Rosa's TurbineBenchmark

Percentage Accurate: 84.5% → 96.7%

Time: 14.0s

Alternatives: 13

Speedup: 1.7×

• 52.8% 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: 84.5% 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: 96.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -2 \cdot 10^{+115}:\\ \;\;\;\;t\_0 + \left(\left(r \cdot \left(r \cdot w\right)\right) \cdot \left(w \cdot -0.25\right) + -1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_0 + 3\right) + \frac{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot \left(0.125 \cdot \left(v \cdot 2 - 3\right)\right)}{1 - v}\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= v -2e+115)
     (+ t_0 (+ (* (* r (* r w)) (* w -0.25)) -1.5))
     (-
      (+
       (+ t_0 3.0)
       (/ (* (* (* r w) (* r w)) (* 0.125 (- (* v 2.0) 3.0))) (- 1.0 v)))
      4.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (v <= -2e+115) {
		tmp = t_0 + (((r * (r * w)) * (w * -0.25)) + -1.5);
	} else {
		tmp = ((t_0 + 3.0) + ((((r * w) * (r * w)) * (0.125 * ((v * 2.0) - 3.0))) / (1.0 - v))) - 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 = 2.0d0 / (r * r)
    if (v <= (-2d+115)) then
        tmp = t_0 + (((r * (r * w)) * (w * (-0.25d0))) + (-1.5d0))
    else
        tmp = ((t_0 + 3.0d0) + ((((r * w) * (r * w)) * (0.125d0 * ((v * 2.0d0) - 3.0d0))) / (1.0d0 - v))) - 4.5d0
    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 (v <= -2e+115) {
		tmp = t_0 + (((r * (r * w)) * (w * -0.25)) + -1.5);
	} else {
		tmp = ((t_0 + 3.0) + ((((r * w) * (r * w)) * (0.125 * ((v * 2.0) - 3.0))) / (1.0 - v))) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if v <= -2e+115:
		tmp = t_0 + (((r * (r * w)) * (w * -0.25)) + -1.5)
	else:
		tmp = ((t_0 + 3.0) + ((((r * w) * (r * w)) * (0.125 * ((v * 2.0) - 3.0))) / (1.0 - v))) - 4.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (v <= -2e+115)
		tmp = Float64(t_0 + Float64(Float64(Float64(r * Float64(r * w)) * Float64(w * -0.25)) + -1.5));
	else
		tmp = Float64(Float64(Float64(t_0 + 3.0) + Float64(Float64(Float64(Float64(r * w) * Float64(r * w)) * Float64(0.125 * Float64(Float64(v * 2.0) - 3.0))) / Float64(1.0 - v))) - 4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (v <= -2e+115)
		tmp = t_0 + (((r * (r * w)) * (w * -0.25)) + -1.5);
	else
		tmp = ((t_0 + 3.0) + ((((r * w) * (r * w)) * (0.125 * ((v * 2.0) - 3.0))) / (1.0 - v))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -2e+115], N[(t$95$0 + N[(N[(N[(r * N[(r * w), $MachinePrecision]), $MachinePrecision] * N[(w * -0.25), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$0 + 3.0), $MachinePrecision] + N[(N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] * N[(0.125 * N[(N[(v * 2.0), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -2e115

   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 69.2%

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

   5. Taylor expanded in v around 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 86.2%

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

   7. Simplified 86.2%

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

      1. associate-*r* N/A

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

      2. swap-sqr N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

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

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

      6. *-commutative N/A

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

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

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

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

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

      9. *-lowering-*.f64 99.9%

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

   9. Applied egg-rr 99.9%

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

   if -2e115 < v

   1. Initial program 81.2%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. 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 99.3%

