6%

   \[\leadsto \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
7. Step-by-step derivation

   1. *-commutative 57.6%

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

   2. *-un-lft-identity 57.6%

      \[\leadsto \left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right) \cdot x.re - \color{blue}{\left(1 \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\right)} \cdot x.im \]

   3. distribute-lft-in 57.6%

      \[\leadsto \left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right) \cdot x.re - \color{blue}{\left(1 \cdot \left(x.re \cdot x.im\right) + 1 \cdot \left(x.re \cdot x.im\right)\right)} \cdot x.im \]

   4. distribute-rgt-out 57.6%

      \[\leadsto \left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right) \cdot x.re - \color{blue}{\left(\left(x.re \cdot x.im\right) \cdot \left(1 + 1\right)\right)} \cdot x.im \]

   5. metadata-eval 57.6%

      \[\leadsto \left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right) \cdot x.re - \left(\left(x.re \cdot x.im\right) \cdot \color{blue}{2}\right) \cdot x.im \]

8. Applied egg-rr 57.6%

   \[\leadsto \left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right) \cdot x.re - \color{blue}{\left(\left(x.re \cdot x.im\right) \cdot 2\right)} \cdot x.im \]

9. Taylor expanded in x.im around 0 36.6%

   \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right) + {x.re}^{2} \cdot \left(x.re - 27\right)} \]
10. Step-by-step derivation

    1. sub-neg 36.6%

       \[\leadsto x.im \cdot \left(x.re \cdot \color{blue}{\left(x.re + \left(-27\right)\right)}\right) + {x.re}^{2} \cdot \left(x.re - 27\right) \]

    2. metadata-eval 36.6%

       \[\leadsto x.im \cdot \left(x.re \cdot \left(x.re + \color{blue}{-27}\right)\right) + {x.re}^{2} \cdot \left(x.re - 27\right) \]

    3. associate-*r* 40.9%

       \[\leadsto \color{blue}{\left(x.im \cdot x.re\right) \cdot \left(x.re + -27\right)} + {x.re}^{2} \cdot \left(x.re - 27\right) \]

    4. *-commutative 40.9%

       \[\leadsto \color{blue}{\left(x.re \cdot x.im\right)} \cdot \left(x.re + -27\right) + {x.re}^{2} \cdot \left(x.re - 27\right) \]

    5. sub-neg 40.9%

       \[\leadsto \left(x.re \cdot x.im\right) \cdot \left(x.re + -27\right) + {x.re}^{2} \cdot \color{blue}{\left(x.re + \left(-27\right)\right)} \]

    6. metadata-eval 40.9%

       \[\leadsto \left(x.re \cdot x.im\right) \cdot \left(x.re + -27\right) + {x.re}^{2} \cdot \left(x.re + \color{blue}{-27}\right) \]

    7. distribute-rgt-out 47.5%

       \[\leadsto \color{blue}{\left(x.re + -27\right) \cdot \left(x.re \cdot x.im + {x.re}^{2}\right)} \]

    8. unpow2 47.5%

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

    9. distribute-lft-in 50.2%

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

11. Simplified 50.2%

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

12. Taylor expanded in x.re around inf 51.6%

    \[\leadsto \left(x.re + -27\right) \cdot \left(x.re \cdot \color{blue}{\left(x.re \cdot \left(1 + \frac{x.im}{x.re}\right)\right)}\right) \]
13. Add Preprocessing

Alternative 7: 53.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ \left(x.im\_m + x.re\right) \cdot \left(x.re \cdot \left(x.re + -27\right)\right) \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
(FPCore (x.re x.im_m)
 :precision binary64
 (* (+ x.im_m x.re) (* x.re (+ x.re -27.0))))

C

x.im_m = fabs(x_46_im);
double code(double x_46_re, double x_46_im_m) {
	return (x_46_im_m + x_46_re) * (x_46_re * (x_46_re + -27.0));
}

Fortran

x.im_m = abs(x_46im)
real(8) function code(x_46re, x_46im_m)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    code = (x_46im_m + x_46re) * (x_46re * (x_46re + (-27.0d0)))
end function

Java

x.im_m = Math.abs(x_46_im);
public static double code(double x_46_re, double x_46_im_m) {
	return (x_46_im_m + x_46_re) * (x_46_re * (x_46_re + -27.0));
}

