math.cube on complex, real part

Percentage Accurate: 82.7% → 99.7%

Time: 14.0s

Alternatives: 10

Speedup: 0.4×

• 45.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 82.7% 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: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) - x.im \cdot \left(x.re\_m \cdot x.im + x.re\_m \cdot x.im\right) \leq 0:\\ \;\;\;\;\left(x.re\_m \cdot x.im\right) \cdot \left(x.im \cdot -3\right)\\ \mathbf{else}:\\ \;\;\;\;{x.re\_m}^{3} \cdot \left(1 + -3 \cdot \left(\frac{x.im}{x.re\_m} \cdot \frac{x.im}{x.re\_m}\right)\right)\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<=
       (-
        (* x.re_m (- (* x.re_m x.re_m) (* x.im x.im)))
        (* x.im (+ (* x.re_m x.im) (* x.re_m x.im))))
       0.0)
    (* (* x.re_m x.im) (* x.im -3.0))
    (*
     (pow x.re_m 3.0)
     (+ 1.0 (* -3.0 (* (/ x.im x.re_m) (/ x.im x.re_m))))))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= 0.0) {
		tmp = (x_46_re_m * x_46_im) * (x_46_im * -3.0);
	} else {
		tmp = pow(x_46_re_m, 3.0) * (1.0 + (-3.0 * ((x_46_im / x_46_re_m) * (x_46_im / x_46_re_m))));
	}
	return x_46_re_s * tmp;
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (((x_46re_m * ((x_46re_m * x_46re_m) - (x_46im * x_46im))) - (x_46im * ((x_46re_m * x_46im) + (x_46re_m * x_46im)))) <= 0.0d0) then
        tmp = (x_46re_m * x_46im) * (x_46im * (-3.0d0))
    else
        tmp = (x_46re_m ** 3.0d0) * (1.0d0 + ((-3.0d0) * ((x_46im / x_46re_m) * (x_46im / x_46re_m))))
    end if
    code = x_46re_s * tmp
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= 0.0) {
		tmp = (x_46_re_m * x_46_im) * (x_46_im * -3.0);
	} else {
		tmp = Math.pow(x_46_re_m, 3.0) * (1.0 + (-3.0 * ((x_46_im / x_46_re_m) * (x_46_im / x_46_re_m))));
	}
	return x_46_re_s * tmp;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	tmp = 0
	if ((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= 0.0:
		tmp = (x_46_re_m * x_46_im) * (x_46_im * -3.0)
	else:
		tmp = math.pow(x_46_re_m, 3.0) * (1.0 + (-3.0 * ((x_46_im / x_46_re_m) * (x_46_im / x_46_re_m))))
	return x_46_re_s * tmp

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im))) - Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_re_m * x_46_im)))) <= 0.0)
		tmp = Float64(Float64(x_46_re_m * x_46_im) * Float64(x_46_im * -3.0));
	else
		tmp = Float64((x_46_re_m ^ 3.0) * Float64(1.0 + Float64(-3.0 * Float64(Float64(x_46_im / x_46_re_m) * Float64(x_46_im / x_46_re_m)))));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0;
	if (((x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= 0.0)
		tmp = (x_46_re_m * x_46_im) * (x_46_im * -3.0);
	else
		tmp = (x_46_re_m ^ 3.0) * (1.0 + (-3.0 * ((x_46_im / x_46_re_m) * (x_46_im / x_46_re_m))));
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision], N[(N[Power[x$46$re$95$m, 3.0], $MachinePrecision] * N[(1.0 + N[(-3.0 * N[(N[(x$46$im / x$46$re$95$m), $MachinePrecision] * N[(x$46$im / x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < 0.0

   1. Initial program 90.6%

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

   3. Simplified 86.8%

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

   5. Taylor expanded in x.re around 0 56.9%

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

      1. add-sqr-sqrt 44.0%

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

      2. unpow2 44.0%

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

      3. swap-sqr 53.1%

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

      4. unpow2 53.1%

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

   7. Applied egg-rr 53.1%

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

      1. unpow2 53.1%

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

      2. *-commutative 53.1%

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

      3. associate-*r* 53.2%

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

      4. associate-*r* 53.2%

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

      5. add-sqr-sqrt 66.1%

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

      6. *-commutative 66.1%

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

   9. Applied egg-rr 66.1%

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

       1. associate-*r* 66.2%

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

       2. *-commutative 66.2%

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

       3. add-log-exp 41.7%

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

       4. *-un-lft-identity 41.7%

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

       5. 1-exp 41.7%

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

       6. mul0-lft 41.7%

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

       7. *-commutative 41.7%

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

       8. mul0-lft 41.7%

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

       9. metadata-eval 41.7%

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

       10. distribute-rgt1-in 41.7%

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

       11. exp-sum 41.7%

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

       12. +-commutative 41.7%

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

       13. add-log-exp 66.2%

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

       14. *-commutative 66.2%

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

       15. distribute-rgt-in 66.2%

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

       16. fma-define 66.2%

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

   11. Applied egg-rr 66.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x.im \cdot \left(x.re \cdot -3\right), x.im, x.im \cdot 0\right)} \]
   12. Step-by-step derivation

       1. fma-undefine 66.1%

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

       2. mul0-rgt 66.1%

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

       3. +-rgt-identity 66.1%

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

       4. associate-*r* 66.2%

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

       5. associate-*l* 66.2%

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

       6. *-commutative 66.2%

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

   13. Applied egg-rr 66.2%

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

   if 0.0 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im))

   1. Initial program 63.5%

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

   3. Simplified 60.4%

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

   5. Taylor expanded in x.re around inf 58.7%

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

      1. unpow2 58.7%

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

      2. unpow2 58.7%

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

      3. times-frac 79.5%

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

   7. Applied egg-rr 79.5%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq 0:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.im \cdot -3\right)\\ \mathbf{else}:\\ \;\;\;\;{x.re}^{3} \cdot \left(1 + -3 \cdot \left(\frac{x.im}{x.re} \cdot \frac{x.im}{x.re}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 17:\\ \;\;\;\;{x.re\_m}^{3} + -3 \cdot \left(x.im \cdot \left(x.re\_m \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(\left(x.re\_m - x.im\right) \cdot \left(x.re\_m + x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 17.0)
    (+ (pow x.re_m 3.0) (* -3.0 (* x.im (* x.re_m x.im))))
    (* x.re_m (* (- x.re_m x.im) (+ x.re_m x.im))))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 17.0) {
		tmp = pow(x_46_re_m, 3.0) + (-3.0 * (x_46_im * (x_46_re_m * x_46_im)));
	} else {
		tmp = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im));
	}
	return x_46_re_s * tmp;
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re_m <= 17.0d0) then
        tmp = (x_46re_m ** 3.0d0) + ((-3.0d0) * (x_46im * (x_46re_m * x_46im)))
    else
        tmp = x_46re_m * ((x_46re_m - x_46im) * (x_46re_m + x_46im))
    end if
    code = x_46re_s * tmp
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 17.0) {
		tmp = Math.pow(x_46_re_m, 3.0) + (-3.0 * (x_46_im * (x_46_re_m * x_46_im)));
	} else {
		tmp = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im));
	}
	return x_46_re_s * tmp;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	tmp = 0
	if x_46_re_m <= 17.0:
		tmp = math.pow(x_46_re_m, 3.0) + (-3.0 * (x_46_im * (x_46_re_m * x_46_im)))
	else:
		tmp = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im))
	return x_46_re_s * tmp

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_re_m <= 17.0)
		tmp = Float64((x_46_re_m ^ 3.0) + Float64(-3.0 * Float64(x_46_im * Float64(x_46_re_m * x_46_im))));
	else
		tmp = Float64(x_46_re_m * Float64(Float64(x_46_re_m - x_46_im) * Float64(x_46_re_m + x_46_im)));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0;
	if (x_46_re_m <= 17.0)
		tmp = (x_46_re_m ^ 3.0) + (-3.0 * (x_46_im * (x_46_re_m * x_46_im)));
	else
		tmp = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im));
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 17.0], N[(N[Power[x$46$re$95$m, 3.0], $MachinePrecision] + N[(-3.0 * N[(x$46$im * N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re$95$m * N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(x$46$re$95$m + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x.re < 17

   1. Initial program 79.1%

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

   3. Simplified 76.9%

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

      1. add-sqr-sqrt 46.6%

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

      2. pow2 46.6%

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

      3. *-commutative 46.6%

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

      4. sqrt-prod 24.5%

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

      5. sqrt-prod 16.5%

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

      6. add-sqr-sqrt 31.0%

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

   6. Applied egg-rr 31.0%

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

      1. unpow2 29.4%

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

      2. *-commutative 29.4%

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

      3. associate-*r* 29.5%

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

      4. associate-*r* 29.5%

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

      5. add-sqr-sqrt 68.6%

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

      6. *-commutative 68.6%

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

   8. Applied egg-rr 88.0%

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

   if 17 < x.re

   1. Initial program 72.7%

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

      1. *-commutative 72.7%

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

      2. *-commutative 72.7%

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

      3. flip-+ 0.0%

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

      4. +-inverses 0.0%

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

      5. metadata-eval 0.0%

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

      6. +-inverses 0.0%

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

      7. metadata-eval 0.0%

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

      8. associate-*r/ 0.0%

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

      9. metadata-eval 0.0%

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

      10. metadata-eval 0.0%

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

   4. Applied egg-rr 0.0%

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

   6. Simplified 81.6%

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

      1. difference-of-squares 98.7%

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

      2. *-commutative 98.7%

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

   8. Applied egg-rr 98.7%

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

4. Final simplification 90.9%

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

Alternative 3: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ \begin{array}{l} t_0 := x.re\_m \cdot \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) - x.im \cdot \left(x.re\_m \cdot x.im + x.re\_m \cdot x.im\right)\\ t_1 := x.re\_m \cdot \left(\left(x.re\_m - x.im\right) \cdot \left(x.re\_m + x.im\right)\right)\\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(x.re\_m \cdot x.im\right) \cdot \left(x.im \cdot -3\right)\\ \mathbf{elif}\;t\_0 \leq 10^{+72}:\\ \;\;\;\;t\_1 - x.im \cdot \left(\left(x.re\_m \cdot x.im\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (let* ((t_0
         (-
          (* x.re_m (- (* x.re_m x.re_m) (* x.im x.im)))
          (* x.im (+ (* x.re_m x.im) (* x.re_m x.im)))))
        (t_1 (* x.re_m (* (- x.re_m x.im) (+ x.re_m x.im)))))
   (*
    x.re_s
    (if (<= t_0 (- INFINITY))
      (* (* x.re_m x.im) (* x.im -3.0))
      (if (<= t_0 1e+72) (- t_1 (* x.im (* (* x.re_m x.im) 2.0))) t_1)))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double t_0 = (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)));
	double t_1 = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (x_46_re_m * x_46_im) * (x_46_im * -3.0);
	} else if (t_0 <= 1e+72) {
		tmp = t_1 - (x_46_im * ((x_46_re_m * x_46_im) * 2.0));
	} else {
		tmp = t_1;
	}
	return x_46_re_s * tmp;
}

