Rosa's TurbineBenchmark

Percentage Accurate: 85.0% → 99.4%

Time: 15.0s

Alternatives: 15

Speedup: 1.1×

• 53.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (-
  (-
   (+ 3.0 (/ 2.0 (* r r)))
   (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
  4.5))

C

double code(double v, double w, double r) {
	return ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = ((3.0d0 + (2.0d0 / (r * r))) - (((0.125d0 * (3.0d0 - (2.0d0 * v))) * (((w * w) * r) * r)) / (1.0d0 - v))) - 4.5d0
end function

Java

public static double code(double v, double w, double r) {
	return ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
}

Python

def code(v, w, r):
	return ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5

Julia

function code(v, w, r)
	return Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(Float64(Float64(w * w) * r) * r)) / Float64(1.0 - v))) - 4.5)
end

MATLAB

function tmp = code(v, w, r)
	tmp = ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
end

Wolfram

code[v_, w_, r_] := N[(N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]

Initial Program: 85.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (-
  (-
   (+ 3.0 (/ 2.0 (* r r)))
   (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
  4.5))

C

double code(double v, double w, double r) {
	return ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = ((3.0d0 + (2.0d0 / (r * r))) - (((0.125d0 * (3.0d0 - (2.0d0 * v))) * (((w * w) * r) * r)) / (1.0d0 - v))) - 4.5d0
end function

Java

public static double code(double v, double w, double r) {
	return ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
}

Python

def code(v, w, r):
	return ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5

Julia

function code(v, w, r)
	return Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(Float64(Float64(w * w) * r) * r)) / Float64(1.0 - v))) - 4.5)
end

MATLAB

function tmp = code(v, w, r)
	tmp = ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
end

Wolfram

code[v_, w_, r_] := N[(N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]

Alternative 1: 99.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.375 + v \cdot -0.25\\ t_1 := \frac{2}{r \cdot r}\\ \mathbf{if}\;w \cdot w \leq 6 \cdot 10^{-78}:\\ \;\;\;\;t\_1 + \left(\frac{r \cdot \left(w \cdot \left(w \cdot r\right)\right)}{v + -1} \cdot t\_0 + -1.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \left(w \cdot \left(w \cdot \left(r \cdot r\right)\right)\right) \cdot \frac{t\_0}{v + -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (+ 0.375 (* v -0.25))) (t_1 (/ 2.0 (* r r))))
   (if (<= (* w w) 6e-78)
     (+ t_1 (+ (* (/ (* r (* w (* w r))) (+ v -1.0)) t_0) -1.5))
     (+ t_1 (* (* w (* w (* r r))) (/ t_0 (+ v -1.0)))))))

C

double code(double v, double w, double r) {
	double t_0 = 0.375 + (v * -0.25);
	double t_1 = 2.0 / (r * r);
	double tmp;
	if ((w * w) <= 6e-78) {
		tmp = t_1 + ((((r * (w * (w * r))) / (v + -1.0)) * t_0) + -1.5);
	} else {
		tmp = t_1 + ((w * (w * (r * r))) * (t_0 / (v + -1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.375d0 + (v * (-0.25d0))
    t_1 = 2.0d0 / (r * r)
    if ((w * w) <= 6d-78) then
        tmp = t_1 + ((((r * (w * (w * r))) / (v + (-1.0d0))) * t_0) + (-1.5d0))
    else
        tmp = t_1 + ((w * (w * (r * r))) * (t_0 / (v + (-1.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 0.375 + (v * -0.25);
	double t_1 = 2.0 / (r * r);
	double tmp;
	if ((w * w) <= 6e-78) {
		tmp = t_1 + ((((r * (w * (w * r))) / (v + -1.0)) * t_0) + -1.5);
	} else {
		tmp = t_1 + ((w * (w * (r * r))) * (t_0 / (v + -1.0)));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 0.375 + (v * -0.25)
	t_1 = 2.0 / (r * r)
	tmp = 0
	if (w * w) <= 6e-78:
		tmp = t_1 + ((((r * (w * (w * r))) / (v + -1.0)) * t_0) + -1.5)
	else:
		tmp = t_1 + ((w * (w * (r * r))) * (t_0 / (v + -1.0)))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(0.375 + Float64(v * -0.25))
	t_1 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(w * w) <= 6e-78)
		tmp = Float64(t_1 + Float64(Float64(Float64(Float64(r * Float64(w * Float64(w * r))) / Float64(v + -1.0)) * t_0) + -1.5));
	else
		tmp = Float64(t_1 + Float64(Float64(w * Float64(w * Float64(r * r))) * Float64(t_0 / Float64(v + -1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 0.375 + (v * -0.25);
	t_1 = 2.0 / (r * r);
	tmp = 0.0;
	if ((w * w) <= 6e-78)
		tmp = t_1 + ((((r * (w * (w * r))) / (v + -1.0)) * t_0) + -1.5);
	else
		tmp = t_1 + ((w * (w * (r * r))) * (t_0 / (v + -1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(w * w), $MachinePrecision], 6e-78], N[(t$95$1 + N[(N[(N[(N[(r * N[(w * N[(w * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + -1.5), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(N[(w * N[(w * N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w w) < 5.99999999999999975e-78

   1. Initial program 92.2%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Step-by-step derivation

      1. associate--l- N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)} \]

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{2}{r \cdot r}\right), \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(r \cdot r\right)\right), \left(\color{blue}{3} - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

      7. associate--r+ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \color{blue}{\frac{9}{2}}\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2}\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + 3\right) - \frac{9}{2}\right)\right) \]

      10. associate--l+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(3 - \frac{9}{2}\right)}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \frac{-3}{2}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\frac{-9}{2} + \color{blue}{3}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\left(\mathsf{neg}\left(\frac{9}{2}\right)\right) + 3\right)\right)\right) \]

   3. Simplified 79.0%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\frac{\left(r \cdot r\right) \cdot \left(\left(0.375 + v \cdot -0.25\right) \cdot \left(w \cdot w\right)\right)}{v + -1} + -1.5\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot \left(w \cdot w\right)\right) \cdot \left(r \cdot r\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot \left(\left(w \cdot w\right) \cdot \left(r \cdot r\right)\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right), \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \left(v \cdot \frac{-1}{4}\right)\right), \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \left(r \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(r, \left(\left(w \cdot w\right) \cdot r\right)\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(r, \left(r \cdot \left(w \cdot w\right)\right)\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \left(w \cdot w\right)\right)\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

      11. *-lowering-*.f64 92.3%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, w\right)\right)\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

   6. Applied egg-rr 92.3%

      \[\leadsto \frac{2}{r \cdot r} + \left(\frac{\color{blue}{\left(0.375 + v \cdot -0.25\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}}{v + -1} + -1.5\right) \]
   7. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{v + -1}\right), \frac{-3}{2}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\left(\frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{v + -1} \cdot \left(\frac{3}{8} + v \cdot \frac{-1}{4}\right)\right), \frac{-3}{2}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\left(\frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{v + -1} \cdot \left(3 \cdot \frac{1}{8} + v \cdot \frac{-1}{4}\right)\right), \frac{-3}{2}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\left(\frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{v + -1} \cdot \left(3 \cdot \frac{1}{8} + v \cdot \left(-2 \cdot \frac{1}{8}\right)\right)\right), \frac{-3}{2}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\left(\frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{v + -1} \cdot \left(3 \cdot \frac{1}{8} + \left(v \cdot -2\right) \cdot \frac{1}{8}\right)\right), \frac{-3}{2}\right)\right) \]

      6. distribute-rgt-in N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\left(\frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{v + -1} \cdot \left(\frac{1}{8} \cdot \left(3 + v \cdot -2\right)\right)\right), \frac{-3}{2}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\left(\frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{v + -1} \cdot \left(\frac{1}{8} \cdot \left(3 + -2 \cdot v\right)\right)\right), \frac{-3}{2}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\left(\frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{v + -1} \cdot \left(\frac{1}{8} \cdot \left(3 + \left(\mathsf{neg}\left(2\right)\right) \cdot v\right)\right)\right), \frac{-3}{2}\right)\right) \]

      9. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\left(\frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{v + -1} \cdot \left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right)\right), \frac{-3}{2}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{v + -1}\right), \left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right)\right), \frac{-3}{2}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right), \left(v + -1\right)\right), \left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right)\right), \frac{-3}{2}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \left(r \cdot \left(w \cdot w\right)\right)\right), \left(v + -1\right)\right), \left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right)\right), \frac{-3}{2}\right)\right) \]

      13. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \left(\left(r \cdot w\right) \cdot w\right)\right), \left(v + -1\right)\right), \left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right)\right), \frac{-3}{2}\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \left(w \cdot \left(r \cdot w\right)\right)\right), \left(v + -1\right)\right), \left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right)\right), \frac{-3}{2}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, \left(r \cdot w\right)\right)\right), \left(v + -1\right)\right), \left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right)\right), \frac{-3}{2}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, w\right)\right)\right), \left(v + -1\right)\right), \left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right)\right), \frac{-3}{2}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, w\right)\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right)\right), \frac{-3}{2}\right)\right) \]

      18. cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, w\right)\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \left(\frac{1}{8} \cdot \left(3 + \left(\mathsf{neg}\left(2\right)\right) \cdot v\right)\right)\right), \frac{-3}{2}\right)\right) \]

      19. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, w\right)\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \left(\frac{1}{8} \cdot \left(3 + -2 \cdot v\right)\right)\right), \frac{-3}{2}\right)\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, w\right)\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \left(\frac{1}{8} \cdot \left(3 + v \cdot -2\right)\right)\right), \frac{-3}{2}\right)\right) \]

   8. Applied egg-rr 99.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{\frac{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}{v + -1} \cdot \left(0.375 + v \cdot -0.25\right)} + -1.5\right) \]

   if 5.99999999999999975e-78 < (*.f64 w w)

   1. Initial program 78.5%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Step-by-step derivation

      1. associate--l- N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)} \]

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{2}{r \cdot r}\right), \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(r \cdot r\right)\right), \left(\color{blue}{3} - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

      7. associate--r+ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \color{blue}{\frac{9}{2}}\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2}\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + 3\right) - \frac{9}{2}\right)\right) \]

      10. associate--l+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(3 - \frac{9}{2}\right)}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \frac{-3}{2}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\frac{-9}{2} + \color{blue}{3}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\left(\mathsf{neg}\left(\frac{9}{2}\right)\right) + 3\right)\right)\right) \]

   3. Simplified 73.9%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\frac{\left(r \cdot r\right) \cdot \left(\left(0.375 + v \cdot -0.25\right) \cdot \left(w \cdot w\right)\right)}{v + -1} + -1.5\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot \left(w \cdot w\right)\right) \cdot \left(r \cdot r\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot \left(\left(w \cdot w\right) \cdot \left(r \cdot r\right)\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right), \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \left(v \cdot \frac{-1}{4}\right)\right), \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \left(r \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(r, \left(\left(w \cdot w\right) \cdot r\right)\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(r, \left(r \cdot \left(w \cdot w\right)\right)\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \left(w \cdot w\right)\right)\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

      11. *-lowering-*.f64 78.5%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, w\right)\right)\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

   6. Applied egg-rr 78.5%

      \[\leadsto \frac{2}{r \cdot r} + \left(\frac{\color{blue}{\left(0.375 + v \cdot -0.25\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}}{v + -1} + -1.5\right) \]

   7. Taylor expanded in r around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \color{blue}{\left(\frac{{r}^{2} \cdot \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)}{v - 1}\right)}\right) \]
   8. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\left({r}^{2} \cdot \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \color{blue}{\left(v - 1\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({r}^{2}\right), \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(\color{blue}{v} - 1\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(r \cdot r\right), \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left({w}^{2}\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left(w \cdot w\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{+.f64}\left(\frac{3}{8}, \left(\frac{-1}{4} \cdot v\right)\right)\right)\right), \left(v - 1\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{+.f64}\left(\frac{3}{8}, \left(v \cdot \frac{-1}{4}\right)\right)\right)\right), \left(v - 1\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right)\right)\right), \left(v - 1\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right)\right)\right), \left(v + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right)\right)\right), \left(v + -1\right)\right)\right) \]

      13. +-lowering-+.f64 73.9%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right)\right)\right), \mathsf{+.f64}\left(v, \color{blue}{-1}\right)\right)\right) \]

   9. Simplified 73.9%

      \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\frac{\left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot \left(0.375 + v \cdot -0.25\right)\right)}{v + -1}} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\frac{\left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) \cdot \left(\frac{3}{8} + v \cdot \frac{-1}{4}\right)}{\color{blue}{v} + -1}\right)\right) \]

       2. associate-/l* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) \cdot \color{blue}{\frac{\frac{3}{8} + v \cdot \frac{-1}{4}}{v + -1}}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{*.f64}\left(\left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right), \color{blue}{\left(\frac{\frac{3}{8} + v \cdot \frac{-1}{4}}{v + -1}\right)}\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{*.f64}\left(\left(\left(\left(r \cdot r\right) \cdot w\right) \cdot w\right), \left(\frac{\color{blue}{\frac{3}{8} + v \cdot \frac{-1}{4}}}{v + -1}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(r \cdot r\right) \cdot w\right), w\right), \left(\frac{\color{blue}{\frac{3}{8} + v \cdot \frac{-1}{4}}}{v + -1}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(r \cdot r\right), w\right), w\right), \left(\frac{\color{blue}{\frac{3}{8}} + v \cdot \frac{-1}{4}}{v + -1}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), w\right), w\right), \left(\frac{\frac{3}{8} + v \cdot \frac{-1}{4}}{v + -1}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), w\right), w\right), \mathsf{/.f64}\left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right), \color{blue}{\left(v + -1\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), w\right), w\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \left(v \cdot \frac{-1}{4}\right)\right), \left(\color{blue}{v} + -1\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), w\right), w\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \left(v + -1\right)\right)\right)\right) \]

       11. +-lowering-+.f64 99.9%

           \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), w\right), w\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{+.f64}\left(v, \color{blue}{-1}\right)\right)\right)\right) \]

   11. Applied egg-rr 99.9%

       \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(\left(\left(r \cdot r\right) \cdot w\right) \cdot w\right) \cdot \frac{0.375 + v \cdot -0.25}{v + -1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \cdot w \leq 6 \cdot 10^{-78}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(\frac{r \cdot \left(w \cdot \left(w \cdot r\right)\right)}{v + -1} \cdot \left(0.375 + v \cdot -0.25\right) + -1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(w \cdot \left(w \cdot \left(r \cdot r\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{v + -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 67.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 2 \cdot 10^{-151}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{elif}\;r \leq 0.205:\\ \;\;\;\;\frac{2}{r \cdot r} - \left(r \cdot \left(\left(w \cdot w\right) \cdot r\right)\right) \cdot 0.25\\ \mathbf{elif}\;r \leq 2.4 \cdot 10^{+148}:\\ \;\;\;\;r \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot -0.375 + \frac{-1.5}{r \cdot r}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(r \cdot \left(w \cdot \left(w \cdot r\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{v + -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 2e-151)
   (/ (/ 2.0 r) r)
   (if (<= r 0.205)
     (- (/ 2.0 (* r r)) (* (* r (* (* w w) r)) 0.25))
     (if (<= r 2.4e+148)
       (* r (* r (+ (* (* w w) -0.375) (/ -1.5 (* r r)))))
       (* (* r (* w (* w r))) (/ (+ 0.375 (* v -0.25)) (+ v -1.0)))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 2e-151) {
		tmp = (2.0 / r) / r;
	} else if (r <= 0.205) {
		tmp = (2.0 / (r * r)) - ((r * ((w * w) * r)) * 0.25);
	} else if (r <= 2.4e+148) {
		tmp = r * (r * (((w * w) * -0.375) + (-1.5 / (r * r))));
	} else {
		tmp = (r * (w * (w * r))) * ((0.375 + (v * -0.25)) / (v + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 2d-151) then
        tmp = (2.0d0 / r) / r
    else if (r <= 0.205d0) then
        tmp = (2.0d0 / (r * r)) - ((r * ((w * w) * r)) * 0.25d0)
    else if (r <= 2.4d+148) then
        tmp = r * (r * (((w * w) * (-0.375d0)) + ((-1.5d0) / (r * r))))
    else
        tmp = (r * (w * (w * r))) * ((0.375d0 + (v * (-0.25d0))) / (v + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 2e-151) {
		tmp = (2.0 / r) / r;
	} else if (r <= 0.205) {
		tmp = (2.0 / (r * r)) - ((r * ((w * w) * r)) * 0.25);
	} else if (r <= 2.4e+148) {
		tmp = r * (r * (((w * w) * -0.375) + (-1.5 / (r * r))));
	} else {
		tmp = (r * (w * (w * r))) * ((0.375 + (v * -0.25)) / (v + -1.0));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 2e-151:
		tmp = (2.0 / r) / r
	elif r <= 0.205:
		tmp = (2.0 / (r * r)) - ((r * ((w * w) * r)) * 0.25)
	elif r <= 2.4e+148:
		tmp = r * (r * (((w * w) * -0.375) + (-1.5 / (r * r))))
	else:
		tmp = (r * (w * (w * r))) * ((0.375 + (v * -0.25)) / (v + -1.0))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 2e-151)
		tmp = Float64(Float64(2.0 / r) / r);
	elseif (r <= 0.205)
		tmp = Float64(Float64(2.0 / Float64(r * r)) - Float64(Float64(r * Float64(Float64(w * w) * r)) * 0.25));
	elseif (r <= 2.4e+148)
		tmp = Float64(r * Float64(r * Float64(Float64(Float64(w * w) * -0.375) + Float64(-1.5 / Float64(r * r)))));
	else
		tmp = Float64(Float64(r * Float64(w * Float64(w * r))) * Float64(Float64(0.375 + Float64(v * -0.25)) / Float64(v + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 2e-151)
		tmp = (2.0 / r) / r;
	elseif (r <= 0.205)
		tmp = (2.0 / (r * r)) - ((r * ((w * w) * r)) * 0.25);
	elseif (r <= 2.4e+148)
		tmp = r * (r * (((w * w) * -0.375) + (-1.5 / (r * r))));
	else
		tmp = (r * (w * (w * r))) * ((0.375 + (v * -0.25)) / (v + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 2e-151], N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision], If[LessEqual[r, 0.205], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] - N[(N[(r * N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[r, 2.4e+148], N[(r * N[(r * N[(N[(N[(w * w), $MachinePrecision] * -0.375), $MachinePrecision] + N[(-1.5 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(r * N[(w * N[(w * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if r < 1.9999999999999999e-151

   1. Initial program 84.8%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Step-by-step derivation

      1. associate--l- N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)} \]

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{2}{r \cdot r}\right), \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(r \cdot r\right)\right), \left(\color{blue}{3} - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

      7. associate--r+ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \color{blue}{\frac{9}{2}}\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2}\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + 3\right) - \frac{9}{2}\right)\right) \]

      10. associate--l+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(3 - \frac{9}{2}\right)}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \frac{-3}{2}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\frac{-9}{2} + \color{blue}{3}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\left(\mathsf{neg}\left(\frac{9}{2}\right)\right) + 3\right)\right)\right) \]

   3. Simplified 77.7%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\frac{\left(r \cdot r\right) \cdot \left(\left(0.375 + v \cdot -0.25\right) \cdot \left(w \cdot w\right)\right)}{v + -1} + -1.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left({r}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(r \cdot \color{blue}{r}\right)\right) \]

      3. *-lowering-*.f64 59.7%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, \color{blue}{r}\right)\right) \]

   7. Simplified 59.7%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r}} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{r}}{\color{blue}{r}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{r}\right), \color{blue}{r}\right) \]

      3. /-lowering-/.f64 59.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, r\right), r\right) \]

   9. Applied egg-rr 59.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{r}}{r}} \]

   if 1.9999999999999999e-151 < r < 0.204999999999999988

   1. Initial program 85.6%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Step-by-step derivation

      1. associate--l- N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)} \]

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{2}{r \cdot r}\right), \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(r \cdot r\right)\right), \left(\color{blue}{3} - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

      7. associate--r+ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \color{blue}{\frac{9}{2}}\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2}\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + 3\right) - \frac{9}{2}\right)\right) \]

      10. associate--l+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(3 - \frac{9}{2}\right)}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \frac{-3}{2}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\frac{-9}{2} + \color{blue}{3}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\left(\mathsf{neg}\left(\frac{9}{2}\right)\right) + 3\right)\right)\right) \]

   3. Simplified 76.7%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\frac{\left(r \cdot r\right) \cdot \left(\left(0.375 + v \cdot -0.25\right) \cdot \left(w \cdot w\right)\right)}{v + -1} + -1.5\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot \left(w \cdot w\right)\right) \cdot \left(r \cdot r\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot \left(\left(w \cdot w\right) \cdot \left(r \cdot r\right)\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right), \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \left(v \cdot \frac{-1}{4}\right)\right), \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \left(r \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(r, \left(\left(w \cdot w\right) \cdot r\right)\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(r, \left(r \cdot \left(w \cdot w\right)\right)\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \left(w \cdot w\right)\right)\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

      11. *-lowering-*.f64 85.6%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, w\right)\right)\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

   6. Applied egg-rr 85.6%

      \[\leadsto \frac{2}{r \cdot r} + \left(\frac{\color{blue}{\left(0.375 + v \cdot -0.25\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}}{v + -1} + -1.5\right) \]

   7. Taylor expanded in r around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \color{blue}{\left(\frac{{r}^{2} \cdot \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)}{v - 1}\right)}\right) \]
   8. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\left({r}^{2} \cdot \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \color{blue}{\left(v - 1\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({r}^{2}\right), \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(\color{blue}{v} - 1\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(r \cdot r\right), \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left({w}^{2}\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left(w \cdot w\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{+.f64}\left(\frac{3}{8}, \left(\frac{-1}{4} \cdot v\right)\right)\right)\right), \left(v - 1\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{+.f64}\left(\frac{3}{8}, \left(v \cdot \frac{-1}{4}\right)\right)\right)\right), \left(v - 1\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right)\right)\right), \left(v - 1\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right)\right)\right), \left(v + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right)\right)\right), \left(v + -1\right)\right)\right) \]

      13. +-lowering-+.f64 76.6%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right)\right)\right), \mathsf{+.f64}\left(v, \color{blue}{-1}\right)\right)\right) \]

   9. Simplified 76.6%

      \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\frac{\left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot \left(0.375 + v \cdot -0.25\right)\right)}{v + -1}} \]
   10. Step-by-step derivation

       1. frac-2neg N/A

          \[\leadsto \frac{2}{r \cdot r} + \frac{\mathsf{neg}\left(\left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot \left(\frac{3}{8} + v \cdot \frac{-1}{4}\right)\right)\right)}{\color{blue}{\mathsf{neg}\left(\left(v + -1\right)\right)}} \]

       2. distribute-frac-neg N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{neg}\left(\frac{\left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot \left(\frac{3}{8} + v \cdot \frac{-1}{4}\right)\right)}{\mathsf{neg}\left(\left(v + -1\right)\right)}\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{neg}\left(\frac{\left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) \cdot \left(\frac{3}{8} + v \cdot \frac{-1}{4}\right)}{\mathsf{neg}\left(\left(v + -1\right)\right)}\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{neg}\left(\frac{\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot \left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right)}{\mathsf{neg}\left(\left(v + -1\right)\right)}\right)\right) \]

       5. unswap-sqr N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{neg}\left(\frac{\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}{\mathsf{neg}\left(\left(v + -1\right)\right)}\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{neg}\left(\frac{\left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)}{\mathsf{neg}\left(\left(v + -1\right)\right)}\right)\right) \]