Python

x.im_m = math.fabs(x_46_im)
def code(x_46_re, x_46_im_m):
	return (x_46_im_m + x_46_re) * (x_46_re * (x_46_re + -27.0))

Julia

x.im_m = abs(x_46_im)
function code(x_46_re, x_46_im_m)
	return Float64(Float64(x_46_im_m + x_46_re) * Float64(x_46_re * Float64(x_46_re + -27.0)))
end

MATLAB

x.im_m = abs(x_46_im);
function tmp = code(x_46_re, x_46_im_m)
	tmp = (x_46_im_m + x_46_re) * (x_46_re * (x_46_re + -27.0));
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
code[x$46$re_, x$46$im$95$m_] := N[(N[(x$46$im$95$m + x$46$re), $MachinePrecision] * N[(x$46$re * N[(x$46$re + -27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.5%

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

   1. difference-of-squares 87.8%

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

4. Applied egg-rr 87.8%

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

6. Simplified 57.6%

   \[\leadsto \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right)} \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
7. Step-by-step derivation

   1. *-commutative 57.6%

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

   2. *-un-lft-identity 57.6%

      \[\leadsto \left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right) \cdot x.re - \color{blue}{\left(1 \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\right)} \cdot x.im \]

   3. distribute-lft-in 57.6%

      \[\leadsto \left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right) \cdot x.re - \color{blue}{\left(1 \cdot \left(x.re \cdot x.im\right) + 1 \cdot \left(x.re \cdot x.im\right)\right)} \cdot x.im \]

   4. distribute-rgt-out 57.6%

      \[\leadsto \left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right) \cdot x.re - \color{blue}{\left(\left(x.re \cdot x.im\right) \cdot \left(1 + 1\right)\right)} \cdot x.im \]

   5. metadata-eval 57.6%

      \[\leadsto \left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right) \cdot x.re - \left(\left(x.re \cdot x.im\right) \cdot \color{blue}{2}\right) \cdot x.im \]

8. Applied egg-rr 57.6%

   \[\leadsto \left(\left(x.im + x.re\right) \cdot \left(x.re + -27\right)\right) \cdot x.re - \color{blue}{\left(\left(x.re \cdot x.im\right) \cdot 2\right)} \cdot x.im \]

9. Taylor expanded in x.im around 0 36.6%

   \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right) + {x.re}^{2} \cdot \left(x.re - 27\right)} \]
10. Step-by-step derivation

    1. sub-neg 36.6%

       \[\leadsto x.im \cdot \left(x.re \cdot \color{blue}{\left(x.re + \left(-27\right)\right)}\right) + {x.re}^{2} \cdot \left(x.re - 27\right) \]

    2. metadata-eval 36.6%

       \[\leadsto x.im \cdot \left(x.re \cdot \left(x.re + \color{blue}{-27}\right)\right) + {x.re}^{2} \cdot \left(x.re - 27\right) \]

    3. associate-*r* 40.9%

       \[\leadsto \color{blue}{\left(x.im \cdot x.re\right) \cdot \left(x.re + -27\right)} + {x.re}^{2} \cdot \left(x.re - 27\right) \]

    4. *-commutative 40.9%

       \[\leadsto \color{blue}{\left(x.re \cdot x.im\right)} \cdot \left(x.re + -27\right) + {x.re}^{2} \cdot \left(x.re - 27\right) \]

    5. sub-neg 40.9%

       \[\leadsto \left(x.re \cdot x.im\right) \cdot \left(x.re + -27\right) + {x.re}^{2} \cdot \color{blue}{\left(x.re + \left(-27\right)\right)} \]

    6. metadata-eval 40.9%

       \[\leadsto \left(x.re \cdot x.im\right) \cdot \left(x.re + -27\right) + {x.re}^{2} \cdot \left(x.re + \color{blue}{-27}\right) \]

    7. distribute-rgt-out 47.5%

       \[\leadsto \color{blue}{\left(x.re + -27\right) \cdot \left(x.re \cdot x.im + {x.re}^{2}\right)} \]

    8. unpow2 47.5%

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

    9. distribute-lft-in 50.2%

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

11. Simplified 50.2%

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

12. Taylor expanded in x.im around 0 36.6%

    \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right) + {x.re}^{2} \cdot \left(x.re - 27\right)} \]
13. Step-by-step derivation

    1. +-commutative 36.6%

       \[\leadsto \color{blue}{{x.re}^{2} \cdot \left(x.re - 27\right) + x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right)} \]