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double t_0 = (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)));
	double t_1 = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = (x_46_re_m * x_46_im) * (x_46_im * -3.0);
	} else if (t_0 <= 1e+72) {
		tmp = t_1 - (x_46_im * ((x_46_re_m * x_46_im) * 2.0));
	} else {
		tmp = t_1;
	}
	return x_46_re_s * tmp;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	t_0 = (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))
	t_1 = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = (x_46_re_m * x_46_im) * (x_46_im * -3.0)
	elif t_0 <= 1e+72:
		tmp = t_1 - (x_46_im * ((x_46_re_m * x_46_im) * 2.0))
	else:
		tmp = t_1
	return x_46_re_s * tmp

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	t_0 = Float64(Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im))) - Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_re_m * x_46_im))))
	t_1 = Float64(x_46_re_m * Float64(Float64(x_46_re_m - x_46_im) * Float64(x_46_re_m + x_46_im)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(x_46_re_m * x_46_im) * Float64(x_46_im * -3.0));
	elseif (t_0 <= 1e+72)
		tmp = Float64(t_1 - Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_im) * 2.0)));
	else
		tmp = t_1;
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	t_0 = (x_46_re_m * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) - (x_46_im * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)));
	t_1 = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = (x_46_re_m * x_46_im) * (x_46_im * -3.0);
	elseif (t_0 <= 1e+72)
		tmp = t_1 - (x_46_im * ((x_46_re_m * x_46_im) * 2.0));
	else
		tmp = t_1;
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := Block[{t$95$0 = N[(N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x$46$re$95$m * N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(x$46$re$95$m + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$re$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+72], N[(t$95$1 - N[(x$46$im * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 80.3%

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

   3. Simplified 72.3%

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

   5. Taylor expanded in x.re around 0 40.4%

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

      1. add-sqr-sqrt 40.0%

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

      2. unpow2 40.0%

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

      3. swap-sqr 59.5%

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

      4. unpow2 59.5%

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

   7. Applied egg-rr 59.5%

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

      1. unpow2 59.5%

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

      2. *-commutative 59.5%

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

      3. associate-*r* 59.5%

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

      4. associate-*r* 59.5%

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

      5. add-sqr-sqrt 60.0%

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

      6. *-commutative 60.0%

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

   9. Applied egg-rr 60.0%

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

       1. associate-*r* 59.9%

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

       2. *-commutative 59.9%

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

       3. add-log-exp 40.1%

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

       4. *-un-lft-identity 40.1%

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

       5. 1-exp 40.1%

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

       6. mul0-lft 40.1%

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

       7. *-commutative 40.1%

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

       8. mul0-lft 40.1%

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

       9. metadata-eval 40.1%

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

       10. distribute-rgt1-in 40.1%

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

       11. exp-sum 40.1%

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

       12. +-commutative 40.1%

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

       13. add-log-exp 59.9%

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

       14. *-commutative 59.9%

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

       15. distribute-rgt-in 59.9%

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

       16. fma-define 59.9%

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

   11. Applied egg-rr 60.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x.im \cdot \left(x.re \cdot -3\right), x.im, x.im \cdot 0\right)} \]
   12. Step-by-step derivation

       1. fma-undefine 60.0%

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

       2. mul0-rgt 60.0%

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

       3. +-rgt-identity 60.0%

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

       4. associate-*r* 59.9%

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

       5. associate-*l* 60.0%

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

       6. *-commutative 60.0%

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

   13. Applied egg-rr 60.0%

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

   if -inf.0 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < 9.99999999999999944e71

   1. Initial program 99.7%

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

      1. difference-of-squares 67.9%

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

      2. *-commutative 67.9%

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

   4. Applied egg-rr 99.7%

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

      1. *-commutative 99.7%

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

      2. *-un-lft-identity 99.7%

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

      3. distribute-lft-in 99.7%

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

      4. distribute-rgt-out 99.7%

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

      5. metadata-eval 99.7%

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

   6. Applied egg-rr 99.7%

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

   if 9.99999999999999944e71 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im))

   1. Initial program 56.2%

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

      1. *-commutative 56.2%

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

      2. *-commutative 56.2%

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

      3. flip-+ 0.0%

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

      4. +-inverses 0.0%

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

      5. metadata-eval 0.0%

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

      6. +-inverses 0.0%

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

      7. metadata-eval 0.0%

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

      8. associate-*r/ 0.0%

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

      9. metadata-eval 0.0%

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

      10. metadata-eval 0.0%

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

   4. Applied egg-rr 0.0%

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

   6. Simplified 59.7%

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

      1. difference-of-squares 84.7%

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

      2. *-commutative 84.7%

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

   8. Applied egg-rr 84.7%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -\infty:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.im \cdot -3\right)\\ \mathbf{elif}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq 10^{+72}:\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) - x.im \cdot \left(\left(x.re \cdot x.im\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.re\_m \leq 2.05 \cdot 10^{-66}:\\ \;\;\;\;\left(x.re\_m \cdot x.im\right) \cdot \left(x.im \cdot -3\right)\\ \mathbf{elif}\;x.re\_m \leq 3.5 \cdot 10^{-30} \lor \neg \left(x.re\_m \leq 5.1 \cdot 10^{-30} \lor \neg \left(x.re\_m \leq 6.4 \cdot 10^{-16}\right) \land \left(x.re\_m \leq 0.000245 \lor \neg \left(x.re\_m \leq 8.8 \cdot 10^{+61}\right) \land x.re\_m \leq 9 \cdot 10^{+61}\right)\right):\\ \;\;\;\;x.re\_m \cdot \left(\left(x.re\_m - x.im\right) \cdot \left(x.re\_m + x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.re\_m \cdot x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (*
  x.re_s
  (if (<= x.re_m 2.05e-66)
    (* (* x.re_m x.im) (* x.im -3.0))
    (if (or (<= x.re_m 3.5e-30)
            (not
             (or (<= x.re_m 5.1e-30)
                 (and (not (<= x.re_m 6.4e-16))
                      (or (<= x.re_m 0.000245)
                          (and (not (<= x.re_m 8.8e+61))
                               (<= x.re_m 9e+61)))))))
      (* x.re_m (* (- x.re_m x.im) (+ x.re_m x.im)))
      (* -3.0 (* x.im (* x.re_m x.im)))))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 2.05e-66) {
		tmp = (x_46_re_m * x_46_im) * (x_46_im * -3.0);
	} else if ((x_46_re_m <= 3.5e-30) || !((x_46_re_m <= 5.1e-30) || (!(x_46_re_m <= 6.4e-16) && ((x_46_re_m <= 0.000245) || (!(x_46_re_m <= 8.8e+61) && (x_46_re_m <= 9e+61)))))) {
		tmp = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im));
	} else {
		tmp = -3.0 * (x_46_im * (x_46_re_m * x_46_im));
	}
	return x_46_re_s * tmp;
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re_m <= 2.05d-66) then
        tmp = (x_46re_m * x_46im) * (x_46im * (-3.0d0))
    else if ((x_46re_m <= 3.5d-30) .or. (.not. (x_46re_m <= 5.1d-30) .or. (.not. (x_46re_m <= 6.4d-16)) .and. (x_46re_m <= 0.000245d0) .or. (.not. (x_46re_m <= 8.8d+61)) .and. (x_46re_m <= 9d+61))) then
        tmp = x_46re_m * ((x_46re_m - x_46im) * (x_46re_m + x_46im))
    else
        tmp = (-3.0d0) * (x_46im * (x_46re_m * x_46im))
    end if
    code = x_46re_s * tmp
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 2.05e-66) {
		tmp = (x_46_re_m * x_46_im) * (x_46_im * -3.0);
	} else if ((x_46_re_m <= 3.5e-30) || !((x_46_re_m <= 5.1e-30) || (!(x_46_re_m <= 6.4e-16) && ((x_46_re_m <= 0.000245) || (!(x_46_re_m <= 8.8e+61) && (x_46_re_m <= 9e+61)))))) {
		tmp = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im));
	} else {
		tmp = -3.0 * (x_46_im * (x_46_re_m * x_46_im));
	}
	return x_46_re_s * tmp;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	tmp = 0
	if x_46_re_m <= 2.05e-66:
		tmp = (x_46_re_m * x_46_im) * (x_46_im * -3.0)
	elif (x_46_re_m <= 3.5e-30) or not ((x_46_re_m <= 5.1e-30) or (not (x_46_re_m <= 6.4e-16) and ((x_46_re_m <= 0.000245) or (not (x_46_re_m <= 8.8e+61) and (x_46_re_m <= 9e+61))))):
		tmp = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im))
	else:
		tmp = -3.0 * (x_46_im * (x_46_re_m * x_46_im))
	return x_46_re_s * tmp