       7. +-commutative N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{neg}\left(\frac{\left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)}{\mathsf{neg}\left(\left(-1 + v\right)\right)}\right)\right) \]

       8. distribute-neg-in N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{neg}\left(\frac{\left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)}{\left(\mathsf{neg}\left(-1\right)\right) + \left(\mathsf{neg}\left(v\right)\right)}\right)\right) \]

       9. metadata-eval N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{neg}\left(\frac{\left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)}{1 + \left(\mathsf{neg}\left(v\right)\right)}\right)\right) \]

       10. sub-neg N/A

           \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{neg}\left(\frac{\left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)}{1 - v}\right)\right) \]

       11. associate-*r/ N/A

           \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{neg}\left(\left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot \left(r \cdot w\right)\right) \cdot \frac{r \cdot w}{1 - v}\right)\right) \]

   11. Applied egg-rr 96.4%

       \[\leadsto \color{blue}{\frac{2}{r \cdot r} - \frac{\left(0.375 + v \cdot -0.25\right) \cdot \left(r \cdot w\right)}{\frac{1 - v}{r \cdot w}}} \]

   12. Taylor expanded in v around inf

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \color{blue}{\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)}\right) \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left({r}^{2} \cdot {w}^{2}\right) \cdot \color{blue}{\frac{1}{4}}\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{*.f64}\left(\left({r}^{2} \cdot {w}^{2}\right), \color{blue}{\frac{1}{4}}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{*.f64}\left(\left(\left(r \cdot r\right) \cdot {w}^{2}\right), \frac{1}{4}\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{*.f64}\left(\left(r \cdot \left(r \cdot {w}^{2}\right)\right), \frac{1}{4}\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \left(r \cdot {w}^{2}\right)\right), \frac{1}{4}\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \left({w}^{2}\right)\right)\right), \frac{1}{4}\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \left(w \cdot w\right)\right)\right), \frac{1}{4}\right)\right) \]

       8. *-lowering-*.f64 85.5%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, w\right)\right)\right), \frac{1}{4}\right)\right) \]

   14. Simplified 85.5%

       \[\leadsto \frac{2}{r \cdot r} - \color{blue}{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot 0.25} \]

   if 0.204999999999999988 < r < 2.39999999999999995e148

   1. Initial program 85.7%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Step-by-step derivation

      1. associate--l- N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)} \]

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{2}{r \cdot r}\right), \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(r \cdot r\right)\right), \left(\color{blue}{3} - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

      7. associate--r+ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \color{blue}{\frac{9}{2}}\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2}\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + 3\right) - \frac{9}{2}\right)\right) \]

      10. associate--l+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(3 - \frac{9}{2}\right)}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \frac{-3}{2}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\frac{-9}{2} + \color{blue}{3}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\left(\mathsf{neg}\left(\frac{9}{2}\right)\right) + 3\right)\right)\right) \]

   3. Simplified 85.7%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\frac{\left(r \cdot r\right) \cdot \left(\left(0.375 + v \cdot -0.25\right) \cdot \left(w \cdot w\right)\right)}{v + -1} + -1.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\color{blue}{\left(\frac{-3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)}, \frac{-3}{2}\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\left(\left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-3}{8}\right), \frac{-3}{2}\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\left({r}^{2} \cdot \left({w}^{2} \cdot \frac{-3}{8}\right)\right), \frac{-3}{2}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\left({r}^{2} \cdot \left(\frac{-3}{8} \cdot {w}^{2}\right)\right), \frac{-3}{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({r}^{2}\right), \left(\frac{-3}{8} \cdot {w}^{2}\right)\right), \frac{-3}{2}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(r \cdot r\right), \left(\frac{-3}{8} \cdot {w}^{2}\right)\right), \frac{-3}{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left(\frac{-3}{8} \cdot {w}^{2}\right)\right), \frac{-3}{2}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left({w}^{2} \cdot \frac{-3}{8}\right)\right), \frac{-3}{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left({w}^{2}\right), \frac{-3}{8}\right)\right), \frac{-3}{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left(w \cdot w\right), \frac{-3}{8}\right)\right), \frac{-3}{2}\right)\right) \]

      10. *-lowering-*.f64 78.9%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \frac{-3}{2}\right)\right) \]

   7. Simplified 78.9%

      \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{\left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot -0.375\right)} + -1.5\right) \]

   8. Taylor expanded in r around inf

      \[\leadsto \color{blue}{{r}^{2} \cdot \left(\frac{-3}{8} \cdot {w}^{2} - \frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)} \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \left(r \cdot r\right) \cdot \left(\color{blue}{\frac{-3}{8} \cdot {w}^{2}} - \frac{3}{2} \cdot \frac{1}{{r}^{2}}\right) \]

      2. associate-*l* N/A

         \[\leadsto r \cdot \color{blue}{\left(r \cdot \left(\frac{-3}{8} \cdot {w}^{2} - \frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(r, \color{blue}{\left(r \cdot \left(\frac{-3}{8} \cdot {w}^{2} - \frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \color{blue}{\left(\frac{-3}{8} \cdot {w}^{2} - \frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)}\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \left(\frac{-3}{8} \cdot {w}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\left(\frac{-3}{8} \cdot {w}^{2}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\left({w}^{2} \cdot \frac{-3}{8}\right), \left(\mathsf{neg}\left(\color{blue}{\frac{3}{2} \cdot \frac{1}{{r}^{2}}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({w}^{2}\right), \frac{-3}{8}\right), \left(\mathsf{neg}\left(\color{blue}{\frac{3}{2} \cdot \frac{1}{{r}^{2}}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(w \cdot w\right), \frac{-3}{8}\right), \left(\mathsf{neg}\left(\color{blue}{\frac{3}{2}} \cdot \frac{1}{{r}^{2}}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right), \left(\mathsf{neg}\left(\color{blue}{\frac{3}{2}} \cdot \frac{1}{{r}^{2}}\right)\right)\right)\right)\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right), \left(\mathsf{neg}\left(\frac{\frac{3}{2} \cdot 1}{{r}^{2}}\right)\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right), \left(\mathsf{neg}\left(\frac{\frac{3}{2}}{{r}^{2}}\right)\right)\right)\right)\right) \]

      13. distribute-neg-frac N/A

          \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right), \left(\frac{\mathsf{neg}\left(\frac{3}{2}\right)}{\color{blue}{{r}^{2}}}\right)\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right), \left(\frac{\frac{-3}{2}}{{\color{blue}{r}}^{2}}\right)\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right), \mathsf{/.f64}\left(\frac{-3}{2}, \color{blue}{\left({r}^{2}\right)}\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right), \mathsf{/.f64}\left(\frac{-3}{2}, \left(r \cdot \color{blue}{r}\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 78.7%

          \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right), \mathsf{/.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(r, \color{blue}{r}\right)\right)\right)\right)\right) \]

   10. Simplified 78.7%

       \[\leadsto \color{blue}{r \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot -0.375 + \frac{-1.5}{r \cdot r}\right)\right)} \]

   if 2.39999999999999995e148 < r

   1. Initial program 87.3%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Step-by-step derivation

      1. associate--l- N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)} \]

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{2}{r \cdot r}\right), \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(r \cdot r\right)\right), \left(\color{blue}{3} - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

      7. associate--r+ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \color{blue}{\frac{9}{2}}\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2}\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + 3\right) - \frac{9}{2}\right)\right) \]

      10. associate--l+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(3 - \frac{9}{2}\right)}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \frac{-3}{2}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\frac{-9}{2} + \color{blue}{3}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\left(\mathsf{neg}\left(\frac{9}{2}\right)\right) + 3\right)\right)\right) \]

   3. Simplified 53.2%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\frac{\left(r \cdot r\right) \cdot \left(\left(0.375 + v \cdot -0.25\right) \cdot \left(w \cdot w\right)\right)}{v + -1} + -1.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in r around inf

      \[\leadsto \color{blue}{\frac{{r}^{2} \cdot \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)}{v - 1}} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left({r}^{2} \cdot \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \color{blue}{\left(v - 1\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({r}^{2}\right), \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(\color{blue}{v} - 1\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(r \cdot r\right), \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left(\left(w \cdot w\right) \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left(w \cdot \left(w \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right)\right), \left(v - 1\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \left(w \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right)\right), \left(v - 1\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right)\right), \left(v - 1\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{3}{8}, \left(\frac{-1}{4} \cdot v\right)\right)\right)\right)\right), \left(v - 1\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(\frac{-1}{4}, v\right)\right)\right)\right)\right), \left(v - 1\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(\frac{-1}{4}, v\right)\right)\right)\right)\right), \left(v + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(\frac{-1}{4}, v\right)\right)\right)\right)\right), \left(v + -1\right)\right) \]

      13. +-lowering-+.f64 53.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(\frac{-1}{4}, v\right)\right)\right)\right)\right), \mathsf{+.f64}\left(v, \color{blue}{-1}\right)\right) \]

   7. Simplified 53.6%

      \[\leadsto \color{blue}{\frac{\left(r \cdot r\right) \cdot \left(w \cdot \left(w \cdot \left(0.375 + -0.25 \cdot v\right)\right)\right)}{v + -1}} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)}{v + -1} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot \left(\frac{3}{8} + v \cdot \frac{-1}{4}\right)\right)}{v + -1} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) \cdot \left(\frac{3}{8} + v \cdot \frac{-1}{4}\right)}{\color{blue}{v} + -1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(\frac{3}{8} + v \cdot \frac{-1}{4}\right)}{v + -1} \]

      5. associate-/l* N/A

         \[\leadsto \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \color{blue}{\frac{\frac{3}{8} + v \cdot \frac{-1}{4}}{v + -1}} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right), \color{blue}{\left(\frac{\frac{3}{8} + v \cdot \frac{-1}{4}}{v + -1}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \left(r \cdot \left(w \cdot w\right)\right)\right), \left(\frac{\color{blue}{\frac{3}{8} + v \cdot \frac{-1}{4}}}{v + -1}\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \left(\left(r \cdot w\right) \cdot w\right)\right), \left(\frac{\frac{3}{8} + \color{blue}{v \cdot \frac{-1}{4}}}{v + -1}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \left(w \cdot \left(r \cdot w\right)\right)\right), \left(\frac{\frac{3}{8} + \color{blue}{v \cdot \frac{-1}{4}}}{v + -1}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, \left(r \cdot w\right)\right)\right), \left(\frac{\frac{3}{8} + \color{blue}{v \cdot \frac{-1}{4}}}{v + -1}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, w\right)\right)\right), \left(\frac{\frac{3}{8} + v \cdot \color{blue}{\frac{-1}{4}}}{v + -1}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, w\right)\right)\right), \left(\frac{3 \cdot \frac{1}{8} + v \cdot \frac{-1}{4}}{v + -1}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, w\right)\right)\right), \left(\frac{3 \cdot \frac{1}{8} + v \cdot \left(-2 \cdot \frac{1}{8}\right)}{v + -1}\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, w\right)\right)\right), \left(\frac{3 \cdot \frac{1}{8} + \left(v \cdot -2\right) \cdot \frac{1}{8}}{v + -1}\right)\right) \]

      15. distribute-rgt-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, w\right)\right)\right), \left(\frac{\frac{1}{8} \cdot \left(3 + v \cdot -2\right)}{\color{blue}{v} + -1}\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, w\right)\right)\right), \left(\frac{\frac{1}{8} \cdot \left(3 + -2 \cdot v\right)}{v + -1}\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, w\right)\right)\right), \left(\frac{\frac{1}{8} \cdot \left(3 + \left(\mathsf{neg}\left(2\right)\right) \cdot v\right)}{v + -1}\right)\right) \]

      18. cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, w\right)\right)\right), \left(\frac{\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)}{v + -1}\right)\right) \]

   9. Applied egg-rr 78.9%

      \[\leadsto \color{blue}{\left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{v + -1}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 2 \cdot 10^{-151}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{elif}\;r \leq 0.205:\\ \;\;\;\;\frac{2}{r \cdot r} - \left(r \cdot \left(\left(w \cdot w\right) \cdot r\right)\right) \cdot 0.25\\ \mathbf{elif}\;r \leq 2.4 \cdot 10^{+148}:\\ \;\;\;\;r \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot -0.375 + \frac{-1.5}{r \cdot r}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(r \cdot \left(w \cdot \left(w \cdot r\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{v + -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 64.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 2 \cdot 10^{-151}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{elif}\;r \leq 0.22:\\ \;\;\;\;\frac{2}{r \cdot r} - \left(r \cdot \left(\left(w \cdot w\right) \cdot r\right)\right) \cdot 0.25\\ \mathbf{elif}\;r \leq 1.35 \cdot 10^{+234}:\\ \;\;\;\;r \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot -0.375 + \frac{-1.5}{r \cdot r}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(r \cdot 0.375\right) \cdot \frac{w \cdot \left(w \cdot r\right)}{v + -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 2e-151)
   (/ (/ 2.0 r) r)
   (if (<= r 0.22)
     (- (/ 2.0 (* r r)) (* (* r (* (* w w) r)) 0.25))
     (if (<= r 1.35e+234)
       (* r (* r (+ (* (* w w) -0.375) (/ -1.5 (* r r)))))
       (* (* r 0.375) (/ (* w (* w r)) (+ v -1.0)))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 2e-151) {
		tmp = (2.0 / r) / r;
	} else if (r <= 0.22) {
		tmp = (2.0 / (r * r)) - ((r * ((w * w) * r)) * 0.25);
	} else if (r <= 1.35e+234) {
		tmp = r * (r * (((w * w) * -0.375) + (-1.5 / (r * r))));
	} else {
		tmp = (r * 0.375) * ((w * (w * r)) / (v + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 2d-151) then
        tmp = (2.0d0 / r) / r
    else if (r <= 0.22d0) then
        tmp = (2.0d0 / (r * r)) - ((r * ((w * w) * r)) * 0.25d0)
    else if (r <= 1.35d+234) then
        tmp = r * (r * (((w * w) * (-0.375d0)) + ((-1.5d0) / (r * r))))
    else
        tmp = (r * 0.375d0) * ((w * (w * r)) / (v + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 2e-151) {
		tmp = (2.0 / r) / r;
	} else if (r <= 0.22) {
		tmp = (2.0 / (r * r)) - ((r * ((w * w) * r)) * 0.25);
	} else if (r <= 1.35e+234) {
		tmp = r * (r * (((w * w) * -0.375) + (-1.5 / (r * r))));
	} else {
		tmp = (r * 0.375) * ((w * (w * r)) / (v + -1.0));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 2e-151:
		tmp = (2.0 / r) / r
	elif r <= 0.22:
		tmp = (2.0 / (r * r)) - ((r * ((w * w) * r)) * 0.25)
	elif r <= 1.35e+234:
		tmp = r * (r * (((w * w) * -0.375) + (-1.5 / (r * r))))
	else:
		tmp = (r * 0.375) * ((w * (w * r)) / (v + -1.0))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 2e-151)
		tmp = Float64(Float64(2.0 / r) / r);
	elseif (r <= 0.22)
		tmp = Float64(Float64(2.0 / Float64(r * r)) - Float64(Float64(r * Float64(Float64(w * w) * r)) * 0.25));
	elseif (r <= 1.35e+234)
		tmp = Float64(r * Float64(r * Float64(Float64(Float64(w * w) * -0.375) + Float64(-1.5 / Float64(r * r)))));
	else
		tmp = Float64(Float64(r * 0.375) * Float64(Float64(w * Float64(w * r)) / Float64(v + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 2e-151)
		tmp = (2.0 / r) / r;
	elseif (r <= 0.22)
		tmp = (2.0 / (r * r)) - ((r * ((w * w) * r)) * 0.25);
	elseif (r <= 1.35e+234)
		tmp = r * (r * (((w * w) * -0.375) + (-1.5 / (r * r))));
	else
		tmp = (r * 0.375) * ((w * (w * r)) / (v + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 2e-151], N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision], If[LessEqual[r, 0.22], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] - N[(N[(r * N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[r, 1.35e+234], N[(r * N[(r * N[(N[(N[(w * w), $MachinePrecision] * -0.375), $MachinePrecision] + N[(-1.5 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(r * 0.375), $MachinePrecision] * N[(N[(w * N[(w * r), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if r < 1.9999999999999999e-151

   1. Initial program 84.8%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Step-by-step derivation

      1. associate--l- N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)} \]

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{2}{r \cdot r}\right), \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(r \cdot r\right)\right), \left(\color{blue}{3} - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

      7. associate--r+ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \color{blue}{\frac{9}{2}}\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2}\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + 3\right) - \frac{9}{2}\right)\right) \]

      10. associate--l+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(3 - \frac{9}{2}\right)}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \frac{-3}{2}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\frac{-9}{2} + \color{blue}{3}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\left(\mathsf{neg}\left(\frac{9}{2}\right)\right) + 3\right)\right)\right) \]

   3. Simplified 77.7%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\frac{\left(r \cdot r\right) \cdot \left(\left(0.375 + v \cdot -0.25\right) \cdot \left(w \cdot w\right)\right)}{v + -1} + -1.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left({r}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(r \cdot \color{blue}{r}\right)\right) \]

      3. *-lowering-*.f64 59.7%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, \color{blue}{r}\right)\right) \]

   7. Simplified 59.7%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r}} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{r}}{\color{blue}{r}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{r}\right), \color{blue}{r}\right) \]

      3. /-lowering-/.f64 59.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, r\right), r\right) \]

   9. Applied egg-rr 59.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{r}}{r}} \]

   if 1.9999999999999999e-151 < r < 0.220000000000000001

   1. Initial program 85.6%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Step-by-step derivation

      1. associate--l- N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)} \]

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{2}{r \cdot r}\right), \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(r \cdot r\right)\right), \left(\color{blue}{3} - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

      7. associate--r+ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \color{blue}{\frac{9}{2}}\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2}\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + 3\right) - \frac{9}{2}\right)\right) \]

      10. associate--l+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(3 - \frac{9}{2}\right)}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \frac{-3}{2}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\frac{-9}{2} + \color{blue}{3}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\left(\mathsf{neg}\left(\frac{9}{2}\right)\right) + 3\right)\right)\right) \]

   3. Simplified 76.7%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\frac{\left(r \cdot r\right) \cdot \left(\left(0.375 + v \cdot -0.25\right) \cdot \left(w \cdot w\right)\right)}{v + -1} + -1.5\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot \left(w \cdot w\right)\right) \cdot \left(r \cdot r\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot \left(\left(w \cdot w\right) \cdot \left(r \cdot r\right)\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right), \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \left(v \cdot \frac{-1}{4}\right)\right), \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \left(r \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(r, \left(\left(w \cdot w\right) \cdot r\right)\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(r, \left(r \cdot \left(w \cdot w\right)\right)\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \left(w \cdot w\right)\right)\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

      11. *-lowering-*.f64 85.6%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, w\right)\right)\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

   6. Applied egg-rr 85.6%

      \[\leadsto \frac{2}{r \cdot r} + \left(\frac{\color{blue}{\left(0.375 + v \cdot -0.25\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}}{v + -1} + -1.5\right) \]

   7. Taylor expanded in r around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \color{blue}{\left(\frac{{r}^{2} \cdot \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)}{v - 1}\right)}\right) \]
   8. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\left({r}^{2} \cdot \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \color{blue}{\left(v - 1\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({r}^{2}\right), \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(\color{blue}{v} - 1\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(r \cdot r\right), \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left({w}^{2}\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left(w \cdot w\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{+.f64}\left(\frac{3}{8}, \left(\frac{-1}{4} \cdot v\right)\right)\right)\right), \left(v - 1\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{+.f64}\left(\frac{3}{8}, \left(v \cdot \frac{-1}{4}\right)\right)\right)\right), \left(v - 1\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right)\right)\right), \left(v - 1\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right)\right)\right), \left(v + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right)\right)\right), \left(v + -1\right)\right)\right) \]

      13. +-lowering-+.f64 76.6%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right)\right)\right), \mathsf{+.f64}\left(v, \color{blue}{-1}\right)\right)\right) \]

   9. Simplified 76.6%

      \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\frac{\left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot \left(0.375 + v \cdot -0.25\right)\right)}{v + -1}} \]
   10. Step-by-step derivation

       1. frac-2neg N/A

          \[\leadsto \frac{2}{r \cdot r} + \frac{\mathsf{neg}\left(\left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot \left(\frac{3}{8} + v \cdot \frac{-1}{4}\right)\right)\right)}{\color{blue}{\mathsf{neg}\left(\left(v + -1\right)\right)}} \]

       2. distribute-frac-neg N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{neg}\left(\frac{\left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot \left(\frac{3}{8} + v \cdot \frac{-1}{4}\right)\right)}{\mathsf{neg}\left(\left(v + -1\right)\right)}\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{neg}\left(\frac{\left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) \cdot \left(\frac{3}{8} + v \cdot \frac{-1}{4}\right)}{\mathsf{neg}\left(\left(v + -1\right)\right)}\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{neg}\left(\frac{\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot \left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right)}{\mathsf{neg}\left(\left(v + -1\right)\right)}\right)\right) \]

       5. unswap-sqr N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{neg}\left(\frac{\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}{\mathsf{neg}\left(\left(v + -1\right)\right)}\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{neg}\left(\frac{\left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)}{\mathsf{neg}\left(\left(v + -1\right)\right)}\right)\right) \]

       7. +-commutative N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{neg}\left(\frac{\left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)}{\mathsf{neg}\left(\left(-1 + v\right)\right)}\right)\right) \]

       8. distribute-neg-in N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{neg}\left(\frac{\left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)}{\left(\mathsf{neg}\left(-1\right)\right) + \left(\mathsf{neg}\left(v\right)\right)}\right)\right) \]

       9. metadata-eval N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{neg}\left(\frac{\left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)}{1 + \left(\mathsf{neg}\left(v\right)\right)}\right)\right) \]

       10. sub-neg N/A

           \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{neg}\left(\frac{\left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)}{1 - v}\right)\right) \]

       11. associate-*r/ N/A

           \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{neg}\left(\left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot \left(r \cdot w\right)\right) \cdot \frac{r \cdot w}{1 - v}\right)\right) \]

   11. Applied egg-rr 96.4%

       \[\leadsto \color{blue}{\frac{2}{r \cdot r} - \frac{\left(0.375 + v \cdot -0.25\right) \cdot \left(r \cdot w\right)}{\frac{1 - v}{r \cdot w}}} \]

   12. Taylor expanded in v around inf

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \color{blue}{\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)}\right) \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left({r}^{2} \cdot {w}^{2}\right) \cdot \color{blue}{\frac{1}{4}}\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{*.f64}\left(\left({r}^{2} \cdot {w}^{2}\right), \color{blue}{\frac{1}{4}}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{*.f64}\left(\left(\left(r \cdot r\right) \cdot {w}^{2}\right), \frac{1}{4}\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{*.f64}\left(\left(r \cdot \left(r \cdot {w}^{2}\right)\right), \frac{1}{4}\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \left(r \cdot {w}^{2}\right)\right), \frac{1}{4}\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \left({w}^{2}\right)\right)\right), \frac{1}{4}\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \left(w \cdot w\right)\right)\right), \frac{1}{4}\right)\right) \]

       8. *-lowering-*.f64 85.5%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, w\right)\right)\right), \frac{1}{4}\right)\right) \]

   14. Simplified 85.5%

       \[\leadsto \frac{2}{r \cdot r} - \color{blue}{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot 0.25} \]

   if 0.220000000000000001 < r < 1.3500000000000001e234

   1. Initial program 88.6%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Step-by-step derivation

      1. associate--l- N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)} \]

