math.cube on complex, imaginary part

Percentage Accurate: 82.6% → 99.8%

Time: 9.6s

Alternatives: 9

Speedup: 1.2×

• 43.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 82.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<=
      (+
       (* x.im (- (* x.re x.re) (* x.im x.im)))
       (* x.re (+ (* x.re x.im) (* x.re x.im))))
      INFINITY)
   (+ (* (+ x.re x.im) (* x.im (- x.re x.im))) (* x.re (* x.re (+ x.im x.im))))
   (*
    x.im
    (* x.im (* x.im (+ -1.0 (* 3.0 (* x.re (/ (/ x.re x.im) x.im)))))))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (((x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im)))) <= ((double) INFINITY)) {
		tmp = ((x_46_re + x_46_im) * (x_46_im * (x_46_re - x_46_im))) + (x_46_re * (x_46_re * (x_46_im + x_46_im)));
	} else {
		tmp = x_46_im * (x_46_im * (x_46_im * (-1.0 + (3.0 * (x_46_re * ((x_46_re / x_46_im) / x_46_im))))));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (((x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im)))) <= Double.POSITIVE_INFINITY) {
		tmp = ((x_46_re + x_46_im) * (x_46_im * (x_46_re - x_46_im))) + (x_46_re * (x_46_re * (x_46_im + x_46_im)));
	} else {
		tmp = x_46_im * (x_46_im * (x_46_im * (-1.0 + (3.0 * (x_46_re * ((x_46_re / x_46_im) / x_46_im))))));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if ((x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im)))) <= math.inf:
		tmp = ((x_46_re + x_46_im) * (x_46_im * (x_46_re - x_46_im))) + (x_46_re * (x_46_re * (x_46_im + x_46_im)))
	else:
		tmp = x_46_im * (x_46_im * (x_46_im * (-1.0 + (3.0 * (x_46_re * ((x_46_re / x_46_im) / x_46_im))))))
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (Float64(Float64(x_46_im * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im))) + Float64(x_46_re * Float64(Float64(x_46_re * x_46_im) + Float64(x_46_re * x_46_im)))) <= Inf)
		tmp = Float64(Float64(Float64(x_46_re + x_46_im) * Float64(x_46_im * Float64(x_46_re - x_46_im))) + Float64(x_46_re * Float64(x_46_re * Float64(x_46_im + x_46_im))));
	else
		tmp = Float64(x_46_im * Float64(x_46_im * Float64(x_46_im * Float64(-1.0 + Float64(3.0 * Float64(x_46_re * Float64(Float64(x_46_re / x_46_im) / x_46_im)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (((x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im)))) <= Inf)
		tmp = ((x_46_re + x_46_im) * (x_46_im * (x_46_re - x_46_im))) + (x_46_re * (x_46_re * (x_46_im + x_46_im)));
	else
		tmp = x_46_im * (x_46_im * (x_46_im * (-1.0 + (3.0 * (x_46_re * ((x_46_re / x_46_im) / x_46_im))))));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 95.3%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. 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 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. *-commutative N/A

