math.cube on complex, imaginary part

Percentage Accurate: 82.3% → 99.5%

Time: 9.5s

Alternatives: 8

Speedup: 1.6×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 82.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ \begin{array}{l} \mathbf{if}\;x.re\_m \leq 2.25 \cdot 10^{-79}:\\ \;\;\;\;{\left(0 - x.im\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \frac{x.im \cdot \frac{x.im}{x.re\_m}}{x.re\_m}\right) \cdot \left(x.re\_m \cdot \left(x.re\_m \cdot x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im)
 :precision binary64
 (if (<= x.re_m 2.25e-79)
   (pow (- 0.0 x.im) 3.0)
   (* (- 3.0 (/ (* x.im (/ x.im x.re_m)) x.re_m)) (* x.re_m (* x.re_m x.im)))))

C

x.re_m = fabs(x_46_re);
double code(double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 2.25e-79) {
		tmp = pow((0.0 - x_46_im), 3.0);
	} else {
		tmp = (3.0 - ((x_46_im * (x_46_im / x_46_re_m)) / x_46_re_m)) * (x_46_re_m * (x_46_re_m * x_46_im));
	}
	return tmp;
}

Fortran

x.re_m = abs(x_46re)
real(8) function code(x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re_m <= 2.25d-79) then
        tmp = (0.0d0 - x_46im) ** 3.0d0
    else
        tmp = (3.0d0 - ((x_46im * (x_46im / x_46re_m)) / x_46re_m)) * (x_46re_m * (x_46re_m * x_46im))
    end if
    code = tmp
end function

Java

x.re_m = Math.abs(x_46_re);
public static double code(double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 2.25e-79) {
		tmp = Math.pow((0.0 - x_46_im), 3.0);
	} else {
		tmp = (3.0 - ((x_46_im * (x_46_im / x_46_re_m)) / x_46_re_m)) * (x_46_re_m * (x_46_re_m * x_46_im));
	}
	return tmp;
}

Python

x.re_m = math.fabs(x_46_re)
def code(x_46_re_m, x_46_im):
	tmp = 0
	if x_46_re_m <= 2.25e-79:
		tmp = math.pow((0.0 - x_46_im), 3.0)
	else:
		tmp = (3.0 - ((x_46_im * (x_46_im / x_46_re_m)) / x_46_re_m)) * (x_46_re_m * (x_46_re_m * x_46_im))
	return tmp

Julia

x.re_m = abs(x_46_re)
function code(x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_re_m <= 2.25e-79)
		tmp = Float64(0.0 - x_46_im) ^ 3.0;
	else
		tmp = Float64(Float64(3.0 - Float64(Float64(x_46_im * Float64(x_46_im / x_46_re_m)) / x_46_re_m)) * Float64(x_46_re_m * Float64(x_46_re_m * x_46_im)));
	end
	return tmp
end

MATLAB

x.re_m = abs(x_46_re);
function tmp_2 = code(x_46_re_m, x_46_im)
	tmp = 0.0;
	if (x_46_re_m <= 2.25e-79)
		tmp = (0.0 - x_46_im) ^ 3.0;
	else
		tmp = (3.0 - ((x_46_im * (x_46_im / x_46_re_m)) / x_46_re_m)) * (x_46_re_m * (x_46_re_m * x_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
code[x$46$re$95$m_, x$46$im_] := If[LessEqual[x$46$re$95$m, 2.25e-79], N[Power[N[(0.0 - x$46$im), $MachinePrecision], 3.0], $MachinePrecision], N[(N[(3.0 - N[(N[(x$46$im * N[(x$46$im / x$46$re$95$m), $MachinePrecision]), $MachinePrecision] / x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 2.2500000000000001e-79

   1. Initial program 88.3%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-out N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \color{blue}{\left(\left(x.re + x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right) \]

      8. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right) - \color{blue}{x.im \cdot x.im}\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right), \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(x.re \cdot \left(\left(x.re + x.re\right) + x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(\left(x.re + x.re\right) + x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      12. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(2 \cdot x.re + x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      13. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(\left(2 + 1\right) \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(3 \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(x.re \cdot 3\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      17. *-lowering-*.f64 91.5%

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

   3. Simplified 91.5%

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

   5. Taylor expanded in x.im around inf

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left({x.im}^{3}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{{x.im}^{3}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left({x.im}^{3}\right)}\right) \]

      4. cube-mult N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \left(x.im \cdot \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \left(x.im \cdot {x.im}^{\color{blue}{2}}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x.im, \color{blue}{\left({x.im}^{2}\right)}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x.im, \left(x.im \cdot \color{blue}{x.im}\right)\right)\right) \]

      8. *-lowering-*.f64 71.8%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

   7. Simplified 71.8%

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

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(x.im \cdot \left(x.im \cdot x.im\right)\right) \]

      2. cube-unmult N/A

         \[\leadsto \mathsf{neg}\left({x.im}^{3}\right) \]

      3. cube-neg N/A

         \[\leadsto {\left(\mathsf{neg}\left(x.im\right)\right)}^{\color{blue}{3}} \]

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\mathsf{neg}\left(x.im\right)\right), \color{blue}{3}\right) \]

      5. neg-sub0 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(0 - x.im\right), 3\right) \]

      6. --lowering--.f64 71.9%

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, x.im\right), 3\right) \]

   9. Applied egg-rr 71.9%

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

   if 2.2500000000000001e-79 < x.re

   1. Initial program 78.1%

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

      1. difference-of-squares N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right) \cdot x.im\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(x.re, x.im\right)}, \mathsf{*.f64}\left(x.im, x.re\right)\right), x.re\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.im\right), \mathsf{*.f64}\left(x.im, x.re\right)\right)}, x.re\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x.re + x.im\right), \left(\left(x.re - x.im\right) \cdot x.im\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.im\right), \mathsf{*.f64}\left(x.im, x.re\right)\right)}, x.re\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x.im + x.re\right), \left(\left(x.re - x.im\right) \cdot x.im\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(x.re, x.im\right)}, \mathsf{*.f64}\left(x.im, x.re\right)\right), x.re\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x.im, x.re\right), \left(\left(x.re - x.im\right) \cdot x.im\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(x.re, x.im\right)}, \mathsf{*.f64}\left(x.im, x.re\right)\right), x.re\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(\left(x.re - x.im\right), x.im\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.im\right), \color{blue}{\mathsf{*.f64}\left(x.im, x.re\right)}\right), x.re\right)\right) \]

      7. --lowering--.f64 91.3%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(x.re, x.im\right), x.im\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.im\right), \mathsf{*.f64}\left(\color{blue}{x.im}, x.re\right)\right), x.re\right)\right) \]