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

   4. Applied egg-rr 99.3%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}}{1 - v}\right) - 4.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -2 \cdot 10^{+115}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(\left(r \cdot \left(r \cdot w\right)\right) \cdot \left(w \cdot -0.25\right) + -1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{2}{r \cdot r} + 3\right) + \frac{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot \left(0.125 \cdot \left(v \cdot 2 - 3\right)\right)}{1 - v}\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;w \cdot w \leq 10^{+223}:\\ \;\;\;\;t\_0 + \left(-1.5 + \frac{r \cdot \left(w \cdot \left(\left(r \cdot w\right) \cdot \left(0.375 + v \cdot -0.25\right)\right)\right)}{v + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \left(-1.5 + \left(w \cdot \left(\left(r \cdot r\right) \cdot w\right)\right) \cdot -0.375\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= (* w w) 1e+223)
     (+
      t_0
      (+ -1.5 (/ (* r (* w (* (* r w) (+ 0.375 (* v -0.25))))) (+ v -1.0))))
     (+ t_0 (+ -1.5 (* (* w (* (* r r) w)) -0.375))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((w * w) <= 1e+223) {
		tmp = t_0 + (-1.5 + ((r * (w * ((r * w) * (0.375 + (v * -0.25))))) / (v + -1.0)));
	} else {
		tmp = t_0 + (-1.5 + ((w * ((r * r) * w)) * -0.375));
	}
	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 ((w * w) <= 1d+223) then
        tmp = t_0 + ((-1.5d0) + ((r * (w * ((r * w) * (0.375d0 + (v * (-0.25d0)))))) / (v + (-1.0d0))))
    else
        tmp = t_0 + ((-1.5d0) + ((w * ((r * r) * w)) * (-0.375d0)))
    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 ((w * w) <= 1e+223) {
		tmp = t_0 + (-1.5 + ((r * (w * ((r * w) * (0.375 + (v * -0.25))))) / (v + -1.0)));
	} else {
		tmp = t_0 + (-1.5 + ((w * ((r * r) * w)) * -0.375));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (w * w) <= 1e+223:
		tmp = t_0 + (-1.5 + ((r * (w * ((r * w) * (0.375 + (v * -0.25))))) / (v + -1.0)))
	else:
		tmp = t_0 + (-1.5 + ((w * ((r * r) * w)) * -0.375))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(w * w) <= 1e+223)
		tmp = Float64(t_0 + Float64(-1.5 + Float64(Float64(r * Float64(w * Float64(Float64(r * w) * Float64(0.375 + Float64(v * -0.25))))) / Float64(v + -1.0))));
	else
		tmp = Float64(t_0 + Float64(-1.5 + Float64(Float64(w * Float64(Float64(r * r) * w)) * -0.375)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if ((w * w) <= 1e+223)
		tmp = t_0 + (-1.5 + ((r * (w * ((r * w) * (0.375 + (v * -0.25))))) / (v + -1.0)));
	else
		tmp = t_0 + (-1.5 + ((w * ((r * r) * w)) * -0.375));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(w * w), $MachinePrecision], 1e+223], N[(t$95$0 + N[(-1.5 + N[(N[(r * N[(w * N[(N[(r * w), $MachinePrecision] * N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 + N[(N[(w * N[(N[(r * r), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w w) < 1.00000000000000005e223

   1. Initial program 95.0%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 80.3%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

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

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

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

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

      12. *-lowering-*.f64 95.0%

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

   6. Applied egg-rr 95.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 99.3%

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

   8. Applied egg-rr 99.3%

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

   if 1.00000000000000005e223 < (*.f64 w w)

   1. Initial program 59.5%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 56.3%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

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

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

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

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

      12. *-lowering-*.f64 58.4%

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

   6. Applied egg-rr 58.4%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 80.4%

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

   8. Applied egg-rr 80.4%

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

   9. Taylor expanded in v around 0

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

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

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

       4. *-lowering-*.f64 74.1%

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

   11. Simplified 74.1%

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

   12. Taylor expanded in v around 0

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

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

       3. *-commutative N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

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

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

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

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

       8. unpow2 N/A

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

       9. *-lowering-*.f64 97.4%

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

   14. Simplified 97.4%

       \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{\left(w \cdot \left(w \cdot \left(r \cdot r\right)\right)\right) \cdot -0.375} + -1.5\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \cdot w \leq 10^{+223}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \frac{r \cdot \left(w \cdot \left(\left(r \cdot w\right) \cdot \left(0.375 + v \cdot -0.25\right)\right)\right)}{v + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \left(w \cdot \left(\left(r \cdot r\right) \cdot w\right)\right) \cdot -0.375\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;w \cdot w \leq 10^{+114}:\\ \;\;\;\;t\_0 + \left(-1.5 + \frac{r \cdot \left(\left(r \cdot w\right) \cdot \left(w \cdot \left(0.375 + v \cdot -0.25\right)\right)\right)}{v + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \left(-1.5 + \left(w \cdot \left(\left(r \cdot r\right) \cdot w\right)\right) \cdot -0.375\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= (* w w) 1e+114)
     (+
      t_0
      (+ -1.5 (/ (* r (* (* r w) (* w (+ 0.375 (* v -0.25))))) (+ v -1.0))))
     (+ t_0 (+ -1.5 (* (* w (* (* r r) w)) -0.375))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((w * w) <= 1e+114) {
		tmp = t_0 + (-1.5 + ((r * ((r * w) * (w * (0.375 + (v * -0.25))))) / (v + -1.0)));
	} else {
		tmp = t_0 + (-1.5 + ((w * ((r * r) * w)) * -0.375));
	}
	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 ((w * w) <= 1d+114) then
        tmp = t_0 + ((-1.5d0) + ((r * ((r * w) * (w * (0.375d0 + (v * (-0.25d0)))))) / (v + (-1.0d0))))
    else
        tmp = t_0 + ((-1.5d0) + ((w * ((r * r) * w)) * (-0.375d0)))
    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 ((w * w) <= 1e+114) {
		tmp = t_0 + (-1.5 + ((r * ((r * w) * (w * (0.375 + (v * -0.25))))) / (v + -1.0)));
	} else {
		tmp = t_0 + (-1.5 + ((w * ((r * r) * w)) * -0.375));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (w * w) <= 1e+114:
		tmp = t_0 + (-1.5 + ((r * ((r * w) * (w * (0.375 + (v * -0.25))))) / (v + -1.0)))
	else:
		tmp = t_0 + (-1.5 + ((w * ((r * r) * w)) * -0.375))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(w * w) <= 1e+114)
		tmp = Float64(t_0 + Float64(-1.5 + Float64(Float64(r * Float64(Float64(r * w) * Float64(w * Float64(0.375 + Float64(v * -0.25))))) / Float64(v + -1.0))));
	else
		tmp = Float64(t_0 + Float64(-1.5 + Float64(Float64(w * Float64(Float64(r * r) * w)) * -0.375)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if ((w * w) <= 1e+114)
		tmp = t_0 + (-1.5 + ((r * ((r * w) * (w * (0.375 + (v * -0.25))))) / (v + -1.0)));
	else
		tmp = t_0 + (-1.5 + ((w * ((r * r) * w)) * -0.375));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(w * w), $MachinePrecision], 1e+114], N[(t$95$0 + N[(-1.5 + N[(N[(r * N[(N[(r * w), $MachinePrecision] * N[(w * N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 + N[(N[(w * N[(N[(r * r), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w w) < 1e114