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{2}{r \cdot r}\right), \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(r \cdot r\right)\right), \left(\color{blue}{3} - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

      7. associate--r+ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \color{blue}{\frac{9}{2}}\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2}\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + 3\right) - \frac{9}{2}\right)\right) \]

      10. associate--l+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(3 - \frac{9}{2}\right)}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \frac{-3}{2}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\frac{-9}{2} + \color{blue}{3}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\left(\mathsf{neg}\left(\frac{9}{2}\right)\right) + 3\right)\right)\right) \]

   3. Simplified 74.9%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\frac{\left(r \cdot r\right) \cdot \left(\left(0.375 + v \cdot -0.25\right) \cdot \left(w \cdot w\right)\right)}{v + -1} + -1.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\color{blue}{\left(\frac{-3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)}, \frac{-3}{2}\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\left(\left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-3}{8}\right), \frac{-3}{2}\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\left({r}^{2} \cdot \left({w}^{2} \cdot \frac{-3}{8}\right)\right), \frac{-3}{2}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\left({r}^{2} \cdot \left(\frac{-3}{8} \cdot {w}^{2}\right)\right), \frac{-3}{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({r}^{2}\right), \left(\frac{-3}{8} \cdot {w}^{2}\right)\right), \frac{-3}{2}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(r \cdot r\right), \left(\frac{-3}{8} \cdot {w}^{2}\right)\right), \frac{-3}{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left(\frac{-3}{8} \cdot {w}^{2}\right)\right), \frac{-3}{2}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left({w}^{2} \cdot \frac{-3}{8}\right)\right), \frac{-3}{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left({w}^{2}\right), \frac{-3}{8}\right)\right), \frac{-3}{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left(w \cdot w\right), \frac{-3}{8}\right)\right), \frac{-3}{2}\right)\right) \]

      10. *-lowering-*.f64 69.1%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \frac{-3}{2}\right)\right) \]

   7. Simplified 69.1%

      \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{\left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot -0.375\right)} + -1.5\right) \]

   8. Taylor expanded in r around inf

      \[\leadsto \color{blue}{{r}^{2} \cdot \left(\frac{-3}{8} \cdot {w}^{2} - \frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)} \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \left(r \cdot r\right) \cdot \left(\color{blue}{\frac{-3}{8} \cdot {w}^{2}} - \frac{3}{2} \cdot \frac{1}{{r}^{2}}\right) \]

      2. associate-*l* N/A

         \[\leadsto r \cdot \color{blue}{\left(r \cdot \left(\frac{-3}{8} \cdot {w}^{2} - \frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(r, \color{blue}{\left(r \cdot \left(\frac{-3}{8} \cdot {w}^{2} - \frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \color{blue}{\left(\frac{-3}{8} \cdot {w}^{2} - \frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)}\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \left(\frac{-3}{8} \cdot {w}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\left(\frac{-3}{8} \cdot {w}^{2}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\left({w}^{2} \cdot \frac{-3}{8}\right), \left(\mathsf{neg}\left(\color{blue}{\frac{3}{2} \cdot \frac{1}{{r}^{2}}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({w}^{2}\right), \frac{-3}{8}\right), \left(\mathsf{neg}\left(\color{blue}{\frac{3}{2} \cdot \frac{1}{{r}^{2}}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(w \cdot w\right), \frac{-3}{8}\right), \left(\mathsf{neg}\left(\color{blue}{\frac{3}{2}} \cdot \frac{1}{{r}^{2}}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right), \left(\mathsf{neg}\left(\color{blue}{\frac{3}{2}} \cdot \frac{1}{{r}^{2}}\right)\right)\right)\right)\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right), \left(\mathsf{neg}\left(\frac{\frac{3}{2} \cdot 1}{{r}^{2}}\right)\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right), \left(\mathsf{neg}\left(\frac{\frac{3}{2}}{{r}^{2}}\right)\right)\right)\right)\right) \]

      13. distribute-neg-frac N/A

          \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right), \left(\frac{\mathsf{neg}\left(\frac{3}{2}\right)}{\color{blue}{{r}^{2}}}\right)\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right), \left(\frac{\frac{-3}{2}}{{\color{blue}{r}}^{2}}\right)\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right), \mathsf{/.f64}\left(\frac{-3}{2}, \color{blue}{\left({r}^{2}\right)}\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right), \mathsf{/.f64}\left(\frac{-3}{2}, \left(r \cdot \color{blue}{r}\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 71.4%

          \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right), \mathsf{/.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(r, \color{blue}{r}\right)\right)\right)\right)\right) \]

   10. Simplified 71.4%

       \[\leadsto \color{blue}{r \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot -0.375 + \frac{-1.5}{r \cdot r}\right)\right)} \]

   if 1.3500000000000001e234 < r

   1. Initial program 67.5%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Step-by-step derivation

      1. associate--l- N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)} \]

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{2}{r \cdot r}\right), \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(r \cdot r\right)\right), \left(\color{blue}{3} - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

      7. associate--r+ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \color{blue}{\frac{9}{2}}\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2}\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + 3\right) - \frac{9}{2}\right)\right) \]

      10. associate--l+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(3 - \frac{9}{2}\right)}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \frac{-3}{2}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\frac{-9}{2} + \color{blue}{3}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\left(\mathsf{neg}\left(\frac{9}{2}\right)\right) + 3\right)\right)\right) \]

   3. Simplified 51.1%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\frac{\left(r \cdot r\right) \cdot \left(\left(0.375 + v \cdot -0.25\right) \cdot \left(w \cdot w\right)\right)}{v + -1} + -1.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in r around inf

      \[\leadsto \color{blue}{\frac{{r}^{2} \cdot \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)}{v - 1}} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left({r}^{2} \cdot \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \color{blue}{\left(v - 1\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({r}^{2}\right), \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(\color{blue}{v} - 1\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(r \cdot r\right), \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left(\left(w \cdot w\right) \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left(w \cdot \left(w \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right)\right), \left(v - 1\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \left(w \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right)\right), \left(v - 1\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right)\right), \left(v - 1\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{3}{8}, \left(\frac{-1}{4} \cdot v\right)\right)\right)\right)\right), \left(v - 1\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(\frac{-1}{4}, v\right)\right)\right)\right)\right), \left(v - 1\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(\frac{-1}{4}, v\right)\right)\right)\right)\right), \left(v + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(\frac{-1}{4}, v\right)\right)\right)\right)\right), \left(v + -1\right)\right) \]

      13. +-lowering-+.f64 51.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(\frac{-1}{4}, v\right)\right)\right)\right)\right), \mathsf{+.f64}\left(v, \color{blue}{-1}\right)\right) \]

   7. Simplified 51.9%

      \[\leadsto \color{blue}{\frac{\left(r \cdot r\right) \cdot \left(w \cdot \left(w \cdot \left(0.375 + -0.25 \cdot v\right)\right)\right)}{v + -1}} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)}{v + -1} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot \left(\frac{3}{8} + v \cdot \frac{-1}{4}\right)\right)}{v + -1} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) \cdot \left(\frac{3}{8} + v \cdot \frac{-1}{4}\right)}{\color{blue}{v} + -1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(\frac{3}{8} + v \cdot \frac{-1}{4}\right)}{v + -1} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{\color{blue}{v} + -1} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot r\right) \cdot \left(r \cdot \left(w \cdot w\right)\right)}{\color{blue}{v} + -1} \]

      7. associate-/l* N/A

         \[\leadsto \left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot r\right) \cdot \color{blue}{\frac{r \cdot \left(w \cdot w\right)}{v + -1}} \]

      8. *-commutative N/A

         \[\leadsto \left(r \cdot \left(\frac{3}{8} + v \cdot \frac{-1}{4}\right)\right) \cdot \frac{\color{blue}{r \cdot \left(w \cdot w\right)}}{v + -1} \]

      9. metadata-eval N/A

         \[\leadsto \left(r \cdot \left(3 \cdot \frac{1}{8} + v \cdot \frac{-1}{4}\right)\right) \cdot \frac{r \cdot \left(\color{blue}{w} \cdot w\right)}{v + -1} \]

      10. metadata-eval N/A

          \[\leadsto \left(r \cdot \left(3 \cdot \frac{1}{8} + v \cdot \left(-2 \cdot \frac{1}{8}\right)\right)\right) \cdot \frac{r \cdot \left(w \cdot w\right)}{v + -1} \]

      11. associate-*l* N/A

          \[\leadsto \left(r \cdot \left(3 \cdot \frac{1}{8} + \left(v \cdot -2\right) \cdot \frac{1}{8}\right)\right) \cdot \frac{r \cdot \left(w \cdot \color{blue}{w}\right)}{v + -1} \]

      12. distribute-rgt-in N/A

          \[\leadsto \left(r \cdot \left(\frac{1}{8} \cdot \left(3 + v \cdot -2\right)\right)\right) \cdot \frac{r \cdot \color{blue}{\left(w \cdot w\right)}}{v + -1} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(r \cdot \left(\frac{1}{8} \cdot \left(3 + v \cdot -2\right)\right)\right), \color{blue}{\left(\frac{r \cdot \left(w \cdot w\right)}{v + -1}\right)}\right) \]

   9. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\left(r \cdot \left(0.375 + v \cdot -0.25\right)\right) \cdot \frac{w \cdot \left(r \cdot w\right)}{v + -1}} \]

   10. Taylor expanded in v around 0

       \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{3}{8} \cdot r\right)}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, w\right)\right), \mathsf{+.f64}\left(v, -1\right)\right)\right) \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(r \cdot \frac{3}{8}\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, w\right)\right)}, \mathsf{+.f64}\left(v, -1\right)\right)\right) \]

       2. *-lowering-*.f64 73.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \frac{3}{8}\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, w\right)\right)}, \mathsf{+.f64}\left(v, -1\right)\right)\right) \]

   12. Simplified 73.1%

       \[\leadsto \color{blue}{\left(r \cdot 0.375\right)} \cdot \frac{w \cdot \left(r \cdot w\right)}{v + -1} \]
3. Recombined 4 regimes into one program.

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 2 \cdot 10^{-151}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{elif}\;r \leq 0.22:\\ \;\;\;\;\frac{2}{r \cdot r} - \left(r \cdot \left(\left(w \cdot w\right) \cdot r\right)\right) \cdot 0.25\\ \mathbf{elif}\;r \leq 1.35 \cdot 10^{+234}:\\ \;\;\;\;r \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot -0.375 + \frac{-1.5}{r \cdot r}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(r \cdot 0.375\right) \cdot \frac{w \cdot \left(w \cdot r\right)}{v + -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 64.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 2 \cdot 10^{-151}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{elif}\;r \leq 580000000000:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot -0.25\right)\\ \mathbf{elif}\;r \leq 1.35 \cdot 10^{+236}:\\ \;\;\;\;r \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot -0.375 + \frac{-1.5}{r \cdot r}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(r \cdot 0.375\right) \cdot \frac{w \cdot \left(w \cdot r\right)}{v + -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 2e-151)
   (/ (/ 2.0 r) r)
   (if (<= r 580000000000.0)
     (+ (/ 2.0 (* r r)) (* (* r r) (* (* w w) -0.25)))
     (if (<= r 1.35e+236)
       (* r (* r (+ (* (* w w) -0.375) (/ -1.5 (* r r)))))
       (* (* r 0.375) (/ (* w (* w r)) (+ v -1.0)))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 2e-151) {
		tmp = (2.0 / r) / r;
	} else if (r <= 580000000000.0) {
		tmp = (2.0 / (r * r)) + ((r * r) * ((w * w) * -0.25));
	} else if (r <= 1.35e+236) {
		tmp = r * (r * (((w * w) * -0.375) + (-1.5 / (r * r))));
	} else {
		tmp = (r * 0.375) * ((w * (w * r)) / (v + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 2d-151) then
        tmp = (2.0d0 / r) / r
    else if (r <= 580000000000.0d0) then
        tmp = (2.0d0 / (r * r)) + ((r * r) * ((w * w) * (-0.25d0)))
    else if (r <= 1.35d+236) then
        tmp = r * (r * (((w * w) * (-0.375d0)) + ((-1.5d0) / (r * r))))
    else
        tmp = (r * 0.375d0) * ((w * (w * r)) / (v + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 2e-151) {
		tmp = (2.0 / r) / r;
	} else if (r <= 580000000000.0) {
		tmp = (2.0 / (r * r)) + ((r * r) * ((w * w) * -0.25));
	} else if (r <= 1.35e+236) {
		tmp = r * (r * (((w * w) * -0.375) + (-1.5 / (r * r))));
	} else {
		tmp = (r * 0.375) * ((w * (w * r)) / (v + -1.0));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 2e-151:
		tmp = (2.0 / r) / r
	elif r <= 580000000000.0:
		tmp = (2.0 / (r * r)) + ((r * r) * ((w * w) * -0.25))
	elif r <= 1.35e+236:
		tmp = r * (r * (((w * w) * -0.375) + (-1.5 / (r * r))))
	else:
		tmp = (r * 0.375) * ((w * (w * r)) / (v + -1.0))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 2e-151)
		tmp = Float64(Float64(2.0 / r) / r);
	elseif (r <= 580000000000.0)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + Float64(Float64(r * r) * Float64(Float64(w * w) * -0.25)));
	elseif (r <= 1.35e+236)
		tmp = Float64(r * Float64(r * Float64(Float64(Float64(w * w) * -0.375) + Float64(-1.5 / Float64(r * r)))));
	else
		tmp = Float64(Float64(r * 0.375) * Float64(Float64(w * Float64(w * r)) / Float64(v + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 2e-151)
		tmp = (2.0 / r) / r;
	elseif (r <= 580000000000.0)
		tmp = (2.0 / (r * r)) + ((r * r) * ((w * w) * -0.25));
	elseif (r <= 1.35e+236)
		tmp = r * (r * (((w * w) * -0.375) + (-1.5 / (r * r))));
	else
		tmp = (r * 0.375) * ((w * (w * r)) / (v + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 2e-151], N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision], If[LessEqual[r, 580000000000.0], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(N[(r * r), $MachinePrecision] * N[(N[(w * w), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[r, 1.35e+236], N[(r * N[(r * N[(N[(N[(w * w), $MachinePrecision] * -0.375), $MachinePrecision] + N[(-1.5 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(r * 0.375), $MachinePrecision] * N[(N[(w * N[(w * r), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if r < 1.9999999999999999e-151

   1. Initial program 84.8%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Step-by-step derivation

      1. associate--l- N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)} \]

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{2}{r \cdot r}\right), \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(r \cdot r\right)\right), \left(\color{blue}{3} - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

      7. associate--r+ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \color{blue}{\frac{9}{2}}\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2}\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + 3\right) - \frac{9}{2}\right)\right) \]

      10. associate--l+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(3 - \frac{9}{2}\right)}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \frac{-3}{2}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\frac{-9}{2} + \color{blue}{3}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\left(\mathsf{neg}\left(\frac{9}{2}\right)\right) + 3\right)\right)\right) \]

   3. Simplified 77.7%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\frac{\left(r \cdot r\right) \cdot \left(\left(0.375 + v \cdot -0.25\right) \cdot \left(w \cdot w\right)\right)}{v + -1} + -1.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left({r}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(r \cdot \color{blue}{r}\right)\right) \]

      3. *-lowering-*.f64 59.7%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, \color{blue}{r}\right)\right) \]

   7. Simplified 59.7%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r}} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{r}}{\color{blue}{r}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{r}\right), \color{blue}{r}\right) \]

      3. /-lowering-/.f64 59.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, r\right), r\right) \]

   9. Applied egg-rr 59.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{r}}{r}} \]

   if 1.9999999999999999e-151 < r < 5.8e11

   1. Initial program 83.7%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Step-by-step derivation

      1. associate--l- N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)} \]

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{2}{r \cdot r}\right), \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(r \cdot r\right)\right), \left(\color{blue}{3} - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

      7. associate--r+ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \color{blue}{\frac{9}{2}}\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2}\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + 3\right) - \frac{9}{2}\right)\right) \]

      10. associate--l+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(3 - \frac{9}{2}\right)}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \frac{-3}{2}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\frac{-9}{2} + \color{blue}{3}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\left(\mathsf{neg}\left(\frac{9}{2}\right)\right) + 3\right)\right)\right) \]

   3. Simplified 75.3%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\frac{\left(r \cdot r\right) \cdot \left(\left(0.375 + v \cdot -0.25\right) \cdot \left(w \cdot w\right)\right)}{v + -1} + -1.5\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot \left(w \cdot w\right)\right) \cdot \left(r \cdot r\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot \left(\left(w \cdot w\right) \cdot \left(r \cdot r\right)\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right), \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \left(v \cdot \frac{-1}{4}\right)\right), \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \left(r \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(r, \left(\left(w \cdot w\right) \cdot r\right)\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(r, \left(r \cdot \left(w \cdot w\right)\right)\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \left(w \cdot w\right)\right)\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

      11. *-lowering-*.f64 83.7%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, w\right)\right)\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

   6. Applied egg-rr 83.7%

      \[\leadsto \frac{2}{r \cdot r} + \left(\frac{\color{blue}{\left(0.375 + v \cdot -0.25\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}}{v + -1} + -1.5\right) \]

   7. Taylor expanded in r around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \color{blue}{\left(\frac{{r}^{2} \cdot \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)}{v - 1}\right)}\right) \]
   8. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\left({r}^{2} \cdot \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \color{blue}{\left(v - 1\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({r}^{2}\right), \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(\color{blue}{v} - 1\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(r \cdot r\right), \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left({w}^{2}\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left(w \cdot w\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{+.f64}\left(\frac{3}{8}, \left(\frac{-1}{4} \cdot v\right)\right)\right)\right), \left(v - 1\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{+.f64}\left(\frac{3}{8}, \left(v \cdot \frac{-1}{4}\right)\right)\right)\right), \left(v - 1\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right)\right)\right), \left(v - 1\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right)\right)\right), \left(v + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right)\right)\right), \left(v + -1\right)\right)\right) \]

      13. +-lowering-+.f64 75.2%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right)\right)\right), \mathsf{+.f64}\left(v, \color{blue}{-1}\right)\right)\right) \]

   9. Simplified 75.2%

      \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\frac{\left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot \left(0.375 + v \cdot -0.25\right)\right)}{v + -1}} \]

   10. Taylor expanded in v around inf

       \[\leadsto \color{blue}{\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + 2 \cdot \frac{1}{{r}^{2}}} \]
   11. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto 2 \cdot \frac{1}{{r}^{2}} + \color{blue}{\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)} \]

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

          \[\leadsto \mathsf{+.f64}\left(\left(2 \cdot \frac{1}{{r}^{2}}\right), \color{blue}{\left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)}\right) \]

       3. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(\left(\frac{2 \cdot 1}{{r}^{2}}\right), \left(\color{blue}{\frac{-1}{4}} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\left(\frac{2}{{r}^{2}}\right), \left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left({r}^{2}\right)\right), \left(\color{blue}{\frac{-1}{4}} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(r \cdot r\right)\right), \left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left({r}^{2} \cdot {w}^{2}\right) \cdot \color{blue}{\frac{-1}{4}}\right)\right) \]

       9. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left({r}^{2} \cdot \color{blue}{\left({w}^{2} \cdot \frac{-1}{4}\right)}\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left({r}^{2} \cdot \left(\frac{-1}{4} \cdot \color{blue}{{w}^{2}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{*.f64}\left(\left({r}^{2}\right), \color{blue}{\left(\frac{-1}{4} \cdot {w}^{2}\right)}\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{*.f64}\left(\left(r \cdot r\right), \left(\color{blue}{\frac{-1}{4}} \cdot {w}^{2}\right)\right)\right) \]

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

           \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left(\color{blue}{\frac{-1}{4}} \cdot {w}^{2}\right)\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left({w}^{2} \cdot \color{blue}{\frac{-1}{4}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left({w}^{2}\right), \color{blue}{\frac{-1}{4}}\right)\right)\right) \]

       16. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left(w \cdot w\right), \frac{-1}{4}\right)\right)\right) \]

       17. *-lowering-*.f64 84.0%

           \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-1}{4}\right)\right)\right) \]

   12. Simplified 84.0%

       \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot -0.25\right)} \]

   if 5.8e11 < r < 1.3500000000000001e236

   1. Initial program 90.1%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Step-by-step derivation

      1. associate--l- N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)} \]

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{2}{r \cdot r}\right), \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(r \cdot r\right)\right), \left(\color{blue}{3} - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

      7. associate--r+ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \color{blue}{\frac{9}{2}}\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2}\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + 3\right) - \frac{9}{2}\right)\right) \]

      10. associate--l+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(3 - \frac{9}{2}\right)}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \frac{-3}{2}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\frac{-9}{2} + \color{blue}{3}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\left(\mathsf{neg}\left(\frac{9}{2}\right)\right) + 3\right)\right)\right) \]

   3. Simplified 75.9%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\frac{\left(r \cdot r\right) \cdot \left(\left(0.375 + v \cdot -0.25\right) \cdot \left(w \cdot w\right)\right)}{v + -1} + -1.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\color{blue}{\left(\frac{-3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)}, \frac{-3}{2}\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\left(\left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-3}{8}\right), \frac{-3}{2}\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\left({r}^{2} \cdot \left({w}^{2} \cdot \frac{-3}{8}\right)\right), \frac{-3}{2}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\left({r}^{2} \cdot \left(\frac{-3}{8} \cdot {w}^{2}\right)\right), \frac{-3}{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({r}^{2}\right), \left(\frac{-3}{8} \cdot {w}^{2}\right)\right), \frac{-3}{2}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(r \cdot r\right), \left(\frac{-3}{8} \cdot {w}^{2}\right)\right), \frac{-3}{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left(\frac{-3}{8} \cdot {w}^{2}\right)\right), \frac{-3}{2}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left({w}^{2} \cdot \frac{-3}{8}\right)\right), \frac{-3}{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left({w}^{2}\right), \frac{-3}{8}\right)\right), \frac{-3}{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left(w \cdot w\right), \frac{-3}{8}\right)\right), \frac{-3}{2}\right)\right) \]

      10. *-lowering-*.f64 69.5%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \frac{-3}{2}\right)\right) \]

   7. Simplified 69.5%

      \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{\left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot -0.375\right)} + -1.5\right) \]

   8. Taylor expanded in r around inf

      \[\leadsto \color{blue}{{r}^{2} \cdot \left(\frac{-3}{8} \cdot {w}^{2} - \frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)} \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \left(r \cdot r\right) \cdot \left(\color{blue}{\frac{-3}{8} \cdot {w}^{2}} - \frac{3}{2} \cdot \frac{1}{{r}^{2}}\right) \]

      2. associate-*l* N/A

         \[\leadsto r \cdot \color{blue}{\left(r \cdot \left(\frac{-3}{8} \cdot {w}^{2} - \frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(r, \color{blue}{\left(r \cdot \left(\frac{-3}{8} \cdot {w}^{2} - \frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \color{blue}{\left(\frac{-3}{8} \cdot {w}^{2} - \frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)}\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \left(\frac{-3}{8} \cdot {w}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\left(\frac{-3}{8} \cdot {w}^{2}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\left({w}^{2} \cdot \frac{-3}{8}\right), \left(\mathsf{neg}\left(\color{blue}{\frac{3}{2} \cdot \frac{1}{{r}^{2}}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({w}^{2}\right), \frac{-3}{8}\right), \left(\mathsf{neg}\left(\color{blue}{\frac{3}{2} \cdot \frac{1}{{r}^{2}}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(w \cdot w\right), \frac{-3}{8}\right), \left(\mathsf{neg}\left(\color{blue}{\frac{3}{2}} \cdot \frac{1}{{r}^{2}}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right), \left(\mathsf{neg}\left(\color{blue}{\frac{3}{2}} \cdot \frac{1}{{r}^{2}}\right)\right)\right)\right)\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right), \left(\mathsf{neg}\left(\frac{\frac{3}{2} \cdot 1}{{r}^{2}}\right)\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right), \left(\mathsf{neg}\left(\frac{\frac{3}{2}}{{r}^{2}}\right)\right)\right)\right)\right) \]