   4. Applied egg-rr 91.3%

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

   5. Taylor expanded in x.re around inf

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-+l+ N/A

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

      5. +-commutative N/A

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

   7. Simplified 83.6%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(3 - \frac{x.im \cdot x.im}{x.re \cdot x.re}\right), \color{blue}{\left(\left(x.re \cdot x.re\right) \cdot x.im\right)}\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(3, \left(\frac{x.im \cdot x.im}{x.re \cdot x.re}\right)\right), \left(\color{blue}{\left(x.re \cdot x.re\right)} \cdot x.im\right)\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(3, \left(\frac{\frac{x.im \cdot x.im}{x.re}}{x.re}\right)\right), \left(\left(x.re \cdot \color{blue}{x.re}\right) \cdot x.im\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(3, \mathsf{/.f64}\left(\left(\frac{x.im \cdot x.im}{x.re}\right), x.re\right)\right), \left(\left(x.re \cdot \color{blue}{x.re}\right) \cdot x.im\right)\right) \]

      7. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(3, \mathsf{/.f64}\left(\left(x.im \cdot \frac{x.im}{x.re}\right), x.re\right)\right), \left(\left(x.re \cdot x.re\right) \cdot x.im\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(3, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x.im, \left(\frac{x.im}{x.re}\right)\right), x.re\right)\right), \left(\left(x.re \cdot x.re\right) \cdot x.im\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(3, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(x.im, x.re\right)\right), x.re\right)\right), \left(\left(x.re \cdot x.re\right) \cdot x.im\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(3, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(x.im, x.re\right)\right), x.re\right)\right), \left(x.re \cdot \color{blue}{\left(x.re \cdot x.im\right)}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(3, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(x.im, x.re\right)\right), x.re\right)\right), \left(x.re \cdot \left(x.im \cdot \color{blue}{x.re}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(3, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(x.im, x.re\right)\right), x.re\right)\right), \mathsf{*.f64}\left(x.re, \color{blue}{\left(x.im \cdot x.re\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(3, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(x.im, x.re\right)\right), x.re\right)\right), \mathsf{*.f64}\left(x.re, \left(x.re \cdot \color{blue}{x.im}\right)\right)\right) \]

      14. *-lowering-*.f64 99.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(3, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(x.im, x.re\right)\right), x.re\right)\right), \mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, \color{blue}{x.im}\right)\right)\right) \]

   9. Applied egg-rr 99.8%

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

Alternative 2: 95.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ \begin{array}{l} t_0 := x.im \cdot \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right)\\ \mathbf{if}\;t\_0 + x.re\_m \cdot \left(x.re\_m \cdot x.im + x.re\_m \cdot x.im\right) \leq 10^{+203}:\\ \;\;\;\;t\_0 + x.im \cdot \left(\left(x.re\_m \cdot x.re\_m\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \frac{x.im \cdot \frac{x.im}{x.re\_m}}{x.re\_m}\right) \cdot \left(x.re\_m \cdot \left(x.re\_m \cdot x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im)
 :precision binary64
 (let* ((t_0 (* x.im (- (* x.re_m x.re_m) (* x.im x.im)))))
   (if (<= (+ t_0 (* x.re_m (+ (* x.re_m x.im) (* x.re_m x.im)))) 1e+203)
     (+ t_0 (* x.im (* (* x.re_m x.re_m) 2.0)))
     (*
      (- 3.0 (/ (* x.im (/ x.im x.re_m)) x.re_m))
      (* x.re_m (* x.re_m x.im))))))

C

x.re_m = fabs(x_46_re);
double code(double x_46_re_m, double x_46_im) {
	double t_0 = x_46_im * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im));
	double tmp;
	if ((t_0 + (x_46_re_m * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= 1e+203) {
		tmp = t_0 + (x_46_im * ((x_46_re_m * x_46_re_m) * 2.0));
	} else {
		tmp = (3.0 - ((x_46_im * (x_46_im / x_46_re_m)) / x_46_re_m)) * (x_46_re_m * (x_46_re_m * x_46_im));
	}
	return tmp;
}

Fortran

x.re_m = abs(x_46re)
real(8) function code(x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x_46im * ((x_46re_m * x_46re_m) - (x_46im * x_46im))
    if ((t_0 + (x_46re_m * ((x_46re_m * x_46im) + (x_46re_m * x_46im)))) <= 1d+203) then
        tmp = t_0 + (x_46im * ((x_46re_m * x_46re_m) * 2.0d0))
    else
        tmp = (3.0d0 - ((x_46im * (x_46im / x_46re_m)) / x_46re_m)) * (x_46re_m * (x_46re_m * x_46im))
    end if
    code = tmp
end function

Java

x.re_m = Math.abs(x_46_re);
public static double code(double x_46_re_m, double x_46_im) {
	double t_0 = x_46_im * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im));
	double tmp;
	if ((t_0 + (x_46_re_m * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= 1e+203) {
		tmp = t_0 + (x_46_im * ((x_46_re_m * x_46_re_m) * 2.0));
	} else {
		tmp = (3.0 - ((x_46_im * (x_46_im / x_46_re_m)) / x_46_re_m)) * (x_46_re_m * (x_46_re_m * x_46_im));
	}
	return tmp;
}

Python

x.re_m = math.fabs(x_46_re)
def code(x_46_re_m, x_46_im):
	t_0 = x_46_im * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))
	tmp = 0
	if (t_0 + (x_46_re_m * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= 1e+203:
		tmp = t_0 + (x_46_im * ((x_46_re_m * x_46_re_m) * 2.0))
	else:
		tmp = (3.0 - ((x_46_im * (x_46_im / x_46_re_m)) / x_46_re_m)) * (x_46_re_m * (x_46_re_m * x_46_im))
	return tmp

Julia

x.re_m = abs(x_46_re)
function code(x_46_re_m, x_46_im)
	t_0 = Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im)))
	tmp = 0.0
	if (Float64(t_0 + Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_re_m * x_46_im)))) <= 1e+203)
		tmp = Float64(t_0 + Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_re_m) * 2.0)));
	else
		tmp = Float64(Float64(3.0 - Float64(Float64(x_46_im * Float64(x_46_im / x_46_re_m)) / x_46_re_m)) * Float64(x_46_re_m * Float64(x_46_re_m * x_46_im)));
	end
	return tmp
end

MATLAB

x.re_m = abs(x_46_re);
function tmp_2 = code(x_46_re_m, x_46_im)
	t_0 = x_46_im * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im));
	tmp = 0.0;
	if ((t_0 + (x_46_re_m * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= 1e+203)
		tmp = t_0 + (x_46_im * ((x_46_re_m * x_46_re_m) * 2.0));
	else
		tmp = (3.0 - ((x_46_im * (x_46_im / x_46_re_m)) / x_46_re_m)) * (x_46_re_m * (x_46_re_m * x_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
code[x$46$re$95$m_, x$46$im_] := Block[{t$95$0 = N[(x$46$im * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 + 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], 1e+203], N[(t$95$0 + N[(x$46$im * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 - N[(N[(x$46$im * N[(x$46$im / x$46$re$95$m), $MachinePrecision]), $MachinePrecision] / x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < 9.9999999999999999e202