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

      2. distribute-rgt-out N/A

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

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

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

      4. +-lowering-+.f64 99.8%

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

   6. Applied egg-rr 99.8%

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

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

   1. Initial program 0.0%

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

      1. +-commutative 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 32.0%

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

   3. Simplified 32.0%

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

   5. Taylor expanded in x.im around inf

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

   7. Simplified 100.0%

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

4. Final simplification 99.8%

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

Alternative 2: 92.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq 5 \cdot 10^{+76}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.re\right) \cdot 3 - x.im \cdot x.im\right)\\ \mathbf{elif}\;x.re \leq 5.7 \cdot 10^{+210}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 3 - \frac{x.im \cdot x.im}{x.re}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.re 5e+76)
   (* x.im (- (* (* x.re x.re) 3.0) (* x.im x.im)))
   (if (<= x.re 5.7e+210)
     (* x.re (* x.im (- (* x.re 3.0) (/ (* x.im x.im) x.re))))
     (* 3.0 (* x.re (* x.re x.im))))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= 5e+76) {
		tmp = x_46_im * (((x_46_re * x_46_re) * 3.0) - (x_46_im * x_46_im));
	} else if (x_46_re <= 5.7e+210) {
		tmp = x_46_re * (x_46_im * ((x_46_re * 3.0) - ((x_46_im * x_46_im) / x_46_re)));
	} else {
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re <= 5d+76) then
        tmp = x_46im * (((x_46re * x_46re) * 3.0d0) - (x_46im * x_46im))
    else if (x_46re <= 5.7d+210) then
        tmp = x_46re * (x_46im * ((x_46re * 3.0d0) - ((x_46im * x_46im) / x_46re)))
    else
        tmp = 3.0d0 * (x_46re * (x_46re * x_46im))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= 5e+76) {
		tmp = x_46_im * (((x_46_re * x_46_re) * 3.0) - (x_46_im * x_46_im));
	} else if (x_46_re <= 5.7e+210) {
		tmp = x_46_re * (x_46_im * ((x_46_re * 3.0) - ((x_46_im * x_46_im) / x_46_re)));
	} else {
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_re <= 5e+76:
		tmp = x_46_im * (((x_46_re * x_46_re) * 3.0) - (x_46_im * x_46_im))
	elif x_46_re <= 5.7e+210:
		tmp = x_46_re * (x_46_im * ((x_46_re * 3.0) - ((x_46_im * x_46_im) / x_46_re)))
	else:
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im))
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_re <= 5e+76)
		tmp = Float64(x_46_im * Float64(Float64(Float64(x_46_re * x_46_re) * 3.0) - Float64(x_46_im * x_46_im)));
	elseif (x_46_re <= 5.7e+210)
		tmp = Float64(x_46_re * Float64(x_46_im * Float64(Float64(x_46_re * 3.0) - Float64(Float64(x_46_im * x_46_im) / x_46_re))));
	else
		tmp = Float64(3.0 * Float64(x_46_re * Float64(x_46_re * x_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_re <= 5e+76)
		tmp = x_46_im * (((x_46_re * x_46_re) * 3.0) - (x_46_im * x_46_im));
	elseif (x_46_re <= 5.7e+210)
		tmp = x_46_re * (x_46_im * ((x_46_re * 3.0) - ((x_46_im * x_46_im) / x_46_re)));
	else
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$re, 5e+76], N[(x$46$im * N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] * 3.0), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 5.7e+210], N[(x$46$re * N[(x$46$im * N[(N[(x$46$re * 3.0), $MachinePrecision] - N[(N[(x$46$im * x$46$im), $MachinePrecision] / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 * N[(x$46$re * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.re < 4.99999999999999991e76

   1. Initial program 90.1%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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 92.4%

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

   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
   5. Step-by-step derivation

      1. associate-*r* N/A

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

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

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

      3. *-lowering-*.f64 92.4%

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

   6. Applied egg-rr 92.4%

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

   if 4.99999999999999991e76 < x.re < 5.6999999999999995e210

   1. Initial program 55.3%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. 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 79.0%

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

   4. Applied egg-rr 79.0%

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

   5. Taylor expanded in x.re around inf

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

   7. Simplified 87.2%

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

   8. Taylor expanded in x.im around 0

      \[\leadsto \mathsf{*.f64}\left(x.re, \color{blue}{\left(x.im \cdot \left(-1 \cdot \frac{{x.im}^{2}}{x.re} + 3 \cdot x.re\right)\right)}\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.im, \color{blue}{\left(-1 \cdot \frac{{x.im}^{2}}{x.re} + 3 \cdot x.re\right)}\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.im, \left(3 \cdot x.re + \color{blue}{-1 \cdot \frac{{x.im}^{2}}{x.re}}\right)\right)\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.im, \left(3 \cdot x.re + \left(\mathsf{neg}\left(\frac{{x.im}^{2}}{x.re}\right)\right)\right)\right)\right) \]

      4. unsub-neg N/A

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

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

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

      6. *-commutative N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 95.6%

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

   10. Simplified 95.6%

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

   if 5.6999999999999995e210 < x.re

   1. Initial program 77.8%

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

      1. +-commutative 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 77.8%

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

   3. Simplified 77.8%

      \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re \cdot 3\right) - x.im \cdot x.im\right)} \]
   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. *-lowering-*.f64 N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 100.0%