      13. distribute-neg-frac N/A

          \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right), \left(\frac{\mathsf{neg}\left(\frac{3}{2}\right)}{\color{blue}{{r}^{2}}}\right)\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right), \left(\frac{\frac{-3}{2}}{{\color{blue}{r}}^{2}}\right)\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right), \mathsf{/.f64}\left(\frac{-3}{2}, \color{blue}{\left({r}^{2}\right)}\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right), \mathsf{/.f64}\left(\frac{-3}{2}, \left(r \cdot \color{blue}{r}\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 71.9%

          \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right), \mathsf{/.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(r, \color{blue}{r}\right)\right)\right)\right)\right) \]

   10. Simplified 71.9%

       \[\leadsto \color{blue}{r \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot -0.375 + \frac{-1.5}{r \cdot r}\right)\right)} \]

   if 1.3500000000000001e236 < r

   1. Initial program 67.5%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Step-by-step derivation

      1. associate--l- N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)} \]

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{2}{r \cdot r}\right), \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(r \cdot r\right)\right), \left(\color{blue}{3} - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

      7. associate--r+ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \color{blue}{\frac{9}{2}}\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2}\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + 3\right) - \frac{9}{2}\right)\right) \]

      10. associate--l+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(3 - \frac{9}{2}\right)}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \frac{-3}{2}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\frac{-9}{2} + \color{blue}{3}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\left(\mathsf{neg}\left(\frac{9}{2}\right)\right) + 3\right)\right)\right) \]

   3. Simplified 51.1%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\frac{\left(r \cdot r\right) \cdot \left(\left(0.375 + v \cdot -0.25\right) \cdot \left(w \cdot w\right)\right)}{v + -1} + -1.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in r around inf

      \[\leadsto \color{blue}{\frac{{r}^{2} \cdot \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)}{v - 1}} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left({r}^{2} \cdot \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \color{blue}{\left(v - 1\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({r}^{2}\right), \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(\color{blue}{v} - 1\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(r \cdot r\right), \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left(\left(w \cdot w\right) \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left(w \cdot \left(w \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right)\right), \left(v - 1\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \left(w \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right)\right), \left(v - 1\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right)\right), \left(v - 1\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{3}{8}, \left(\frac{-1}{4} \cdot v\right)\right)\right)\right)\right), \left(v - 1\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(\frac{-1}{4}, v\right)\right)\right)\right)\right), \left(v - 1\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(\frac{-1}{4}, v\right)\right)\right)\right)\right), \left(v + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(\frac{-1}{4}, v\right)\right)\right)\right)\right), \left(v + -1\right)\right) \]

      13. +-lowering-+.f64 51.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(\frac{-1}{4}, v\right)\right)\right)\right)\right), \mathsf{+.f64}\left(v, \color{blue}{-1}\right)\right) \]

   7. Simplified 51.9%

      \[\leadsto \color{blue}{\frac{\left(r \cdot r\right) \cdot \left(w \cdot \left(w \cdot \left(0.375 + -0.25 \cdot v\right)\right)\right)}{v + -1}} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)}{v + -1} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot \left(\frac{3}{8} + v \cdot \frac{-1}{4}\right)\right)}{v + -1} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) \cdot \left(\frac{3}{8} + v \cdot \frac{-1}{4}\right)}{\color{blue}{v} + -1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(\frac{3}{8} + v \cdot \frac{-1}{4}\right)}{v + -1} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{\color{blue}{v} + -1} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot r\right) \cdot \left(r \cdot \left(w \cdot w\right)\right)}{\color{blue}{v} + -1} \]

      7. associate-/l* N/A

         \[\leadsto \left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot r\right) \cdot \color{blue}{\frac{r \cdot \left(w \cdot w\right)}{v + -1}} \]

      8. *-commutative N/A

         \[\leadsto \left(r \cdot \left(\frac{3}{8} + v \cdot \frac{-1}{4}\right)\right) \cdot \frac{\color{blue}{r \cdot \left(w \cdot w\right)}}{v + -1} \]

      9. metadata-eval N/A

         \[\leadsto \left(r \cdot \left(3 \cdot \frac{1}{8} + v \cdot \frac{-1}{4}\right)\right) \cdot \frac{r \cdot \left(\color{blue}{w} \cdot w\right)}{v + -1} \]

      10. metadata-eval N/A

          \[\leadsto \left(r \cdot \left(3 \cdot \frac{1}{8} + v \cdot \left(-2 \cdot \frac{1}{8}\right)\right)\right) \cdot \frac{r \cdot \left(w \cdot w\right)}{v + -1} \]

      11. associate-*l* N/A

          \[\leadsto \left(r \cdot \left(3 \cdot \frac{1}{8} + \left(v \cdot -2\right) \cdot \frac{1}{8}\right)\right) \cdot \frac{r \cdot \left(w \cdot \color{blue}{w}\right)}{v + -1} \]

      12. distribute-rgt-in N/A

          \[\leadsto \left(r \cdot \left(\frac{1}{8} \cdot \left(3 + v \cdot -2\right)\right)\right) \cdot \frac{r \cdot \color{blue}{\left(w \cdot w\right)}}{v + -1} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(r \cdot \left(\frac{1}{8} \cdot \left(3 + v \cdot -2\right)\right)\right), \color{blue}{\left(\frac{r \cdot \left(w \cdot w\right)}{v + -1}\right)}\right) \]

   9. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\left(r \cdot \left(0.375 + v \cdot -0.25\right)\right) \cdot \frac{w \cdot \left(r \cdot w\right)}{v + -1}} \]

   10. Taylor expanded in v around 0

       \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{3}{8} \cdot r\right)}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, w\right)\right), \mathsf{+.f64}\left(v, -1\right)\right)\right) \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(r \cdot \frac{3}{8}\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, w\right)\right)}, \mathsf{+.f64}\left(v, -1\right)\right)\right) \]

       2. *-lowering-*.f64 73.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \frac{3}{8}\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, w\right)\right)}, \mathsf{+.f64}\left(v, -1\right)\right)\right) \]

   12. Simplified 73.1%

       \[\leadsto \color{blue}{\left(r \cdot 0.375\right)} \cdot \frac{w \cdot \left(r \cdot w\right)}{v + -1} \]
3. Recombined 4 regimes into one program.

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 2 \cdot 10^{-151}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{elif}\;r \leq 580000000000:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot -0.25\right)\\ \mathbf{elif}\;r \leq 1.35 \cdot 10^{+236}:\\ \;\;\;\;r \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot -0.375 + \frac{-1.5}{r \cdot r}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(r \cdot 0.375\right) \cdot \frac{w \cdot \left(w \cdot r\right)}{v + -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.375 + v \cdot -0.25\\ \mathbf{if}\;r \leq 580000000000:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(w \cdot \left(w \cdot \left(r \cdot r\right)\right)\right) \cdot \frac{t\_0}{v + -1}\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \left(\left(w \cdot r\right) \cdot t\_0\right) \cdot \frac{w \cdot r}{v + -1}\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (+ 0.375 (* v -0.25))))
   (if (<= r 580000000000.0)
     (+ (/ 2.0 (* r r)) (* (* w (* w (* r r))) (/ t_0 (+ v -1.0))))
     (- (+ 3.0 (* (* (* w r) t_0) (/ (* w r) (+ v -1.0)))) 4.5))))

C

double code(double v, double w, double r) {
	double t_0 = 0.375 + (v * -0.25);
	double tmp;
	if (r <= 580000000000.0) {
		tmp = (2.0 / (r * r)) + ((w * (w * (r * r))) * (t_0 / (v + -1.0)));
	} else {
		tmp = (3.0 + (((w * r) * t_0) * ((w * r) / (v + -1.0)))) - 4.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.375d0 + (v * (-0.25d0))
    if (r <= 580000000000.0d0) then
        tmp = (2.0d0 / (r * r)) + ((w * (w * (r * r))) * (t_0 / (v + (-1.0d0))))
    else
        tmp = (3.0d0 + (((w * r) * t_0) * ((w * r) / (v + (-1.0d0))))) - 4.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 0.375 + (v * -0.25);
	double tmp;
	if (r <= 580000000000.0) {
		tmp = (2.0 / (r * r)) + ((w * (w * (r * r))) * (t_0 / (v + -1.0)));
	} else {
		tmp = (3.0 + (((w * r) * t_0) * ((w * r) / (v + -1.0)))) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 0.375 + (v * -0.25)
	tmp = 0
	if r <= 580000000000.0:
		tmp = (2.0 / (r * r)) + ((w * (w * (r * r))) * (t_0 / (v + -1.0)))
	else:
		tmp = (3.0 + (((w * r) * t_0) * ((w * r) / (v + -1.0)))) - 4.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(0.375 + Float64(v * -0.25))
	tmp = 0.0
	if (r <= 580000000000.0)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + Float64(Float64(w * Float64(w * Float64(r * r))) * Float64(t_0 / Float64(v + -1.0))));
	else
		tmp = Float64(Float64(3.0 + Float64(Float64(Float64(w * r) * t_0) * Float64(Float64(w * r) / Float64(v + -1.0)))) - 4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 0.375 + (v * -0.25);
	tmp = 0.0;
	if (r <= 580000000000.0)
		tmp = (2.0 / (r * r)) + ((w * (w * (r * r))) * (t_0 / (v + -1.0)));
	else
		tmp = (3.0 + (((w * r) * t_0) * ((w * r) / (v + -1.0)))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 580000000000.0], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(N[(w * N[(w * N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 + N[(N[(N[(w * r), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[(w * r), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 5.8e11

   1. Initial program 84.6%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Step-by-step derivation

      1. associate--l- N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)} \]

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{2}{r \cdot r}\right), \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(r \cdot r\right)\right), \left(\color{blue}{3} - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

      7. associate--r+ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \color{blue}{\frac{9}{2}}\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2}\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + 3\right) - \frac{9}{2}\right)\right) \]

      10. associate--l+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(3 - \frac{9}{2}\right)}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \frac{-3}{2}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\frac{-9}{2} + \color{blue}{3}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\left(\mathsf{neg}\left(\frac{9}{2}\right)\right) + 3\right)\right)\right) \]

   3. Simplified 77.2%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\frac{\left(r \cdot r\right) \cdot \left(\left(0.375 + v \cdot -0.25\right) \cdot \left(w \cdot w\right)\right)}{v + -1} + -1.5\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot \left(w \cdot w\right)\right) \cdot \left(r \cdot r\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot \left(\left(w \cdot w\right) \cdot \left(r \cdot r\right)\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right), \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \left(v \cdot \frac{-1}{4}\right)\right), \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \left(r \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(r, \left(\left(w \cdot w\right) \cdot r\right)\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(r, \left(r \cdot \left(w \cdot w\right)\right)\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \left(w \cdot w\right)\right)\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

      11. *-lowering-*.f64 84.6%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, w\right)\right)\right)\right), \mathsf{+.f64}\left(v, -1\right)\right), \frac{-3}{2}\right)\right) \]

   6. Applied egg-rr 84.6%

      \[\leadsto \frac{2}{r \cdot r} + \left(\frac{\color{blue}{\left(0.375 + v \cdot -0.25\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}}{v + -1} + -1.5\right) \]

   7. Taylor expanded in r around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \color{blue}{\left(\frac{{r}^{2} \cdot \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)}{v - 1}\right)}\right) \]
   8. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\left({r}^{2} \cdot \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \color{blue}{\left(v - 1\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({r}^{2}\right), \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(\color{blue}{v} - 1\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(r \cdot r\right), \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left({w}^{2}\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left(w \cdot w\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{+.f64}\left(\frac{3}{8}, \left(\frac{-1}{4} \cdot v\right)\right)\right)\right), \left(v - 1\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{+.f64}\left(\frac{3}{8}, \left(v \cdot \frac{-1}{4}\right)\right)\right)\right), \left(v - 1\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right)\right)\right), \left(v - 1\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right)\right)\right), \left(v + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right)\right)\right), \left(v + -1\right)\right)\right) \]

      13. +-lowering-+.f64 71.8%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right)\right)\right), \mathsf{+.f64}\left(v, \color{blue}{-1}\right)\right)\right) \]

   9. Simplified 71.8%

      \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\frac{\left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot \left(0.375 + v \cdot -0.25\right)\right)}{v + -1}} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\frac{\left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) \cdot \left(\frac{3}{8} + v \cdot \frac{-1}{4}\right)}{\color{blue}{v} + -1}\right)\right) \]

       2. associate-/l* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) \cdot \color{blue}{\frac{\frac{3}{8} + v \cdot \frac{-1}{4}}{v + -1}}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{*.f64}\left(\left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right), \color{blue}{\left(\frac{\frac{3}{8} + v \cdot \frac{-1}{4}}{v + -1}\right)}\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{*.f64}\left(\left(\left(\left(r \cdot r\right) \cdot w\right) \cdot w\right), \left(\frac{\color{blue}{\frac{3}{8} + v \cdot \frac{-1}{4}}}{v + -1}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(r \cdot r\right) \cdot w\right), w\right), \left(\frac{\color{blue}{\frac{3}{8} + v \cdot \frac{-1}{4}}}{v + -1}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(r \cdot r\right), w\right), w\right), \left(\frac{\color{blue}{\frac{3}{8}} + v \cdot \frac{-1}{4}}{v + -1}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), w\right), w\right), \left(\frac{\frac{3}{8} + v \cdot \frac{-1}{4}}{v + -1}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), w\right), w\right), \mathsf{/.f64}\left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right), \color{blue}{\left(v + -1\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), w\right), w\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \left(v \cdot \frac{-1}{4}\right)\right), \left(\color{blue}{v} + -1\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), w\right), w\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \left(v + -1\right)\right)\right)\right) \]

       11. +-lowering-+.f64 87.0%

           \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), w\right), w\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{+.f64}\left(v, \color{blue}{-1}\right)\right)\right)\right) \]

   11. Applied egg-rr 87.0%

       \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(\left(\left(r \cdot r\right) \cdot w\right) \cdot w\right) \cdot \frac{0.375 + v \cdot -0.25}{v + -1}} \]

   if 5.8e11 < r

   1. Initial program 87.6%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \left(\left(w \cdot w\right) \cdot \left(r \cdot r\right)\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

      2. unswap-sqr N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \mathsf{*.f64}\left(\left(w \cdot r\right), \left(w \cdot r\right)\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \mathsf{*.f64}\left(\left(r \cdot w\right), \left(w \cdot r\right)\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \left(w \cdot r\right)\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \left(r \cdot w\right)\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

      7. *-lowering-*.f64 91.2%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{*.f64}\left(r, w\right)\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

   4. Applied egg-rr 91.2%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}}{1 - v}\right) - 4.5 \]
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \left(\frac{\left(\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)}{1 - v}\right)\right), \frac{9}{2}\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \left(\left(\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(r \cdot w\right)\right) \cdot \frac{r \cdot w}{1 - v}\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\left(\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(r \cdot w\right)\right), \left(\frac{r \cdot w}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right), \left(r \cdot w\right)\right), \left(\frac{r \cdot w}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

      5. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{8} \cdot \left(3 + \left(\mathsf{neg}\left(2\right)\right) \cdot v\right)\right), \left(r \cdot w\right)\right), \left(\frac{r \cdot w}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{8} \cdot \left(3 + -2 \cdot v\right)\right), \left(r \cdot w\right)\right), \left(\frac{r \cdot w}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{8} \cdot \left(3 + v \cdot -2\right)\right), \left(r \cdot w\right)\right), \left(\frac{r \cdot w}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(3 \cdot \frac{1}{8} + \left(v \cdot -2\right) \cdot \frac{1}{8}\right), \left(r \cdot w\right)\right), \left(\frac{r \cdot w}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{3}{8} + \left(v \cdot -2\right) \cdot \frac{1}{8}\right), \left(r \cdot w\right)\right), \left(\frac{r \cdot w}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{3}{8} + v \cdot \left(-2 \cdot \frac{1}{8}\right)\right), \left(r \cdot w\right)\right), \left(\frac{r \cdot w}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right), \left(r \cdot w\right)\right), \left(\frac{r \cdot w}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{3}{8} + \frac{-1}{4} \cdot v\right), \left(r \cdot w\right)\right), \left(\frac{r \cdot w}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \left(\frac{-1}{4} \cdot v\right)\right), \left(r \cdot w\right)\right), \left(\frac{r \cdot w}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \left(v \cdot \frac{-1}{4}\right)\right), \left(r \cdot w\right)\right), \left(\frac{r \cdot w}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \left(r \cdot w\right)\right), \left(\frac{r \cdot w}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(r, w\right)\right), \left(\frac{r \cdot w}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(r, w\right)\right), \mathsf{/.f64}\left(\left(r \cdot w\right), \left(1 - v\right)\right)\right)\right), \frac{9}{2}\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(r, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, w\right), \left(1 - v\right)\right)\right)\right), \frac{9}{2}\right) \]

      19. --lowering--.f64 94.5%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(r, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{\_.f64}\left(1, v\right)\right)\right)\right), \frac{9}{2}\right) \]

   6. Applied egg-rr 94.5%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\left(0.375 + v \cdot -0.25\right) \cdot \left(r \cdot w\right)\right) \cdot \frac{r \cdot w}{1 - v}}\right) - 4.5 \]

   7. Taylor expanded in r around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\color{blue}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(r, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{\_.f64}\left(1, v\right)\right)\right)\right), \frac{9}{2}\right) \]
   8. Step-by-step derivation

   9. Simplified 94.5%

      \[\leadsto \left(\color{blue}{3} - \left(\left(0.375 + v \cdot -0.25\right) \cdot \left(r \cdot w\right)\right) \cdot \frac{r \cdot w}{1 - v}\right) - 4.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 580000000000:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(w \cdot \left(w \cdot \left(r \cdot r\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{v + -1}\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \left(\left(w \cdot r\right) \cdot \left(0.375 + v \cdot -0.25\right)\right) \cdot \frac{w \cdot r}{v + -1}\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 3.9 \cdot 10^{-79}:\\ \;\;\;\;t\_0 + \left(-1.5 + \left(w \cdot \left(w \cdot \left(r \cdot r\right)\right)\right) \cdot -0.375\right)\\ \mathbf{elif}\;r \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;t\_0 + \left(-1.5 + \left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(w \cdot r\right) \cdot \left(0.375 + v \cdot -0.25\right)\right) \cdot \frac{w \cdot r}{v + -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 3.9e-79)
     (+ t_0 (+ -1.5 (* (* w (* w (* r r))) -0.375)))
     (if (<= r 1.32e+154)
       (+ t_0 (+ -1.5 (* (* r r) (* (* w w) -0.25))))
       (* (* (* w r) (+ 0.375 (* v -0.25))) (/ (* w r) (+ v -1.0)))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 3.9e-79) {
		tmp = t_0 + (-1.5 + ((w * (w * (r * r))) * -0.375));
	} else if (r <= 1.32e+154) {
		tmp = t_0 + (-1.5 + ((r * r) * ((w * w) * -0.25)));
	} else {
		tmp = ((w * r) * (0.375 + (v * -0.25))) * ((w * r) / (v + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (r <= 3.9d-79) then
        tmp = t_0 + ((-1.5d0) + ((w * (w * (r * r))) * (-0.375d0)))
    else if (r <= 1.32d+154) then
        tmp = t_0 + ((-1.5d0) + ((r * r) * ((w * w) * (-0.25d0))))
    else
        tmp = ((w * r) * (0.375d0 + (v * (-0.25d0)))) * ((w * r) / (v + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 3.9e-79) {
		tmp = t_0 + (-1.5 + ((w * (w * (r * r))) * -0.375));
	} else if (r <= 1.32e+154) {
		tmp = t_0 + (-1.5 + ((r * r) * ((w * w) * -0.25)));
	} else {
		tmp = ((w * r) * (0.375 + (v * -0.25))) * ((w * r) / (v + -1.0));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if r <= 3.9e-79:
		tmp = t_0 + (-1.5 + ((w * (w * (r * r))) * -0.375))
	elif r <= 1.32e+154:
		tmp = t_0 + (-1.5 + ((r * r) * ((w * w) * -0.25)))
	else:
		tmp = ((w * r) * (0.375 + (v * -0.25))) * ((w * r) / (v + -1.0))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 3.9e-79)
		tmp = Float64(t_0 + Float64(-1.5 + Float64(Float64(w * Float64(w * Float64(r * r))) * -0.375)));
	elseif (r <= 1.32e+154)
		tmp = Float64(t_0 + Float64(-1.5 + Float64(Float64(r * r) * Float64(Float64(w * w) * -0.25))));
	else
		tmp = Float64(Float64(Float64(w * r) * Float64(0.375 + Float64(v * -0.25))) * Float64(Float64(w * r) / Float64(v + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (r <= 3.9e-79)
		tmp = t_0 + (-1.5 + ((w * (w * (r * r))) * -0.375));
	elseif (r <= 1.32e+154)
		tmp = t_0 + (-1.5 + ((r * r) * ((w * w) * -0.25)));
	else
		tmp = ((w * r) * (0.375 + (v * -0.25))) * ((w * r) / (v + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 3.9e-79], N[(t$95$0 + N[(-1.5 + N[(N[(w * N[(w * N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[r, 1.32e+154], N[(t$95$0 + N[(-1.5 + N[(N[(r * r), $MachinePrecision] * N[(N[(w * w), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(w * r), $MachinePrecision] * N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(w * r), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if r < 3.90000000000000006e-79

   1. Initial program 84.3%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Step-by-step derivation

      1. associate--l- N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)} \]

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{2}{r \cdot r}\right), \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(r \cdot r\right)\right), \left(\color{blue}{3} - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

      7. associate--r+ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \color{blue}{\frac{9}{2}}\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2}\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + 3\right) - \frac{9}{2}\right)\right) \]

      10. associate--l+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(3 - \frac{9}{2}\right)}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \frac{-3}{2}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\frac{-9}{2} + \color{blue}{3}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\left(\mathsf{neg}\left(\frac{9}{2}\right)\right) + 3\right)\right)\right) \]

   3. Simplified 76.8%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\frac{\left(r \cdot r\right) \cdot \left(\left(0.375 + v \cdot -0.25\right) \cdot \left(w \cdot w\right)\right)}{v + -1} + -1.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\color{blue}{\left(\frac{-3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)}, \frac{-3}{2}\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\left(\left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-3}{8}\right), \frac{-3}{2}\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\left({r}^{2} \cdot \left({w}^{2} \cdot \frac{-3}{8}\right)\right), \frac{-3}{2}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\left({r}^{2} \cdot \left(\frac{-3}{8} \cdot {w}^{2}\right)\right), \frac{-3}{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({r}^{2}\right), \left(\frac{-3}{8} \cdot {w}^{2}\right)\right), \frac{-3}{2}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(r \cdot r\right), \left(\frac{-3}{8} \cdot {w}^{2}\right)\right), \frac{-3}{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left(\frac{-3}{8} \cdot {w}^{2}\right)\right), \frac{-3}{2}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left({w}^{2} \cdot \frac{-3}{8}\right)\right), \frac{-3}{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left({w}^{2}\right), \frac{-3}{8}\right)\right), \frac{-3}{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left(w \cdot w\right), \frac{-3}{8}\right)\right), \frac{-3}{2}\right)\right) \]