   1. Initial program 93.0%

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

   3. Taylor expanded in x.re around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), x.im\right), \color{blue}{\left(2 \cdot \left(x.im \cdot {x.re}^{2}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), x.im\right), \left(\left(2 \cdot x.im\right) \cdot \color{blue}{{x.re}^{2}}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), x.im\right), \left(\left(x.im \cdot 2\right) \cdot {\color{blue}{x.re}}^{2}\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), x.im\right), \left(x.im \cdot \color{blue}{\left(2 \cdot {x.re}^{2}\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), x.im\right), \mathsf{*.f64}\left(x.im, \color{blue}{\left(2 \cdot {x.re}^{2}\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), x.im\right), \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(2, \color{blue}{\left({x.re}^{2}\right)}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), x.im\right), \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(2, \left(x.re \cdot \color{blue}{x.re}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 93.0%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), x.im\right), \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(x.re, \color{blue}{x.re}\right)\right)\right)\right) \]

   5. Simplified 93.0%

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

   if 9.9999999999999999e202 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re))

   1. Initial program 67.6%

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

      1. difference-of-squares N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right) \cdot x.im\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(x.re, x.im\right)}, \mathsf{*.f64}\left(x.im, x.re\right)\right), x.re\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.im\right), \mathsf{*.f64}\left(x.im, x.re\right)\right)}, x.re\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x.re + x.im\right), \left(\left(x.re - x.im\right) \cdot x.im\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.im\right), \mathsf{*.f64}\left(x.im, x.re\right)\right)}, x.re\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x.im + x.re\right), \left(\left(x.re - x.im\right) \cdot x.im\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(x.re, x.im\right)}, \mathsf{*.f64}\left(x.im, x.re\right)\right), x.re\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x.im, x.re\right), \left(\left(x.re - x.im\right) \cdot x.im\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(x.re, x.im\right)}, \mathsf{*.f64}\left(x.im, x.re\right)\right), x.re\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(\left(x.re - x.im\right), x.im\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.im\right), \color{blue}{\mathsf{*.f64}\left(x.im, x.re\right)}\right), x.re\right)\right) \]

      7. --lowering--.f64 78.9%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(x.re, x.im\right), x.im\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.im\right), \mathsf{*.f64}\left(\color{blue}{x.im}, x.re\right)\right), x.re\right)\right) \]

   4. Applied egg-rr 78.9%

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

   5. Taylor expanded in x.re around inf

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-+l+ N/A

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

      5. +-commutative N/A

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

   7. Simplified 61.1%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(3 - \frac{x.im \cdot x.im}{x.re \cdot x.re}\right), \color{blue}{\left(\left(x.re \cdot x.re\right) \cdot x.im\right)}\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(3, \left(\frac{x.im \cdot x.im}{x.re \cdot x.re}\right)\right), \left(\color{blue}{\left(x.re \cdot x.re\right)} \cdot x.im\right)\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(3, \left(\frac{\frac{x.im \cdot x.im}{x.re}}{x.re}\right)\right), \left(\left(x.re \cdot \color{blue}{x.re}\right) \cdot x.im\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(3, \mathsf{/.f64}\left(\left(\frac{x.im \cdot x.im}{x.re}\right), x.re\right)\right), \left(\left(x.re \cdot \color{blue}{x.re}\right) \cdot x.im\right)\right) \]

      7. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(3, \mathsf{/.f64}\left(\left(x.im \cdot \frac{x.im}{x.re}\right), x.re\right)\right), \left(\left(x.re \cdot x.re\right) \cdot x.im\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(3, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x.im, \left(\frac{x.im}{x.re}\right)\right), x.re\right)\right), \left(\left(x.re \cdot x.re\right) \cdot x.im\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(3, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(x.im, x.re\right)\right), x.re\right)\right), \left(\left(x.re \cdot x.re\right) \cdot x.im\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(3, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(x.im, x.re\right)\right), x.re\right)\right), \left(x.re \cdot \color{blue}{\left(x.re \cdot x.im\right)}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(3, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(x.im, x.re\right)\right), x.re\right)\right), \left(x.re \cdot \left(x.im \cdot \color{blue}{x.re}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(3, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(x.im, x.re\right)\right), x.re\right)\right), \mathsf{*.f64}\left(x.re, \color{blue}{\left(x.im \cdot x.re\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(3, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(x.im, x.re\right)\right), x.re\right)\right), \mathsf{*.f64}\left(x.re, \left(x.re \cdot \color{blue}{x.im}\right)\right)\right) \]

      14. *-lowering-*.f64 94.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(3, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(x.im, x.re\right)\right), x.re\right)\right), \mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, \color{blue}{x.im}\right)\right)\right) \]

   9. Applied egg-rr 94.7%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq 10^{+203}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.im \cdot \left(\left(x.re \cdot x.re\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \frac{x.im \cdot \frac{x.im}{x.re}}{x.re}\right) \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ \begin{array}{l} \mathbf{if}\;x.re\_m \leq 100:\\ \;\;\;\;x.im \cdot \left(x.re\_m \cdot \left(x.re\_m \cdot 3\right) - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \frac{x.im \cdot \frac{x.im}{x.re\_m}}{x.re\_m}\right) \cdot \left(x.re\_m \cdot \left(x.re\_m \cdot x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im)
 :precision binary64
 (if (<= x.re_m 100.0)
   (* x.im (- (* x.re_m (* x.re_m 3.0)) (* x.im x.im)))
   (* (- 3.0 (/ (* x.im (/ x.im x.re_m)) x.re_m)) (* x.re_m (* x.re_m x.im)))))

C

x.re_m = fabs(x_46_re);
double code(double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 100.0) {
		tmp = x_46_im * ((x_46_re_m * (x_46_re_m * 3.0)) - (x_46_im * x_46_im));
	} else {
		tmp = (3.0 - ((x_46_im * (x_46_im / x_46_re_m)) / x_46_re_m)) * (x_46_re_m * (x_46_re_m * x_46_im));
	}
	return tmp;
}

Fortran

x.re_m = abs(x_46re)
real(8) function code(x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re_m <= 100.0d0) then
        tmp = x_46im * ((x_46re_m * (x_46re_m * 3.0d0)) - (x_46im * x_46im))
    else
        tmp = (3.0d0 - ((x_46im * (x_46im / x_46re_m)) / x_46re_m)) * (x_46re_m * (x_46re_m * x_46im))
    end if
    code = tmp
end function