   1. Initial program 94.7%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 83.2%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

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

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

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

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

      12. *-lowering-*.f64 94.8%

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

   6. Applied egg-rr 94.8%

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

      1. associate-*r* N/A

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

      2. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      7. *-lowering-*.f64 99.8%

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

   8. Applied egg-rr 99.8%

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

   if 1e114 < (*.f64 w w)

   1. Initial program 67.2%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 58.0%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

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

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

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

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

      12. *-lowering-*.f64 66.3%

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

   6. Applied egg-rr 66.3%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 83.7%

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

   8. Applied egg-rr 83.7%

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

   9. Taylor expanded in v around 0

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

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

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

       4. *-lowering-*.f64 76.2%

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

   11. Simplified 76.2%

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

   12. Taylor expanded in v around 0

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

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

       3. *-commutative N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

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

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

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

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

       8. unpow2 N/A

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

       9. *-lowering-*.f64 96.6%

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

   14. Simplified 96.6%

       \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{\left(w \cdot \left(w \cdot \left(r \cdot r\right)\right)\right) \cdot -0.375} + -1.5\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \cdot w \leq 10^{+114}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \frac{r \cdot \left(\left(r \cdot w\right) \cdot \left(w \cdot \left(0.375 + v \cdot -0.25\right)\right)\right)}{v + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \left(w \cdot \left(\left(r \cdot r\right) \cdot w\right)\right) \cdot -0.375\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := t\_0 + \left(-1.5 + \left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot -0.25\right)\right)\\ \mathbf{if}\;v \leq -2.1 \cdot 10^{+28}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;v \leq 7 \cdot 10^{-10}:\\ \;\;\;\;t\_0 + \left(-1.5 + \frac{r \cdot \left(w \cdot \left(r \cdot \left(w \cdot 0.375\right)\right)\right)}{v + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1 (+ t_0 (+ -1.5 (* (* r w) (* (* r w) -0.25))))))
   (if (<= v -2.1e+28)
     t_1
     (if (<= v 7e-10)
       (+ t_0 (+ -1.5 (/ (* r (* w (* r (* w 0.375)))) (+ v -1.0))))
       t_1))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = t_0 + (-1.5 + ((r * w) * ((r * w) * -0.25)));
	double tmp;
	if (v <= -2.1e+28) {
		tmp = t_1;
	} else if (v <= 7e-10) {
		tmp = t_0 + (-1.5 + ((r * (w * (r * (w * 0.375)))) / (v + -1.0)));
	} else {
		tmp = t_1;
	}
	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 = 2.0d0 / (r * r)
    t_1 = t_0 + ((-1.5d0) + ((r * w) * ((r * w) * (-0.25d0))))
    if (v <= (-2.1d+28)) then
        tmp = t_1
    else if (v <= 7d-10) then
        tmp = t_0 + ((-1.5d0) + ((r * (w * (r * (w * 0.375d0)))) / (v + (-1.0d0))))
    else
        tmp = t_1
    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 t_1 = t_0 + (-1.5 + ((r * w) * ((r * w) * -0.25)));
	double tmp;
	if (v <= -2.1e+28) {
		tmp = t_1;
	} else if (v <= 7e-10) {
		tmp = t_0 + (-1.5 + ((r * (w * (r * (w * 0.375)))) / (v + -1.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	t_1 = t_0 + (-1.5 + ((r * w) * ((r * w) * -0.25)))
	tmp = 0
	if v <= -2.1e+28:
		tmp = t_1
	elif v <= 7e-10:
		tmp = t_0 + (-1.5 + ((r * (w * (r * (w * 0.375)))) / (v + -1.0)))
	else:
		tmp = t_1
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(t_0 + Float64(-1.5 + Float64(Float64(r * w) * Float64(Float64(r * w) * -0.25))))
	tmp = 0.0
	if (v <= -2.1e+28)
		tmp = t_1;
	elseif (v <= 7e-10)
		tmp = Float64(t_0 + Float64(-1.5 + Float64(Float64(r * Float64(w * Float64(r * Float64(w * 0.375)))) / Float64(v + -1.0))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	t_1 = t_0 + (-1.5 + ((r * w) * ((r * w) * -0.25)));
	tmp = 0.0;
	if (v <= -2.1e+28)
		tmp = t_1;
	elseif (v <= 7e-10)
		tmp = t_0 + (-1.5 + ((r * (w * (r * (w * 0.375)))) / (v + -1.0)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + N[(-1.5 + N[(N[(r * w), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -2.1e+28], t$95$1, If[LessEqual[v, 7e-10], N[(t$95$0 + N[(-1.5 + N[(N[(r * N[(w * N[(r * N[(w * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if v < -2.09999999999999989e28 or 6.99999999999999961e-10 < v