      10. *-lowering-*.f64 77.8%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \frac{-3}{2}\right)\right) \]

   7. Simplified 77.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{\left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot -0.375\right)} + -1.5\right) \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\left(\left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) \cdot \frac{-3}{8}\right), \frac{-3}{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right), \frac{-3}{8}\right), \frac{-3}{2}\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\left(\left(r \cdot r\right) \cdot w\right) \cdot w\right), \frac{-3}{8}\right), \frac{-3}{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(r \cdot r\right) \cdot w\right), w\right), \frac{-3}{8}\right), \frac{-3}{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(r \cdot r\right), w\right), w\right), \frac{-3}{8}\right), \frac{-3}{2}\right)\right) \]

      6. *-lowering-*.f64 88.9%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), w\right), w\right), \frac{-3}{8}\right), \frac{-3}{2}\right)\right) \]

   9. Applied egg-rr 88.9%

      \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{\left(\left(\left(r \cdot r\right) \cdot w\right) \cdot w\right) \cdot -0.375} + -1.5\right) \]

   if 3.90000000000000006e-79 < r < 1.31999999999999998e154

   1. Initial program 86.9%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Step-by-step derivation

      1. associate--l- N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)} \]

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{2}{r \cdot r}\right), \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(r \cdot r\right)\right), \left(\color{blue}{3} - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

      7. associate--r+ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \color{blue}{\frac{9}{2}}\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2}\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + 3\right) - \frac{9}{2}\right)\right) \]

      10. associate--l+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(3 - \frac{9}{2}\right)}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \frac{-3}{2}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\frac{-9}{2} + \color{blue}{3}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\left(\mathsf{neg}\left(\frac{9}{2}\right)\right) + 3\right)\right)\right) \]

   3. Simplified 85.1%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\frac{\left(r \cdot r\right) \cdot \left(\left(0.375 + v \cdot -0.25\right) \cdot \left(w \cdot w\right)\right)}{v + -1} + -1.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\color{blue}{\left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)}, \frac{-3}{2}\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\left(\left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-1}{4}\right), \frac{-3}{2}\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\left({r}^{2} \cdot \left({w}^{2} \cdot \frac{-1}{4}\right)\right), \frac{-3}{2}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\left({r}^{2} \cdot \left(\frac{-1}{4} \cdot {w}^{2}\right)\right), \frac{-3}{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({r}^{2}\right), \left(\frac{-1}{4} \cdot {w}^{2}\right)\right), \frac{-3}{2}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(r \cdot r\right), \left(\frac{-1}{4} \cdot {w}^{2}\right)\right), \frac{-3}{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left(\frac{-1}{4} \cdot {w}^{2}\right)\right), \frac{-3}{2}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left({w}^{2} \cdot \frac{-1}{4}\right)\right), \frac{-3}{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left({w}^{2}\right), \frac{-1}{4}\right)\right), \frac{-3}{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left(w \cdot w\right), \frac{-1}{4}\right)\right), \frac{-3}{2}\right)\right) \]

      10. *-lowering-*.f64 93.1%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-1}{4}\right)\right), \frac{-3}{2}\right)\right) \]

   7. Simplified 93.1%

      \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{\left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot -0.25\right)} + -1.5\right) \]

   if 1.31999999999999998e154 < r

   1. Initial program 90.1%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Step-by-step derivation

      1. associate--l- N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)} \]

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{2}{r \cdot r}\right), \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(r \cdot r\right)\right), \left(\color{blue}{3} - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

      7. associate--r+ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \color{blue}{\frac{9}{2}}\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2}\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + 3\right) - \frac{9}{2}\right)\right) \]

      10. associate--l+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(3 - \frac{9}{2}\right)}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \frac{-3}{2}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\frac{-9}{2} + \color{blue}{3}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\left(\mathsf{neg}\left(\frac{9}{2}\right)\right) + 3\right)\right)\right) \]

   3. Simplified 51.0%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\frac{\left(r \cdot r\right) \cdot \left(\left(0.375 + v \cdot -0.25\right) \cdot \left(w \cdot w\right)\right)}{v + -1} + -1.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in r around inf

      \[\leadsto \color{blue}{\frac{{r}^{2} \cdot \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)}{v - 1}} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left({r}^{2} \cdot \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \color{blue}{\left(v - 1\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({r}^{2}\right), \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(\color{blue}{v} - 1\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(r \cdot r\right), \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left(\left(w \cdot w\right) \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left(w \cdot \left(w \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right)\right), \left(v - 1\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \left(w \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right)\right), \left(v - 1\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right)\right), \left(v - 1\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{3}{8}, \left(\frac{-1}{4} \cdot v\right)\right)\right)\right)\right), \left(v - 1\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(\frac{-1}{4}, v\right)\right)\right)\right)\right), \left(v - 1\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(\frac{-1}{4}, v\right)\right)\right)\right)\right), \left(v + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(\frac{-1}{4}, v\right)\right)\right)\right)\right), \left(v + -1\right)\right) \]

      13. +-lowering-+.f64 51.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(\frac{-1}{4}, v\right)\right)\right)\right)\right), \mathsf{+.f64}\left(v, \color{blue}{-1}\right)\right) \]

   7. Simplified 51.3%

      \[\leadsto \color{blue}{\frac{\left(r \cdot r\right) \cdot \left(w \cdot \left(w \cdot \left(0.375 + -0.25 \cdot v\right)\right)\right)}{v + -1}} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)}{v + -1} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot \left(\frac{3}{8} + v \cdot \frac{-1}{4}\right)\right)}{v + -1} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) \cdot \left(\frac{3}{8} + v \cdot \frac{-1}{4}\right)}{\color{blue}{v} + -1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(\frac{3}{8} + v \cdot \frac{-1}{4}\right)}{v + -1} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{\color{blue}{v} + -1} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot r\right) \cdot \left(r \cdot \left(w \cdot w\right)\right)}{\color{blue}{v} + -1} \]

      7. associate-/l* N/A

         \[\leadsto \left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot r\right) \cdot \color{blue}{\frac{r \cdot \left(w \cdot w\right)}{v + -1}} \]

      8. *-commutative N/A

         \[\leadsto \left(r \cdot \left(\frac{3}{8} + v \cdot \frac{-1}{4}\right)\right) \cdot \frac{\color{blue}{r \cdot \left(w \cdot w\right)}}{v + -1} \]

      9. metadata-eval N/A

         \[\leadsto \left(r \cdot \left(3 \cdot \frac{1}{8} + v \cdot \frac{-1}{4}\right)\right) \cdot \frac{r \cdot \left(\color{blue}{w} \cdot w\right)}{v + -1} \]

      10. metadata-eval N/A

          \[\leadsto \left(r \cdot \left(3 \cdot \frac{1}{8} + v \cdot \left(-2 \cdot \frac{1}{8}\right)\right)\right) \cdot \frac{r \cdot \left(w \cdot w\right)}{v + -1} \]

      11. associate-*l* N/A

          \[\leadsto \left(r \cdot \left(3 \cdot \frac{1}{8} + \left(v \cdot -2\right) \cdot \frac{1}{8}\right)\right) \cdot \frac{r \cdot \left(w \cdot \color{blue}{w}\right)}{v + -1} \]

      12. distribute-rgt-in N/A

          \[\leadsto \left(r \cdot \left(\frac{1}{8} \cdot \left(3 + v \cdot -2\right)\right)\right) \cdot \frac{r \cdot \color{blue}{\left(w \cdot w\right)}}{v + -1} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(r \cdot \left(\frac{1}{8} \cdot \left(3 + v \cdot -2\right)\right)\right), \color{blue}{\left(\frac{r \cdot \left(w \cdot w\right)}{v + -1}\right)}\right) \]

   9. Applied egg-rr 75.5%

      \[\leadsto \color{blue}{\left(r \cdot \left(0.375 + v \cdot -0.25\right)\right) \cdot \frac{w \cdot \left(r \cdot w\right)}{v + -1}} \]
   10. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto \left(r \cdot \left(\frac{3}{8} + v \cdot \frac{-1}{4}\right)\right) \cdot \left(w \cdot \color{blue}{\frac{r \cdot w}{v + -1}}\right) \]

       2. associate-*r* N/A

          \[\leadsto \left(\left(r \cdot \left(\frac{3}{8} + v \cdot \frac{-1}{4}\right)\right) \cdot w\right) \cdot \color{blue}{\frac{r \cdot w}{v + -1}} \]

       3. *-commutative N/A

          \[\leadsto \left(\left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot r\right) \cdot w\right) \cdot \frac{\color{blue}{r} \cdot w}{v + -1} \]

       4. associate-*r* N/A

          \[\leadsto \left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot \left(r \cdot w\right)\right) \cdot \frac{\color{blue}{r \cdot w}}{v + -1} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot \left(r \cdot w\right)\right), \color{blue}{\left(\frac{r \cdot w}{v + -1}\right)}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right), \left(r \cdot w\right)\right), \left(\frac{\color{blue}{r \cdot w}}{v + -1}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \left(v \cdot \frac{-1}{4}\right)\right), \left(r \cdot w\right)\right), \left(\frac{\color{blue}{r} \cdot w}{v + -1}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \left(r \cdot w\right)\right), \left(\frac{r \cdot w}{v + -1}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(r, w\right)\right), \left(\frac{r \cdot \color{blue}{w}}{v + -1}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(r, w\right)\right), \mathsf{/.f64}\left(\left(r \cdot w\right), \color{blue}{\left(v + -1\right)}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(r, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, w\right), \left(\color{blue}{v} + -1\right)\right)\right) \]

       12. +-lowering-+.f64 75.8%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(r, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{+.f64}\left(v, \color{blue}{-1}\right)\right)\right) \]

   11. Applied egg-rr 75.8%

       \[\leadsto \color{blue}{\left(\left(0.375 + v \cdot -0.25\right) \cdot \left(r \cdot w\right)\right) \cdot \frac{r \cdot w}{v + -1}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 3.9 \cdot 10^{-79}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \left(w \cdot \left(w \cdot \left(r \cdot r\right)\right)\right) \cdot -0.375\right)\\ \mathbf{elif}\;r \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(w \cdot r\right) \cdot \left(0.375 + v \cdot -0.25\right)\right) \cdot \frac{w \cdot r}{v + -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 2 \cdot 10^{-151}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{elif}\;r \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(w \cdot r\right) \cdot \left(0.375 + v \cdot -0.25\right)\right) \cdot \frac{w \cdot r}{v + -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 2e-151)
   (/ (/ 2.0 r) r)
   (if (<= r 1.32e+154)
     (+ (/ 2.0 (* r r)) (+ -1.5 (* (* r r) (* (* w w) -0.25))))
     (* (* (* w r) (+ 0.375 (* v -0.25))) (/ (* w r) (+ v -1.0))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 2e-151) {
		tmp = (2.0 / r) / r;
	} else if (r <= 1.32e+154) {
		tmp = (2.0 / (r * r)) + (-1.5 + ((r * r) * ((w * w) * -0.25)));
	} else {
		tmp = ((w * r) * (0.375 + (v * -0.25))) * ((w * r) / (v + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 2d-151) then
        tmp = (2.0d0 / r) / r
    else if (r <= 1.32d+154) then
        tmp = (2.0d0 / (r * r)) + ((-1.5d0) + ((r * r) * ((w * w) * (-0.25d0))))
    else
        tmp = ((w * r) * (0.375d0 + (v * (-0.25d0)))) * ((w * r) / (v + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 2e-151) {
		tmp = (2.0 / r) / r;
	} else if (r <= 1.32e+154) {
		tmp = (2.0 / (r * r)) + (-1.5 + ((r * r) * ((w * w) * -0.25)));
	} else {
		tmp = ((w * r) * (0.375 + (v * -0.25))) * ((w * r) / (v + -1.0));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 2e-151:
		tmp = (2.0 / r) / r
	elif r <= 1.32e+154:
		tmp = (2.0 / (r * r)) + (-1.5 + ((r * r) * ((w * w) * -0.25)))
	else:
		tmp = ((w * r) * (0.375 + (v * -0.25))) * ((w * r) / (v + -1.0))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 2e-151)
		tmp = Float64(Float64(2.0 / r) / r);
	elseif (r <= 1.32e+154)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + Float64(-1.5 + Float64(Float64(r * r) * Float64(Float64(w * w) * -0.25))));
	else
		tmp = Float64(Float64(Float64(w * r) * Float64(0.375 + Float64(v * -0.25))) * Float64(Float64(w * r) / Float64(v + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 2e-151)
		tmp = (2.0 / r) / r;
	elseif (r <= 1.32e+154)
		tmp = (2.0 / (r * r)) + (-1.5 + ((r * r) * ((w * w) * -0.25)));
	else
		tmp = ((w * r) * (0.375 + (v * -0.25))) * ((w * r) / (v + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 2e-151], N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision], If[LessEqual[r, 1.32e+154], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(-1.5 + N[(N[(r * r), $MachinePrecision] * N[(N[(w * w), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(w * r), $MachinePrecision] * N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(w * r), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if r < 1.9999999999999999e-151

   1. Initial program 84.8%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Step-by-step derivation

      1. associate--l- N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)} \]

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{2}{r \cdot r}\right), \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(r \cdot r\right)\right), \left(\color{blue}{3} - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

      7. associate--r+ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \color{blue}{\frac{9}{2}}\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2}\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + 3\right) - \frac{9}{2}\right)\right) \]

      10. associate--l+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(3 - \frac{9}{2}\right)}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \frac{-3}{2}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\frac{-9}{2} + \color{blue}{3}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\left(\mathsf{neg}\left(\frac{9}{2}\right)\right) + 3\right)\right)\right) \]

   3. Simplified 77.7%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\frac{\left(r \cdot r\right) \cdot \left(\left(0.375 + v \cdot -0.25\right) \cdot \left(w \cdot w\right)\right)}{v + -1} + -1.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left({r}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(r \cdot \color{blue}{r}\right)\right) \]

      3. *-lowering-*.f64 59.7%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, \color{blue}{r}\right)\right) \]

   7. Simplified 59.7%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r}} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{r}}{\color{blue}{r}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{r}\right), \color{blue}{r}\right) \]

      3. /-lowering-/.f64 59.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, r\right), r\right) \]

   9. Applied egg-rr 59.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{r}}{r}} \]

   if 1.9999999999999999e-151 < r < 1.31999999999999998e154

   1. Initial program 84.9%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Step-by-step derivation

      1. associate--l- N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)} \]

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{2}{r \cdot r}\right), \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(r \cdot r\right)\right), \left(\color{blue}{3} - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

      7. associate--r+ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \color{blue}{\frac{9}{2}}\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2}\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + 3\right) - \frac{9}{2}\right)\right) \]

      10. associate--l+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(3 - \frac{9}{2}\right)}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \frac{-3}{2}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\frac{-9}{2} + \color{blue}{3}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\left(\mathsf{neg}\left(\frac{9}{2}\right)\right) + 3\right)\right)\right) \]

   3. Simplified 80.7%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\frac{\left(r \cdot r\right) \cdot \left(\left(0.375 + v \cdot -0.25\right) \cdot \left(w \cdot w\right)\right)}{v + -1} + -1.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\color{blue}{\left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)}, \frac{-3}{2}\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\left(\left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-1}{4}\right), \frac{-3}{2}\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\left({r}^{2} \cdot \left({w}^{2} \cdot \frac{-1}{4}\right)\right), \frac{-3}{2}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\left({r}^{2} \cdot \left(\frac{-1}{4} \cdot {w}^{2}\right)\right), \frac{-3}{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({r}^{2}\right), \left(\frac{-1}{4} \cdot {w}^{2}\right)\right), \frac{-3}{2}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(r \cdot r\right), \left(\frac{-1}{4} \cdot {w}^{2}\right)\right), \frac{-3}{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left(\frac{-1}{4} \cdot {w}^{2}\right)\right), \frac{-3}{2}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left({w}^{2} \cdot \frac{-1}{4}\right)\right), \frac{-3}{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left({w}^{2}\right), \frac{-1}{4}\right)\right), \frac{-3}{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left(w \cdot w\right), \frac{-1}{4}\right)\right), \frac{-3}{2}\right)\right) \]

      10. *-lowering-*.f64 89.4%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-1}{4}\right)\right), \frac{-3}{2}\right)\right) \]

   7. Simplified 89.4%

      \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{\left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot -0.25\right)} + -1.5\right) \]

   if 1.31999999999999998e154 < r

   1. Initial program 90.1%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Step-by-step derivation

      1. associate--l- N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)} \]

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{2}{r \cdot r}\right), \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(r \cdot r\right)\right), \left(\color{blue}{3} - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

      7. associate--r+ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \color{blue}{\frac{9}{2}}\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2}\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + 3\right) - \frac{9}{2}\right)\right) \]

      10. associate--l+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(3 - \frac{9}{2}\right)}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \frac{-3}{2}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\frac{-9}{2} + \color{blue}{3}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\left(\mathsf{neg}\left(\frac{9}{2}\right)\right) + 3\right)\right)\right) \]

   3. Simplified 51.0%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\frac{\left(r \cdot r\right) \cdot \left(\left(0.375 + v \cdot -0.25\right) \cdot \left(w \cdot w\right)\right)}{v + -1} + -1.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in r around inf

      \[\leadsto \color{blue}{\frac{{r}^{2} \cdot \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)}{v - 1}} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left({r}^{2} \cdot \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \color{blue}{\left(v - 1\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({r}^{2}\right), \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(\color{blue}{v} - 1\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(r \cdot r\right), \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left(\left(w \cdot w\right) \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left(w \cdot \left(w \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right)\right), \left(v - 1\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \left(w \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right)\right), \left(v - 1\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right)\right), \left(v - 1\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{3}{8}, \left(\frac{-1}{4} \cdot v\right)\right)\right)\right)\right), \left(v - 1\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(\frac{-1}{4}, v\right)\right)\right)\right)\right), \left(v - 1\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(\frac{-1}{4}, v\right)\right)\right)\right)\right), \left(v + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(\frac{-1}{4}, v\right)\right)\right)\right)\right), \left(v + -1\right)\right) \]

      13. +-lowering-+.f64 51.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(\frac{-1}{4}, v\right)\right)\right)\right)\right), \mathsf{+.f64}\left(v, \color{blue}{-1}\right)\right) \]

   7. Simplified 51.3%

      \[\leadsto \color{blue}{\frac{\left(r \cdot r\right) \cdot \left(w \cdot \left(w \cdot \left(0.375 + -0.25 \cdot v\right)\right)\right)}{v + -1}} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)}{v + -1} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot \left(\frac{3}{8} + v \cdot \frac{-1}{4}\right)\right)}{v + -1} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) \cdot \left(\frac{3}{8} + v \cdot \frac{-1}{4}\right)}{\color{blue}{v} + -1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(\frac{3}{8} + v \cdot \frac{-1}{4}\right)}{v + -1} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{\color{blue}{v} + -1} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot r\right) \cdot \left(r \cdot \left(w \cdot w\right)\right)}{\color{blue}{v} + -1} \]

      7. associate-/l* N/A

         \[\leadsto \left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot r\right) \cdot \color{blue}{\frac{r \cdot \left(w \cdot w\right)}{v + -1}} \]

      8. *-commutative N/A

         \[\leadsto \left(r \cdot \left(\frac{3}{8} + v \cdot \frac{-1}{4}\right)\right) \cdot \frac{\color{blue}{r \cdot \left(w \cdot w\right)}}{v + -1} \]

      9. metadata-eval N/A

         \[\leadsto \left(r \cdot \left(3 \cdot \frac{1}{8} + v \cdot \frac{-1}{4}\right)\right) \cdot \frac{r \cdot \left(\color{blue}{w} \cdot w\right)}{v + -1} \]

      10. metadata-eval N/A

          \[\leadsto \left(r \cdot \left(3 \cdot \frac{1}{8} + v \cdot \left(-2 \cdot \frac{1}{8}\right)\right)\right) \cdot \frac{r \cdot \left(w \cdot w\right)}{v + -1} \]

      11. associate-*l* N/A

          \[\leadsto \left(r \cdot \left(3 \cdot \frac{1}{8} + \left(v \cdot -2\right) \cdot \frac{1}{8}\right)\right) \cdot \frac{r \cdot \left(w \cdot \color{blue}{w}\right)}{v + -1} \]

      12. distribute-rgt-in N/A

          \[\leadsto \left(r \cdot \left(\frac{1}{8} \cdot \left(3 + v \cdot -2\right)\right)\right) \cdot \frac{r \cdot \color{blue}{\left(w \cdot w\right)}}{v + -1} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(r \cdot \left(\frac{1}{8} \cdot \left(3 + v \cdot -2\right)\right)\right), \color{blue}{\left(\frac{r \cdot \left(w \cdot w\right)}{v + -1}\right)}\right) \]

   9. Applied egg-rr 75.5%

      \[\leadsto \color{blue}{\left(r \cdot \left(0.375 + v \cdot -0.25\right)\right) \cdot \frac{w \cdot \left(r \cdot w\right)}{v + -1}} \]
   10. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto \left(r \cdot \left(\frac{3}{8} + v \cdot \frac{-1}{4}\right)\right) \cdot \left(w \cdot \color{blue}{\frac{r \cdot w}{v + -1}}\right) \]

       2. associate-*r* N/A

          \[\leadsto \left(\left(r \cdot \left(\frac{3}{8} + v \cdot \frac{-1}{4}\right)\right) \cdot w\right) \cdot \color{blue}{\frac{r \cdot w}{v + -1}} \]

       3. *-commutative N/A

          \[\leadsto \left(\left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot r\right) \cdot w\right) \cdot \frac{\color{blue}{r} \cdot w}{v + -1} \]

       4. associate-*r* N/A

          \[\leadsto \left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot \left(r \cdot w\right)\right) \cdot \frac{\color{blue}{r \cdot w}}{v + -1} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot \left(r \cdot w\right)\right), \color{blue}{\left(\frac{r \cdot w}{v + -1}\right)}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right), \left(r \cdot w\right)\right), \left(\frac{\color{blue}{r \cdot w}}{v + -1}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \left(v \cdot \frac{-1}{4}\right)\right), \left(r \cdot w\right)\right), \left(\frac{\color{blue}{r} \cdot w}{v + -1}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \left(r \cdot w\right)\right), \left(\frac{r \cdot w}{v + -1}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(r, w\right)\right), \left(\frac{r \cdot \color{blue}{w}}{v + -1}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(r, w\right)\right), \mathsf{/.f64}\left(\left(r \cdot w\right), \color{blue}{\left(v + -1\right)}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(r, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, w\right), \left(\color{blue}{v} + -1\right)\right)\right) \]

       12. +-lowering-+.f64 75.8%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(r, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{+.f64}\left(v, \color{blue}{-1}\right)\right)\right) \]