Java

x.re_m = Math.abs(x_46_re);
public static double code(double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 100.0) {
		tmp = x_46_im * ((x_46_re_m * (x_46_re_m * 3.0)) - (x_46_im * x_46_im));
	} else {
		tmp = (3.0 - ((x_46_im * (x_46_im / x_46_re_m)) / x_46_re_m)) * (x_46_re_m * (x_46_re_m * x_46_im));
	}
	return tmp;
}

Python

x.re_m = math.fabs(x_46_re)
def code(x_46_re_m, x_46_im):
	tmp = 0
	if x_46_re_m <= 100.0:
		tmp = x_46_im * ((x_46_re_m * (x_46_re_m * 3.0)) - (x_46_im * x_46_im))
	else:
		tmp = (3.0 - ((x_46_im * (x_46_im / x_46_re_m)) / x_46_re_m)) * (x_46_re_m * (x_46_re_m * x_46_im))
	return tmp

Julia

x.re_m = abs(x_46_re)
function code(x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_re_m <= 100.0)
		tmp = Float64(x_46_im * Float64(Float64(x_46_re_m * Float64(x_46_re_m * 3.0)) - Float64(x_46_im * x_46_im)));
	else
		tmp = Float64(Float64(3.0 - Float64(Float64(x_46_im * Float64(x_46_im / x_46_re_m)) / x_46_re_m)) * Float64(x_46_re_m * Float64(x_46_re_m * x_46_im)));
	end
	return tmp
end

MATLAB

x.re_m = abs(x_46_re);
function tmp_2 = code(x_46_re_m, x_46_im)
	tmp = 0.0;
	if (x_46_re_m <= 100.0)
		tmp = x_46_im * ((x_46_re_m * (x_46_re_m * 3.0)) - (x_46_im * x_46_im));
	else
		tmp = (3.0 - ((x_46_im * (x_46_im / x_46_re_m)) / x_46_re_m)) * (x_46_re_m * (x_46_re_m * x_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
code[x$46$re$95$m_, x$46$im_] := If[LessEqual[x$46$re$95$m, 100.0], N[(x$46$im * N[(N[(x$46$re$95$m * N[(x$46$re$95$m * 3.0), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 - N[(N[(x$46$im * N[(x$46$im / x$46$re$95$m), $MachinePrecision]), $MachinePrecision] / x$46$re$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 100

   1. Initial program 89.1%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-out N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \color{blue}{\left(\left(x.re + x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right) \]

      8. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right) - \color{blue}{x.im \cdot x.im}\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right), \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(x.re \cdot \left(\left(x.re + x.re\right) + x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(\left(x.re + x.re\right) + x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      12. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(2 \cdot x.re + x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      13. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(\left(2 + 1\right) \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(3 \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(x.re \cdot 3\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      17. *-lowering-*.f64 92.1%

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

   3. Simplified 92.1%

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

   if 100 < x.re

   1. Initial program 72.8%

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

      1. difference-of-squares N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right) \cdot x.im\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(x.re, x.im\right)}, \mathsf{*.f64}\left(x.im, x.re\right)\right), x.re\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.im\right), \mathsf{*.f64}\left(x.im, x.re\right)\right)}, x.re\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x.re + x.im\right), \left(\left(x.re - x.im\right) \cdot x.im\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.im\right), \mathsf{*.f64}\left(x.im, x.re\right)\right)}, x.re\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x.im + x.re\right), \left(\left(x.re - x.im\right) \cdot x.im\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(x.re, x.im\right)}, \mathsf{*.f64}\left(x.im, x.re\right)\right), x.re\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x.im, x.re\right), \left(\left(x.re - x.im\right) \cdot x.im\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(x.re, x.im\right)}, \mathsf{*.f64}\left(x.im, x.re\right)\right), x.re\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(\left(x.re - x.im\right), x.im\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.im\right), \color{blue}{\mathsf{*.f64}\left(x.im, x.re\right)}\right), x.re\right)\right) \]

      7. --lowering--.f64 89.3%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(x.re, x.im\right), x.im\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.im\right), \mathsf{*.f64}\left(\color{blue}{x.im}, x.re\right)\right), x.re\right)\right) \]

   4. Applied egg-rr 89.3%

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

   5. Taylor expanded in x.re around inf

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-+l+ N/A

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

      5. +-commutative N/A

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

   7. Simplified 81.4%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(3 - \frac{x.im \cdot x.im}{x.re \cdot x.re}\right), \color{blue}{\left(\left(x.re \cdot x.re\right) \cdot x.im\right)}\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(3, \left(\frac{x.im \cdot x.im}{x.re \cdot x.re}\right)\right), \left(\color{blue}{\left(x.re \cdot x.re\right)} \cdot x.im\right)\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(3, \left(\frac{\frac{x.im \cdot x.im}{x.re}}{x.re}\right)\right), \left(\left(x.re \cdot \color{blue}{x.re}\right) \cdot x.im\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(3, \mathsf{/.f64}\left(\left(\frac{x.im \cdot x.im}{x.re}\right), x.re\right)\right), \left(\left(x.re \cdot \color{blue}{x.re}\right) \cdot x.im\right)\right) \]

      7. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(3, \mathsf{/.f64}\left(\left(x.im \cdot \frac{x.im}{x.re}\right), x.re\right)\right), \left(\left(x.re \cdot x.re\right) \cdot x.im\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(3, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x.im, \left(\frac{x.im}{x.re}\right)\right), x.re\right)\right), \left(\left(x.re \cdot x.re\right) \cdot x.im\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(3, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(x.im, x.re\right)\right), x.re\right)\right), \left(\left(x.re \cdot x.re\right) \cdot x.im\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(3, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(x.im, x.re\right)\right), x.re\right)\right), \left(x.re \cdot \color{blue}{\left(x.re \cdot x.im\right)}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(3, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(x.im, x.re\right)\right), x.re\right)\right), \left(x.re \cdot \left(x.im \cdot \color{blue}{x.re}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(3, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(x.im, x.re\right)\right), x.re\right)\right), \mathsf{*.f64}\left(x.re, \color{blue}{\left(x.im \cdot x.re\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(3, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(x.im, x.re\right)\right), x.re\right)\right), \mathsf{*.f64}\left(x.re, \left(x.re \cdot \color{blue}{x.im}\right)\right)\right) \]

      14. *-lowering-*.f64 99.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(3, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(x.im, x.re\right)\right), x.re\right)\right), \mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, \color{blue}{x.im}\right)\right)\right) \]

   9. Applied egg-rr 99.8%

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

Alternative 4: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ \begin{array}{l} \mathbf{if}\;x.re\_m \leq 2.8 \cdot 10^{+57}:\\ \;\;\;\;x.im \cdot \left(x.re\_m \cdot \left(x.re\_m \cdot 3\right) - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot \left(x.im \cdot \left(3 - \frac{x.im \cdot \frac{x.im}{x.re\_m}}{x.re\_m}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im)
 :precision binary64
 (if (<= x.re_m 2.8e+57)
   (* x.im (- (* x.re_m (* x.re_m 3.0)) (* x.im x.im)))
   (* x.re_m (* x.re_m (* x.im (- 3.0 (/ (* x.im (/ x.im x.re_m)) x.re_m)))))))