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

   7. Simplified 100.0%

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

      1. associate-*r* N/A

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

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

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

      3. *-lowering-*.f64 100.0%

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

   9. Applied egg-rr 100.0%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 5 \cdot 10^{+76}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.re\right) \cdot 3 - x.im \cdot x.im\right)\\ \mathbf{elif}\;x.re \leq 5.7 \cdot 10^{+210}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 3 - \frac{x.im \cdot x.im}{x.re}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq 2.2 \cdot 10^{-28}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 3 - \frac{x.im \cdot x.im}{x.re}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.im \cdot \left(-1 + 3 \cdot \left(x.re \cdot \frac{\frac{x.re}{x.im}}{x.im}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im 2.2e-28)
   (* x.re (* x.im (- (* x.re 3.0) (/ (* x.im x.im) x.re))))
   (*
    x.im
    (* x.im (* x.im (+ -1.0 (* 3.0 (* x.re (/ (/ x.re x.im) x.im)))))))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 2.2e-28) {
		tmp = x_46_re * (x_46_im * ((x_46_re * 3.0) - ((x_46_im * x_46_im) / x_46_re)));
	} else {
		tmp = x_46_im * (x_46_im * (x_46_im * (-1.0 + (3.0 * (x_46_re * ((x_46_re / x_46_im) / x_46_im))))));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46im <= 2.2d-28) then
        tmp = x_46re * (x_46im * ((x_46re * 3.0d0) - ((x_46im * x_46im) / x_46re)))
    else
        tmp = x_46im * (x_46im * (x_46im * ((-1.0d0) + (3.0d0 * (x_46re * ((x_46re / x_46im) / x_46im))))))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 2.2e-28) {
		tmp = x_46_re * (x_46_im * ((x_46_re * 3.0) - ((x_46_im * x_46_im) / x_46_re)));
	} else {
		tmp = x_46_im * (x_46_im * (x_46_im * (-1.0 + (3.0 * (x_46_re * ((x_46_re / x_46_im) / x_46_im))))));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= 2.2e-28:
		tmp = x_46_re * (x_46_im * ((x_46_re * 3.0) - ((x_46_im * x_46_im) / x_46_re)))
	else:
		tmp = x_46_im * (x_46_im * (x_46_im * (-1.0 + (3.0 * (x_46_re * ((x_46_re / x_46_im) / x_46_im))))))
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= 2.2e-28)
		tmp = Float64(x_46_re * Float64(x_46_im * Float64(Float64(x_46_re * 3.0) - Float64(Float64(x_46_im * x_46_im) / x_46_re))));
	else
		tmp = Float64(x_46_im * Float64(x_46_im * Float64(x_46_im * Float64(-1.0 + Float64(3.0 * Float64(x_46_re * Float64(Float64(x_46_re / x_46_im) / x_46_im)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_im <= 2.2e-28)
		tmp = x_46_re * (x_46_im * ((x_46_re * 3.0) - ((x_46_im * x_46_im) / x_46_re)));
	else
		tmp = x_46_im * (x_46_im * (x_46_im * (-1.0 + (3.0 * (x_46_re * ((x_46_re / x_46_im) / x_46_im))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, 2.2e-28], N[(x$46$re * N[(x$46$im * N[(N[(x$46$re * 3.0), $MachinePrecision] - N[(N[(x$46$im * x$46$im), $MachinePrecision] / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(x$46$im * N[(x$46$im * N[(-1.0 + N[(3.0 * N[(x$46$re * N[(N[(x$46$re / x$46$im), $MachinePrecision] / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 2.19999999999999996e-28

   1. Initial program 87.1%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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 94.3%

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

   4. Applied egg-rr 94.3%

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

   5. Taylor expanded in x.re around inf

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

   7. Simplified 74.6%

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

   8. Taylor expanded in x.im around 0

      \[\leadsto \mathsf{*.f64}\left(x.re, \color{blue}{\left(x.im \cdot \left(-1 \cdot \frac{{x.im}^{2}}{x.re} + 3 \cdot x.re\right)\right)}\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.im, \color{blue}{\left(-1 \cdot \frac{{x.im}^{2}}{x.re} + 3 \cdot x.re\right)}\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.im, \left(3 \cdot x.re + \color{blue}{-1 \cdot \frac{{x.im}^{2}}{x.re}}\right)\right)\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x.re, \mathsf{*.f64}\left(x.im, \left(3 \cdot x.re + \left(\mathsf{neg}\left(\frac{{x.im}^{2}}{x.re}\right)\right)\right)\right)\right) \]