   1. Initial program 81.7%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 69.3%

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

   5. Taylor expanded in 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 80.1%

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

   7. Simplified 80.1%

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

      1. associate-*r* N/A

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

      2. swap-sqr N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

      7. *-lowering-*.f64 99.8%

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

   9. Applied egg-rr 99.8%

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

   if -2.09999999999999989e28 < v < 6.99999999999999961e-10

   1. Initial program 81.9%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 73.7%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

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

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

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

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

      12. *-lowering-*.f64 81.9%

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

   6. Applied egg-rr 81.9%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 95.9%

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

   8. Applied egg-rr 95.9%

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

   9. Taylor expanded in v around 0

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

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

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

       4. *-lowering-*.f64 95.8%

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

   11. Simplified 95.8%

       \[\leadsto \frac{2}{r \cdot r} + \left(\frac{\left(\color{blue}{\left(r \cdot \left(w \cdot 0.375\right)\right)} \cdot w\right) \cdot r}{v + -1} + -1.5\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -2.1 \cdot 10^{+28}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot -0.25\right)\right)\\ \mathbf{elif}\;v \leq 7 \cdot 10^{-10}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \frac{r \cdot \left(w \cdot \left(r \cdot \left(w \cdot 0.375\right)\right)\right)}{v + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot -0.25\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.25 \cdot \left(w \cdot w\right)\\ t_1 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 2.15 \cdot 10^{-95}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;r \leq 4000000000:\\ \;\;\;\;t\_1 + \left(-1.5 + \left(r \cdot r\right) \cdot t\_0\right)\\ \mathbf{elif}\;r \leq 9.2 \cdot 10^{+234}:\\ \;\;\;\;-1.5 + r \cdot \left(r \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;-0.375 \cdot \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* -0.25 (* w w))) (t_1 (/ 2.0 (* r r))))
   (if (<= r 2.15e-95)
     t_1
     (if (<= r 4000000000.0)
       (+ t_1 (+ -1.5 (* (* r r) t_0)))
       (if (<= r 9.2e+234)
         (+ -1.5 (* r (* r t_0)))
         (* -0.375 (* r (* w (* r w)))))))))

C

double code(double v, double w, double r) {
	double t_0 = -0.25 * (w * w);
	double t_1 = 2.0 / (r * r);
	double tmp;
	if (r <= 2.15e-95) {
		tmp = t_1;
	} else if (r <= 4000000000.0) {
		tmp = t_1 + (-1.5 + ((r * r) * t_0));
	} else if (r <= 9.2e+234) {
		tmp = -1.5 + (r * (r * t_0));
	} else {
		tmp = -0.375 * (r * (w * (r * w)));
	}
	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 = (-0.25d0) * (w * w)
    t_1 = 2.0d0 / (r * r)
    if (r <= 2.15d-95) then
        tmp = t_1
    else if (r <= 4000000000.0d0) then
        tmp = t_1 + ((-1.5d0) + ((r * r) * t_0))
    else if (r <= 9.2d+234) then
        tmp = (-1.5d0) + (r * (r * t_0))
    else
        tmp = (-0.375d0) * (r * (w * (r * w)))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = -0.25 * (w * w);
	double t_1 = 2.0 / (r * r);
	double tmp;
	if (r <= 2.15e-95) {
		tmp = t_1;
	} else if (r <= 4000000000.0) {
		tmp = t_1 + (-1.5 + ((r * r) * t_0));
	} else if (r <= 9.2e+234) {
		tmp = -1.5 + (r * (r * t_0));
	} else {
		tmp = -0.375 * (r * (w * (r * w)));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = -0.25 * (w * w)
	t_1 = 2.0 / (r * r)
	tmp = 0
	if r <= 2.15e-95:
		tmp = t_1
	elif r <= 4000000000.0:
		tmp = t_1 + (-1.5 + ((r * r) * t_0))
	elif r <= 9.2e+234:
		tmp = -1.5 + (r * (r * t_0))
	else:
		tmp = -0.375 * (r * (w * (r * w)))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(-0.25 * Float64(w * w))
	t_1 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 2.15e-95)
		tmp = t_1;
	elseif (r <= 4000000000.0)
		tmp = Float64(t_1 + Float64(-1.5 + Float64(Float64(r * r) * t_0)));
	elseif (r <= 9.2e+234)
		tmp = Float64(-1.5 + Float64(r * Float64(r * t_0)));
	else
		tmp = Float64(-0.375 * Float64(r * Float64(w * Float64(r * w))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = -0.25 * (w * w);
	t_1 = 2.0 / (r * r);
	tmp = 0.0;
	if (r <= 2.15e-95)
		tmp = t_1;
	elseif (r <= 4000000000.0)
		tmp = t_1 + (-1.5 + ((r * r) * t_0));
	elseif (r <= 9.2e+234)
		tmp = -1.5 + (r * (r * t_0));
	else
		tmp = -0.375 * (r * (w * (r * w)));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(-0.25 * N[(w * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 2.15e-95], t$95$1, If[LessEqual[r, 4000000000.0], N[(t$95$1 + N[(-1.5 + N[(N[(r * r), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[r, 9.2e+234], N[(-1.5 + N[(r * N[(r * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.375 * N[(r * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if r < 2.14999999999999999e-95