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_re_m <= 2.05e-66)
		tmp = Float64(Float64(x_46_re_m * x_46_im) * Float64(x_46_im * -3.0));
	elseif ((x_46_re_m <= 3.5e-30) || !((x_46_re_m <= 5.1e-30) || (!(x_46_re_m <= 6.4e-16) && ((x_46_re_m <= 0.000245) || (!(x_46_re_m <= 8.8e+61) && (x_46_re_m <= 9e+61))))))
		tmp = Float64(x_46_re_m * Float64(Float64(x_46_re_m - x_46_im) * Float64(x_46_re_m + x_46_im)));
	else
		tmp = Float64(-3.0 * Float64(x_46_im * Float64(x_46_re_m * x_46_im)));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = 0.0;
	if (x_46_re_m <= 2.05e-66)
		tmp = (x_46_re_m * x_46_im) * (x_46_im * -3.0);
	elseif ((x_46_re_m <= 3.5e-30) || ~(((x_46_re_m <= 5.1e-30) || (~((x_46_re_m <= 6.4e-16)) && ((x_46_re_m <= 0.000245) || (~((x_46_re_m <= 8.8e+61)) && (x_46_re_m <= 9e+61)))))))
		tmp = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m + x_46_im));
	else
		tmp = -3.0 * (x_46_im * (x_46_re_m * x_46_im));
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * If[LessEqual[x$46$re$95$m, 2.05e-66], N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x$46$re$95$m, 3.5e-30], N[Not[Or[LessEqual[x$46$re$95$m, 5.1e-30], And[N[Not[LessEqual[x$46$re$95$m, 6.4e-16]], $MachinePrecision], Or[LessEqual[x$46$re$95$m, 0.000245], And[N[Not[LessEqual[x$46$re$95$m, 8.8e+61]], $MachinePrecision], LessEqual[x$46$re$95$m, 9e+61]]]]]], $MachinePrecision]], N[(x$46$re$95$m * N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(x$46$re$95$m + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-3.0 * N[(x$46$im * N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x.re < 2.04999999999999999e-66

   1. Initial program 78.6%

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

   3. Simplified 76.3%

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

   5. Taylor expanded in x.re around 0 57.6%

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

      1. add-sqr-sqrt 21.2%

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

      2. unpow2 21.2%

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

      3. swap-sqr 27.5%

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

      4. unpow2 27.5%

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

   7. Applied egg-rr 27.5%

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

      1. unpow2 27.5%

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

      2. *-commutative 27.5%

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

      3. associate-*r* 27.5%

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

      4. associate-*r* 27.5%

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

      5. add-sqr-sqrt 68.6%

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

      6. *-commutative 68.6%

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

   9. Applied egg-rr 68.6%

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

       1. associate-*r* 68.6%

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

       2. *-commutative 68.6%

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

       3. add-log-exp 36.6%

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

       4. *-un-lft-identity 36.6%

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

       5. 1-exp 36.6%

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

       6. mul0-lft 36.6%

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

       7. *-commutative 36.6%

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

       8. mul0-lft 36.6%

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

       9. metadata-eval 36.6%

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

       10. distribute-rgt1-in 36.6%

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

       11. exp-sum 36.6%

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

       12. +-commutative 36.6%

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

       13. add-log-exp 68.6%

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

       14. *-commutative 68.6%

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

       15. distribute-rgt-in 68.6%

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

       16. fma-define 68.6%

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

   11. Applied egg-rr 68.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x.im \cdot \left(x.re \cdot -3\right), x.im, x.im \cdot 0\right)} \]
   12. Step-by-step derivation

       1. fma-undefine 68.6%

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

       2. mul0-rgt 68.6%

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

       3. +-rgt-identity 68.6%

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

       4. associate-*r* 68.6%

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

       5. associate-*l* 68.7%

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

       6. *-commutative 68.7%

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

   13. Applied egg-rr 68.7%

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

   if 2.04999999999999999e-66 < x.re < 3.5000000000000003e-30 or 5.09999999999999972e-30 < x.re < 6.40000000000000046e-16 or 2.4499999999999999e-4 < x.re < 8.8000000000000001e61 or 9e61 < x.re

   1. Initial program 75.2%

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

      1. *-commutative 75.2%

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

      2. *-commutative 75.2%

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

      3. flip-+ 0.0%

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

      4. +-inverses 0.0%

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

      5. metadata-eval 0.0%

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

      6. +-inverses 0.0%

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

      7. metadata-eval 0.0%

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

      8. associate-*r/ 0.0%

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

      9. metadata-eval 0.0%

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

      10. metadata-eval 0.0%

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

   4. Applied egg-rr 0.0%

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

   6. Simplified 81.7%

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

      1. difference-of-squares 97.3%

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

      2. *-commutative 97.3%

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

   8. Applied egg-rr 97.3%

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

   if 3.5000000000000003e-30 < x.re < 5.09999999999999972e-30 or 6.40000000000000046e-16 < x.re < 2.4499999999999999e-4 or 8.8000000000000001e61 < x.re < 9e61

   1. Initial program 54.3%

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

   3. Simplified 53.5%

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

   5. Taylor expanded in x.re around 0 54.3%

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

      1. add-sqr-sqrt 53.0%

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

      2. unpow2 53.0%

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

      3. swap-sqr 100.0%

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

      4. unpow2 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. unpow2 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

      5. add-sqr-sqrt 100.0%

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

      6. *-commutative 100.0%

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

   9. Applied egg-rr 100.0%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 2.05 \cdot 10^{-66}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.im \cdot -3\right)\\ \mathbf{elif}\;x.re \leq 3.5 \cdot 10^{-30} \lor \neg \left(x.re \leq 5.1 \cdot 10^{-30} \lor \neg \left(x.re \leq 6.4 \cdot 10^{-16}\right) \land \left(x.re \leq 0.000245 \lor \neg \left(x.re \leq 8.8 \cdot 10^{+61}\right) \land x.re \leq 9 \cdot 10^{+61}\right)\right):\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ \begin{array}{l} t_0 := \left(x.re\_m \cdot x.im\right) \cdot \left(x.im \cdot -3\right)\\ t_1 := x.re\_m \cdot \left(\left(x.re\_m - x.im\right) \cdot \left(x.re\_m - x.im\right)\right)\\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.im \leq 245000000000:\\ \;\;\;\;\left(x.re\_m - x.im\right) \cdot \left(x.re\_m \cdot \left(x.re\_m - x.im\right)\right)\\ \mathbf{elif}\;x.im \leq 1.8 \cdot 10^{+82}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x.im \leq 10^{+109}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x.im \leq 1.42 \cdot 10^{+147}:\\ \;\;\;\;x.im \cdot \left(x.re\_m \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{elif}\;x.im \leq 1.05 \cdot 10^{+148} \lor \neg \left(x.im \leq 9 \cdot 10^{+181}\right) \land x.im \leq 9.5 \cdot 10^{+181}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (let* ((t_0 (* (* x.re_m x.im) (* x.im -3.0)))
        (t_1 (* x.re_m (* (- x.re_m x.im) (- x.re_m x.im)))))
   (*
    x.re_s
    (if (<= x.im 245000000000.0)
      (* (- x.re_m x.im) (* x.re_m (- x.re_m x.im)))
      (if (<= x.im 1.8e+82)
        t_0
        (if (<= x.im 1e+109)
          t_1
          (if (<= x.im 1.42e+147)
            (* x.im (* x.re_m (* x.im -3.0)))
            (if (or (<= x.im 1.05e+148)
                    (and (not (<= x.im 9e+181)) (<= x.im 9.5e+181)))
              t_1
              t_0))))))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double t_0 = (x_46_re_m * x_46_im) * (x_46_im * -3.0);
	double t_1 = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m - x_46_im));
	double tmp;
	if (x_46_im <= 245000000000.0) {
		tmp = (x_46_re_m - x_46_im) * (x_46_re_m * (x_46_re_m - x_46_im));
	} else if (x_46_im <= 1.8e+82) {
		tmp = t_0;
	} else if (x_46_im <= 1e+109) {
		tmp = t_1;
	} else if (x_46_im <= 1.42e+147) {
		tmp = x_46_im * (x_46_re_m * (x_46_im * -3.0));
	} else if ((x_46_im <= 1.05e+148) || (!(x_46_im <= 9e+181) && (x_46_im <= 9.5e+181))) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return x_46_re_s * tmp;
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x_46re_m * x_46im) * (x_46im * (-3.0d0))
    t_1 = x_46re_m * ((x_46re_m - x_46im) * (x_46re_m - x_46im))
    if (x_46im <= 245000000000.0d0) then
        tmp = (x_46re_m - x_46im) * (x_46re_m * (x_46re_m - x_46im))
    else if (x_46im <= 1.8d+82) then
        tmp = t_0
    else if (x_46im <= 1d+109) then
        tmp = t_1
    else if (x_46im <= 1.42d+147) then
        tmp = x_46im * (x_46re_m * (x_46im * (-3.0d0)))
    else if ((x_46im <= 1.05d+148) .or. (.not. (x_46im <= 9d+181)) .and. (x_46im <= 9.5d+181)) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = x_46re_s * tmp
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double t_0 = (x_46_re_m * x_46_im) * (x_46_im * -3.0);
	double t_1 = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m - x_46_im));
	double tmp;
	if (x_46_im <= 245000000000.0) {
		tmp = (x_46_re_m - x_46_im) * (x_46_re_m * (x_46_re_m - x_46_im));
	} else if (x_46_im <= 1.8e+82) {
		tmp = t_0;
	} else if (x_46_im <= 1e+109) {
		tmp = t_1;
	} else if (x_46_im <= 1.42e+147) {
		tmp = x_46_im * (x_46_re_m * (x_46_im * -3.0));
	} else if ((x_46_im <= 1.05e+148) || (!(x_46_im <= 9e+181) && (x_46_im <= 9.5e+181))) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return x_46_re_s * tmp;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	t_0 = (x_46_re_m * x_46_im) * (x_46_im * -3.0)
	t_1 = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m - x_46_im))
	tmp = 0
	if x_46_im <= 245000000000.0:
		tmp = (x_46_re_m - x_46_im) * (x_46_re_m * (x_46_re_m - x_46_im))
	elif x_46_im <= 1.8e+82:
		tmp = t_0
	elif x_46_im <= 1e+109:
		tmp = t_1
	elif x_46_im <= 1.42e+147:
		tmp = x_46_im * (x_46_re_m * (x_46_im * -3.0))
	elif (x_46_im <= 1.05e+148) or (not (x_46_im <= 9e+181) and (x_46_im <= 9.5e+181)):
		tmp = t_1
	else:
		tmp = t_0
	return x_46_re_s * tmp