      4. unsub-neg N/A

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

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

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

      6. *-commutative N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 96.3%

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

   10. Simplified 96.3%

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

   if 2.19999999999999996e-28 < x.im

   1. Initial program 82.0%

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

      1. +-commutative 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 89.1%

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

   3. Simplified 89.1%

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

   5. Taylor expanded in x.im around inf

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

   7. Simplified 99.9%

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

Alternative 4: 90.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq 6.8 \cdot 10^{+101}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.re\right) \cdot 3 - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.re 6.8e+101)
   (* x.im (- (* (* x.re x.re) 3.0) (* x.im x.im)))
   (* 3.0 (* x.re (* x.re x.im)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= 6.8e+101) {
		tmp = x_46_im * (((x_46_re * x_46_re) * 3.0) - (x_46_im * x_46_im));
	} else {
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re <= 6.8d+101) then
        tmp = x_46im * (((x_46re * x_46re) * 3.0d0) - (x_46im * x_46im))
    else
        tmp = 3.0d0 * (x_46re * (x_46re * x_46im))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= 6.8e+101) {
		tmp = x_46_im * (((x_46_re * x_46_re) * 3.0) - (x_46_im * x_46_im));
	} else {
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_re <= 6.8e+101:
		tmp = x_46_im * (((x_46_re * x_46_re) * 3.0) - (x_46_im * x_46_im))
	else:
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im))
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_re <= 6.8e+101)
		tmp = Float64(x_46_im * Float64(Float64(Float64(x_46_re * x_46_re) * 3.0) - Float64(x_46_im * x_46_im)));
	else
		tmp = Float64(3.0 * Float64(x_46_re * Float64(x_46_re * x_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_re <= 6.8e+101)
		tmp = x_46_im * (((x_46_re * x_46_re) * 3.0) - (x_46_im * x_46_im));
	else
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$re, 6.8e+101], N[(x$46$im * N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] * 3.0), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 * N[(x$46$re * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 6.80000000000000034e101

   1. Initial program 88.9%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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 92.5%

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

   3. Simplified 92.5%

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

      1. associate-*r* N/A

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

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

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

      3. *-lowering-*.f64 92.6%

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

   6. Applied egg-rr 92.6%

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

   if 6.80000000000000034e101 < x.re

   1. Initial program 69.1%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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 69.1%

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

   3. Simplified 69.1%

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

   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. *-lowering-*.f64 N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 82.3%

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

   7. Simplified 82.3%

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

      1. associate-*r* N/A

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

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

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

      3. *-lowering-*.f64 94.6%

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

   9. Applied egg-rr 94.6%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 6.8 \cdot 10^{+101}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.re\right) \cdot 3 - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq 6.8 \cdot 10^{+101}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.re \cdot 3\right) - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.re 6.8e+101)
   (* x.im (- (* x.re (* x.re 3.0)) (* x.im x.im)))
   (* 3.0 (* x.re (* x.re x.im)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= 6.8e+101) {
		tmp = x_46_im * ((x_46_re * (x_46_re * 3.0)) - (x_46_im * x_46_im));
	} else {
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re <= 6.8d+101) then
        tmp = x_46im * ((x_46re * (x_46re * 3.0d0)) - (x_46im * x_46im))
    else
        tmp = 3.0d0 * (x_46re * (x_46re * x_46im))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= 6.8e+101) {
		tmp = x_46_im * ((x_46_re * (x_46_re * 3.0)) - (x_46_im * x_46_im));
	} else {
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_re <= 6.8e+101:
		tmp = x_46_im * ((x_46_re * (x_46_re * 3.0)) - (x_46_im * x_46_im))
	else:
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im))
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_re <= 6.8e+101)
		tmp = Float64(x_46_im * Float64(Float64(x_46_re * Float64(x_46_re * 3.0)) - Float64(x_46_im * x_46_im)));
	else
		tmp = Float64(3.0 * Float64(x_46_re * Float64(x_46_re * x_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_re <= 6.8e+101)
		tmp = x_46_im * ((x_46_re * (x_46_re * 3.0)) - (x_46_im * x_46_im));
	else
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$re, 6.8e+101], N[(x$46$im * N[(N[(x$46$re * N[(x$46$re * 3.0), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 * N[(x$46$re * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 6.80000000000000034e101

   1. Initial program 88.9%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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 92.5%

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

   3. Simplified 92.5%

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

   if 6.80000000000000034e101 < x.re

   1. Initial program 69.1%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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 69.1%

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

   3. Simplified 69.1%

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

   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. *-lowering-*.f64 N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 82.3%