   1. Initial program 77.2%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 65.8%

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

   5. Taylor expanded in 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 52.4%

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

   7. Simplified 52.4%

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

   if 2.14999999999999999e-95 < r < 4e9

   1. Initial program 91.7%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 91.6%

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

   5. Taylor expanded in 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 85.4%

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

   7. Simplified 85.4%

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

   if 4e9 < r < 9.2000000000000004e234

   1. Initial program 100.0%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 89.3%

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

   5. Taylor expanded in 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 87.1%

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

   7. Simplified 87.1%

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

      1. associate-*r* N/A

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

      2. swap-sqr N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

      7. *-lowering-*.f64 95.6%

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

   9. Applied egg-rr 95.6%

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

   10. Taylor expanded in r around inf

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

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

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

       2. unpow2 N/A

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

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

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

       4. sub-neg N/A

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

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

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

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

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

       7. unpow2 N/A

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

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

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

       9. associate-*r/ N/A

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

       10. metadata-eval N/A

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

       11. distribute-neg-frac N/A

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

       12. metadata-eval N/A

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

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

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

       14. unpow2 N/A

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

       15. *-lowering-*.f64 87.1%

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

   12. Simplified 87.1%

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

       1. distribute-rgt-in N/A

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

       2. div-inv N/A

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

       3. associate-*l* N/A

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

       4. inv-pow N/A

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

       5. pow-plus N/A

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

       6. metadata-eval N/A

          \[\leadsto \left(\frac{-1}{4} \cdot \left(w \cdot w\right)\right) \cdot \left(r \cdot r\right) + \frac{-3}{2} \cdot {\left(r \cdot r\right)}^{0} \]

       7. metadata-eval N/A

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

       8. metadata-eval N/A

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

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

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

       10. *-commutative N/A

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

       11. associate-*l* N/A

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

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

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

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

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

       14. *-commutative N/A

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

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

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

       16. *-lowering-*.f64 95.7%

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

   14. Applied egg-rr 95.7%

       \[\leadsto \color{blue}{r \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot -0.25\right)\right) + -1.5} \]

   if 9.2000000000000004e234 < r

   1. Initial program 76.8%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 62.5%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

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

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

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

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

      12. *-lowering-*.f64 76.8%

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

   6. Applied egg-rr 76.8%

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

   7. Taylor expanded in r around inf

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

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

      6. unpow2 N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

      13. +-lowering-+.f64 62.5%

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

   9. Simplified 62.5%

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

   10. Taylor expanded in v around 0

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

       1. *-commutative N/A

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

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

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

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

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 76.1%

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

   12. Simplified 76.1%

       \[\leadsto \color{blue}{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot -0.375} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

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

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

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

       3. *-lowering-*.f64 87.5%

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

   14. Applied egg-rr 87.5%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 2.15 \cdot 10^{-95}:\\ \;\;\;\;\frac{2}{r \cdot r}\\ \mathbf{elif}\;r \leq 4000000000:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \left(r \cdot r\right) \cdot \left(-0.25 \cdot \left(w \cdot w\right)\right)\right)\\ \mathbf{elif}\;r \leq 9.2 \cdot 10^{+234}:\\ \;\;\;\;-1.5 + r \cdot \left(r \cdot \left(-0.25 \cdot \left(w \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.375 \cdot \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 0.0066:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{elif}\;r \leq 4.7 \cdot 10^{+235}:\\ \;\;\;\;-1.5 + r \cdot \left(r \cdot \left(-0.25 \cdot \left(w \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.375 \cdot \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 0.0066)
   (+ (/ 2.0 (* r r)) -1.5)
   (if (<= r 4.7e+235)
     (+ -1.5 (* r (* r (* -0.25 (* w w)))))
     (* -0.375 (* r (* w (* r w)))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 0.0066) {
		tmp = (2.0 / (r * r)) + -1.5;
	} else if (r <= 4.7e+235) {
		tmp = -1.5 + (r * (r * (-0.25 * (w * w))));
	} else {
		tmp = -0.375 * (r * (w * (r * w)));
	}
	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 <= 0.0066d0) then
        tmp = (2.0d0 / (r * r)) + (-1.5d0)
    else if (r <= 4.7d+235) then
        tmp = (-1.5d0) + (r * (r * ((-0.25d0) * (w * w))))
    else
        tmp = (-0.375d0) * (r * (w * (r * w)))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 0.0066) {
		tmp = (2.0 / (r * r)) + -1.5;
	} else if (r <= 4.7e+235) {
		tmp = -1.5 + (r * (r * (-0.25 * (w * w))));
	} else {
		tmp = -0.375 * (r * (w * (r * w)));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 0.0066:
		tmp = (2.0 / (r * r)) + -1.5
	elif r <= 4.7e+235:
		tmp = -1.5 + (r * (r * (-0.25 * (w * w))))
	else:
		tmp = -0.375 * (r * (w * (r * w)))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 0.0066)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + -1.5);
	elseif (r <= 4.7e+235)
		tmp = Float64(-1.5 + Float64(r * Float64(r * Float64(-0.25 * Float64(w * w)))));
	else
		tmp = Float64(-0.375 * Float64(r * Float64(w * Float64(r * w))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 0.0066)
		tmp = (2.0 / (r * r)) + -1.5;
	elseif (r <= 4.7e+235)
		tmp = -1.5 + (r * (r * (-0.25 * (w * w))));
	else
		tmp = -0.375 * (r * (w * (r * w)));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 0.0066], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision], If[LessEqual[r, 4.7e+235], N[(-1.5 + N[(r * N[(r * N[(-0.25 * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.375 * N[(r * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if r < 0.0066