   11. Applied egg-rr 75.8%

       \[\leadsto \color{blue}{\left(\left(0.375 + v \cdot -0.25\right) \cdot \left(r \cdot w\right)\right) \cdot \frac{r \cdot w}{v + -1}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 2 \cdot 10^{-151}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{elif}\;r \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(w \cdot r\right) \cdot \left(0.375 + v \cdot -0.25\right)\right) \cdot \frac{w \cdot r}{v + -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 2 \cdot 10^{-151}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{elif}\;r \leq 2.55 \cdot 10^{+148}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot -0.375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(r \cdot \left(w \cdot \left(w \cdot r\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{v + -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 2e-151)
   (/ (/ 2.0 r) r)
   (if (<= r 2.55e+148)
     (+ (/ 2.0 (* r r)) (+ -1.5 (* (* r r) (* (* w w) -0.375))))
     (* (* r (* w (* w r))) (/ (+ 0.375 (* v -0.25)) (+ v -1.0))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 2e-151) {
		tmp = (2.0 / r) / r;
	} else if (r <= 2.55e+148) {
		tmp = (2.0 / (r * r)) + (-1.5 + ((r * r) * ((w * w) * -0.375)));
	} else {
		tmp = (r * (w * (w * r))) * ((0.375 + (v * -0.25)) / (v + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 2d-151) then
        tmp = (2.0d0 / r) / r
    else if (r <= 2.55d+148) then
        tmp = (2.0d0 / (r * r)) + ((-1.5d0) + ((r * r) * ((w * w) * (-0.375d0))))
    else
        tmp = (r * (w * (w * r))) * ((0.375d0 + (v * (-0.25d0))) / (v + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 2e-151) {
		tmp = (2.0 / r) / r;
	} else if (r <= 2.55e+148) {
		tmp = (2.0 / (r * r)) + (-1.5 + ((r * r) * ((w * w) * -0.375)));
	} else {
		tmp = (r * (w * (w * r))) * ((0.375 + (v * -0.25)) / (v + -1.0));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 2e-151:
		tmp = (2.0 / r) / r
	elif r <= 2.55e+148:
		tmp = (2.0 / (r * r)) + (-1.5 + ((r * r) * ((w * w) * -0.375)))
	else:
		tmp = (r * (w * (w * r))) * ((0.375 + (v * -0.25)) / (v + -1.0))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 2e-151)
		tmp = Float64(Float64(2.0 / r) / r);
	elseif (r <= 2.55e+148)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + Float64(-1.5 + Float64(Float64(r * r) * Float64(Float64(w * w) * -0.375))));
	else
		tmp = Float64(Float64(r * Float64(w * Float64(w * r))) * Float64(Float64(0.375 + Float64(v * -0.25)) / Float64(v + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 2e-151)
		tmp = (2.0 / r) / r;
	elseif (r <= 2.55e+148)
		tmp = (2.0 / (r * r)) + (-1.5 + ((r * r) * ((w * w) * -0.375)));
	else
		tmp = (r * (w * (w * r))) * ((0.375 + (v * -0.25)) / (v + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 2e-151], N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision], If[LessEqual[r, 2.55e+148], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(-1.5 + N[(N[(r * r), $MachinePrecision] * N[(N[(w * w), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(r * N[(w * N[(w * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if r < 1.9999999999999999e-151

   1. Initial program 84.8%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Step-by-step derivation

      1. associate--l- N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)} \]

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{2}{r \cdot r}\right), \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(r \cdot r\right)\right), \left(\color{blue}{3} - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

      7. associate--r+ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \color{blue}{\frac{9}{2}}\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2}\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + 3\right) - \frac{9}{2}\right)\right) \]

      10. associate--l+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(3 - \frac{9}{2}\right)}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \frac{-3}{2}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\frac{-9}{2} + \color{blue}{3}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\left(\mathsf{neg}\left(\frac{9}{2}\right)\right) + 3\right)\right)\right) \]

   3. Simplified 77.7%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\frac{\left(r \cdot r\right) \cdot \left(\left(0.375 + v \cdot -0.25\right) \cdot \left(w \cdot w\right)\right)}{v + -1} + -1.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left({r}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(r \cdot \color{blue}{r}\right)\right) \]

      3. *-lowering-*.f64 59.7%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, \color{blue}{r}\right)\right) \]

   7. Simplified 59.7%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r}} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{r}}{\color{blue}{r}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{r}\right), \color{blue}{r}\right) \]

      3. /-lowering-/.f64 59.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, r\right), r\right) \]

   9. Applied egg-rr 59.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{r}}{r}} \]

   if 1.9999999999999999e-151 < r < 2.54999999999999993e148

   1. Initial program 85.7%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Step-by-step derivation

      1. associate--l- N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)} \]

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{2}{r \cdot r}\right), \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(r \cdot r\right)\right), \left(\color{blue}{3} - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

      7. associate--r+ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \color{blue}{\frac{9}{2}}\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2}\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + 3\right) - \frac{9}{2}\right)\right) \]

      10. associate--l+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(3 - \frac{9}{2}\right)}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \frac{-3}{2}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\frac{-9}{2} + \color{blue}{3}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\left(\mathsf{neg}\left(\frac{9}{2}\right)\right) + 3\right)\right)\right) \]

   3. Simplified 81.2%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\frac{\left(r \cdot r\right) \cdot \left(\left(0.375 + v \cdot -0.25\right) \cdot \left(w \cdot w\right)\right)}{v + -1} + -1.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\color{blue}{\left(\frac{-3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)}, \frac{-3}{2}\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\left(\left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-3}{8}\right), \frac{-3}{2}\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\left({r}^{2} \cdot \left({w}^{2} \cdot \frac{-3}{8}\right)\right), \frac{-3}{2}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\left({r}^{2} \cdot \left(\frac{-3}{8} \cdot {w}^{2}\right)\right), \frac{-3}{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({r}^{2}\right), \left(\frac{-3}{8} \cdot {w}^{2}\right)\right), \frac{-3}{2}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(r \cdot r\right), \left(\frac{-3}{8} \cdot {w}^{2}\right)\right), \frac{-3}{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left(\frac{-3}{8} \cdot {w}^{2}\right)\right), \frac{-3}{2}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left({w}^{2} \cdot \frac{-3}{8}\right)\right), \frac{-3}{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left({w}^{2}\right), \frac{-3}{8}\right)\right), \frac{-3}{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left(w \cdot w\right), \frac{-3}{8}\right)\right), \frac{-3}{2}\right)\right) \]

      10. *-lowering-*.f64 79.8%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \frac{-3}{2}\right)\right) \]

   7. Simplified 79.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{\left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot -0.375\right)} + -1.5\right) \]

   if 2.54999999999999993e148 < r

   1. Initial program 87.3%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Step-by-step derivation

      1. associate--l- N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)} \]

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{2}{r \cdot r}\right), \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(r \cdot r\right)\right), \left(\color{blue}{3} - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

      7. associate--r+ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \color{blue}{\frac{9}{2}}\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2}\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + 3\right) - \frac{9}{2}\right)\right) \]

      10. associate--l+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(3 - \frac{9}{2}\right)}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \frac{-3}{2}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\frac{-9}{2} + \color{blue}{3}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\left(\mathsf{neg}\left(\frac{9}{2}\right)\right) + 3\right)\right)\right) \]

   3. Simplified 53.2%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\frac{\left(r \cdot r\right) \cdot \left(\left(0.375 + v \cdot -0.25\right) \cdot \left(w \cdot w\right)\right)}{v + -1} + -1.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in r around inf

      \[\leadsto \color{blue}{\frac{{r}^{2} \cdot \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)}{v - 1}} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left({r}^{2} \cdot \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \color{blue}{\left(v - 1\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({r}^{2}\right), \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(\color{blue}{v} - 1\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(r \cdot r\right), \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left(\left(w \cdot w\right) \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left(w \cdot \left(w \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right)\right), \left(v - 1\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \left(w \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right)\right), \left(v - 1\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right)\right), \left(v - 1\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{3}{8}, \left(\frac{-1}{4} \cdot v\right)\right)\right)\right)\right), \left(v - 1\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(\frac{-1}{4}, v\right)\right)\right)\right)\right), \left(v - 1\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(\frac{-1}{4}, v\right)\right)\right)\right)\right), \left(v + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(\frac{-1}{4}, v\right)\right)\right)\right)\right), \left(v + -1\right)\right) \]

      13. +-lowering-+.f64 53.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(\frac{-1}{4}, v\right)\right)\right)\right)\right), \mathsf{+.f64}\left(v, \color{blue}{-1}\right)\right) \]

   7. Simplified 53.6%

      \[\leadsto \color{blue}{\frac{\left(r \cdot r\right) \cdot \left(w \cdot \left(w \cdot \left(0.375 + -0.25 \cdot v\right)\right)\right)}{v + -1}} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)}{v + -1} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot \left(\frac{3}{8} + v \cdot \frac{-1}{4}\right)\right)}{v + -1} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) \cdot \left(\frac{3}{8} + v \cdot \frac{-1}{4}\right)}{\color{blue}{v} + -1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(\frac{3}{8} + v \cdot \frac{-1}{4}\right)}{v + -1} \]

      5. associate-/l* N/A

         \[\leadsto \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \color{blue}{\frac{\frac{3}{8} + v \cdot \frac{-1}{4}}{v + -1}} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right), \color{blue}{\left(\frac{\frac{3}{8} + v \cdot \frac{-1}{4}}{v + -1}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \left(r \cdot \left(w \cdot w\right)\right)\right), \left(\frac{\color{blue}{\frac{3}{8} + v \cdot \frac{-1}{4}}}{v + -1}\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \left(\left(r \cdot w\right) \cdot w\right)\right), \left(\frac{\frac{3}{8} + \color{blue}{v \cdot \frac{-1}{4}}}{v + -1}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \left(w \cdot \left(r \cdot w\right)\right)\right), \left(\frac{\frac{3}{8} + \color{blue}{v \cdot \frac{-1}{4}}}{v + -1}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, \left(r \cdot w\right)\right)\right), \left(\frac{\frac{3}{8} + \color{blue}{v \cdot \frac{-1}{4}}}{v + -1}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, w\right)\right)\right), \left(\frac{\frac{3}{8} + v \cdot \color{blue}{\frac{-1}{4}}}{v + -1}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, w\right)\right)\right), \left(\frac{3 \cdot \frac{1}{8} + v \cdot \frac{-1}{4}}{v + -1}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, w\right)\right)\right), \left(\frac{3 \cdot \frac{1}{8} + v \cdot \left(-2 \cdot \frac{1}{8}\right)}{v + -1}\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, w\right)\right)\right), \left(\frac{3 \cdot \frac{1}{8} + \left(v \cdot -2\right) \cdot \frac{1}{8}}{v + -1}\right)\right) \]

      15. distribute-rgt-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, w\right)\right)\right), \left(\frac{\frac{1}{8} \cdot \left(3 + v \cdot -2\right)}{\color{blue}{v} + -1}\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, w\right)\right)\right), \left(\frac{\frac{1}{8} \cdot \left(3 + -2 \cdot v\right)}{v + -1}\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, w\right)\right)\right), \left(\frac{\frac{1}{8} \cdot \left(3 + \left(\mathsf{neg}\left(2\right)\right) \cdot v\right)}{v + -1}\right)\right) \]

      18. cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, w\right)\right)\right), \left(\frac{\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)}{v + -1}\right)\right) \]

   9. Applied egg-rr 78.9%

      \[\leadsto \color{blue}{\left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{v + -1}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 2 \cdot 10^{-151}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{elif}\;r \leq 2.55 \cdot 10^{+148}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot -0.375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(r \cdot \left(w \cdot \left(w \cdot r\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{v + -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 92.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 10500000000000:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \left(w \cdot \left(w \cdot \left(r \cdot r\right)\right)\right) \cdot -0.375\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \left(\left(w \cdot r\right) \cdot \left(0.375 + v \cdot -0.25\right)\right) \cdot \frac{w \cdot r}{v + -1}\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 10500000000000.0)
   (+ (/ 2.0 (* r r)) (+ -1.5 (* (* w (* w (* r r))) -0.375)))
   (-
    (+ 3.0 (* (* (* w r) (+ 0.375 (* v -0.25))) (/ (* w r) (+ v -1.0))))
    4.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 10500000000000.0) {
		tmp = (2.0 / (r * r)) + (-1.5 + ((w * (w * (r * r))) * -0.375));
	} else {
		tmp = (3.0 + (((w * r) * (0.375 + (v * -0.25))) * ((w * r) / (v + -1.0)))) - 4.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 10500000000000.0d0) then
        tmp = (2.0d0 / (r * r)) + ((-1.5d0) + ((w * (w * (r * r))) * (-0.375d0)))
    else
        tmp = (3.0d0 + (((w * r) * (0.375d0 + (v * (-0.25d0)))) * ((w * r) / (v + (-1.0d0))))) - 4.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 10500000000000.0) {
		tmp = (2.0 / (r * r)) + (-1.5 + ((w * (w * (r * r))) * -0.375));
	} else {
		tmp = (3.0 + (((w * r) * (0.375 + (v * -0.25))) * ((w * r) / (v + -1.0)))) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 10500000000000.0:
		tmp = (2.0 / (r * r)) + (-1.5 + ((w * (w * (r * r))) * -0.375))
	else:
		tmp = (3.0 + (((w * r) * (0.375 + (v * -0.25))) * ((w * r) / (v + -1.0)))) - 4.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 10500000000000.0)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + Float64(-1.5 + Float64(Float64(w * Float64(w * Float64(r * r))) * -0.375)));
	else
		tmp = Float64(Float64(3.0 + Float64(Float64(Float64(w * r) * Float64(0.375 + Float64(v * -0.25))) * Float64(Float64(w * r) / Float64(v + -1.0)))) - 4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 10500000000000.0)
		tmp = (2.0 / (r * r)) + (-1.5 + ((w * (w * (r * r))) * -0.375));
	else
		tmp = (3.0 + (((w * r) * (0.375 + (v * -0.25))) * ((w * r) / (v + -1.0)))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 10500000000000.0], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(-1.5 + N[(N[(w * N[(w * N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 + N[(N[(N[(w * r), $MachinePrecision] * N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(w * r), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 1.05e13

   1. Initial program 84.6%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Step-by-step derivation

      1. associate--l- N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)} \]

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{2}{r \cdot r}\right), \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(r \cdot r\right)\right), \left(\color{blue}{3} - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

      7. associate--r+ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \color{blue}{\frac{9}{2}}\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2}\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + 3\right) - \frac{9}{2}\right)\right) \]

      10. associate--l+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(3 - \frac{9}{2}\right)}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \frac{-3}{2}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\frac{-9}{2} + \color{blue}{3}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\left(\mathsf{neg}\left(\frac{9}{2}\right)\right) + 3\right)\right)\right) \]

   3. Simplified 77.2%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\frac{\left(r \cdot r\right) \cdot \left(\left(0.375 + v \cdot -0.25\right) \cdot \left(w \cdot w\right)\right)}{v + -1} + -1.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\color{blue}{\left(\frac{-3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)}, \frac{-3}{2}\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\left(\left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-3}{8}\right), \frac{-3}{2}\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\left({r}^{2} \cdot \left({w}^{2} \cdot \frac{-3}{8}\right)\right), \frac{-3}{2}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\left({r}^{2} \cdot \left(\frac{-3}{8} \cdot {w}^{2}\right)\right), \frac{-3}{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({r}^{2}\right), \left(\frac{-3}{8} \cdot {w}^{2}\right)\right), \frac{-3}{2}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(r \cdot r\right), \left(\frac{-3}{8} \cdot {w}^{2}\right)\right), \frac{-3}{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left(\frac{-3}{8} \cdot {w}^{2}\right)\right), \frac{-3}{2}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left({w}^{2} \cdot \frac{-3}{8}\right)\right), \frac{-3}{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left({w}^{2}\right), \frac{-3}{8}\right)\right), \frac{-3}{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left(w \cdot w\right), \frac{-3}{8}\right)\right), \frac{-3}{2}\right)\right) \]

      10. *-lowering-*.f64 77.9%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \frac{-3}{2}\right)\right) \]

   7. Simplified 77.9%

      \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{\left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot -0.375\right)} + -1.5\right) \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\left(\left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) \cdot \frac{-3}{8}\right), \frac{-3}{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right), \frac{-3}{8}\right), \frac{-3}{2}\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\left(\left(r \cdot r\right) \cdot w\right) \cdot w\right), \frac{-3}{8}\right), \frac{-3}{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(r \cdot r\right) \cdot w\right), w\right), \frac{-3}{8}\right), \frac{-3}{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(r \cdot r\right), w\right), w\right), \frac{-3}{8}\right), \frac{-3}{2}\right)\right) \]

      6. *-lowering-*.f64 88.6%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), w\right), w\right), \frac{-3}{8}\right), \frac{-3}{2}\right)\right) \]

   9. Applied egg-rr 88.6%

      \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{\left(\left(\left(r \cdot r\right) \cdot w\right) \cdot w\right) \cdot -0.375} + -1.5\right) \]

   if 1.05e13 < r

   1. Initial program 87.6%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \left(\left(w \cdot w\right) \cdot \left(r \cdot r\right)\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

      2. unswap-sqr N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \mathsf{*.f64}\left(\left(w \cdot r\right), \left(w \cdot r\right)\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \mathsf{*.f64}\left(\left(r \cdot w\right), \left(w \cdot r\right)\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \left(w \cdot r\right)\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \left(r \cdot w\right)\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

      7. *-lowering-*.f64 91.2%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{*.f64}\left(r, w\right)\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

   4. Applied egg-rr 91.2%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}}{1 - v}\right) - 4.5 \]
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \left(\frac{\left(\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)}{1 - v}\right)\right), \frac{9}{2}\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \left(\left(\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(r \cdot w\right)\right) \cdot \frac{r \cdot w}{1 - v}\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\left(\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(r \cdot w\right)\right), \left(\frac{r \cdot w}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right), \left(r \cdot w\right)\right), \left(\frac{r \cdot w}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

      5. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{8} \cdot \left(3 + \left(\mathsf{neg}\left(2\right)\right) \cdot v\right)\right), \left(r \cdot w\right)\right), \left(\frac{r \cdot w}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{8} \cdot \left(3 + -2 \cdot v\right)\right), \left(r \cdot w\right)\right), \left(\frac{r \cdot w}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{8} \cdot \left(3 + v \cdot -2\right)\right), \left(r \cdot w\right)\right), \left(\frac{r \cdot w}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(3 \cdot \frac{1}{8} + \left(v \cdot -2\right) \cdot \frac{1}{8}\right), \left(r \cdot w\right)\right), \left(\frac{r \cdot w}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{3}{8} + \left(v \cdot -2\right) \cdot \frac{1}{8}\right), \left(r \cdot w\right)\right), \left(\frac{r \cdot w}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{3}{8} + v \cdot \left(-2 \cdot \frac{1}{8}\right)\right), \left(r \cdot w\right)\right), \left(\frac{r \cdot w}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right), \left(r \cdot w\right)\right), \left(\frac{r \cdot w}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{3}{8} + \frac{-1}{4} \cdot v\right), \left(r \cdot w\right)\right), \left(\frac{r \cdot w}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \left(\frac{-1}{4} \cdot v\right)\right), \left(r \cdot w\right)\right), \left(\frac{r \cdot w}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \left(v \cdot \frac{-1}{4}\right)\right), \left(r \cdot w\right)\right), \left(\frac{r \cdot w}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \left(r \cdot w\right)\right), \left(\frac{r \cdot w}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(r, w\right)\right), \left(\frac{r \cdot w}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(r, w\right)\right), \mathsf{/.f64}\left(\left(r \cdot w\right), \left(1 - v\right)\right)\right)\right), \frac{9}{2}\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(r, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, w\right), \left(1 - v\right)\right)\right)\right), \frac{9}{2}\right) \]

      19. --lowering--.f64 94.5%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(r, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{\_.f64}\left(1, v\right)\right)\right)\right), \frac{9}{2}\right) \]

   6. Applied egg-rr 94.5%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\left(0.375 + v \cdot -0.25\right) \cdot \left(r \cdot w\right)\right) \cdot \frac{r \cdot w}{1 - v}}\right) - 4.5 \]

   7. Taylor expanded in r around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\color{blue}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(r, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{\_.f64}\left(1, v\right)\right)\right)\right), \frac{9}{2}\right) \]
   8. Step-by-step derivation

   9. Simplified 94.5%

      \[\leadsto \left(\color{blue}{3} - \left(\left(0.375 + v \cdot -0.25\right) \cdot \left(r \cdot w\right)\right) \cdot \frac{r \cdot w}{1 - v}\right) - 4.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 10500000000000:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \left(w \cdot \left(w \cdot \left(r \cdot r\right)\right)\right) \cdot -0.375\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \left(\left(w \cdot r\right) \cdot \left(0.375 + v \cdot -0.25\right)\right) \cdot \frac{w \cdot r}{v + -1}\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 0.00175:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{elif}\;r \leq 3.2 \cdot 10^{+235}:\\ \;\;\;\;r \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot -0.375 + \frac{-1.5}{r \cdot r}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(r \cdot 0.375\right) \cdot \frac{w \cdot \left(w \cdot r\right)}{v + -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 0.00175)
   (+ (/ 2.0 (* r r)) -1.5)
   (if (<= r 3.2e+235)
     (* r (* r (+ (* (* w w) -0.375) (/ -1.5 (* r r)))))
     (* (* r 0.375) (/ (* w (* w r)) (+ v -1.0))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 0.00175) {
		tmp = (2.0 / (r * r)) + -1.5;
	} else if (r <= 3.2e+235) {
		tmp = r * (r * (((w * w) * -0.375) + (-1.5 / (r * r))));
	} else {
		tmp = (r * 0.375) * ((w * (w * r)) / (v + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 0.00175d0) then
        tmp = (2.0d0 / (r * r)) + (-1.5d0)
    else if (r <= 3.2d+235) then
        tmp = r * (r * (((w * w) * (-0.375d0)) + ((-1.5d0) / (r * r))))
    else
        tmp = (r * 0.375d0) * ((w * (w * r)) / (v + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 0.00175) {
		tmp = (2.0 / (r * r)) + -1.5;
	} else if (r <= 3.2e+235) {
		tmp = r * (r * (((w * w) * -0.375) + (-1.5 / (r * r))));
	} else {
		tmp = (r * 0.375) * ((w * (w * r)) / (v + -1.0));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 0.00175:
		tmp = (2.0 / (r * r)) + -1.5
	elif r <= 3.2e+235:
		tmp = r * (r * (((w * w) * -0.375) + (-1.5 / (r * r))))
	else:
		tmp = (r * 0.375) * ((w * (w * r)) / (v + -1.0))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 0.00175)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + -1.5);
	elseif (r <= 3.2e+235)
		tmp = Float64(r * Float64(r * Float64(Float64(Float64(w * w) * -0.375) + Float64(-1.5 / Float64(r * r)))));
	else
		tmp = Float64(Float64(r * 0.375) * Float64(Float64(w * Float64(w * r)) / Float64(v + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 0.00175)
		tmp = (2.0 / (r * r)) + -1.5;
	elseif (r <= 3.2e+235)
		tmp = r * (r * (((w * w) * -0.375) + (-1.5 / (r * r))));
	else
		tmp = (r * 0.375) * ((w * (w * r)) / (v + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 0.00175], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision], If[LessEqual[r, 3.2e+235], N[(r * N[(r * N[(N[(N[(w * w), $MachinePrecision] * -0.375), $MachinePrecision] + N[(-1.5 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(r * 0.375), $MachinePrecision] * N[(N[(w * N[(w * r), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if r < 0.00175000000000000004

   1. Initial program 85.0%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Step-by-step derivation

      1. associate--l- N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)} \]

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{2}{r \cdot r}\right), \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(r \cdot r\right)\right), \left(\color{blue}{3} - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

      7. associate--r+ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \color{blue}{\frac{9}{2}}\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2}\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + 3\right) - \frac{9}{2}\right)\right) \]

      10. associate--l+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(3 - \frac{9}{2}\right)}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \frac{-3}{2}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\frac{-9}{2} + \color{blue}{3}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\left(\mathsf{neg}\left(\frac{9}{2}\right)\right) + 3\right)\right)\right) \]