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

   7. Simplified 82.3%

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

      1. associate-*r* N/A

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

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

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

      3. *-lowering-*.f64 94.6%

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

   9. Applied egg-rr 94.6%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 6.8 \cdot 10^{+101}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.re \cdot 3\right) - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq 4.2 \cdot 10^{-23}:\\ \;\;\;\;\left(x.im \cdot x.im\right) \cdot \left(0 - x.im\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.re 4.2e-23)
   (* (* x.im x.im) (- 0.0 x.im))
   (* 3.0 (* x.re (* x.re x.im)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= 4.2e-23) {
		tmp = (x_46_im * x_46_im) * (0.0 - x_46_im);
	} else {
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re <= 4.2d-23) then
        tmp = (x_46im * x_46im) * (0.0d0 - x_46im)
    else
        tmp = 3.0d0 * (x_46re * (x_46re * x_46im))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= 4.2e-23) {
		tmp = (x_46_im * x_46_im) * (0.0 - x_46_im);
	} else {
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_re <= 4.2e-23:
		tmp = (x_46_im * x_46_im) * (0.0 - x_46_im)
	else:
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im))
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_re <= 4.2e-23)
		tmp = Float64(Float64(x_46_im * x_46_im) * Float64(0.0 - x_46_im));
	else
		tmp = Float64(3.0 * Float64(x_46_re * Float64(x_46_re * x_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_re <= 4.2e-23)
		tmp = (x_46_im * x_46_im) * (0.0 - x_46_im);
	else
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$re, 4.2e-23], N[(N[(x$46$im * x$46$im), $MachinePrecision] * N[(0.0 - x$46$im), $MachinePrecision]), $MachinePrecision], N[(3.0 * N[(x$46$re * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 4.2000000000000002e-23

   1. Initial program 90.1%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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 91.6%

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

   3. Simplified 91.6%

      \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re \cdot 3\right) - x.im \cdot x.im\right)} \]
   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 63.5%

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

   7. Simplified 63.5%

      \[\leadsto \color{blue}{0 - x.im \cdot \left(x.im \cdot x.im\right)} \]
   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. neg-lowering-neg.f64 N/A

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

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

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

      4. *-lowering-*.f64 63.5%

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

   9. Applied egg-rr 63.5%

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

   if 4.2000000000000002e-23 < x.re

   1. Initial program 73.8%

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

      1. +-commutative 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 81.5%

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

   3. Simplified 81.5%

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

   5. Taylor expanded in x.im around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 71.9%

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

   7. Simplified 71.9%

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

      1. associate-*r* N/A

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

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

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

      3. *-lowering-*.f64 79.3%

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

   9. Applied egg-rr 79.3%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 4.2 \cdot 10^{-23}:\\ \;\;\;\;\left(x.im \cdot x.im\right) \cdot \left(0 - x.im\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq 2.9 \cdot 10^{-24}:\\ \;\;\;\;\left(x.im \cdot x.im\right) \cdot \left(0 - x.im\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\left(x.re \cdot x.re\right) \cdot x.im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.re 2.9e-24)
   (* (* x.im x.im) (- 0.0 x.im))
   (* 3.0 (* (* x.re x.re) x.im))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= 2.9e-24) {
		tmp = (x_46_im * x_46_im) * (0.0 - x_46_im);
	} else {
		tmp = 3.0 * ((x_46_re * x_46_re) * x_46_im);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re <= 2.9d-24) then
        tmp = (x_46im * x_46im) * (0.0d0 - x_46im)
    else
        tmp = 3.0d0 * ((x_46re * x_46re) * x_46im)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= 2.9e-24) {
		tmp = (x_46_im * x_46_im) * (0.0 - x_46_im);
	} else {
		tmp = 3.0 * ((x_46_re * x_46_re) * x_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_re <= 2.9e-24:
		tmp = (x_46_im * x_46_im) * (0.0 - x_46_im)
	else:
		tmp = 3.0 * ((x_46_re * x_46_re) * x_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_re <= 2.9e-24)
		tmp = Float64(Float64(x_46_im * x_46_im) * Float64(0.0 - x_46_im));
	else
		tmp = Float64(3.0 * Float64(Float64(x_46_re * x_46_re) * x_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_re <= 2.9e-24)
		tmp = (x_46_im * x_46_im) * (0.0 - x_46_im);
	else
		tmp = 3.0 * ((x_46_re * x_46_re) * x_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$re, 2.9e-24], N[(N[(x$46$im * x$46$im), $MachinePrecision] * N[(0.0 - x$46$im), $MachinePrecision]), $MachinePrecision], N[(3.0 * N[(N[(x$46$re * x$46$re), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 2.8999999999999999e-24