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	t_0 = Float64(Float64(x_46_re_m * x_46_im) * Float64(x_46_im * -3.0))
	t_1 = Float64(x_46_re_m * Float64(Float64(x_46_re_m - x_46_im) * Float64(x_46_re_m - x_46_im)))
	tmp = 0.0
	if (x_46_im <= 245000000000.0)
		tmp = Float64(Float64(x_46_re_m - x_46_im) * Float64(x_46_re_m * Float64(x_46_re_m - x_46_im)));
	elseif (x_46_im <= 1.8e+82)
		tmp = t_0;
	elseif (x_46_im <= 1e+109)
		tmp = t_1;
	elseif (x_46_im <= 1.42e+147)
		tmp = Float64(x_46_im * Float64(x_46_re_m * Float64(x_46_im * -3.0)));
	elseif ((x_46_im <= 1.05e+148) || (!(x_46_im <= 9e+181) && (x_46_im <= 9.5e+181)))
		tmp = t_1;
	else
		tmp = t_0;
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	t_0 = (x_46_re_m * x_46_im) * (x_46_im * -3.0);
	t_1 = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m - x_46_im));
	tmp = 0.0;
	if (x_46_im <= 245000000000.0)
		tmp = (x_46_re_m - x_46_im) * (x_46_re_m * (x_46_re_m - x_46_im));
	elseif (x_46_im <= 1.8e+82)
		tmp = t_0;
	elseif (x_46_im <= 1e+109)
		tmp = t_1;
	elseif (x_46_im <= 1.42e+147)
		tmp = x_46_im * (x_46_re_m * (x_46_im * -3.0));
	elseif ((x_46_im <= 1.05e+148) || (~((x_46_im <= 9e+181)) && (x_46_im <= 9.5e+181)))
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := Block[{t$95$0 = N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x$46$re$95$m * N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(x$46$re$95$m - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$re$95$s * If[LessEqual[x$46$im, 245000000000.0], N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(x$46$re$95$m * N[(x$46$re$95$m - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 1.8e+82], t$95$0, If[LessEqual[x$46$im, 1e+109], t$95$1, If[LessEqual[x$46$im, 1.42e+147], N[(x$46$im * N[(x$46$re$95$m * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x$46$im, 1.05e+148], And[N[Not[LessEqual[x$46$im, 9e+181]], $MachinePrecision], LessEqual[x$46$im, 9.5e+181]]], t$95$1, t$95$0]]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if x.im < 2.45e11

   1. Initial program 83.2%

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

      1. difference-of-squares 82.3%

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

      2. *-commutative 82.3%

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

   4. Applied egg-rr 86.9%

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

      1. *-commutative 86.9%

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

      2. *-un-lft-identity 86.9%

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

      3. distribute-lft-in 86.9%

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

      4. distribute-rgt-out 86.9%

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

      5. metadata-eval 86.9%

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

   6. Applied egg-rr 86.9%

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

      1. cancel-sign-sub-inv 86.9%

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

      2. associate-*l* 92.1%

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

      3. *-commutative 92.1%

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

      4. fma-define 92.1%

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

      5. +-commutative 92.1%

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

      6. associate-*l* 92.1%

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

   8. Applied egg-rr 92.1%

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

      1. fma-undefine 92.1%

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

      2. +-commutative 92.1%

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

   10. Applied egg-rr 63.5%

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

       1. mul0-rgt 63.5%

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

       2. mul0-rgt 63.5%

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

       3. +-lft-identity 63.5%

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

       4. *-commutative 63.5%

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

       5. associate-*l* 63.5%

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

   12. Simplified 63.5%

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

       1. associate-*r* 63.5%

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

       2. sub-neg 63.5%

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

       3. distribute-lft-in 52.2%

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

   14. Applied egg-rr 52.2%

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

       1. distribute-lft-out 63.5%

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

       2. sub-neg 63.5%

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

   16. Simplified 63.5%

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

   if 2.45e11 < x.im < 1.80000000000000007e82 or 1.04999999999999999e148 < x.im < 9e181 or 9.50000000000000032e181 < x.im

   1. Initial program 57.2%

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

   3. Simplified 49.2%

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

   5. Taylor expanded in x.re around 0 59.1%

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

      1. add-sqr-sqrt 40.7%

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

      2. unpow2 40.7%

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

      3. swap-sqr 55.7%

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

      4. unpow2 55.7%

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

   7. Applied egg-rr 55.7%

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

      1. unpow2 55.7%

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

      2. *-commutative 55.7%

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

      3. associate-*r* 55.8%

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

      4. associate-*r* 55.7%

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

      5. add-sqr-sqrt 79.8%

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

      6. *-commutative 79.8%

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

   9. Applied egg-rr 79.8%

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

       1. associate-*r* 79.8%

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

       2. *-commutative 79.8%

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

       3. add-log-exp 41.9%

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

       4. *-un-lft-identity 41.9%

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

       5. 1-exp 41.9%

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

       6. mul0-lft 41.9%

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

       7. *-commutative 41.9%

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

       8. mul0-lft 41.9%

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

       9. metadata-eval 41.9%

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

       10. distribute-rgt1-in 41.9%

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

       11. exp-sum 41.9%

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

       12. +-commutative 41.9%

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

       13. add-log-exp 79.8%

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

       14. *-commutative 79.8%

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

       15. distribute-rgt-in 79.8%

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

       16. fma-define 79.8%

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

   11. Applied egg-rr 79.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x.im \cdot \left(x.re \cdot -3\right), x.im, x.im \cdot 0\right)} \]
   12. Step-by-step derivation

       1. fma-undefine 79.9%

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

       2. mul0-rgt 79.9%

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

       3. +-rgt-identity 79.9%

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

       4. associate-*r* 79.8%

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

       5. associate-*l* 79.9%

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

       6. *-commutative 79.9%

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

   13. Applied egg-rr 79.9%

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

   if 1.80000000000000007e82 < x.im < 9.99999999999999982e108 or 1.42e147 < x.im < 1.04999999999999999e148 or 9e181 < x.im < 9.50000000000000032e181

   1. Initial program 39.7%

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

      1. difference-of-squares 67.1%

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

      2. *-commutative 67.1%

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

   4. Applied egg-rr 39.7%

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

      1. *-commutative 39.7%

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

      2. *-un-lft-identity 39.7%

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

      3. distribute-lft-in 39.7%

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

      4. distribute-rgt-out 39.7%

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

      5. metadata-eval 39.7%

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

   6. Applied egg-rr 39.7%

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

      1. cancel-sign-sub-inv 39.7%

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

      2. associate-*l* 39.7%

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

      3. *-commutative 39.7%

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

      4. fma-define 39.7%

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

      5. +-commutative 39.7%

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

      6. associate-*l* 39.7%

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

   8. Applied egg-rr 39.7%

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

      1. fma-undefine 39.7%

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

      2. +-commutative 39.7%

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

   10. Applied egg-rr 60.6%

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

       1. mul0-rgt 60.6%

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

       2. mul0-rgt 60.6%

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

       3. +-lft-identity 60.6%

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

       4. *-commutative 60.6%

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

       5. associate-*l* 60.6%

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

   12. Simplified 60.6%

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

   if 9.99999999999999982e108 < x.im < 1.42e147

   1. Initial program 85.7%

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

   3. Simplified 71.2%

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

   5. Taylor expanded in x.re around 0 85.7%

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

      1. add-sqr-sqrt 28.6%

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

      2. unpow2 28.6%

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

      3. swap-sqr 28.6%

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

      4. unpow2 28.6%

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

   7. Applied egg-rr 28.6%

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

      1. unpow2 28.6%

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

      2. *-commutative 28.6%

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

      3. associate-*r* 28.6%

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

      4. associate-*r* 28.6%

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

      5. add-sqr-sqrt 85.5%

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

      6. *-commutative 85.5%

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

   9. Applied egg-rr 85.5%

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

       1. associate-*r* 85.3%

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

       2. *-commutative 85.3%

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

       3. add-log-exp 31.0%

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

       4. *-un-lft-identity 31.0%

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

       5. 1-exp 31.0%

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

       6. mul0-lft 31.0%

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

       7. *-commutative 31.0%

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

       8. mul0-lft 31.0%

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

       9. metadata-eval 31.0%

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

       10. distribute-rgt1-in 31.0%

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

       11. exp-sum 31.0%

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

       12. +-commutative 31.0%

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

       13. add-log-exp 85.3%

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

       14. *-commutative 85.3%

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

       15. distribute-rgt-in 85.3%

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

       16. fma-define 85.3%

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

   11. Applied egg-rr 85.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x.im \cdot \left(x.re \cdot -3\right), x.im, x.im \cdot 0\right)} \]
   12. Step-by-step derivation