    2. sub-neg 36.6%

       \[\leadsto {x.re}^{2} \cdot \color{blue}{\left(x.re + \left(-27\right)\right)} + x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right) \]

    3. metadata-eval 36.6%

       \[\leadsto {x.re}^{2} \cdot \left(x.re + \color{blue}{-27}\right) + x.im \cdot \left(x.re \cdot \left(x.re - 27\right)\right) \]

    4. sub-neg 36.6%

       \[\leadsto {x.re}^{2} \cdot \left(x.re + -27\right) + x.im \cdot \left(x.re \cdot \color{blue}{\left(x.re + \left(-27\right)\right)}\right) \]

    5. metadata-eval 36.6%

       \[\leadsto {x.re}^{2} \cdot \left(x.re + -27\right) + x.im \cdot \left(x.re \cdot \left(x.re + \color{blue}{-27}\right)\right) \]

    6. associate-*r* 40.9%

       \[\leadsto {x.re}^{2} \cdot \left(x.re + -27\right) + \color{blue}{\left(x.im \cdot x.re\right) \cdot \left(x.re + -27\right)} \]

    7. *-commutative 40.9%

       \[\leadsto {x.re}^{2} \cdot \left(x.re + -27\right) + \color{blue}{\left(x.re \cdot x.im\right)} \cdot \left(x.re + -27\right) \]

    8. distribute-rgt-in 47.5%

       \[\leadsto \color{blue}{\left(x.re + -27\right) \cdot \left({x.re}^{2} + x.re \cdot x.im\right)} \]

    9. unpow2 47.5%

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

    10. distribute-lft-in 50.2%

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

    11. associate-*r* 50.2%

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

    12. *-commutative 50.2%

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

14. Simplified 50.2%

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

15. Final simplification 50.2%

    \[\leadsto \left(x.im + x.re\right) \cdot \left(x.re \cdot \left(x.re + -27\right)\right) \]
16. Add Preprocessing

Alternative 8: 2.8% accurate, 19.0× speedup

Math

\[\begin{array}{l} x.im_m = \left|x.im\right| \\ 8 \end{array} \]

FPCore

x.im_m = (fabs.f64 x.im)
(FPCore (x.re x.im_m) :precision binary64 8.0)

C

x.im_m = fabs(x_46_im);
double code(double x_46_re, double x_46_im_m) {
	return 8.0;
}

Fortran

x.im_m = abs(x_46im)
real(8) function code(x_46re, x_46im_m)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im_m
    code = 8.0d0
end function

Java

x.im_m = Math.abs(x_46_im);
public static double code(double x_46_re, double x_46_im_m) {
	return 8.0;
}

Python

x.im_m = math.fabs(x_46_im)
def code(x_46_re, x_46_im_m):
	return 8.0

Julia

x.im_m = abs(x_46_im)
function code(x_46_re, x_46_im_m)
	return 8.0
end

MATLAB

x.im_m = abs(x_46_im);
function tmp = code(x_46_re, x_46_im_m)
	tmp = 8.0;
end

Wolfram

x.im_m = N[Abs[x$46$im], $MachinePrecision]
code[x$46$re_, x$46$im$95$m_] := 8.0

Derivation

1. Initial program 83.5%

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

3. Simplified 81.5%

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

   1. flip-+ 18.1%

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

   2. unpow-prod-down 18.1%

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

   3. div-sub 18.1%

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

   4. pow2 18.1%

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

   5. pow-pow 18.1%

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

   6. metadata-eval 18.1%

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

   7. *-commutative 18.1%

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

   8. associate-*r* 18.1%

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

   9. associate-*l* 18.1%

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

   10. pow2 18.1%

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

6. Applied egg-rr 11.1%

   \[\leadsto \color{blue}{\frac{{x.re}^{6}}{{x.re}^{3} - {x.im}^{2} \cdot \left(-3 \cdot x.re\right)} - \frac{{x.re}^{2} \cdot \left({x.im}^{4} \cdot 9\right)}{{x.re}^{3} - {x.im}^{2} \cdot \left(-3 \cdot x.re\right)}} \]

8. Simplified 2.7%

   \[\leadsto \color{blue}{8} \]
9. Add Preprocessing

Developer Target 1: 87.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024179 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (* (* x.re x.re) (- x.re x.im)) (* (* x.re x.im) (- x.re (* 3 x.im)))))

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