   1. Initial program 90.1%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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 91.6%

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

   3. Simplified 91.6%

      \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re \cdot 3\right) - x.im \cdot x.im\right)} \]
   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 63.5%

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

   7. Simplified 63.5%

      \[\leadsto \color{blue}{0 - x.im \cdot \left(x.im \cdot x.im\right)} \]
   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. neg-lowering-neg.f64 N/A

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

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

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

      4. *-lowering-*.f64 63.5%

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

   9. Applied egg-rr 63.5%

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

   if 2.8999999999999999e-24 < x.re

   1. Initial program 73.8%

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

      1. +-commutative 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 81.5%

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

   3. Simplified 81.5%

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

   5. Taylor expanded in x.im around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 71.9%

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

   7. Simplified 71.9%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 2.9 \cdot 10^{-24}:\\ \;\;\;\;\left(x.im \cdot x.im\right) \cdot \left(0 - x.im\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\left(x.re \cdot x.re\right) \cdot x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq 8.4 \cdot 10^{+211}:\\ \;\;\;\;\left(x.im \cdot x.im\right) \cdot \left(0 - x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot x.im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.re 8.4e+211) (* (* x.im x.im) (- 0.0 x.im)) (* x.im (* x.im x.im))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= 8.4e+211) {
		tmp = (x_46_im * x_46_im) * (0.0 - x_46_im);
	} else {
		tmp = x_46_im * (x_46_im * x_46_im);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re <= 8.4d+211) then
        tmp = (x_46im * x_46im) * (0.0d0 - x_46im)
    else
        tmp = x_46im * (x_46im * x_46im)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= 8.4e+211) {
		tmp = (x_46_im * x_46_im) * (0.0 - x_46_im);
	} else {
		tmp = x_46_im * (x_46_im * x_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_re <= 8.4e+211:
		tmp = (x_46_im * x_46_im) * (0.0 - x_46_im)
	else:
		tmp = x_46_im * (x_46_im * x_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_re <= 8.4e+211)
		tmp = Float64(Float64(x_46_im * x_46_im) * Float64(0.0 - x_46_im));
	else
		tmp = Float64(x_46_im * Float64(x_46_im * x_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_re <= 8.4e+211)
		tmp = (x_46_im * x_46_im) * (0.0 - x_46_im);
	else
		tmp = x_46_im * (x_46_im * x_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$re, 8.4e+211], N[(N[(x$46$im * x$46$im), $MachinePrecision] * N[(0.0 - x$46$im), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 8.3999999999999999e211

   1. Initial program 86.6%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. 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 89.9%

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

   3. Simplified 89.9%

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

   5. Taylor expanded in x.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 56.6%

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

   7. Simplified 56.6%

      \[\leadsto \color{blue}{0 - x.im \cdot \left(x.im \cdot x.im\right)} \]
   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. neg-lowering-neg.f64 N/A

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

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

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

      4. *-lowering-*.f64 56.6%

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

   9. Applied egg-rr 56.6%

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

   if 8.3999999999999999e211 < x.re

   1. Initial program 77.8%

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

      1. +-commutative 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 77.8%

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

   3. Simplified 77.8%

      \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re \cdot 3\right) - x.im \cdot x.im\right)} \]
   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 0.7%

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

   7. Simplified 0.7%

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

      1. flip3-- N/A

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

      2. metadata-eval N/A

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

      3. sub0-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left({\left(x.im \cdot \left(x.im \cdot x.im\right)\right)}^{3}\right)}{\color{blue}{0 \cdot 0} + \left(\left(x.im \cdot \left(x.im \cdot x.im\right)\right) \cdot \left(x.im \cdot \left(x.im \cdot x.im\right)\right) + 0 \cdot \left(x.im \cdot \left(x.im \cdot x.im\right)\right)\right)} \]

      4. cube-neg N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left(x.im \cdot \left(x.im \cdot x.im\right)\right)\right)}^{3}}{\color{blue}{0 \cdot 0} + \left(\left(x.im \cdot \left(x.im \cdot x.im\right)\right) \cdot \left(x.im \cdot \left(x.im \cdot x.im\right)\right) + 0 \cdot \left(x.im \cdot \left(x.im \cdot x.im\right)\right)\right)} \]