C

x.re_m = fabs(x_46_re);
double code(double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 2.8e+57) {
		tmp = x_46_im * ((x_46_re_m * (x_46_re_m * 3.0)) - (x_46_im * x_46_im));
	} else {
		tmp = x_46_re_m * (x_46_re_m * (x_46_im * (3.0 - ((x_46_im * (x_46_im / x_46_re_m)) / x_46_re_m))));
	}
	return tmp;
}

Fortran

x.re_m = abs(x_46re)
real(8) function code(x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re_m <= 2.8d+57) then
        tmp = x_46im * ((x_46re_m * (x_46re_m * 3.0d0)) - (x_46im * x_46im))
    else
        tmp = x_46re_m * (x_46re_m * (x_46im * (3.0d0 - ((x_46im * (x_46im / x_46re_m)) / x_46re_m))))
    end if
    code = tmp
end function

Java

x.re_m = Math.abs(x_46_re);
public static double code(double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 2.8e+57) {
		tmp = x_46_im * ((x_46_re_m * (x_46_re_m * 3.0)) - (x_46_im * x_46_im));
	} else {
		tmp = x_46_re_m * (x_46_re_m * (x_46_im * (3.0 - ((x_46_im * (x_46_im / x_46_re_m)) / x_46_re_m))));
	}
	return tmp;
}

Python

x.re_m = math.fabs(x_46_re)
def code(x_46_re_m, x_46_im):
	tmp = 0
	if x_46_re_m <= 2.8e+57:
		tmp = x_46_im * ((x_46_re_m * (x_46_re_m * 3.0)) - (x_46_im * x_46_im))
	else:
		tmp = x_46_re_m * (x_46_re_m * (x_46_im * (3.0 - ((x_46_im * (x_46_im / x_46_re_m)) / x_46_re_m))))
	return tmp

Julia

x.re_m = abs(x_46_re)
function code(x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_re_m <= 2.8e+57)
		tmp = Float64(x_46_im * Float64(Float64(x_46_re_m * Float64(x_46_re_m * 3.0)) - Float64(x_46_im * x_46_im)));
	else
		tmp = Float64(x_46_re_m * Float64(x_46_re_m * Float64(x_46_im * Float64(3.0 - Float64(Float64(x_46_im * Float64(x_46_im / x_46_re_m)) / x_46_re_m)))));
	end
	return tmp
end

MATLAB

x.re_m = abs(x_46_re);
function tmp_2 = code(x_46_re_m, x_46_im)
	tmp = 0.0;
	if (x_46_re_m <= 2.8e+57)
		tmp = x_46_im * ((x_46_re_m * (x_46_re_m * 3.0)) - (x_46_im * x_46_im));
	else
		tmp = x_46_re_m * (x_46_re_m * (x_46_im * (3.0 - ((x_46_im * (x_46_im / x_46_re_m)) / x_46_re_m))));
	end
	tmp_2 = tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
code[x$46$re$95$m_, x$46$im_] := If[LessEqual[x$46$re$95$m, 2.8e+57], N[(x$46$im * N[(N[(x$46$re$95$m * N[(x$46$re$95$m * 3.0), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re$95$m * N[(x$46$re$95$m * N[(x$46$im * N[(3.0 - N[(N[(x$46$im * N[(x$46$im / x$46$re$95$m), $MachinePrecision]), $MachinePrecision] / x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 2.8e57

   1. Initial program 89.5%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-out N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \color{blue}{\left(\left(x.re + x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right) \]

      8. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right) - \color{blue}{x.im \cdot x.im}\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right), \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(x.re \cdot \left(\left(x.re + x.re\right) + x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(\left(x.re + x.re\right) + x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      12. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(2 \cdot x.re + x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      13. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(\left(2 + 1\right) \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(3 \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(x.re \cdot 3\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      17. *-lowering-*.f64 92.4%

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

   3. Simplified 92.4%

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

   if 2.8e57 < x.re

   1. Initial program 69.0%

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

      1. difference-of-squares N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right) \cdot x.im\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(x.re, x.im\right)}, \mathsf{*.f64}\left(x.im, x.re\right)\right), x.re\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.im\right), \mathsf{*.f64}\left(x.im, x.re\right)\right)}, x.re\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x.re + x.im\right), \left(\left(x.re - x.im\right) \cdot x.im\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.im\right), \mathsf{*.f64}\left(x.im, x.re\right)\right)}, x.re\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x.im + x.re\right), \left(\left(x.re - x.im\right) \cdot x.im\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(x.re, x.im\right)}, \mathsf{*.f64}\left(x.im, x.re\right)\right), x.re\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x.im, x.re\right), \left(\left(x.re - x.im\right) \cdot x.im\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(x.re, x.im\right)}, \mathsf{*.f64}\left(x.im, x.re\right)\right), x.re\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(\left(x.re - x.im\right), x.im\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.im\right), \color{blue}{\mathsf{*.f64}\left(x.im, x.re\right)}\right), x.re\right)\right) \]

      7. --lowering--.f64 87.8%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(x.re, x.im\right), x.im\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.im\right), \mathsf{*.f64}\left(\color{blue}{x.im}, x.re\right)\right), x.re\right)\right) \]

   4. Applied egg-rr 87.8%

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

   5. Taylor expanded in x.re around inf

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-+l+ N/A

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

      5. +-commutative N/A

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

   7. Simplified 78.8%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x.re \cdot \left(x.im \cdot \left(3 - \frac{x.im \cdot x.im}{x.re \cdot x.re}\right)\right)\right), \color{blue}{x.re}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, \left(x.im \cdot \left(3 - \frac{x.im \cdot x.im}{x.re \cdot x.re}\right)\right)\right), x.re\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.im, \left(3 - \frac{x.im \cdot x.im}{x.re \cdot x.re}\right)\right)\right), x.re\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(3, \left(\frac{x.im \cdot x.im}{x.re \cdot x.re}\right)\right)\right)\right), x.re\right) \]

      7. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(3, \left(\frac{\frac{x.im \cdot x.im}{x.re}}{x.re}\right)\right)\right)\right), x.re\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(3, \mathsf{/.f64}\left(\left(\frac{x.im \cdot x.im}{x.re}\right), x.re\right)\right)\right)\right), x.re\right) \]

      9. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(3, \mathsf{/.f64}\left(\left(x.im \cdot \frac{x.im}{x.re}\right), x.re\right)\right)\right)\right), x.re\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(3, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x.im, \left(\frac{x.im}{x.re}\right)\right), x.re\right)\right)\right)\right), x.re\right) \]