   1. Initial program 78.4%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 68.1%

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

   5. Taylor expanded in r around 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 67.3%

      \[\leadsto \frac{2}{r \cdot r} + \color{blue}{-1.5} \]

   if 0.0066 < r < 4.6999999999999999e235

   1. Initial program 99.9%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 90.2%

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

   5. Taylor expanded in v around 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 84.5%

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

   7. Simplified 84.5%

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

      1. associate-*r* N/A

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

      2. swap-sqr N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

      7. *-lowering-*.f64 92.1%

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

   9. Applied egg-rr 92.1%

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

   10. Taylor expanded in r around inf

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

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

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

       2. unpow2 N/A

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

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

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

       4. sub-neg N/A

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

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

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

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

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

       7. unpow2 N/A

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

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

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

       9. associate-*r/ N/A

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

       10. metadata-eval N/A

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

       11. distribute-neg-frac N/A

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

       12. metadata-eval N/A

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

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

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

       14. unpow2 N/A

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

       15. *-lowering-*.f64 84.4%

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

   12. Simplified 84.4%

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

       1. distribute-rgt-in N/A

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

       2. div-inv N/A

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

       3. associate-*l* N/A

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

       4. inv-pow N/A

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

       5. pow-plus N/A

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

       6. metadata-eval N/A

          \[\leadsto \left(\frac{-1}{4} \cdot \left(w \cdot w\right)\right) \cdot \left(r \cdot r\right) + \frac{-3}{2} \cdot {\left(r \cdot r\right)}^{0} \]

       7. metadata-eval N/A

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

       8. metadata-eval N/A

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

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

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

       10. *-commutative N/A

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

       11. associate-*l* N/A

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

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

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

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

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

       14. *-commutative N/A

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

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

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

       16. *-lowering-*.f64 92.2%

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

   14. Applied egg-rr 92.2%

       \[\leadsto \color{blue}{r \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot -0.25\right)\right) + -1.5} \]

   if 4.6999999999999999e235 < r

   1. Initial program 76.8%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 62.5%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

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

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

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

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

      12. *-lowering-*.f64 76.8%

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

   6. Applied egg-rr 76.8%

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

   7. Taylor expanded in r around inf

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

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

      6. unpow2 N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

      13. +-lowering-+.f64 62.5%

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

   9. Simplified 62.5%

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

   10. Taylor expanded in v around 0

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

       1. *-commutative N/A

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

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

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

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

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 76.1%

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

   12. Simplified 76.1%

       \[\leadsto \color{blue}{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot -0.375} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

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

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

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

       3. *-lowering-*.f64 87.5%

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

   14. Applied egg-rr 87.5%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 0.0066:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{elif}\;r \leq 4.7 \cdot 10^{+235}:\\ \;\;\;\;-1.5 + r \cdot \left(r \cdot \left(-0.25 \cdot \left(w \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.375 \cdot \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{2}{r \cdot r} + \left(-1.5 + \left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot -0.25\right)\right) \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (+ (/ 2.0 (* r r)) (+ -1.5 (* (* r w) (* (* r w) -0.25)))))

C

double code(double v, double w, double r) {
	return (2.0 / (r * r)) + (-1.5 + ((r * w) * ((r * w) * -0.25)));
}

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) + ((r * w) * ((r * w) * (-0.25d0))))
end function

Java

public static double code(double v, double w, double r) {
	return (2.0 / (r * r)) + (-1.5 + ((r * w) * ((r * w) * -0.25)));
}

Python

def code(v, w, r):
	return (2.0 / (r * r)) + (-1.5 + ((r * w) * ((r * w) * -0.25)))

Julia

function code(v, w, r)
	return Float64(Float64(2.0 / Float64(r * r)) + Float64(-1.5 + Float64(Float64(r * w) * Float64(Float64(r * w) * -0.25))))
end

MATLAB

function tmp = code(v, w, r)
	tmp = (2.0 / (r * r)) + (-1.5 + ((r * w) * ((r * w) * -0.25)));
end