      5. sub0-neg N/A

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

      6. sqr-pow N/A

         \[\leadsto \frac{{\left(0 - x.im \cdot \left(x.im \cdot x.im\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(0 - x.im \cdot \left(x.im \cdot x.im\right)\right)}^{\left(\frac{3}{2}\right)}}{\color{blue}{0 \cdot 0} + \left(\left(x.im \cdot \left(x.im \cdot x.im\right)\right) \cdot \left(x.im \cdot \left(x.im \cdot x.im\right)\right) + 0 \cdot \left(x.im \cdot \left(x.im \cdot x.im\right)\right)\right)} \]

      7. pow-prod-down N/A

         \[\leadsto \frac{{\left(\left(0 - x.im \cdot \left(x.im \cdot x.im\right)\right) \cdot \left(0 - x.im \cdot \left(x.im \cdot x.im\right)\right)\right)}^{\left(\frac{3}{2}\right)}}{\color{blue}{0 \cdot 0} + \left(\left(x.im \cdot \left(x.im \cdot x.im\right)\right) \cdot \left(x.im \cdot \left(x.im \cdot x.im\right)\right) + 0 \cdot \left(x.im \cdot \left(x.im \cdot x.im\right)\right)\right)} \]

      8. sub0-neg N/A

         \[\leadsto \frac{{\left(\left(\mathsf{neg}\left(x.im \cdot \left(x.im \cdot x.im\right)\right)\right) \cdot \left(0 - x.im \cdot \left(x.im \cdot x.im\right)\right)\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(\left(x.im \cdot \left(x.im \cdot x.im\right)\right) \cdot \left(x.im \cdot \left(x.im \cdot x.im\right)\right) + 0 \cdot \left(x.im \cdot \left(x.im \cdot x.im\right)\right)\right)} \]

      9. sub0-neg N/A

         \[\leadsto \frac{{\left(\left(\mathsf{neg}\left(x.im \cdot \left(x.im \cdot x.im\right)\right)\right) \cdot \left(\mathsf{neg}\left(x.im \cdot \left(x.im \cdot x.im\right)\right)\right)\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(\left(x.im \cdot \left(x.im \cdot x.im\right)\right) \cdot \left(x.im \cdot \left(x.im \cdot x.im\right)\right) + 0 \cdot \left(x.im \cdot \left(x.im \cdot x.im\right)\right)\right)} \]

      10. sqr-neg N/A

          \[\leadsto \frac{{\left(\left(x.im \cdot \left(x.im \cdot x.im\right)\right) \cdot \left(x.im \cdot \left(x.im \cdot x.im\right)\right)\right)}^{\left(\frac{3}{2}\right)}}{\color{blue}{0} \cdot 0 + \left(\left(x.im \cdot \left(x.im \cdot x.im\right)\right) \cdot \left(x.im \cdot \left(x.im \cdot x.im\right)\right) + 0 \cdot \left(x.im \cdot \left(x.im \cdot x.im\right)\right)\right)} \]

      11. pow-prod-down N/A

          \[\leadsto \frac{{\left(x.im \cdot \left(x.im \cdot x.im\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(x.im \cdot \left(x.im \cdot x.im\right)\right)}^{\left(\frac{3}{2}\right)}}{\color{blue}{0 \cdot 0} + \left(\left(x.im \cdot \left(x.im \cdot x.im\right)\right) \cdot \left(x.im \cdot \left(x.im \cdot x.im\right)\right) + 0 \cdot \left(x.im \cdot \left(x.im \cdot x.im\right)\right)\right)} \]

      12. sqr-pow N/A

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

      13. cube-unmult N/A

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

      14. pow-pow N/A

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

      15. metadata-eval N/A

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

   9. Applied egg-rr 29.9%

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

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 8.4 \cdot 10^{+211}:\\ \;\;\;\;\left(x.im \cdot x.im\right) \cdot \left(0 - x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 20.9% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.0%

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

   1. +-commutative 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 89.0%

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

3. Simplified 89.0%

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

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 52.7%

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

7. Simplified 52.7%

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

   1. flip3-- N/A

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

   2. metadata-eval N/A

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

   3. sub0-neg N/A

      \[\leadsto \frac{\mathsf{neg}\left({\left(x.im \cdot \left(x.im \cdot x.im\right)\right)}^{3}\right)}{\color{blue}{0 \cdot 0} + \left(\left(x.im \cdot \left(x.im \cdot x.im\right)\right) \cdot \left(x.im \cdot \left(x.im \cdot x.im\right)\right) + 0 \cdot \left(x.im \cdot \left(x.im \cdot x.im\right)\right)\right)} \]