       1. fma-undefine 85.5%

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

       2. mul0-rgt 85.5%

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

       3. +-rgt-identity 85.5%

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

       4. associate-*r* 85.3%

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

       5. *-commutative 85.3%

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

       6. associate-*l* 85.5%

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

   13. Applied egg-rr 85.5%

       \[\leadsto \color{blue}{\left(x.re \cdot \left(x.im \cdot -3\right)\right) \cdot x.im} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 245000000000:\\ \;\;\;\;\left(x.re - x.im\right) \cdot \left(x.re \cdot \left(x.re - x.im\right)\right)\\ \mathbf{elif}\;x.im \leq 1.8 \cdot 10^{+82}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.im \cdot -3\right)\\ \mathbf{elif}\;x.im \leq 10^{+109}:\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re - x.im\right)\right)\\ \mathbf{elif}\;x.im \leq 1.42 \cdot 10^{+147}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{elif}\;x.im \leq 1.05 \cdot 10^{+148} \lor \neg \left(x.im \leq 9 \cdot 10^{+181}\right) \land x.im \leq 9.5 \cdot 10^{+181}:\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re - x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.im \cdot -3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ \begin{array}{l} t_0 := \left(x.re\_m \cdot x.im\right) \cdot \left(x.im \cdot -3\right)\\ t_1 := x.re\_m \cdot \left(\left(x.re\_m - x.im\right) \cdot \left(x.re\_m - x.im\right)\right)\\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.im \leq 750000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x.im \leq 8.2 \cdot 10^{+81}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x.im \leq 5 \cdot 10^{+108}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x.im \leq 1.42 \cdot 10^{+147}:\\ \;\;\;\;x.im \cdot \left(x.re\_m \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{elif}\;x.im \leq 1.1 \cdot 10^{+148} \lor \neg \left(x.im \leq 9 \cdot 10^{+181}\right) \land x.im \leq 9.5 \cdot 10^{+181}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (let* ((t_0 (* (* x.re_m x.im) (* x.im -3.0)))
        (t_1 (* x.re_m (* (- x.re_m x.im) (- x.re_m x.im)))))
   (*
    x.re_s
    (if (<= x.im 750000000.0)
      t_1
      (if (<= x.im 8.2e+81)
        t_0
        (if (<= x.im 5e+108)
          t_1
          (if (<= x.im 1.42e+147)
            (* x.im (* x.re_m (* x.im -3.0)))
            (if (or (<= x.im 1.1e+148)
                    (and (not (<= x.im 9e+181)) (<= x.im 9.5e+181)))
              t_1
              t_0))))))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double t_0 = (x_46_re_m * x_46_im) * (x_46_im * -3.0);
	double t_1 = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m - x_46_im));
	double tmp;
	if (x_46_im <= 750000000.0) {
		tmp = t_1;
	} else if (x_46_im <= 8.2e+81) {
		tmp = t_0;
	} else if (x_46_im <= 5e+108) {
		tmp = t_1;
	} else if (x_46_im <= 1.42e+147) {
		tmp = x_46_im * (x_46_re_m * (x_46_im * -3.0));
	} else if ((x_46_im <= 1.1e+148) || (!(x_46_im <= 9e+181) && (x_46_im <= 9.5e+181))) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return x_46_re_s * tmp;
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x_46re_m * x_46im) * (x_46im * (-3.0d0))
    t_1 = x_46re_m * ((x_46re_m - x_46im) * (x_46re_m - x_46im))
    if (x_46im <= 750000000.0d0) then
        tmp = t_1
    else if (x_46im <= 8.2d+81) then
        tmp = t_0
    else if (x_46im <= 5d+108) then
        tmp = t_1
    else if (x_46im <= 1.42d+147) then
        tmp = x_46im * (x_46re_m * (x_46im * (-3.0d0)))
    else if ((x_46im <= 1.1d+148) .or. (.not. (x_46im <= 9d+181)) .and. (x_46im <= 9.5d+181)) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = x_46re_s * tmp
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double t_0 = (x_46_re_m * x_46_im) * (x_46_im * -3.0);
	double t_1 = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m - x_46_im));
	double tmp;
	if (x_46_im <= 750000000.0) {
		tmp = t_1;
	} else if (x_46_im <= 8.2e+81) {
		tmp = t_0;
	} else if (x_46_im <= 5e+108) {
		tmp = t_1;
	} else if (x_46_im <= 1.42e+147) {
		tmp = x_46_im * (x_46_re_m * (x_46_im * -3.0));
	} else if ((x_46_im <= 1.1e+148) || (!(x_46_im <= 9e+181) && (x_46_im <= 9.5e+181))) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return x_46_re_s * tmp;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	t_0 = (x_46_re_m * x_46_im) * (x_46_im * -3.0)
	t_1 = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m - x_46_im))
	tmp = 0
	if x_46_im <= 750000000.0:
		tmp = t_1
	elif x_46_im <= 8.2e+81:
		tmp = t_0
	elif x_46_im <= 5e+108:
		tmp = t_1
	elif x_46_im <= 1.42e+147:
		tmp = x_46_im * (x_46_re_m * (x_46_im * -3.0))
	elif (x_46_im <= 1.1e+148) or (not (x_46_im <= 9e+181) and (x_46_im <= 9.5e+181)):
		tmp = t_1
	else:
		tmp = t_0
	return x_46_re_s * tmp

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	t_0 = Float64(Float64(x_46_re_m * x_46_im) * Float64(x_46_im * -3.0))
	t_1 = Float64(x_46_re_m * Float64(Float64(x_46_re_m - x_46_im) * Float64(x_46_re_m - x_46_im)))
	tmp = 0.0
	if (x_46_im <= 750000000.0)
		tmp = t_1;
	elseif (x_46_im <= 8.2e+81)
		tmp = t_0;
	elseif (x_46_im <= 5e+108)
		tmp = t_1;
	elseif (x_46_im <= 1.42e+147)
		tmp = Float64(x_46_im * Float64(x_46_re_m * Float64(x_46_im * -3.0)));
	elseif ((x_46_im <= 1.1e+148) || (!(x_46_im <= 9e+181) && (x_46_im <= 9.5e+181)))
		tmp = t_1;
	else
		tmp = t_0;
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	t_0 = (x_46_re_m * x_46_im) * (x_46_im * -3.0);
	t_1 = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m - x_46_im));
	tmp = 0.0;
	if (x_46_im <= 750000000.0)
		tmp = t_1;
	elseif (x_46_im <= 8.2e+81)
		tmp = t_0;
	elseif (x_46_im <= 5e+108)
		tmp = t_1;
	elseif (x_46_im <= 1.42e+147)
		tmp = x_46_im * (x_46_re_m * (x_46_im * -3.0));
	elseif ((x_46_im <= 1.1e+148) || (~((x_46_im <= 9e+181)) && (x_46_im <= 9.5e+181)))
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := Block[{t$95$0 = N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x$46$re$95$m * N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(x$46$re$95$m - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$re$95$s * If[LessEqual[x$46$im, 750000000.0], t$95$1, If[LessEqual[x$46$im, 8.2e+81], t$95$0, If[LessEqual[x$46$im, 5e+108], t$95$1, If[LessEqual[x$46$im, 1.42e+147], N[(x$46$im * N[(x$46$re$95$m * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x$46$im, 1.1e+148], And[N[Not[LessEqual[x$46$im, 9e+181]], $MachinePrecision], LessEqual[x$46$im, 9.5e+181]]], t$95$1, t$95$0]]]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.im < 7.5e8 or 8.20000000000000024e81 < x.im < 4.99999999999999991e108 or 1.42e147 < x.im < 1.0999999999999999e148 or 9e181 < x.im < 9.50000000000000032e181

   1. Initial program 82.2%

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

      1. difference-of-squares 81.9%

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

      2. *-commutative 81.9%

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

   4. Applied egg-rr 85.7%

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

      1. *-commutative 85.7%

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

      2. *-un-lft-identity 85.7%

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

      3. distribute-lft-in 85.7%

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

      4. distribute-rgt-out 85.7%

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

      5. metadata-eval 85.7%

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

   6. Applied egg-rr 85.7%

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

      1. cancel-sign-sub-inv 85.7%

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

      2. associate-*l* 90.8%

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

      3. *-commutative 90.8%

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

      4. fma-define 90.8%

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

      5. +-commutative 90.8%

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

      6. associate-*l* 90.8%

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

   8. Applied egg-rr 90.8%

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

      1. fma-undefine 90.8%

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

      2. +-commutative 90.8%

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

   10. Applied egg-rr 63.4%

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

       1. mul0-rgt 63.4%

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

       2. mul0-rgt 63.4%

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

       3. +-lft-identity 63.4%

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

       4. *-commutative 63.4%

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

       5. associate-*l* 63.4%

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

   12. Simplified 63.4%

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

   if 7.5e8 < x.im < 8.20000000000000024e81 or 1.0999999999999999e148 < x.im < 9e181 or 9.50000000000000032e181 < x.im

   1. Initial program 57.2%

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

   3. Simplified 49.2%

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

   5. Taylor expanded in x.re around 0 59.1%

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

      1. add-sqr-sqrt 40.7%

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

      2. unpow2 40.7%

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

      3. swap-sqr 55.7%

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

      4. unpow2 55.7%

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

   7. Applied egg-rr 55.7%

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

      1. unpow2 55.7%

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

      2. *-commutative 55.7%

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

      3. associate-*r* 55.8%

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

      4. associate-*r* 55.7%

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

      5. add-sqr-sqrt 79.8%

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

      6. *-commutative 79.8%

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

   9. Applied egg-rr 79.8%

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

       1. associate-*r* 79.8%

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

       2. *-commutative 79.8%

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

       3. add-log-exp 41.9%

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

       4. *-un-lft-identity 41.9%

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

       5. 1-exp 41.9%

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

       6. mul0-lft 41.9%

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

       7. *-commutative 41.9%

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

       8. mul0-lft 41.9%

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

       9. metadata-eval 41.9%

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

       10. distribute-rgt1-in 41.9%

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

       11. exp-sum 41.9%

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

       12. +-commutative 41.9%

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

       13. add-log-exp 79.8%

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

       14. *-commutative 79.8%

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

       15. distribute-rgt-in 79.8%

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

       16. fma-define 79.8%

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

   11. Applied egg-rr 79.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x.im \cdot \left(x.re \cdot -3\right), x.im, x.im \cdot 0\right)} \]
   12. Step-by-step derivation