   3. Simplified 77.5%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\frac{\left(r \cdot r\right) \cdot \left(\left(0.375 + v \cdot -0.25\right) \cdot \left(w \cdot w\right)\right)}{v + -1} + -1.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in r around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \color{blue}{\frac{-3}{2}}\right) \]
   6. Step-by-step derivation

   7. Simplified 70.0%

      \[\leadsto \frac{2}{r \cdot r} + \color{blue}{-1.5} \]

   if 0.00175000000000000004 < r < 3.20000000000000006e235

   1. Initial program 88.6%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Step-by-step derivation

      1. associate--l- N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)} \]

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{2}{r \cdot r}\right), \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(r \cdot r\right)\right), \left(\color{blue}{3} - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

      7. associate--r+ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \color{blue}{\frac{9}{2}}\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2}\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + 3\right) - \frac{9}{2}\right)\right) \]

      10. associate--l+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(3 - \frac{9}{2}\right)}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \frac{-3}{2}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\frac{-9}{2} + \color{blue}{3}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\left(\mathsf{neg}\left(\frac{9}{2}\right)\right) + 3\right)\right)\right) \]

   3. Simplified 74.9%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\frac{\left(r \cdot r\right) \cdot \left(\left(0.375 + v \cdot -0.25\right) \cdot \left(w \cdot w\right)\right)}{v + -1} + -1.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\color{blue}{\left(\frac{-3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)}, \frac{-3}{2}\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\left(\left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-3}{8}\right), \frac{-3}{2}\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\left({r}^{2} \cdot \left({w}^{2} \cdot \frac{-3}{8}\right)\right), \frac{-3}{2}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\left({r}^{2} \cdot \left(\frac{-3}{8} \cdot {w}^{2}\right)\right), \frac{-3}{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({r}^{2}\right), \left(\frac{-3}{8} \cdot {w}^{2}\right)\right), \frac{-3}{2}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(r \cdot r\right), \left(\frac{-3}{8} \cdot {w}^{2}\right)\right), \frac{-3}{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left(\frac{-3}{8} \cdot {w}^{2}\right)\right), \frac{-3}{2}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left({w}^{2} \cdot \frac{-3}{8}\right)\right), \frac{-3}{2}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left({w}^{2}\right), \frac{-3}{8}\right)\right), \frac{-3}{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left(w \cdot w\right), \frac{-3}{8}\right)\right), \frac{-3}{2}\right)\right) \]

      10. *-lowering-*.f64 69.1%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \frac{-3}{2}\right)\right) \]

   7. Simplified 69.1%

      \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{\left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot -0.375\right)} + -1.5\right) \]

   8. Taylor expanded in r around inf

      \[\leadsto \color{blue}{{r}^{2} \cdot \left(\frac{-3}{8} \cdot {w}^{2} - \frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)} \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \left(r \cdot r\right) \cdot \left(\color{blue}{\frac{-3}{8} \cdot {w}^{2}} - \frac{3}{2} \cdot \frac{1}{{r}^{2}}\right) \]

      2. associate-*l* N/A

         \[\leadsto r \cdot \color{blue}{\left(r \cdot \left(\frac{-3}{8} \cdot {w}^{2} - \frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(r, \color{blue}{\left(r \cdot \left(\frac{-3}{8} \cdot {w}^{2} - \frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \color{blue}{\left(\frac{-3}{8} \cdot {w}^{2} - \frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)}\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \left(\frac{-3}{8} \cdot {w}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\left(\frac{-3}{8} \cdot {w}^{2}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\left({w}^{2} \cdot \frac{-3}{8}\right), \left(\mathsf{neg}\left(\color{blue}{\frac{3}{2} \cdot \frac{1}{{r}^{2}}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({w}^{2}\right), \frac{-3}{8}\right), \left(\mathsf{neg}\left(\color{blue}{\frac{3}{2} \cdot \frac{1}{{r}^{2}}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(w \cdot w\right), \frac{-3}{8}\right), \left(\mathsf{neg}\left(\color{blue}{\frac{3}{2}} \cdot \frac{1}{{r}^{2}}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right), \left(\mathsf{neg}\left(\color{blue}{\frac{3}{2}} \cdot \frac{1}{{r}^{2}}\right)\right)\right)\right)\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right), \left(\mathsf{neg}\left(\frac{\frac{3}{2} \cdot 1}{{r}^{2}}\right)\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right), \left(\mathsf{neg}\left(\frac{\frac{3}{2}}{{r}^{2}}\right)\right)\right)\right)\right) \]

      13. distribute-neg-frac N/A

          \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right), \left(\frac{\mathsf{neg}\left(\frac{3}{2}\right)}{\color{blue}{{r}^{2}}}\right)\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right), \left(\frac{\frac{-3}{2}}{{\color{blue}{r}}^{2}}\right)\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right), \mathsf{/.f64}\left(\frac{-3}{2}, \color{blue}{\left({r}^{2}\right)}\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right), \mathsf{/.f64}\left(\frac{-3}{2}, \left(r \cdot \color{blue}{r}\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 71.4%

          \[\leadsto \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right), \mathsf{/.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(r, \color{blue}{r}\right)\right)\right)\right)\right) \]

   10. Simplified 71.4%

       \[\leadsto \color{blue}{r \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot -0.375 + \frac{-1.5}{r \cdot r}\right)\right)} \]

   if 3.20000000000000006e235 < r

   1. Initial program 67.5%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Step-by-step derivation

      1. associate--l- N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)} \]

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{2}{r \cdot r}\right), \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(r \cdot r\right)\right), \left(\color{blue}{3} - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

      7. associate--r+ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \color{blue}{\frac{9}{2}}\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2}\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + 3\right) - \frac{9}{2}\right)\right) \]

      10. associate--l+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(3 - \frac{9}{2}\right)}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \frac{-3}{2}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\frac{-9}{2} + \color{blue}{3}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\left(\mathsf{neg}\left(\frac{9}{2}\right)\right) + 3\right)\right)\right) \]

   3. Simplified 51.1%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\frac{\left(r \cdot r\right) \cdot \left(\left(0.375 + v \cdot -0.25\right) \cdot \left(w \cdot w\right)\right)}{v + -1} + -1.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in r around inf

      \[\leadsto \color{blue}{\frac{{r}^{2} \cdot \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)}{v - 1}} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left({r}^{2} \cdot \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \color{blue}{\left(v - 1\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({r}^{2}\right), \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(\color{blue}{v} - 1\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(r \cdot r\right), \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left(\left(w \cdot w\right) \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left(w \cdot \left(w \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right)\right), \left(v - 1\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \left(w \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right)\right), \left(v - 1\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right)\right), \left(v - 1\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{3}{8}, \left(\frac{-1}{4} \cdot v\right)\right)\right)\right)\right), \left(v - 1\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(\frac{-1}{4}, v\right)\right)\right)\right)\right), \left(v - 1\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(\frac{-1}{4}, v\right)\right)\right)\right)\right), \left(v + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(\frac{-1}{4}, v\right)\right)\right)\right)\right), \left(v + -1\right)\right) \]

      13. +-lowering-+.f64 51.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(\frac{-1}{4}, v\right)\right)\right)\right)\right), \mathsf{+.f64}\left(v, \color{blue}{-1}\right)\right) \]

   7. Simplified 51.9%

      \[\leadsto \color{blue}{\frac{\left(r \cdot r\right) \cdot \left(w \cdot \left(w \cdot \left(0.375 + -0.25 \cdot v\right)\right)\right)}{v + -1}} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)}{v + -1} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot \left(\frac{3}{8} + v \cdot \frac{-1}{4}\right)\right)}{v + -1} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) \cdot \left(\frac{3}{8} + v \cdot \frac{-1}{4}\right)}{\color{blue}{v} + -1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(\frac{3}{8} + v \cdot \frac{-1}{4}\right)}{v + -1} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{\color{blue}{v} + -1} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot r\right) \cdot \left(r \cdot \left(w \cdot w\right)\right)}{\color{blue}{v} + -1} \]

      7. associate-/l* N/A

         \[\leadsto \left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot r\right) \cdot \color{blue}{\frac{r \cdot \left(w \cdot w\right)}{v + -1}} \]

      8. *-commutative N/A

         \[\leadsto \left(r \cdot \left(\frac{3}{8} + v \cdot \frac{-1}{4}\right)\right) \cdot \frac{\color{blue}{r \cdot \left(w \cdot w\right)}}{v + -1} \]

      9. metadata-eval N/A

         \[\leadsto \left(r \cdot \left(3 \cdot \frac{1}{8} + v \cdot \frac{-1}{4}\right)\right) \cdot \frac{r \cdot \left(\color{blue}{w} \cdot w\right)}{v + -1} \]

      10. metadata-eval N/A

          \[\leadsto \left(r \cdot \left(3 \cdot \frac{1}{8} + v \cdot \left(-2 \cdot \frac{1}{8}\right)\right)\right) \cdot \frac{r \cdot \left(w \cdot w\right)}{v + -1} \]

      11. associate-*l* N/A

          \[\leadsto \left(r \cdot \left(3 \cdot \frac{1}{8} + \left(v \cdot -2\right) \cdot \frac{1}{8}\right)\right) \cdot \frac{r \cdot \left(w \cdot \color{blue}{w}\right)}{v + -1} \]

      12. distribute-rgt-in N/A

          \[\leadsto \left(r \cdot \left(\frac{1}{8} \cdot \left(3 + v \cdot -2\right)\right)\right) \cdot \frac{r \cdot \color{blue}{\left(w \cdot w\right)}}{v + -1} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(r \cdot \left(\frac{1}{8} \cdot \left(3 + v \cdot -2\right)\right)\right), \color{blue}{\left(\frac{r \cdot \left(w \cdot w\right)}{v + -1}\right)}\right) \]

   9. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\left(r \cdot \left(0.375 + v \cdot -0.25\right)\right) \cdot \frac{w \cdot \left(r \cdot w\right)}{v + -1}} \]

   10. Taylor expanded in v around 0

       \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{3}{8} \cdot r\right)}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, w\right)\right), \mathsf{+.f64}\left(v, -1\right)\right)\right) \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(r \cdot \frac{3}{8}\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, w\right)\right)}, \mathsf{+.f64}\left(v, -1\right)\right)\right) \]

       2. *-lowering-*.f64 73.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \frac{3}{8}\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, w\right)\right)}, \mathsf{+.f64}\left(v, -1\right)\right)\right) \]

   12. Simplified 73.1%

       \[\leadsto \color{blue}{\left(r \cdot 0.375\right)} \cdot \frac{w \cdot \left(r \cdot w\right)}{v + -1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 0.00175:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{elif}\;r \leq 3.2 \cdot 10^{+235}:\\ \;\;\;\;r \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot -0.375 + \frac{-1.5}{r \cdot r}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(r \cdot 0.375\right) \cdot \frac{w \cdot \left(w \cdot r\right)}{v + -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 5.1 \cdot 10^{+57}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{elif}\;r \leq 4 \cdot 10^{+235}:\\ \;\;\;\;-0.25 \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(r \cdot 0.375\right) \cdot \frac{w \cdot \left(w \cdot r\right)}{v + -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 5.1e+57)
   (+ (/ 2.0 (* r r)) -1.5)
   (if (<= r 4e+235)
     (* -0.25 (* r (* (* w w) r)))
     (* (* r 0.375) (/ (* w (* w r)) (+ v -1.0))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 5.1e+57) {
		tmp = (2.0 / (r * r)) + -1.5;
	} else if (r <= 4e+235) {
		tmp = -0.25 * (r * ((w * w) * r));
	} else {
		tmp = (r * 0.375) * ((w * (w * r)) / (v + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 5.1d+57) then
        tmp = (2.0d0 / (r * r)) + (-1.5d0)
    else if (r <= 4d+235) then
        tmp = (-0.25d0) * (r * ((w * w) * r))
    else
        tmp = (r * 0.375d0) * ((w * (w * r)) / (v + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 5.1e+57) {
		tmp = (2.0 / (r * r)) + -1.5;
	} else if (r <= 4e+235) {
		tmp = -0.25 * (r * ((w * w) * r));
	} else {
		tmp = (r * 0.375) * ((w * (w * r)) / (v + -1.0));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 5.1e+57:
		tmp = (2.0 / (r * r)) + -1.5
	elif r <= 4e+235:
		tmp = -0.25 * (r * ((w * w) * r))
	else:
		tmp = (r * 0.375) * ((w * (w * r)) / (v + -1.0))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 5.1e+57)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + -1.5);
	elseif (r <= 4e+235)
		tmp = Float64(-0.25 * Float64(r * Float64(Float64(w * w) * r)));
	else
		tmp = Float64(Float64(r * 0.375) * Float64(Float64(w * Float64(w * r)) / Float64(v + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 5.1e+57)
		tmp = (2.0 / (r * r)) + -1.5;
	elseif (r <= 4e+235)
		tmp = -0.25 * (r * ((w * w) * r));
	else
		tmp = (r * 0.375) * ((w * (w * r)) / (v + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 5.1e+57], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision], If[LessEqual[r, 4e+235], N[(-0.25 * N[(r * N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(r * 0.375), $MachinePrecision] * N[(N[(w * N[(w * r), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if r < 5.10000000000000023e57

   1. Initial program 85.4%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Step-by-step derivation

      1. associate--l- N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)} \]

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{2}{r \cdot r}\right), \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(r \cdot r\right)\right), \left(\color{blue}{3} - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

      7. associate--r+ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \color{blue}{\frac{9}{2}}\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2}\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + 3\right) - \frac{9}{2}\right)\right) \]

      10. associate--l+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(3 - \frac{9}{2}\right)}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \frac{-3}{2}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\frac{-9}{2} + \color{blue}{3}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\left(\mathsf{neg}\left(\frac{9}{2}\right)\right) + 3\right)\right)\right) \]

   3. Simplified 78.3%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\frac{\left(r \cdot r\right) \cdot \left(\left(0.375 + v \cdot -0.25\right) \cdot \left(w \cdot w\right)\right)}{v + -1} + -1.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in r around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \color{blue}{\frac{-3}{2}}\right) \]
   6. Step-by-step derivation

   7. Simplified 69.0%

      \[\leadsto \frac{2}{r \cdot r} + \color{blue}{-1.5} \]

   if 5.10000000000000023e57 < r < 4.0000000000000002e235

   1. Initial program 87.5%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Step-by-step derivation

      1. associate--l- N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)} \]

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{2}{r \cdot r}\right), \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(r \cdot r\right)\right), \left(\color{blue}{3} - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

      7. associate--r+ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \color{blue}{\frac{9}{2}}\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2}\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + 3\right) - \frac{9}{2}\right)\right) \]

      10. associate--l+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(3 - \frac{9}{2}\right)}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \frac{-3}{2}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\frac{-9}{2} + \color{blue}{3}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\left(\mathsf{neg}\left(\frac{9}{2}\right)\right) + 3\right)\right)\right) \]

   3. Simplified 69.6%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\frac{\left(r \cdot r\right) \cdot \left(\left(0.375 + v \cdot -0.25\right) \cdot \left(w \cdot w\right)\right)}{v + -1} + -1.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in r around inf

      \[\leadsto \color{blue}{\frac{{r}^{2} \cdot \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)}{v - 1}} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left({r}^{2} \cdot \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \color{blue}{\left(v - 1\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({r}^{2}\right), \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(\color{blue}{v} - 1\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(r \cdot r\right), \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left(\left(w \cdot w\right) \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left(w \cdot \left(w \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right)\right), \left(v - 1\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \left(w \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right)\right), \left(v - 1\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right)\right), \left(v - 1\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{3}{8}, \left(\frac{-1}{4} \cdot v\right)\right)\right)\right)\right), \left(v - 1\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(\frac{-1}{4}, v\right)\right)\right)\right)\right), \left(v - 1\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(\frac{-1}{4}, v\right)\right)\right)\right)\right), \left(v + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(\frac{-1}{4}, v\right)\right)\right)\right)\right), \left(v + -1\right)\right) \]

      13. +-lowering-+.f64 54.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(\frac{-1}{4}, v\right)\right)\right)\right)\right), \mathsf{+.f64}\left(v, \color{blue}{-1}\right)\right) \]

   7. Simplified 54.1%

      \[\leadsto \color{blue}{\frac{\left(r \cdot r\right) \cdot \left(w \cdot \left(w \cdot \left(0.375 + -0.25 \cdot v\right)\right)\right)}{v + -1}} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)}{v + -1} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot \left(\frac{3}{8} + v \cdot \frac{-1}{4}\right)\right)}{v + -1} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) \cdot \left(\frac{3}{8} + v \cdot \frac{-1}{4}\right)}{\color{blue}{v} + -1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(\frac{3}{8} + v \cdot \frac{-1}{4}\right)}{v + -1} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{\color{blue}{v} + -1} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot r\right) \cdot \left(r \cdot \left(w \cdot w\right)\right)}{\color{blue}{v} + -1} \]

      7. associate-/l* N/A

         \[\leadsto \left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot r\right) \cdot \color{blue}{\frac{r \cdot \left(w \cdot w\right)}{v + -1}} \]

      8. *-commutative N/A

         \[\leadsto \left(r \cdot \left(\frac{3}{8} + v \cdot \frac{-1}{4}\right)\right) \cdot \frac{\color{blue}{r \cdot \left(w \cdot w\right)}}{v + -1} \]

      9. metadata-eval N/A

         \[\leadsto \left(r \cdot \left(3 \cdot \frac{1}{8} + v \cdot \frac{-1}{4}\right)\right) \cdot \frac{r \cdot \left(\color{blue}{w} \cdot w\right)}{v + -1} \]

      10. metadata-eval N/A

          \[\leadsto \left(r \cdot \left(3 \cdot \frac{1}{8} + v \cdot \left(-2 \cdot \frac{1}{8}\right)\right)\right) \cdot \frac{r \cdot \left(w \cdot w\right)}{v + -1} \]

      11. associate-*l* N/A

          \[\leadsto \left(r \cdot \left(3 \cdot \frac{1}{8} + \left(v \cdot -2\right) \cdot \frac{1}{8}\right)\right) \cdot \frac{r \cdot \left(w \cdot \color{blue}{w}\right)}{v + -1} \]

      12. distribute-rgt-in N/A

          \[\leadsto \left(r \cdot \left(\frac{1}{8} \cdot \left(3 + v \cdot -2\right)\right)\right) \cdot \frac{r \cdot \color{blue}{\left(w \cdot w\right)}}{v + -1} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(r \cdot \left(\frac{1}{8} \cdot \left(3 + v \cdot -2\right)\right)\right), \color{blue}{\left(\frac{r \cdot \left(w \cdot w\right)}{v + -1}\right)}\right) \]

   9. Applied egg-rr 63.7%

      \[\leadsto \color{blue}{\left(r \cdot \left(0.375 + v \cdot -0.25\right)\right) \cdot \frac{w \cdot \left(r \cdot w\right)}{v + -1}} \]

   10. Taylor expanded in v around inf

       \[\leadsto \color{blue}{\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)} \]
   11. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{4}, \color{blue}{\left({r}^{2} \cdot {w}^{2}\right)}\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{4}, \left(\left(r \cdot r\right) \cdot {\color{blue}{w}}^{2}\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{4}, \left(r \cdot \color{blue}{\left(r \cdot {w}^{2}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(r, \color{blue}{\left(r \cdot {w}^{2}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \color{blue}{\left({w}^{2}\right)}\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \left(w \cdot \color{blue}{w}\right)\right)\right)\right) \]

       7. *-lowering-*.f64 65.6%

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, \color{blue}{w}\right)\right)\right)\right) \]

   12. Simplified 65.6%

       \[\leadsto \color{blue}{-0.25 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)} \]

   if 4.0000000000000002e235 < r

   1. Initial program 67.5%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Step-by-step derivation

      1. associate--l- N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)} \]

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{2}{r \cdot r}\right), \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(r \cdot r\right)\right), \left(\color{blue}{3} - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

      7. associate--r+ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \color{blue}{\frac{9}{2}}\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2}\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + 3\right) - \frac{9}{2}\right)\right) \]

      10. associate--l+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(3 - \frac{9}{2}\right)}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \frac{-3}{2}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\frac{-9}{2} + \color{blue}{3}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\left(\mathsf{neg}\left(\frac{9}{2}\right)\right) + 3\right)\right)\right) \]

   3. Simplified 51.1%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\frac{\left(r \cdot r\right) \cdot \left(\left(0.375 + v \cdot -0.25\right) \cdot \left(w \cdot w\right)\right)}{v + -1} + -1.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in r around inf

      \[\leadsto \color{blue}{\frac{{r}^{2} \cdot \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)}{v - 1}} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left({r}^{2} \cdot \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \color{blue}{\left(v - 1\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({r}^{2}\right), \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(\color{blue}{v} - 1\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(r \cdot r\right), \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left(\left(w \cdot w\right) \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left(w \cdot \left(w \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right)\right), \left(v - 1\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \left(w \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right)\right), \left(v - 1\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right)\right), \left(v - 1\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{3}{8}, \left(\frac{-1}{4} \cdot v\right)\right)\right)\right)\right), \left(v - 1\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(\frac{-1}{4}, v\right)\right)\right)\right)\right), \left(v - 1\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(\frac{-1}{4}, v\right)\right)\right)\right)\right), \left(v + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(\frac{-1}{4}, v\right)\right)\right)\right)\right), \left(v + -1\right)\right) \]

      13. +-lowering-+.f64 51.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(\frac{-1}{4}, v\right)\right)\right)\right)\right), \mathsf{+.f64}\left(v, \color{blue}{-1}\right)\right) \]

   7. Simplified 51.9%

      \[\leadsto \color{blue}{\frac{\left(r \cdot r\right) \cdot \left(w \cdot \left(w \cdot \left(0.375 + -0.25 \cdot v\right)\right)\right)}{v + -1}} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)}{v + -1} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot \left(\frac{3}{8} + v \cdot \frac{-1}{4}\right)\right)}{v + -1} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) \cdot \left(\frac{3}{8} + v \cdot \frac{-1}{4}\right)}{\color{blue}{v} + -1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(\frac{3}{8} + v \cdot \frac{-1}{4}\right)}{v + -1} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{\color{blue}{v} + -1} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot r\right) \cdot \left(r \cdot \left(w \cdot w\right)\right)}{\color{blue}{v} + -1} \]

      7. associate-/l* N/A

         \[\leadsto \left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot r\right) \cdot \color{blue}{\frac{r \cdot \left(w \cdot w\right)}{v + -1}} \]

      8. *-commutative N/A

         \[\leadsto \left(r \cdot \left(\frac{3}{8} + v \cdot \frac{-1}{4}\right)\right) \cdot \frac{\color{blue}{r \cdot \left(w \cdot w\right)}}{v + -1} \]

      9. metadata-eval N/A

         \[\leadsto \left(r \cdot \left(3 \cdot \frac{1}{8} + v \cdot \frac{-1}{4}\right)\right) \cdot \frac{r \cdot \left(\color{blue}{w} \cdot w\right)}{v + -1} \]

      10. metadata-eval N/A

          \[\leadsto \left(r \cdot \left(3 \cdot \frac{1}{8} + v \cdot \left(-2 \cdot \frac{1}{8}\right)\right)\right) \cdot \frac{r \cdot \left(w \cdot w\right)}{v + -1} \]

      11. associate-*l* N/A

          \[\leadsto \left(r \cdot \left(3 \cdot \frac{1}{8} + \left(v \cdot -2\right) \cdot \frac{1}{8}\right)\right) \cdot \frac{r \cdot \left(w \cdot \color{blue}{w}\right)}{v + -1} \]