   4. cube-neg N/A

      \[\leadsto \frac{{\left(\mathsf{neg}\left(x.im \cdot \left(x.im \cdot x.im\right)\right)\right)}^{3}}{\color{blue}{0 \cdot 0} + \left(\left(x.im \cdot \left(x.im \cdot x.im\right)\right) \cdot \left(x.im \cdot \left(x.im \cdot x.im\right)\right) + 0 \cdot \left(x.im \cdot \left(x.im \cdot x.im\right)\right)\right)} \]

   5. sub0-neg N/A

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

   6. sqr-pow N/A

      \[\leadsto \frac{{\left(0 - x.im \cdot \left(x.im \cdot x.im\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(0 - x.im \cdot \left(x.im \cdot x.im\right)\right)}^{\left(\frac{3}{2}\right)}}{\color{blue}{0 \cdot 0} + \left(\left(x.im \cdot \left(x.im \cdot x.im\right)\right) \cdot \left(x.im \cdot \left(x.im \cdot x.im\right)\right) + 0 \cdot \left(x.im \cdot \left(x.im \cdot x.im\right)\right)\right)} \]

   7. pow-prod-down N/A

      \[\leadsto \frac{{\left(\left(0 - x.im \cdot \left(x.im \cdot x.im\right)\right) \cdot \left(0 - x.im \cdot \left(x.im \cdot x.im\right)\right)\right)}^{\left(\frac{3}{2}\right)}}{\color{blue}{0 \cdot 0} + \left(\left(x.im \cdot \left(x.im \cdot x.im\right)\right) \cdot \left(x.im \cdot \left(x.im \cdot x.im\right)\right) + 0 \cdot \left(x.im \cdot \left(x.im \cdot x.im\right)\right)\right)} \]

   8. sub0-neg N/A

      \[\leadsto \frac{{\left(\left(\mathsf{neg}\left(x.im \cdot \left(x.im \cdot x.im\right)\right)\right) \cdot \left(0 - x.im \cdot \left(x.im \cdot x.im\right)\right)\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(\left(x.im \cdot \left(x.im \cdot x.im\right)\right) \cdot \left(x.im \cdot \left(x.im \cdot x.im\right)\right) + 0 \cdot \left(x.im \cdot \left(x.im \cdot x.im\right)\right)\right)} \]

   9. sub0-neg N/A

      \[\leadsto \frac{{\left(\left(\mathsf{neg}\left(x.im \cdot \left(x.im \cdot x.im\right)\right)\right) \cdot \left(\mathsf{neg}\left(x.im \cdot \left(x.im \cdot x.im\right)\right)\right)\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(\left(x.im \cdot \left(x.im \cdot x.im\right)\right) \cdot \left(x.im \cdot \left(x.im \cdot x.im\right)\right) + 0 \cdot \left(x.im \cdot \left(x.im \cdot x.im\right)\right)\right)} \]

   10. sqr-neg N/A

       \[\leadsto \frac{{\left(\left(x.im \cdot \left(x.im \cdot x.im\right)\right) \cdot \left(x.im \cdot \left(x.im \cdot x.im\right)\right)\right)}^{\left(\frac{3}{2}\right)}}{\color{blue}{0} \cdot 0 + \left(\left(x.im \cdot \left(x.im \cdot x.im\right)\right) \cdot \left(x.im \cdot \left(x.im \cdot x.im\right)\right) + 0 \cdot \left(x.im \cdot \left(x.im \cdot x.im\right)\right)\right)} \]

   11. pow-prod-down N/A

       \[\leadsto \frac{{\left(x.im \cdot \left(x.im \cdot x.im\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(x.im \cdot \left(x.im \cdot x.im\right)\right)}^{\left(\frac{3}{2}\right)}}{\color{blue}{0 \cdot 0} + \left(\left(x.im \cdot \left(x.im \cdot x.im\right)\right) \cdot \left(x.im \cdot \left(x.im \cdot x.im\right)\right) + 0 \cdot \left(x.im \cdot \left(x.im \cdot x.im\right)\right)\right)} \]

   12. sqr-pow N/A

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

   13. cube-unmult N/A

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

   14. pow-pow N/A

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

   15. metadata-eval N/A

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

9. Applied egg-rr 20.5%

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

10. Final simplification 20.5%

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

Developer Target 1: 91.2% 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 2024161 
(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)))