       1. fma-undefine 79.9%

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

       2. mul0-rgt 79.9%

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

       3. +-rgt-identity 79.9%

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

       4. associate-*r* 79.8%

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

       5. associate-*l* 79.9%

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

       6. *-commutative 79.9%

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

   13. Applied egg-rr 79.9%

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

   if 4.99999999999999991e108 < x.im < 1.42e147

   1. Initial program 85.7%

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

   3. Simplified 71.2%

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

   5. Taylor expanded in x.re around 0 85.7%

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

      1. add-sqr-sqrt 28.6%

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

      2. unpow2 28.6%

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

      3. swap-sqr 28.6%

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

      4. unpow2 28.6%

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

   7. Applied egg-rr 28.6%

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

      1. unpow2 28.6%

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

      2. *-commutative 28.6%

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

      3. associate-*r* 28.6%

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

      4. associate-*r* 28.6%

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

      5. add-sqr-sqrt 85.5%

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

      6. *-commutative 85.5%

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

   9. Applied egg-rr 85.5%

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

       1. associate-*r* 85.3%

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

       2. *-commutative 85.3%

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

       3. add-log-exp 31.0%

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

       4. *-un-lft-identity 31.0%

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

       5. 1-exp 31.0%

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

       6. mul0-lft 31.0%

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

       7. *-commutative 31.0%

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

       8. mul0-lft 31.0%

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

       9. metadata-eval 31.0%

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

       10. distribute-rgt1-in 31.0%

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

       11. exp-sum 31.0%

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

       12. +-commutative 31.0%

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

       13. add-log-exp 85.3%

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

       14. *-commutative 85.3%

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

       15. distribute-rgt-in 85.3%

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

       16. fma-define 85.3%

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

   11. Applied egg-rr 85.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x.im \cdot \left(x.re \cdot -3\right), x.im, x.im \cdot 0\right)} \]
   12. Step-by-step derivation

       1. fma-undefine 85.5%

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

       2. mul0-rgt 85.5%

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

       3. +-rgt-identity 85.5%

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

       4. associate-*r* 85.3%

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

       5. *-commutative 85.3%

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

       6. associate-*l* 85.5%

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

   13. Applied egg-rr 85.5%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 750000000:\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re - x.im\right)\right)\\ \mathbf{elif}\;x.im \leq 8.2 \cdot 10^{+81}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.im \cdot -3\right)\\ \mathbf{elif}\;x.im \leq 5 \cdot 10^{+108}:\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re - x.im\right)\right)\\ \mathbf{elif}\;x.im \leq 1.42 \cdot 10^{+147}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{elif}\;x.im \leq 1.1 \cdot 10^{+148} \lor \neg \left(x.im \leq 9 \cdot 10^{+181}\right) \land x.im \leq 9.5 \cdot 10^{+181}:\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re - x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.im \cdot -3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ \begin{array}{l} t_0 := \left(x.re\_m \cdot x.im\right) \cdot \left(x.im \cdot -3\right)\\ t_1 := x.re\_m \cdot \left(\left(x.re\_m - x.im\right) \cdot \left(x.re\_m - x.im\right)\right)\\ x.re\_s \cdot \begin{array}{l} \mathbf{if}\;x.im \leq 5800000000:\\ \;\;\;\;\left(x.re\_m - x.im\right) \cdot \left(x.re\_m \cdot \left(x.re\_m - x.im\right)\right)\\ \mathbf{elif}\;x.im \leq 2 \cdot 10^{+82}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x.im \leq 5.2 \cdot 10^{+108}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x.im \leq 1.42 \cdot 10^{+147}:\\ \;\;\;\;x.im \cdot \left(x.re\_m \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{elif}\;x.im \leq 1.05 \cdot 10^{+148}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x.im \leq 9 \cdot 10^{+181}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x.im \leq 1.45 \cdot 10^{+182}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re\_m \cdot -3\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (let* ((t_0 (* (* x.re_m x.im) (* x.im -3.0)))
        (t_1 (* x.re_m (* (- x.re_m x.im) (- x.re_m x.im)))))
   (*
    x.re_s
    (if (<= x.im 5800000000.0)
      (* (- x.re_m x.im) (* x.re_m (- x.re_m x.im)))
      (if (<= x.im 2e+82)
        t_0
        (if (<= x.im 5.2e+108)
          t_1
          (if (<= x.im 1.42e+147)
            (* x.im (* x.re_m (* x.im -3.0)))
            (if (<= x.im 1.05e+148)
              t_1
              (if (<= x.im 9e+181)
                t_0
                (if (<= x.im 1.45e+182)
                  t_1
                  (* x.im (* x.im (* x.re_m -3.0)))))))))))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double t_0 = (x_46_re_m * x_46_im) * (x_46_im * -3.0);
	double t_1 = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m - x_46_im));
	double tmp;
	if (x_46_im <= 5800000000.0) {
		tmp = (x_46_re_m - x_46_im) * (x_46_re_m * (x_46_re_m - x_46_im));
	} else if (x_46_im <= 2e+82) {
		tmp = t_0;
	} else if (x_46_im <= 5.2e+108) {
		tmp = t_1;
	} else if (x_46_im <= 1.42e+147) {
		tmp = x_46_im * (x_46_re_m * (x_46_im * -3.0));
	} else if (x_46_im <= 1.05e+148) {
		tmp = t_1;
	} else if (x_46_im <= 9e+181) {
		tmp = t_0;
	} else if (x_46_im <= 1.45e+182) {
		tmp = t_1;
	} else {
		tmp = x_46_im * (x_46_im * (x_46_re_m * -3.0));
	}
	return x_46_re_s * tmp;
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x_46re_m * x_46im) * (x_46im * (-3.0d0))
    t_1 = x_46re_m * ((x_46re_m - x_46im) * (x_46re_m - x_46im))
    if (x_46im <= 5800000000.0d0) then
        tmp = (x_46re_m - x_46im) * (x_46re_m * (x_46re_m - x_46im))
    else if (x_46im <= 2d+82) then
        tmp = t_0
    else if (x_46im <= 5.2d+108) then
        tmp = t_1
    else if (x_46im <= 1.42d+147) then
        tmp = x_46im * (x_46re_m * (x_46im * (-3.0d0)))
    else if (x_46im <= 1.05d+148) then
        tmp = t_1
    else if (x_46im <= 9d+181) then
        tmp = t_0
    else if (x_46im <= 1.45d+182) then
        tmp = t_1
    else
        tmp = x_46im * (x_46im * (x_46re_m * (-3.0d0)))
    end if
    code = x_46re_s * tmp
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	double t_0 = (x_46_re_m * x_46_im) * (x_46_im * -3.0);
	double t_1 = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m - x_46_im));
	double tmp;
	if (x_46_im <= 5800000000.0) {
		tmp = (x_46_re_m - x_46_im) * (x_46_re_m * (x_46_re_m - x_46_im));
	} else if (x_46_im <= 2e+82) {
		tmp = t_0;
	} else if (x_46_im <= 5.2e+108) {
		tmp = t_1;
	} else if (x_46_im <= 1.42e+147) {
		tmp = x_46_im * (x_46_re_m * (x_46_im * -3.0));
	} else if (x_46_im <= 1.05e+148) {
		tmp = t_1;
	} else if (x_46_im <= 9e+181) {
		tmp = t_0;
	} else if (x_46_im <= 1.45e+182) {
		tmp = t_1;
	} else {
		tmp = x_46_im * (x_46_im * (x_46_re_m * -3.0));
	}
	return x_46_re_s * tmp;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	t_0 = (x_46_re_m * x_46_im) * (x_46_im * -3.0)
	t_1 = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m - x_46_im))
	tmp = 0
	if x_46_im <= 5800000000.0:
		tmp = (x_46_re_m - x_46_im) * (x_46_re_m * (x_46_re_m - x_46_im))
	elif x_46_im <= 2e+82:
		tmp = t_0
	elif x_46_im <= 5.2e+108:
		tmp = t_1
	elif x_46_im <= 1.42e+147:
		tmp = x_46_im * (x_46_re_m * (x_46_im * -3.0))
	elif x_46_im <= 1.05e+148:
		tmp = t_1
	elif x_46_im <= 9e+181:
		tmp = t_0
	elif x_46_im <= 1.45e+182:
		tmp = t_1
	else:
		tmp = x_46_im * (x_46_im * (x_46_re_m * -3.0))
	return x_46_re_s * tmp