Wolfram

code[v_, w_, r_] := N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(-1.5 + N[(N[(r * w), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.8%

   \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
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 71.4%

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

5. Taylor expanded in 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 74.9%

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

7. Simplified 74.9%

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

   1. associate-*r* N/A

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

   2. swap-sqr N/A

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

   3. associate-*l* N/A

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

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

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

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

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

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

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

   7. *-lowering-*.f64 93.8%

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

9. Applied egg-rr 93.8%

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

10. Final simplification 93.8%

    \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 + \left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot -0.25\right)\right) \]
11. Add Preprocessing

Alternative 8: 66.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 2.8 \cdot 10^{+71}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{else}:\\ \;\;\;\;-0.375 \cdot \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 2.8e+71) (+ (/ 2.0 (* r r)) -1.5) (* -0.375 (* r (* w (* r w))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 2.8e+71) {
		tmp = (2.0 / (r * r)) + -1.5;
	} else {
		tmp = -0.375 * (r * (w * (r * w)));
	}
	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 <= 2.8d+71) then
        tmp = (2.0d0 / (r * r)) + (-1.5d0)
    else
        tmp = (-0.375d0) * (r * (w * (r * w)))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 2.8e+71) {
		tmp = (2.0 / (r * r)) + -1.5;
	} else {
		tmp = -0.375 * (r * (w * (r * w)));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 2.8e+71:
		tmp = (2.0 / (r * r)) + -1.5
	else:
		tmp = -0.375 * (r * (w * (r * w)))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 2.8e+71)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + -1.5);
	else
		tmp = Float64(-0.375 * Float64(r * Float64(w * Float64(r * w))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 2.8e+71)
		tmp = (2.0 / (r * r)) + -1.5;
	else
		tmp = -0.375 * (r * (w * (r * w)));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 2.8e+71], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision], N[(-0.375 * N[(r * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 2.80000000000000002e71

   1. Initial program 79.7%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 70.0%

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

   5. Taylor expanded in r around 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 66.5%

      \[\leadsto \frac{2}{r \cdot r} + \color{blue}{-1.5} \]

   if 2.80000000000000002e71 < r

   1. Initial program 92.6%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 78.5%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

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

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

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

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

      12. *-lowering-*.f64 92.6%

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

   6. Applied egg-rr 92.6%

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

   7. Taylor expanded in r around inf

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

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

      6. unpow2 N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

      13. +-lowering-+.f64 66.9%

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

   9. Simplified 66.9%

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

   10. Taylor expanded in v around 0

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

       1. *-commutative N/A

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

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

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

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

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 69.8%

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

   12. Simplified 69.8%

       \[\leadsto \color{blue}{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot -0.375} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

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

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

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

       3. *-lowering-*.f64 73.6%

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

   14. Applied egg-rr 73.6%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 2.8 \cdot 10^{+71}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{else}:\\ \;\;\;\;-0.375 \cdot \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 1.05 \cdot 10^{+71}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{else}:\\ \;\;\;\;-0.375 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 1.05e+71) (+ (/ 2.0 (* r r)) -1.5) (* -0.375 (* r (* r (* w w))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.05e+71) {
		tmp = (2.0 / (r * r)) + -1.5;
	} else {
		tmp = -0.375 * (r * (r * (w * w)));
	}
	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 <= 1.05d+71) then
        tmp = (2.0d0 / (r * r)) + (-1.5d0)
    else
        tmp = (-0.375d0) * (r * (r * (w * w)))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.05e+71) {
		tmp = (2.0 / (r * r)) + -1.5;
	} else {
		tmp = -0.375 * (r * (r * (w * w)));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 1.05e+71:
		tmp = (2.0 / (r * r)) + -1.5
	else:
		tmp = -0.375 * (r * (r * (w * w)))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 1.05e+71)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + -1.5);
	else
		tmp = Float64(-0.375 * Float64(r * Float64(r * Float64(w * w))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 1.05e+71)
		tmp = (2.0 / (r * r)) + -1.5;
	else
		tmp = -0.375 * (r * (r * (w * w)));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 1.05e+71], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision], N[(-0.375 * N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 1.04999999999999995e71

   1. Initial program 79.7%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 70.0%

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

   5. Taylor expanded in r around 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 66.5%

      \[\leadsto \frac{2}{r \cdot r} + \color{blue}{-1.5} \]

   if 1.04999999999999995e71 < r

   1. Initial program 92.6%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 78.5%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