      11. /-lowering-/.f64 99.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(3, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(x.im, x.re\right)\right), x.re\right)\right)\right)\right), x.re\right) \]

   9. Applied egg-rr 99.8%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 2.8 \cdot 10^{+57}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.re \cdot 3\right) - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot \left(3 - \frac{x.im \cdot \frac{x.im}{x.re}}{x.re}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ \begin{array}{l} \mathbf{if}\;x.re\_m \leq 7.8 \cdot 10^{+153}:\\ \;\;\;\;x.im \cdot \left(x.re\_m \cdot \left(x.re\_m \cdot 3\right) - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re\_m \cdot x.im\right) \cdot \left(x.re\_m \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im)
 :precision binary64
 (if (<= x.re_m 7.8e+153)
   (* x.im (- (* x.re_m (* x.re_m 3.0)) (* x.im x.im)))
   (* (* x.re_m x.im) (* x.re_m 3.0))))

C

x.re_m = fabs(x_46_re);
double code(double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 7.8e+153) {
		tmp = x_46_im * ((x_46_re_m * (x_46_re_m * 3.0)) - (x_46_im * x_46_im));
	} else {
		tmp = (x_46_re_m * x_46_im) * (x_46_re_m * 3.0);
	}
	return tmp;
}

Fortran

x.re_m = abs(x_46re)
real(8) function code(x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re_m <= 7.8d+153) then
        tmp = x_46im * ((x_46re_m * (x_46re_m * 3.0d0)) - (x_46im * x_46im))
    else
        tmp = (x_46re_m * x_46im) * (x_46re_m * 3.0d0)
    end if
    code = tmp
end function

Java

x.re_m = Math.abs(x_46_re);
public static double code(double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 7.8e+153) {
		tmp = x_46_im * ((x_46_re_m * (x_46_re_m * 3.0)) - (x_46_im * x_46_im));
	} else {
		tmp = (x_46_re_m * x_46_im) * (x_46_re_m * 3.0);
	}
	return tmp;
}

Python

x.re_m = math.fabs(x_46_re)
def code(x_46_re_m, x_46_im):
	tmp = 0
	if x_46_re_m <= 7.8e+153:
		tmp = x_46_im * ((x_46_re_m * (x_46_re_m * 3.0)) - (x_46_im * x_46_im))
	else:
		tmp = (x_46_re_m * x_46_im) * (x_46_re_m * 3.0)
	return tmp

Julia

x.re_m = abs(x_46_re)
function code(x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_re_m <= 7.8e+153)
		tmp = Float64(x_46_im * Float64(Float64(x_46_re_m * Float64(x_46_re_m * 3.0)) - Float64(x_46_im * x_46_im)));
	else
		tmp = Float64(Float64(x_46_re_m * x_46_im) * Float64(x_46_re_m * 3.0));
	end
	return tmp
end

MATLAB

x.re_m = abs(x_46_re);
function tmp_2 = code(x_46_re_m, x_46_im)
	tmp = 0.0;
	if (x_46_re_m <= 7.8e+153)
		tmp = x_46_im * ((x_46_re_m * (x_46_re_m * 3.0)) - (x_46_im * x_46_im));
	else
		tmp = (x_46_re_m * x_46_im) * (x_46_re_m * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
code[x$46$re$95$m_, x$46$im_] := If[LessEqual[x$46$re$95$m, 7.8e+153], N[(x$46$im * N[(N[(x$46$re$95$m * N[(x$46$re$95$m * 3.0), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] * N[(x$46$re$95$m * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 7.79999999999999966e153

   1. Initial program 88.2%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-out N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \color{blue}{\left(\left(x.re + x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right) \]

      8. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right) - \color{blue}{x.im \cdot x.im}\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right), \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(x.re \cdot \left(\left(x.re + x.re\right) + x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(\left(x.re + x.re\right) + x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      12. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(2 \cdot x.re + x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      13. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(\left(2 + 1\right) \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(3 \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(x.re \cdot 3\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      17. *-lowering-*.f64 93.0%

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

   3. Simplified 93.0%

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

   if 7.79999999999999966e153 < x.re

   1. Initial program 63.9%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-out N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \color{blue}{\left(\left(x.re + x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right) \]

      8. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right) - \color{blue}{x.im \cdot x.im}\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right), \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(x.re \cdot \left(\left(x.re + x.re\right) + x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(\left(x.re + x.re\right) + x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      12. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(2 \cdot x.re + x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      13. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(\left(2 + 1\right) \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(3 \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(x.re \cdot 3\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      17. *-lowering-*.f64 63.9%

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

   3. Simplified 63.9%

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

   5. Taylor expanded in x.re around inf

      \[\leadsto \mathsf{*.f64}\left(x.im, \color{blue}{\left(3 \cdot {x.re}^{2}\right)}\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(3, \color{blue}{\left({x.re}^{2}\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(3, \left(x.re \cdot \color{blue}{x.re}\right)\right)\right) \]

      3. *-lowering-*.f64 70.8%

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(x.re, \color{blue}{x.re}\right)\right)\right) \]

   7. Simplified 70.8%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x.re \cdot 3\right), \color{blue}{\left(x.im \cdot x.re\right)}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, 3\right), \left(\color{blue}{x.im} \cdot x.re\right)\right) \]

      8. *-lowering-*.f64 96.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, 3\right), \mathsf{*.f64}\left(x.im, \color{blue}{x.re}\right)\right) \]

   9. Applied egg-rr 96.4%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 7.8 \cdot 10^{+153}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.re \cdot 3\right) - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ \begin{array}{l} \mathbf{if}\;x.re\_m \leq 10^{+28}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(0 - x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \left(3 \cdot \left(x.re\_m \cdot x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im)
 :precision binary64
 (if (<= x.re_m 1e+28)
   (* x.im (* x.im (- 0.0 x.im)))
   (* x.re_m (* 3.0 (* x.re_m x.im)))))

C

x.re_m = fabs(x_46_re);
double code(double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 1e+28) {
		tmp = x_46_im * (x_46_im * (0.0 - x_46_im));
	} else {
		tmp = x_46_re_m * (3.0 * (x_46_re_m * x_46_im));
	}
	return tmp;
}

Fortran

x.re_m = abs(x_46re)
real(8) function code(x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re_m <= 1d+28) then
        tmp = x_46im * (x_46im * (0.0d0 - x_46im))
    else
        tmp = x_46re_m * (3.0d0 * (x_46re_m * x_46im))
    end if
    code = tmp
end function

Java

x.re_m = Math.abs(x_46_re);
public static double code(double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 1e+28) {
		tmp = x_46_im * (x_46_im * (0.0 - x_46_im));
	} else {
		tmp = x_46_re_m * (3.0 * (x_46_re_m * x_46_im));
	}
	return tmp;
}