      12. distribute-rgt-in N/A

          \[\leadsto \left(r \cdot \left(\frac{1}{8} \cdot \left(3 + v \cdot -2\right)\right)\right) \cdot \frac{r \cdot \color{blue}{\left(w \cdot w\right)}}{v + -1} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(r \cdot \left(\frac{1}{8} \cdot \left(3 + v \cdot -2\right)\right)\right), \color{blue}{\left(\frac{r \cdot \left(w \cdot w\right)}{v + -1}\right)}\right) \]

   9. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\left(r \cdot \left(0.375 + v \cdot -0.25\right)\right) \cdot \frac{w \cdot \left(r \cdot w\right)}{v + -1}} \]

   10. Taylor expanded in v around 0

       \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{3}{8} \cdot r\right)}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, w\right)\right), \mathsf{+.f64}\left(v, -1\right)\right)\right) \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(r \cdot \frac{3}{8}\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, w\right)\right)}, \mathsf{+.f64}\left(v, -1\right)\right)\right) \]

       2. *-lowering-*.f64 73.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \frac{3}{8}\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, w\right)\right)}, \mathsf{+.f64}\left(v, -1\right)\right)\right) \]

   12. Simplified 73.1%

       \[\leadsto \color{blue}{\left(r \cdot 0.375\right)} \cdot \frac{w \cdot \left(r \cdot w\right)}{v + -1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 5.1 \cdot 10^{+57}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{elif}\;r \leq 4 \cdot 10^{+235}:\\ \;\;\;\;-0.25 \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(r \cdot 0.375\right) \cdot \frac{w \cdot \left(w \cdot r\right)}{v + -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 65.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 5.1 \cdot 10^{+57}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 5.1e+57) (+ (/ 2.0 (* r r)) -1.5) (* -0.25 (* r (* (* w w) r)))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 5.1e+57) {
		tmp = (2.0 / (r * r)) + -1.5;
	} else {
		tmp = -0.25 * (r * ((w * w) * r));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 5.1d+57) then
        tmp = (2.0d0 / (r * r)) + (-1.5d0)
    else
        tmp = (-0.25d0) * (r * ((w * w) * r))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 5.1e+57) {
		tmp = (2.0 / (r * r)) + -1.5;
	} else {
		tmp = -0.25 * (r * ((w * w) * r));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 5.1e+57:
		tmp = (2.0 / (r * r)) + -1.5
	else:
		tmp = -0.25 * (r * ((w * w) * r))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 5.1e+57)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + -1.5);
	else
		tmp = Float64(-0.25 * Float64(r * Float64(Float64(w * w) * r)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 5.1e+57)
		tmp = (2.0 / (r * r)) + -1.5;
	else
		tmp = -0.25 * (r * ((w * w) * r));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 5.1e+57], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision], N[(-0.25 * N[(r * N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 5.10000000000000023e57

   1. Initial program 85.4%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Step-by-step derivation

      1. associate--l- N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)} \]

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{2}{r \cdot r}\right), \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(r \cdot r\right)\right), \left(\color{blue}{3} - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

      7. associate--r+ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \color{blue}{\frac{9}{2}}\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2}\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + 3\right) - \frac{9}{2}\right)\right) \]

      10. associate--l+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(3 - \frac{9}{2}\right)}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \frac{-3}{2}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\frac{-9}{2} + \color{blue}{3}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\left(\mathsf{neg}\left(\frac{9}{2}\right)\right) + 3\right)\right)\right) \]

   3. Simplified 78.3%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\frac{\left(r \cdot r\right) \cdot \left(\left(0.375 + v \cdot -0.25\right) \cdot \left(w \cdot w\right)\right)}{v + -1} + -1.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in r around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \color{blue}{\frac{-3}{2}}\right) \]
   6. Step-by-step derivation

   7. Simplified 69.0%

      \[\leadsto \frac{2}{r \cdot r} + \color{blue}{-1.5} \]

   if 5.10000000000000023e57 < r

   1. Initial program 84.8%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Step-by-step derivation

      1. associate--l- N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)} \]

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{2}{r \cdot r}\right), \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(r \cdot r\right)\right), \left(\color{blue}{3} - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

      7. associate--r+ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \color{blue}{\frac{9}{2}}\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2}\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + 3\right) - \frac{9}{2}\right)\right) \]

      10. associate--l+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(3 - \frac{9}{2}\right)}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \frac{-3}{2}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\frac{-9}{2} + \color{blue}{3}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\left(\mathsf{neg}\left(\frac{9}{2}\right)\right) + 3\right)\right)\right) \]

   3. Simplified 67.1%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\frac{\left(r \cdot r\right) \cdot \left(\left(0.375 + v \cdot -0.25\right) \cdot \left(w \cdot w\right)\right)}{v + -1} + -1.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in r around inf

      \[\leadsto \color{blue}{\frac{{r}^{2} \cdot \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)}{v - 1}} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left({r}^{2} \cdot \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \color{blue}{\left(v - 1\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({r}^{2}\right), \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(\color{blue}{v} - 1\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(r \cdot r\right), \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left({w}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left(\left(w \cdot w\right) \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right), \left(v - 1\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left(w \cdot \left(w \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right)\right), \left(v - 1\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \left(w \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right)\right), \left(v - 1\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)\right)\right), \left(v - 1\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{3}{8}, \left(\frac{-1}{4} \cdot v\right)\right)\right)\right)\right), \left(v - 1\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(\frac{-1}{4}, v\right)\right)\right)\right)\right), \left(v - 1\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(\frac{-1}{4}, v\right)\right)\right)\right)\right), \left(v + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(\frac{-1}{4}, v\right)\right)\right)\right)\right), \left(v + -1\right)\right) \]

      13. +-lowering-+.f64 53.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(\frac{-1}{4}, v\right)\right)\right)\right)\right), \mathsf{+.f64}\left(v, \color{blue}{-1}\right)\right) \]

   7. Simplified 53.8%

      \[\leadsto \color{blue}{\frac{\left(r \cdot r\right) \cdot \left(w \cdot \left(w \cdot \left(0.375 + -0.25 \cdot v\right)\right)\right)}{v + -1}} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)\right)}{v + -1} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot \left(\frac{3}{8} + v \cdot \frac{-1}{4}\right)\right)}{v + -1} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) \cdot \left(\frac{3}{8} + v \cdot \frac{-1}{4}\right)}{\color{blue}{v} + -1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(\frac{3}{8} + v \cdot \frac{-1}{4}\right)}{v + -1} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{\color{blue}{v} + -1} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot r\right) \cdot \left(r \cdot \left(w \cdot w\right)\right)}{\color{blue}{v} + -1} \]

      7. associate-/l* N/A

         \[\leadsto \left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot r\right) \cdot \color{blue}{\frac{r \cdot \left(w \cdot w\right)}{v + -1}} \]

      8. *-commutative N/A

         \[\leadsto \left(r \cdot \left(\frac{3}{8} + v \cdot \frac{-1}{4}\right)\right) \cdot \frac{\color{blue}{r \cdot \left(w \cdot w\right)}}{v + -1} \]

      9. metadata-eval N/A

         \[\leadsto \left(r \cdot \left(3 \cdot \frac{1}{8} + v \cdot \frac{-1}{4}\right)\right) \cdot \frac{r \cdot \left(\color{blue}{w} \cdot w\right)}{v + -1} \]

      10. metadata-eval N/A

          \[\leadsto \left(r \cdot \left(3 \cdot \frac{1}{8} + v \cdot \left(-2 \cdot \frac{1}{8}\right)\right)\right) \cdot \frac{r \cdot \left(w \cdot w\right)}{v + -1} \]

      11. associate-*l* N/A

          \[\leadsto \left(r \cdot \left(3 \cdot \frac{1}{8} + \left(v \cdot -2\right) \cdot \frac{1}{8}\right)\right) \cdot \frac{r \cdot \left(w \cdot \color{blue}{w}\right)}{v + -1} \]

      12. distribute-rgt-in N/A

          \[\leadsto \left(r \cdot \left(\frac{1}{8} \cdot \left(3 + v \cdot -2\right)\right)\right) \cdot \frac{r \cdot \color{blue}{\left(w \cdot w\right)}}{v + -1} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(r \cdot \left(\frac{1}{8} \cdot \left(3 + v \cdot -2\right)\right)\right), \color{blue}{\left(\frac{r \cdot \left(w \cdot w\right)}{v + -1}\right)}\right) \]

   9. Applied egg-rr 68.6%

      \[\leadsto \color{blue}{\left(r \cdot \left(0.375 + v \cdot -0.25\right)\right) \cdot \frac{w \cdot \left(r \cdot w\right)}{v + -1}} \]

   10. Taylor expanded in v around inf

       \[\leadsto \color{blue}{\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)} \]
   11. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{4}, \color{blue}{\left({r}^{2} \cdot {w}^{2}\right)}\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{4}, \left(\left(r \cdot r\right) \cdot {\color{blue}{w}}^{2}\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{4}, \left(r \cdot \color{blue}{\left(r \cdot {w}^{2}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(r, \color{blue}{\left(r \cdot {w}^{2}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \color{blue}{\left({w}^{2}\right)}\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \left(w \cdot \color{blue}{w}\right)\right)\right)\right) \]

       7. *-lowering-*.f64 64.0%

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, \color{blue}{w}\right)\right)\right)\right) \]

   12. Simplified 64.0%

       \[\leadsto \color{blue}{-0.25 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 5.1 \cdot 10^{+57}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 50.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 0.066:\\ \;\;\;\;\frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;-1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 (if (<= r 0.066) (/ 2.0 (* r r)) -1.5))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 0.066) {
		tmp = 2.0 / (r * r);
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 0.066d0) then
        tmp = 2.0d0 / (r * r)
    else
        tmp = -1.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 0.066) {
		tmp = 2.0 / (r * r);
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 0.066:
		tmp = 2.0 / (r * r)
	else:
		tmp = -1.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 0.066)
		tmp = Float64(2.0 / Float64(r * r));
	else
		tmp = -1.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 0.066)
		tmp = 2.0 / (r * r);
	else
		tmp = -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 0.066], N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision], -1.5]

Derivation

1. Split input into 2 regimes

2. if r < 0.066000000000000003

   1. Initial program 85.0%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Step-by-step derivation

      1. associate--l- N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)} \]

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{2}{r \cdot r}\right), \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(r \cdot r\right)\right), \left(\color{blue}{3} - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

      7. associate--r+ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \color{blue}{\frac{9}{2}}\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2}\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + 3\right) - \frac{9}{2}\right)\right) \]

      10. associate--l+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(3 - \frac{9}{2}\right)}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \frac{-3}{2}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\frac{-9}{2} + \color{blue}{3}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\left(\mathsf{neg}\left(\frac{9}{2}\right)\right) + 3\right)\right)\right) \]

   3. Simplified 77.5%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\frac{\left(r \cdot r\right) \cdot \left(\left(0.375 + v \cdot -0.25\right) \cdot \left(w \cdot w\right)\right)}{v + -1} + -1.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left({r}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(r \cdot \color{blue}{r}\right)\right) \]

      3. *-lowering-*.f64 62.5%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, \color{blue}{r}\right)\right) \]

   7. Simplified 62.5%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r}} \]

   if 0.066000000000000003 < r

   1. Initial program 86.3%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \left(\left(w \cdot w\right) \cdot \left(r \cdot r\right)\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

      2. unswap-sqr N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \mathsf{*.f64}\left(\left(w \cdot r\right), \left(w \cdot r\right)\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \mathsf{*.f64}\left(\left(r \cdot w\right), \left(w \cdot r\right)\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \left(w \cdot r\right)\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \left(r \cdot w\right)\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

      7. *-lowering-*.f64 89.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{*.f64}\left(r, w\right)\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

   4. Applied egg-rr 89.8%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}}{1 - v}\right) - 4.5 \]
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \left(\frac{\left(\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)}{1 - v}\right)\right), \frac{9}{2}\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \left(\left(\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(r \cdot w\right)\right) \cdot \frac{r \cdot w}{1 - v}\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\left(\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(r \cdot w\right)\right), \left(\frac{r \cdot w}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right), \left(r \cdot w\right)\right), \left(\frac{r \cdot w}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

      5. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{8} \cdot \left(3 + \left(\mathsf{neg}\left(2\right)\right) \cdot v\right)\right), \left(r \cdot w\right)\right), \left(\frac{r \cdot w}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{8} \cdot \left(3 + -2 \cdot v\right)\right), \left(r \cdot w\right)\right), \left(\frac{r \cdot w}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{8} \cdot \left(3 + v \cdot -2\right)\right), \left(r \cdot w\right)\right), \left(\frac{r \cdot w}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

      8. distribute-rgt-in N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(3 \cdot \frac{1}{8} + \left(v \cdot -2\right) \cdot \frac{1}{8}\right), \left(r \cdot w\right)\right), \left(\frac{r \cdot w}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{3}{8} + \left(v \cdot -2\right) \cdot \frac{1}{8}\right), \left(r \cdot w\right)\right), \left(\frac{r \cdot w}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{3}{8} + v \cdot \left(-2 \cdot \frac{1}{8}\right)\right), \left(r \cdot w\right)\right), \left(\frac{r \cdot w}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right), \left(r \cdot w\right)\right), \left(\frac{r \cdot w}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{3}{8} + \frac{-1}{4} \cdot v\right), \left(r \cdot w\right)\right), \left(\frac{r \cdot w}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \left(\frac{-1}{4} \cdot v\right)\right), \left(r \cdot w\right)\right), \left(\frac{r \cdot w}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \left(v \cdot \frac{-1}{4}\right)\right), \left(r \cdot w\right)\right), \left(\frac{r \cdot w}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \left(r \cdot w\right)\right), \left(\frac{r \cdot w}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(r, w\right)\right), \left(\frac{r \cdot w}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(r, w\right)\right), \mathsf{/.f64}\left(\left(r \cdot w\right), \left(1 - v\right)\right)\right)\right), \frac{9}{2}\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(r, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, w\right), \left(1 - v\right)\right)\right)\right), \frac{9}{2}\right) \]

      19. --lowering--.f64 93.0%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(r, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{\_.f64}\left(1, v\right)\right)\right)\right), \frac{9}{2}\right) \]

   6. Applied egg-rr 93.0%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\left(0.375 + v \cdot -0.25\right) \cdot \left(r \cdot w\right)\right) \cdot \frac{r \cdot w}{1 - v}}\right) - 4.5 \]

   7. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-3}{2} \cdot {r}^{2}}{{r}^{2}}} \]
   8. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(2 + \frac{-3}{2} \cdot {r}^{2}\right), \color{blue}{\left({r}^{2}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{-3}{2} \cdot {r}^{2}\right)\right), \left({\color{blue}{r}}^{2}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left({r}^{2} \cdot \frac{-3}{2}\right)\right), \left({r}^{2}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(\left(r \cdot r\right) \cdot \frac{-3}{2}\right)\right), \left({r}^{2}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(r \cdot \left(r \cdot \frac{-3}{2}\right)\right)\right), \left({r}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(r, \left(r \cdot \frac{-3}{2}\right)\right)\right), \left({r}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \frac{-3}{2}\right)\right)\right), \left({r}^{2}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \frac{-3}{2}\right)\right)\right), \left(r \cdot \color{blue}{r}\right)\right) \]

      9. *-lowering-*.f64 22.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \frac{-3}{2}\right)\right)\right), \mathsf{*.f64}\left(r, \color{blue}{r}\right)\right) \]

   9. Simplified 22.2%

      \[\leadsto \color{blue}{\frac{2 + r \cdot \left(r \cdot -1.5\right)}{r \cdot r}} \]

   10. Taylor expanded in r around inf

       \[\leadsto \color{blue}{\frac{-3}{2}} \]
   11. Step-by-step derivation

   12. Simplified 32.2%

       \[\leadsto \color{blue}{-1.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 56.8% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \frac{2}{r \cdot r} + -1.5 \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 (+ (/ 2.0 (* r r)) -1.5))

C

double code(double v, double w, double r) {
	return (2.0 / (r * r)) + -1.5;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = (2.0d0 / (r * r)) + (-1.5d0)
end function

Java

public static double code(double v, double w, double r) {
	return (2.0 / (r * r)) + -1.5;
}

Python

def code(v, w, r):
	return (2.0 / (r * r)) + -1.5

Julia

function code(v, w, r)
	return Float64(Float64(2.0 / Float64(r * r)) + -1.5)
end

MATLAB

function tmp = code(v, w, r)
	tmp = (2.0 / (r * r)) + -1.5;
end

Wolfram

code[v_, w_, r_] := N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision]

Derivation

1. Initial program 85.3%

   \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
2. Step-by-step derivation

   1. associate--l- N/A

      \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

   2. +-commutative N/A

      \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

   3. associate--l+ N/A

      \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)} \]

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

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{2}{r \cdot r}\right), \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)}\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(r \cdot r\right)\right), \left(\color{blue}{3} - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

   7. associate--r+ N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \color{blue}{\frac{9}{2}}\right)\right) \]

   8. sub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2}\right)\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + 3\right) - \frac{9}{2}\right)\right) \]

   10. associate--l+ N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(3 - \frac{9}{2}\right)}\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \frac{-3}{2}\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\frac{-9}{2} + \color{blue}{3}\right)\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\left(\mathsf{neg}\left(\frac{9}{2}\right)\right) + 3\right)\right)\right) \]

3. Simplified 76.4%

   \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\frac{\left(r \cdot r\right) \cdot \left(\left(0.375 + v \cdot -0.25\right) \cdot \left(w \cdot w\right)\right)}{v + -1} + -1.5\right)} \]
4. Add Preprocessing

5. Taylor expanded in r around 0

   \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \color{blue}{\frac{-3}{2}}\right) \]
6. Step-by-step derivation

7. Simplified 61.7%

   \[\leadsto \frac{2}{r \cdot r} + \color{blue}{-1.5} \]
8. Add Preprocessing

Alternative 15: 13.6% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ -1.5 \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 -1.5)

C

double code(double v, double w, double r) {
	return -1.5;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = -1.5d0
end function

Java

public static double code(double v, double w, double r) {
	return -1.5;
}

Python

def code(v, w, r):
	return -1.5

Julia

function code(v, w, r)
	return -1.5
end

MATLAB

function tmp = code(v, w, r)
	tmp = -1.5;
end

Wolfram

code[v_, w_, r_] := -1.5

Derivation

1. Initial program 85.3%

   \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \left(\left(w \cdot w\right) \cdot \left(r \cdot r\right)\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

   2. unswap-sqr N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \mathsf{*.f64}\left(\left(w \cdot r\right), \left(w \cdot r\right)\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \mathsf{*.f64}\left(\left(r \cdot w\right), \left(w \cdot r\right)\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \left(w \cdot r\right)\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \left(r \cdot w\right)\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

   7. *-lowering-*.f64 96.2%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{*.f64}\left(r, w\right)\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

4. Applied egg-rr 96.2%

   \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}}{1 - v}\right) - 4.5 \]
5. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \left(\frac{\left(\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)}{1 - v}\right)\right), \frac{9}{2}\right) \]

   2. associate-/l* N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \left(\left(\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(r \cdot w\right)\right) \cdot \frac{r \cdot w}{1 - v}\right)\right), \frac{9}{2}\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\left(\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(r \cdot w\right)\right), \left(\frac{r \cdot w}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right), \left(r \cdot w\right)\right), \left(\frac{r \cdot w}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

   5. cancel-sign-sub-inv N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{8} \cdot \left(3 + \left(\mathsf{neg}\left(2\right)\right) \cdot v\right)\right), \left(r \cdot w\right)\right), \left(\frac{r \cdot w}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{8} \cdot \left(3 + -2 \cdot v\right)\right), \left(r \cdot w\right)\right), \left(\frac{r \cdot w}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{8} \cdot \left(3 + v \cdot -2\right)\right), \left(r \cdot w\right)\right), \left(\frac{r \cdot w}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

   8. distribute-rgt-in N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(3 \cdot \frac{1}{8} + \left(v \cdot -2\right) \cdot \frac{1}{8}\right), \left(r \cdot w\right)\right), \left(\frac{r \cdot w}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{3}{8} + \left(v \cdot -2\right) \cdot \frac{1}{8}\right), \left(r \cdot w\right)\right), \left(\frac{r \cdot w}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

   10. associate-*l* N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{3}{8} + v \cdot \left(-2 \cdot \frac{1}{8}\right)\right), \left(r \cdot w\right)\right), \left(\frac{r \cdot w}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right), \left(r \cdot w\right)\right), \left(\frac{r \cdot w}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{3}{8} + \frac{-1}{4} \cdot v\right), \left(r \cdot w\right)\right), \left(\frac{r \cdot w}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

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

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \left(\frac{-1}{4} \cdot v\right)\right), \left(r \cdot w\right)\right), \left(\frac{r \cdot w}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \left(v \cdot \frac{-1}{4}\right)\right), \left(r \cdot w\right)\right), \left(\frac{r \cdot w}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

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

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \left(r \cdot w\right)\right), \left(\frac{r \cdot w}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

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

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(r, w\right)\right), \left(\frac{r \cdot w}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

   17. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(r, w\right)\right), \mathsf{/.f64}\left(\left(r \cdot w\right), \left(1 - v\right)\right)\right)\right), \frac{9}{2}\right) \]

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

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(r, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, w\right), \left(1 - v\right)\right)\right)\right), \frac{9}{2}\right) \]

   19. --lowering--.f64 97.6%

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(r, w\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{\_.f64}\left(1, v\right)\right)\right)\right), \frac{9}{2}\right) \]

6. Applied egg-rr 97.6%

   \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\left(0.375 + v \cdot -0.25\right) \cdot \left(r \cdot w\right)\right) \cdot \frac{r \cdot w}{1 - v}}\right) - 4.5 \]

7. Taylor expanded in r around 0

   \[\leadsto \color{blue}{\frac{2 + \frac{-3}{2} \cdot {r}^{2}}{{r}^{2}}} \]
8. Step-by-step derivation

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

      \[\leadsto \mathsf{/.f64}\left(\left(2 + \frac{-3}{2} \cdot {r}^{2}\right), \color{blue}{\left({r}^{2}\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{-3}{2} \cdot {r}^{2}\right)\right), \left({\color{blue}{r}}^{2}\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left({r}^{2} \cdot \frac{-3}{2}\right)\right), \left({r}^{2}\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(\left(r \cdot r\right) \cdot \frac{-3}{2}\right)\right), \left({r}^{2}\right)\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(r \cdot \left(r \cdot \frac{-3}{2}\right)\right)\right), \left({r}^{2}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(r, \left(r \cdot \frac{-3}{2}\right)\right)\right), \left({r}^{2}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \frac{-3}{2}\right)\right)\right), \left({r}^{2}\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \frac{-3}{2}\right)\right)\right), \left(r \cdot \color{blue}{r}\right)\right) \]

   9. *-lowering-*.f64 58.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \frac{-3}{2}\right)\right)\right), \mathsf{*.f64}\left(r, \color{blue}{r}\right)\right) \]

9. Simplified 58.0%

   \[\leadsto \color{blue}{\frac{2 + r \cdot \left(r \cdot -1.5\right)}{r \cdot r}} \]

10. Taylor expanded in r around inf

    \[\leadsto \color{blue}{\frac{-3}{2}} \]
11. Step-by-step derivation

12. Simplified 13.8%

    \[\leadsto \color{blue}{-1.5} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024139 
(FPCore (v w r)
  :name "Rosa's TurbineBenchmark"
  :precision binary64
  (- (- (+ 3.0 (/ 2.0 (* r r))) (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v))) 4.5))