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	t_0 = Float64(Float64(x_46_re_m * x_46_im) * Float64(x_46_im * -3.0))
	t_1 = Float64(x_46_re_m * Float64(Float64(x_46_re_m - x_46_im) * Float64(x_46_re_m - x_46_im)))
	tmp = 0.0
	if (x_46_im <= 5800000000.0)
		tmp = Float64(Float64(x_46_re_m - x_46_im) * Float64(x_46_re_m * Float64(x_46_re_m - x_46_im)));
	elseif (x_46_im <= 2e+82)
		tmp = t_0;
	elseif (x_46_im <= 5.2e+108)
		tmp = t_1;
	elseif (x_46_im <= 1.42e+147)
		tmp = Float64(x_46_im * Float64(x_46_re_m * Float64(x_46_im * -3.0)));
	elseif (x_46_im <= 1.05e+148)
		tmp = t_1;
	elseif (x_46_im <= 9e+181)
		tmp = t_0;
	elseif (x_46_im <= 1.45e+182)
		tmp = t_1;
	else
		tmp = Float64(x_46_im * Float64(x_46_im * Float64(x_46_re_m * -3.0)));
	end
	return Float64(x_46_re_s * tmp)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp_2 = code(x_46_re_s, x_46_re_m, x_46_im)
	t_0 = (x_46_re_m * x_46_im) * (x_46_im * -3.0);
	t_1 = x_46_re_m * ((x_46_re_m - x_46_im) * (x_46_re_m - x_46_im));
	tmp = 0.0;
	if (x_46_im <= 5800000000.0)
		tmp = (x_46_re_m - x_46_im) * (x_46_re_m * (x_46_re_m - x_46_im));
	elseif (x_46_im <= 2e+82)
		tmp = t_0;
	elseif (x_46_im <= 5.2e+108)
		tmp = t_1;
	elseif (x_46_im <= 1.42e+147)
		tmp = x_46_im * (x_46_re_m * (x_46_im * -3.0));
	elseif (x_46_im <= 1.05e+148)
		tmp = t_1;
	elseif (x_46_im <= 9e+181)
		tmp = t_0;
	elseif (x_46_im <= 1.45e+182)
		tmp = t_1;
	else
		tmp = x_46_im * (x_46_im * (x_46_re_m * -3.0));
	end
	tmp_2 = x_46_re_s * tmp;
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := Block[{t$95$0 = N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x$46$re$95$m * N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(x$46$re$95$m - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$re$95$s * If[LessEqual[x$46$im, 5800000000.0], N[(N[(x$46$re$95$m - x$46$im), $MachinePrecision] * N[(x$46$re$95$m * N[(x$46$re$95$m - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 2e+82], t$95$0, If[LessEqual[x$46$im, 5.2e+108], t$95$1, If[LessEqual[x$46$im, 1.42e+147], N[(x$46$im * N[(x$46$re$95$m * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 1.05e+148], t$95$1, If[LessEqual[x$46$im, 9e+181], t$95$0, If[LessEqual[x$46$im, 1.45e+182], t$95$1, N[(x$46$im * N[(x$46$im * N[(x$46$re$95$m * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]), $MachinePrecision]]]

Derivation

1. Split input into 5 regimes

2. if x.im < 5.8e9

   1. Initial program 83.2%

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

      1. difference-of-squares 82.3%

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

      2. *-commutative 82.3%

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

   4. Applied egg-rr 86.9%

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

      1. *-commutative 86.9%

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

      2. *-un-lft-identity 86.9%

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

      3. distribute-lft-in 86.9%

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

      4. distribute-rgt-out 86.9%

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

      5. metadata-eval 86.9%

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

   6. Applied egg-rr 86.9%

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

      1. cancel-sign-sub-inv 86.9%

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

      2. associate-*l* 92.1%

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

      3. *-commutative 92.1%

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

      4. fma-define 92.1%

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

      5. +-commutative 92.1%

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

      6. associate-*l* 92.1%

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

   8. Applied egg-rr 92.1%

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

      1. fma-undefine 92.1%

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

      2. +-commutative 92.1%

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

   10. Applied egg-rr 63.5%

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

       1. mul0-rgt 63.5%

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

       2. mul0-rgt 63.5%

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

       3. +-lft-identity 63.5%

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

       4. *-commutative 63.5%

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

       5. associate-*l* 63.5%

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

   12. Simplified 63.5%

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

       1. associate-*r* 63.5%

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

       2. sub-neg 63.5%

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

       3. distribute-lft-in 52.2%

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

   14. Applied egg-rr 52.2%

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

       1. distribute-lft-out 63.5%

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

       2. sub-neg 63.5%

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

   16. Simplified 63.5%

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

   if 5.8e9 < x.im < 1.9999999999999999e82 or 1.04999999999999999e148 < x.im < 9e181

   1. Initial program 80.1%

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

   3. Simplified 80.1%

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

   5. Taylor expanded in x.re around 0 60.0%

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

      1. add-sqr-sqrt 45.0%

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

      2. unpow2 45.0%

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

      3. swap-sqr 54.5%

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

      4. unpow2 54.5%

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

   7. Applied egg-rr 54.5%

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

      1. unpow2 54.5%

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

      2. *-commutative 54.5%

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

      3. associate-*r* 54.6%

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

      4. associate-*r* 54.6%

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

      5. add-sqr-sqrt 69.7%

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

      6. *-commutative 69.7%

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

   9. Applied egg-rr 69.7%

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

       1. associate-*r* 69.7%

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

       2. *-commutative 69.7%

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

       3. add-log-exp 17.2%

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

       4. *-un-lft-identity 17.2%

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

       5. 1-exp 17.2%

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

       6. mul0-lft 17.2%

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

       7. *-commutative 17.2%

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

       8. mul0-lft 17.2%

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

       9. metadata-eval 17.2%

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

       10. distribute-rgt1-in 17.2%

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

       11. exp-sum 17.2%

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

       12. +-commutative 17.2%

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

       13. add-log-exp 69.7%

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

       14. *-commutative 69.7%

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

       15. distribute-rgt-in 69.7%

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

       16. fma-define 69.7%

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

   11. Applied egg-rr 69.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x.im \cdot \left(x.re \cdot -3\right), x.im, x.im \cdot 0\right)} \]
   12. Step-by-step derivation

       1. fma-undefine 69.7%

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

       2. mul0-rgt 69.7%

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

       3. +-rgt-identity 69.7%

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

       4. associate-*r* 69.7%

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

       5. associate-*l* 70.0%

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

       6. *-commutative 70.0%

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

   13. Applied egg-rr 70.0%

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

   if 1.9999999999999999e82 < x.im < 5.2000000000000005e108 or 1.42e147 < x.im < 1.04999999999999999e148 or 9e181 < x.im < 1.4499999999999999e182

   1. Initial program 39.7%

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

      1. difference-of-squares 67.1%

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

      2. *-commutative 67.1%

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

   4. Applied egg-rr 39.7%

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

      1. *-commutative 39.7%

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

      2. *-un-lft-identity 39.7%

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

      3. distribute-lft-in 39.7%

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

      4. distribute-rgt-out 39.7%

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

      5. metadata-eval 39.7%

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

   6. Applied egg-rr 39.7%

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

      1. cancel-sign-sub-inv 39.7%

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

      2. associate-*l* 39.7%

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

      3. *-commutative 39.7%

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

      4. fma-define 39.7%

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

      5. +-commutative 39.7%

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

      6. associate-*l* 39.7%

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

   8. Applied egg-rr 39.7%

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

      1. fma-undefine 39.7%

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

      2. +-commutative 39.7%

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

   10. Applied egg-rr 60.6%

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

       1. mul0-rgt 60.6%

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

       2. mul0-rgt 60.6%

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

       3. +-lft-identity 60.6%

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

       4. *-commutative 60.6%

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

       5. associate-*l* 60.6%

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

   12. Simplified 60.6%

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

   if 5.2000000000000005e108 < x.im < 1.42e147

   1. Initial program 85.7%

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

   3. Simplified 71.2%

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

   5. Taylor expanded in x.re around 0 85.7%

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

      1. add-sqr-sqrt 28.6%

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

      2. unpow2 28.6%

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

      3. swap-sqr 28.6%

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

      4. unpow2 28.6%

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

   7. Applied egg-rr 28.6%

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

      1. unpow2 28.6%

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

      2. *-commutative 28.6%

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

      3. associate-*r* 28.6%

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

      4. associate-*r* 28.6%

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

      5. add-sqr-sqrt 85.5%

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

      6. *-commutative 85.5%

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

   9. Applied egg-rr 85.5%

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

       1. associate-*r* 85.3%

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

       2. *-commutative 85.3%

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

       3. add-log-exp 31.0%

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

       4. *-un-lft-identity 31.0%

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

       5. 1-exp 31.0%

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

       6. mul0-lft 31.0%

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

       7. *-commutative 31.0%

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

       8. mul0-lft 31.0%

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

       9. metadata-eval 31.0%

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

       10. distribute-rgt1-in 31.0%

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

       11. exp-sum 31.0%

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

       12. +-commutative 31.0%

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

       13. add-log-exp 85.3%

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

       14. *-commutative 85.3%

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

       15. distribute-rgt-in 85.3%

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

       16. fma-define 85.3%

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

   11. Applied egg-rr 85.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x.im \cdot \left(x.re \cdot -3\right), x.im, x.im \cdot 0\right)} \]
   12. Step-by-step derivation

       1. fma-undefine 85.5%

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

       2. mul0-rgt 85.5%

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

       3. +-rgt-identity 85.5%

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

       4. associate-*r* 85.3%

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

       5. *-commutative 85.3%

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

       6. associate-*l* 85.5%

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

   13. Applied egg-rr 85.5%

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

   if 1.4499999999999999e182 < x.im

   1. Initial program 41.9%

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

   3. Simplified 28.5%

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

   5. Taylor expanded in x.re around 0 58.5%

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

      1. add-sqr-sqrt 37.8%

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

      2. unpow2 37.8%

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

      3. swap-sqr 56.5%

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

      4. unpow2 56.5%

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

   7. Applied egg-rr 56.5%

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

      1. unpow2 56.5%

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

      2. *-commutative 56.5%

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

      3. associate-*r* 56.6%

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

      4. associate-*r* 56.5%

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

      5. add-sqr-sqrt 86.5%

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

      6. *-commutative 86.5%

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

   9. Applied egg-rr 86.5%

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

       1. associate-*r* 86.6%

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

       2. *-commutative 86.6%

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

       3. add-log-exp 58.4%

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

       4. *-un-lft-identity 58.4%

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

       5. 1-exp 58.4%

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

       6. mul0-lft 58.4%

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

       7. *-commutative 58.4%

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

       8. mul0-lft 58.4%

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

       9. metadata-eval 58.4%

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

       10. distribute-rgt1-in 58.4%

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

       11. exp-sum 58.4%

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

       12. +-commutative 58.4%

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

       13. add-log-exp 86.6%

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

       14. *-commutative 86.6%

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

       15. distribute-rgt-in 86.6%

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

       16. fma-define 86.6%

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

   11. Applied egg-rr 86.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x.im \cdot \left(x.re \cdot -3\right), x.im, x.im \cdot 0\right)} \]
   12. Step-by-step derivation

       1. fma-undefine 86.6%

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

       2. mul0-rgt 86.6%

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

       3. +-rgt-identity 86.6%

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

       4. associate-*r* 86.6%

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

       5. *-commutative 86.6%

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

       6. associate-*l* 86.6%

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

   13. Applied egg-rr 86.6%

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

       1. add-log-exp 58.4%

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

       2. *-un-lft-identity 58.4%

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

       3. log-prod 58.4%

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

       4. metadata-eval 58.4%

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

       5. add-log-exp 86.6%

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

       6. associate-*r* 86.6%

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

       7. *-commutative 86.6%

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

       8. associate-*l* 86.6%

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

   15. Applied egg-rr 86.6%

       \[\leadsto \color{blue}{\left(0 + x.im \cdot \left(x.re \cdot -3\right)\right)} \cdot x.im \]
3. Recombined 5 regimes into one program.