Python

x.re_m = math.fabs(x_46_re)
def code(x_46_re_m, x_46_im):
	tmp = 0
	if x_46_re_m <= 1e+28:
		tmp = x_46_im * (x_46_im * (0.0 - x_46_im))
	else:
		tmp = x_46_re_m * (3.0 * (x_46_re_m * x_46_im))
	return tmp

Julia

x.re_m = abs(x_46_re)
function code(x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_re_m <= 1e+28)
		tmp = Float64(x_46_im * Float64(x_46_im * Float64(0.0 - x_46_im)));
	else
		tmp = Float64(x_46_re_m * Float64(3.0 * Float64(x_46_re_m * x_46_im)));
	end
	return tmp
end

MATLAB

x.re_m = abs(x_46_re);
function tmp_2 = code(x_46_re_m, x_46_im)
	tmp = 0.0;
	if (x_46_re_m <= 1e+28)
		tmp = x_46_im * (x_46_im * (0.0 - x_46_im));
	else
		tmp = x_46_re_m * (3.0 * (x_46_re_m * x_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
code[x$46$re$95$m_, x$46$im_] := If[LessEqual[x$46$re$95$m, 1e+28], N[(x$46$im * N[(x$46$im * N[(0.0 - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re$95$m * N[(3.0 * N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 9.99999999999999958e27

   1. Initial program 89.2%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-out N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \color{blue}{\left(\left(x.re + x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right) \]

      8. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right) - \color{blue}{x.im \cdot x.im}\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right), \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(x.re \cdot \left(\left(x.re + x.re\right) + x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(\left(x.re + x.re\right) + x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      12. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(2 \cdot x.re + x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      13. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(\left(2 + 1\right) \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(3 \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(x.re \cdot 3\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      17. *-lowering-*.f64 92.2%

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

   3. Simplified 92.2%

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

   5. Taylor expanded in x.im around inf

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left({x.im}^{3}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{{x.im}^{3}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left({x.im}^{3}\right)}\right) \]

      4. cube-mult N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \left(x.im \cdot \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \left(x.im \cdot {x.im}^{\color{blue}{2}}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x.im, \color{blue}{\left({x.im}^{2}\right)}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x.im, \left(x.im \cdot \color{blue}{x.im}\right)\right)\right) \]

      8. *-lowering-*.f64 71.1%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

   7. Simplified 71.1%

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

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(x.im \cdot \left(x.im \cdot x.im\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{neg}\left(\left(x.im \cdot x.im\right) \cdot x.im\right) \]

      3. distribute-lft-neg-in N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right), \color{blue}{x.im}\right) \]

      5. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{neg.f64}\left(\left(x.im \cdot x.im\right)\right), x.im\right) \]

      6. *-lowering-*.f64 71.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right)\right), x.im\right) \]

   9. Applied egg-rr 71.1%

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

   if 9.99999999999999958e27 < x.re

   1. Initial program 71.3%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-out N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \color{blue}{\left(\left(x.re + x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right) \]

      8. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right) - \color{blue}{x.im \cdot x.im}\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right), \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(x.re \cdot \left(\left(x.re + x.re\right) + x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(\left(x.re + x.re\right) + x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      12. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(2 \cdot x.re + x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      13. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(\left(2 + 1\right) \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(3 \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(x.re \cdot 3\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      17. *-lowering-*.f64 80.4%

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

   3. Simplified 80.4%

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

   5. Taylor expanded in x.im around 0

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.re, \color{blue}{\left(\left(3 \cdot x.im\right) \cdot x.re\right)}\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(x.re, \left(3 \cdot \color{blue}{\left(x.im \cdot x.re\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(3, \color{blue}{\left(x.im \cdot x.re\right)}\right)\right) \]

      8. *-lowering-*.f64 86.9%

         \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(x.im, \color{blue}{x.re}\right)\right)\right) \]

   7. Simplified 86.9%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 10^{+28}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(0 - x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(3 \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ \begin{array}{l} \mathbf{if}\;x.re\_m \leq 1.3 \cdot 10^{+28}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(0 - x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(3 \cdot \left(x.re\_m \cdot x.re\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im)
 :precision binary64
 (if (<= x.re_m 1.3e+28)
   (* x.im (* x.im (- 0.0 x.im)))
   (* x.im (* 3.0 (* x.re_m x.re_m)))))

C

x.re_m = fabs(x_46_re);
double code(double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 1.3e+28) {
		tmp = x_46_im * (x_46_im * (0.0 - x_46_im));
	} else {
		tmp = x_46_im * (3.0 * (x_46_re_m * x_46_re_m));
	}
	return tmp;
}

Fortran

x.re_m = abs(x_46re)
real(8) function code(x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re_m <= 1.3d+28) then
        tmp = x_46im * (x_46im * (0.0d0 - x_46im))
    else
        tmp = x_46im * (3.0d0 * (x_46re_m * x_46re_m))
    end if
    code = tmp
end function

Java

x.re_m = Math.abs(x_46_re);
public static double code(double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 1.3e+28) {
		tmp = x_46_im * (x_46_im * (0.0 - x_46_im));
	} else {
		tmp = x_46_im * (3.0 * (x_46_re_m * x_46_re_m));
	}
	return tmp;
}

Python

x.re_m = math.fabs(x_46_re)
def code(x_46_re_m, x_46_im):
	tmp = 0
	if x_46_re_m <= 1.3e+28:
		tmp = x_46_im * (x_46_im * (0.0 - x_46_im))
	else:
		tmp = x_46_im * (3.0 * (x_46_re_m * x_46_re_m))
	return tmp

Julia

x.re_m = abs(x_46_re)
function code(x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_re_m <= 1.3e+28)
		tmp = Float64(x_46_im * Float64(x_46_im * Float64(0.0 - x_46_im)));
	else
		tmp = Float64(x_46_im * Float64(3.0 * Float64(x_46_re_m * x_46_re_m)));
	end
	return tmp
end

MATLAB

x.re_m = abs(x_46_re);
function tmp_2 = code(x_46_re_m, x_46_im)
	tmp = 0.0;
	if (x_46_re_m <= 1.3e+28)
		tmp = x_46_im * (x_46_im * (0.0 - x_46_im));
	else
		tmp = x_46_im * (3.0 * (x_46_re_m * x_46_re_m));
	end
	tmp_2 = tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
code[x$46$re$95$m_, x$46$im_] := If[LessEqual[x$46$re$95$m, 1.3e+28], N[(x$46$im * N[(x$46$im * N[(0.0 - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(3.0 * N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.3000000000000001e28

   1. Initial program 89.2%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-out N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \color{blue}{\left(\left(x.re + x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right) \]

      8. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right) - \color{blue}{x.im \cdot x.im}\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right), \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(x.re \cdot \left(\left(x.re + x.re\right) + x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(\left(x.re + x.re\right) + x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      12. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(2 \cdot x.re + x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      13. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(\left(2 + 1\right) \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(3 \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(x.re \cdot 3\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      17. *-lowering-*.f64 92.2%