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

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

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

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

      12. *-lowering-*.f64 92.6%

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

   6. Applied egg-rr 92.6%

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

   7. Taylor expanded in r around inf

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

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

      6. unpow2 N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

      13. +-lowering-+.f64 66.9%

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

   9. Simplified 66.9%

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

   10. Taylor expanded in v around 0

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

       1. *-commutative N/A

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

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

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

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

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 69.8%

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

   12. Simplified 69.8%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 1.05 \cdot 10^{+71}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{else}:\\ \;\;\;\;-0.375 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 9 \cdot 10^{+70}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{else}:\\ \;\;\;\;\left(r \cdot r\right) \cdot \left(-0.25 \cdot \left(w \cdot w\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 9e+70) (+ (/ 2.0 (* r r)) -1.5) (* (* r r) (* -0.25 (* w w)))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 9e+70) {
		tmp = (2.0 / (r * r)) + -1.5;
	} else {
		tmp = (r * r) * (-0.25 * (w * w));
	}
	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 <= 9d+70) then
        tmp = (2.0d0 / (r * r)) + (-1.5d0)
    else
        tmp = (r * r) * ((-0.25d0) * (w * w))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 9e+70) {
		tmp = (2.0 / (r * r)) + -1.5;
	} else {
		tmp = (r * r) * (-0.25 * (w * w));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 9e+70:
		tmp = (2.0 / (r * r)) + -1.5
	else:
		tmp = (r * r) * (-0.25 * (w * w))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 9e+70)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + -1.5);
	else
		tmp = Float64(Float64(r * r) * Float64(-0.25 * Float64(w * w)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 9e+70)
		tmp = (2.0 / (r * r)) + -1.5;
	else
		tmp = (r * r) * (-0.25 * (w * w));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 9e+70], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision], N[(N[(r * r), $MachinePrecision] * N[(-0.25 * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 8.9999999999999999e70

   1. Initial program 79.7%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 70.0%

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

   5. Taylor expanded in r around 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 66.5%

      \[\leadsto \frac{2}{r \cdot r} + \color{blue}{-1.5} \]

   if 8.9999999999999999e70 < r

   1. Initial program 92.6%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 78.5%

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

   5. Taylor expanded in 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 78.4%

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

   7. Simplified 78.4%

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

   8. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 66.8%

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

   10. Simplified 66.8%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 9 \cdot 10^{+70}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{else}:\\ \;\;\;\;\left(r \cdot r\right) \cdot \left(-0.25 \cdot \left(w \cdot w\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 50.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 0.0066:\\ \;\;\;\;\frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;-1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 (if (<= r 0.0066) (/ 2.0 (* r r)) -1.5))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 0.0066) {
		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 <= 0.0066d0) 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 <= 0.0066) {
		tmp = 2.0 / (r * r);
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 0.0066:
		tmp = 2.0 / (r * r)
	else:
		tmp = -1.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 0.0066)
		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 <= 0.0066)
		tmp = 2.0 / (r * r);
	else
		tmp = -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 0.0066], N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision], -1.5]

Derivation

1. Split input into 2 regimes

2. if r < 0.0066

   1. Initial program 78.4%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 68.1%

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

   5. Taylor expanded in r around 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 53.9%

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

   7. Simplified 53.9%

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

   if 0.0066 < r

   1. Initial program 94.4%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 83.6%

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

   5. Taylor expanded in 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 79.2%

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

   7. Simplified 79.2%

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

      1. associate-*r* N/A

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

      2. swap-sqr N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

      7. *-lowering-*.f64 89.5%

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

   9. Applied egg-rr 89.5%

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

   10. Taylor expanded in r around inf

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

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

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

       2. unpow2 N/A

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

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

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

       4. sub-neg N/A

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

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

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

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

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

       7. unpow2 N/A

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

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

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

       9. associate-*r/ N/A

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

       10. metadata-eval N/A

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

       11. distribute-neg-frac N/A

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

       12. metadata-eval N/A

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

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

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

       14. unpow2 N/A

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

       15. *-lowering-*.f64 79.2%

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

   12. Simplified 79.2%

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

   13. Taylor expanded in r around 0

       \[\leadsto \color{blue}{\frac{-3}{2}} \]
   14. Step-by-step derivation

   15. Simplified 30.4%

       \[\leadsto \color{blue}{-1.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 57.5% 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 81.8%

   \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
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 71.4%

   \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\frac{\left(r \cdot r\right) \cdot \left(\left(0.375 + v \cdot -0.25\right) \cdot \left(w \cdot w\right)\right)}{v + -1} + -1.5\right)} \]
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 59.5%

   \[\leadsto \frac{2}{r \cdot r} + \color{blue}{-1.5} \]
8. Add Preprocessing

Alternative 13: 14.0% 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 81.8%

   \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
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 71.4%

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

5. Taylor expanded in 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 74.9%

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

7. Simplified 74.9%

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

   1. associate-*r* N/A

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

   2. swap-sqr N/A

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

   3. associate-*l* N/A

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

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

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

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

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

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

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

   7. *-lowering-*.f64 93.8%

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

9. Applied egg-rr 93.8%

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

10. Taylor expanded in r around inf

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

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

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

    2. unpow2 N/A

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

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

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

    4. sub-neg N/A

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

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

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

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

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

    7. unpow2 N/A

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

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

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

    9. associate-*r/ N/A

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

    10. metadata-eval N/A

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

    11. distribute-neg-frac N/A

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

    12. metadata-eval N/A

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

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

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

    14. unpow2 N/A

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

    15. *-lowering-*.f64 45.7%

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

12. Simplified 45.7%

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

13. Taylor expanded in r around 0

    \[\leadsto \color{blue}{\frac{-3}{2}} \]
14. Step-by-step derivation

15. Simplified 17.9%

    \[\leadsto \color{blue}{-1.5} \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024149 
(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))