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 5800000000:\\ \;\;\;\;\left(x.re - x.im\right) \cdot \left(x.re \cdot \left(x.re - x.im\right)\right)\\ \mathbf{elif}\;x.im \leq 2 \cdot 10^{+82}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.im \cdot -3\right)\\ \mathbf{elif}\;x.im \leq 5.2 \cdot 10^{+108}:\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re - x.im\right)\right)\\ \mathbf{elif}\;x.im \leq 1.42 \cdot 10^{+147}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{elif}\;x.im \leq 1.05 \cdot 10^{+148}:\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re - x.im\right)\right)\\ \mathbf{elif}\;x.im \leq 9 \cdot 10^{+181}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.im \cdot -3\right)\\ \mathbf{elif}\;x.im \leq 1.45 \cdot 10^{+182}:\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re - x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 55.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \left(\left(x.re\_m \cdot x.im\right) \cdot \left(x.im \cdot -3\right)\right) \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (* x.re_s (* (* x.re_m x.im) (* x.im -3.0))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	return x_46_re_s * ((x_46_re_m * x_46_im) * (x_46_im * -3.0));
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    code = x_46re_s * ((x_46re_m * x_46im) * (x_46im * (-3.0d0)))
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	return x_46_re_s * ((x_46_re_m * x_46_im) * (x_46_im * -3.0));
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	return x_46_re_s * ((x_46_re_m * x_46_im) * (x_46_im * -3.0))

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	return Float64(x_46_re_s * Float64(Float64(x_46_re_m * x_46_im) * Float64(x_46_im * -3.0)))
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = x_46_re_s * ((x_46_re_m * x_46_im) * (x_46_im * -3.0));
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.4%

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

3. Simplified 73.9%

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

5. Taylor expanded in x.re around 0 49.8%

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

   1. add-sqr-sqrt 24.7%

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

   2. unpow2 24.7%

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

   3. swap-sqr 29.4%

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

   4. unpow2 29.4%

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

7. Applied egg-rr 29.4%

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

   1. unpow2 29.4%

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

   2. *-commutative 29.4%

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

   3. associate-*r* 29.4%

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

   4. associate-*r* 29.4%

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

   5. add-sqr-sqrt 57.8%

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

   6. *-commutative 57.8%

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

9. Applied egg-rr 57.8%

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

    1. associate-*r* 57.8%

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

    2. *-commutative 57.8%

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

    3. add-log-exp 34.2%

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

    4. *-un-lft-identity 34.2%

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

    5. 1-exp 34.2%

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

    6. mul0-lft 34.2%

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

    7. *-commutative 34.2%

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

    8. mul0-lft 34.2%

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

    9. metadata-eval 34.2%

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

    10. distribute-rgt1-in 34.2%

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

    11. exp-sum 34.2%

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

    12. +-commutative 34.2%

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

    13. add-log-exp 57.8%

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

    14. *-commutative 57.8%

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

    15. distribute-rgt-in 57.8%

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

    16. fma-define 57.8%

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

11. Applied egg-rr 57.8%

    \[\leadsto \color{blue}{\mathsf{fma}\left(x.im \cdot \left(x.re \cdot -3\right), x.im, x.im \cdot 0\right)} \]
12. Step-by-step derivation

    1. fma-undefine 57.8%

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

    2. mul0-rgt 57.8%

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

    3. +-rgt-identity 57.8%

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

    4. associate-*r* 57.8%

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

    5. associate-*l* 57.9%

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

    6. *-commutative 57.9%

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

13. Applied egg-rr 57.9%

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

14. Final simplification 57.9%

    \[\leadsto \left(x.re \cdot x.im\right) \cdot \left(x.im \cdot -3\right) \]
15. Add Preprocessing

Alternative 9: 55.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot \left(-3 \cdot \left(x.im \cdot \left(x.re\_m \cdot x.im\right)\right)\right) \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im)
 :precision binary64
 (* x.re_s (* -3.0 (* x.im (* x.re_m x.im)))))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	return x_46_re_s * (-3.0 * (x_46_im * (x_46_re_m * x_46_im)));
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    code = x_46re_s * ((-3.0d0) * (x_46im * (x_46re_m * x_46im)))
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	return x_46_re_s * (-3.0 * (x_46_im * (x_46_re_m * x_46_im)));
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	return x_46_re_s * (-3.0 * (x_46_im * (x_46_re_m * x_46_im)))

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	return Float64(x_46_re_s * Float64(-3.0 * Float64(x_46_im * Float64(x_46_re_m * x_46_im))))
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = x_46_re_s * (-3.0 * (x_46_im * (x_46_re_m * x_46_im)));
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * N[(-3.0 * N[(x$46$im * N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.4%

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

3. Simplified 73.9%

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

5. Taylor expanded in x.re around 0 49.8%

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

   1. add-sqr-sqrt 24.7%

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

   2. unpow2 24.7%

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

   3. swap-sqr 29.4%

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

   4. unpow2 29.4%

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

7. Applied egg-rr 29.4%

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

   1. unpow2 29.4%

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

   2. *-commutative 29.4%

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

   3. associate-*r* 29.4%

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

   4. associate-*r* 29.4%

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

   5. add-sqr-sqrt 57.8%

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

   6. *-commutative 57.8%

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

9. Applied egg-rr 57.8%

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

10. Final simplification 57.8%

    \[\leadsto -3 \cdot \left(x.im \cdot \left(x.re \cdot x.im\right)\right) \]
11. Add Preprocessing

Alternative 10: 3.6% accurate, 19.0× speedup

Math

\[\begin{array}{l} x.re\_m = \left|x.re\right| \\ x.re\_s = \mathsf{copysign}\left(1, x.re\right) \\ x.re\_s \cdot x.im \end{array} \]

FPCore

x.re\_m = (fabs.f64 x.re)
x.re\_s = (copysign.f64 #s(literal 1 binary64) x.re)
(FPCore (x.re_s x.re_m x.im) :precision binary64 (* x.re_s x.im))

C

x.re\_m = fabs(x_46_re);
x.re\_s = copysign(1.0, x_46_re);
double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	return x_46_re_s * x_46_im;
}

Fortran

x.re\_m = abs(x_46re)
x.re\_s = copysign(1.0d0, x_46re)
real(8) function code(x_46re_s, x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_s
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    code = x_46re_s * x_46im
end function

Java

x.re\_m = Math.abs(x_46_re);
x.re\_s = Math.copySign(1.0, x_46_re);
public static double code(double x_46_re_s, double x_46_re_m, double x_46_im) {
	return x_46_re_s * x_46_im;
}

Python

x.re\_m = math.fabs(x_46_re)
x.re\_s = math.copysign(1.0, x_46_re)
def code(x_46_re_s, x_46_re_m, x_46_im):
	return x_46_re_s * x_46_im

Julia

x.re\_m = abs(x_46_re)
x.re\_s = copysign(1.0, x_46_re)
function code(x_46_re_s, x_46_re_m, x_46_im)
	return Float64(x_46_re_s * x_46_im)
end

MATLAB

x.re\_m = abs(x_46_re);
x.re\_s = sign(x_46_re) * abs(1.0);
function tmp = code(x_46_re_s, x_46_re_m, x_46_im)
	tmp = x_46_re_s * x_46_im;
end

Wolfram

x.re\_m = N[Abs[x$46$re], $MachinePrecision]
x.re\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$re$95$s_, x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$s * x$46$im), $MachinePrecision]

Derivation

1. Initial program 77.4%

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

   1. difference-of-squares 77.7%

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

   2. *-commutative 77.7%

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

4. Applied egg-rr 82.1%

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

   1. associate-*l* 90.1%

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

   2. fma-neg 90.1%

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

   3. *-commutative 90.1%

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

   4. distribute-rgt-neg-in 90.1%

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

   5. *-commutative 90.1%

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

   6. flip-+ 0.0%

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

   7. *-commutative 0.0%

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

   8. +-inverses 0.0%

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

   9. +-inverses 0.0%

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

   10. *-commutative 0.0%

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

   11. distribute-neg-frac2 0.0%

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

6. Applied egg-rr 0.0%

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

8. Simplified 64.1%

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

9. Taylor expanded in x.re around 0 3.2%

   \[\leadsto \color{blue}{x.im} \]
10. Add Preprocessing

Developer target: 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 2024107 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64

  :alt
  (+ (* (* x.re x.re) (- x.re x.im)) (* (* x.re x.im) (- x.re (* 3.0 x.im))))

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