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

   3. Simplified 92.2%

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

   5. Taylor expanded in x.im around inf

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left({x.im}^{3}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{{x.im}^{3}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left({x.im}^{3}\right)}\right) \]

      4. cube-mult N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \left(x.im \cdot \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \left(x.im \cdot {x.im}^{\color{blue}{2}}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x.im, \color{blue}{\left({x.im}^{2}\right)}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x.im, \left(x.im \cdot \color{blue}{x.im}\right)\right)\right) \]

      8. *-lowering-*.f64 71.1%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

   7. Simplified 71.1%

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

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(x.im \cdot \left(x.im \cdot x.im\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{neg}\left(\left(x.im \cdot x.im\right) \cdot x.im\right) \]

      3. distribute-lft-neg-in N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right), \color{blue}{x.im}\right) \]

      5. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{neg.f64}\left(\left(x.im \cdot x.im\right)\right), x.im\right) \]

      6. *-lowering-*.f64 71.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right)\right), x.im\right) \]

   9. Applied egg-rr 71.1%

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

   if 1.3000000000000001e28 < x.re

   1. Initial program 71.3%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-out N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \color{blue}{\left(\left(x.re + x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right) \]

      8. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right) - \color{blue}{x.im \cdot x.im}\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right), \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(x.re \cdot \left(\left(x.re + x.re\right) + x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(\left(x.re + x.re\right) + x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

      12. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(2 \cdot x.re + x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      13. distribute-lft1-in N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(\left(2 + 1\right) \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(3 \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(x.re \cdot 3\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

      17. *-lowering-*.f64 80.4%

          \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

   3. Simplified 80.4%

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

   5. Taylor expanded in x.re around inf

      \[\leadsto \mathsf{*.f64}\left(x.im, \color{blue}{\left(3 \cdot {x.re}^{2}\right)}\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(3, \color{blue}{\left({x.re}^{2}\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(3, \left(x.re \cdot \color{blue}{x.re}\right)\right)\right) \]

      3. *-lowering-*.f64 73.0%

         \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(x.re, \color{blue}{x.re}\right)\right)\right) \]

   7. Simplified 73.0%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 1.3 \cdot 10^{+28}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(0 - x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(3 \cdot \left(x.re \cdot x.re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 59.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.im \cdot \left(x.im \cdot \left(0 - x.im\right)\right) \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im) :precision binary64 (* x.im (* x.im (- 0.0 x.im))))

C

x.re_m = fabs(x_46_re);
double code(double x_46_re_m, double x_46_im) {
	return x_46_im * (x_46_im * (0.0 - x_46_im));
}

Fortran

x.re_m = abs(x_46re)
real(8) function code(x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    code = x_46im * (x_46im * (0.0d0 - x_46im))
end function

Java

x.re_m = Math.abs(x_46_re);
public static double code(double x_46_re_m, double x_46_im) {
	return x_46_im * (x_46_im * (0.0 - x_46_im));
}

Python

x.re_m = math.fabs(x_46_re)
def code(x_46_re_m, x_46_im):
	return x_46_im * (x_46_im * (0.0 - x_46_im))

Julia

x.re_m = abs(x_46_re)
function code(x_46_re_m, x_46_im)
	return Float64(x_46_im * Float64(x_46_im * Float64(0.0 - x_46_im)))
end

MATLAB

x.re_m = abs(x_46_re);
function tmp = code(x_46_re_m, x_46_im)
	tmp = x_46_im * (x_46_im * (0.0 - x_46_im));
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
code[x$46$re$95$m_, x$46$im_] := N[(x$46$im * N[(x$46$im * N[(0.0 - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.5%

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. distribute-lft-out N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. distribute-lft-out N/A

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

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x.im, \color{blue}{\left(\left(x.re + x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)}\right) \]

   8. associate-+r- N/A

      \[\leadsto \mathsf{*.f64}\left(x.im, \left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right) - \color{blue}{x.im \cdot x.im}\right)\right) \]

   9. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right), \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

   10. distribute-rgt-out N/A

       \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\left(x.re \cdot \left(\left(x.re + x.re\right) + x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(\left(x.re + x.re\right) + x.re\right)\right), \left(\color{blue}{x.im} \cdot x.im\right)\right)\right) \]

   12. count-2 N/A

       \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(2 \cdot x.re + x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

   13. distribute-lft1-in N/A

       \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(\left(2 + 1\right) \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(3 \cdot x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \left(x.re \cdot 3\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

   16. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \left(x.im \cdot x.im\right)\right)\right) \]

   17. *-lowering-*.f64 89.7%

       \[\leadsto \mathsf{*.f64}\left(x.im, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.re, 3\right)\right), \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

3. Simplified 89.7%

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

5. Taylor expanded in x.im around inf

   \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left({x.im}^{3}\right) \]

   2. neg-sub0 N/A

      \[\leadsto 0 - \color{blue}{{x.im}^{3}} \]

   3. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left({x.im}^{3}\right)}\right) \]

   4. cube-mult N/A

      \[\leadsto \mathsf{\_.f64}\left(0, \left(x.im \cdot \color{blue}{\left(x.im \cdot x.im\right)}\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{\_.f64}\left(0, \left(x.im \cdot {x.im}^{\color{blue}{2}}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x.im, \color{blue}{\left({x.im}^{2}\right)}\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x.im, \left(x.im \cdot \color{blue}{x.im}\right)\right)\right) \]

   8. *-lowering-*.f64 59.2%

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x.im, \mathsf{*.f64}\left(x.im, \color{blue}{x.im}\right)\right)\right) \]

7. Simplified 59.2%

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

   1. sub0-neg N/A

      \[\leadsto \mathsf{neg}\left(x.im \cdot \left(x.im \cdot x.im\right)\right) \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{neg}\left(\left(x.im \cdot x.im\right) \cdot x.im\right) \]

   3. distribute-lft-neg-in N/A

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

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{neg}\left(x.im \cdot x.im\right)\right), \color{blue}{x.im}\right) \]

   5. neg-lowering-neg.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{neg.f64}\left(\left(x.im \cdot x.im\right)\right), x.im\right) \]

   6. *-lowering-*.f64 59.2%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right)\right), x.im\right) \]

9. Applied egg-rr 59.2%

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

10. Final simplification 59.2%

    \[\leadsto x.im \cdot \left(x.im \cdot \left(0 - x.im\right)\right) \]
11. Add Preprocessing

Developer Target 1: 91.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double x_46_re, double x_46_im) {
	return ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = ((x_46re * x_46im) * (2.0d0 * x_46re)) + ((x_46im * (x_46re - x_46im)) * (x_46re + x_46im))
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024160 
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (* (* x.re x.im) (* 2 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im))))

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